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2 = 8I1 + 4I2 2 = 8(1.5) + 4I2 2 = 12 + 4I2 4I2 = −10 I2 = −2.5 For a 3x3, this process is iterated as follows: Equation 2 would be solved for I3 and this would be substituted back into equation 1 yielding a new equation (let’s call it A) with only I1 and I2 terms. Similarly, Equation 3 would be solved for I3 and this would be substituted back into equation 2 yielding a new equation (let’s call it B) with only I1 and I2 terms. Equations A and B now make a 2x2 with I1 and I2 as the unknowns and can be solved as outlined above. This would yield values for I1 and I2 which could then be substituted into one of the three original equations to obtain I3. While the substitution method is perfectly valid for an arbitrarily sized system, it proves cumbersome as the system gets larger. Gauss-Jordan Elimination Gauss-Jordan Elimination In some respects, Gauss-Jordan is similar to substitution but it tends to involve less overhead for larger systems and thus is generally preferred. This method involves multiplying one equation by a constant such that when it is subtracted from another equation, one of the unknown terms disappears. The process is then iterated for as many unknowns exist in the system. Using the same example from before: 10 = 20I1 + 8I2 2 = 8I1 + 4I2 Multiply the second equation by the ratio of the coefficients for I2 (8/4 = 2). 2 = 8I1 + 4I2 4 = 16I1 + 8I2 Subtract this new equation from the first equation. The I2 terms will cancel leaving just I1. 10 = 20I1 + 8I2 4 = 16I1 + 8I2 6 = 4I1 I1 = 1.5 Substitute this result back into one of the original equations to obtain I2. For a 3x3, iterate as follows: Using equations 1 and 2, multiply equation 2 by the ratio of the coefficients for I3. Subtract this equation from equation 1 to generate a new equation (let’s call it equation A) that only has I1 and I2 as unknowns. Using equations 2 and 3, multiply equation 3 by the ratio of the coefficients for I3. Subtract this equation from equation 2 to generate a new equation (let’s call it equation B) that only has I1 and I2 as unknowns. Equations A and B now make a 2x2 with I1 and I2 as the unknowns and can be solved as outlined previously. This would yield values for I1 and I2 which could then be substituted into one of the three original equations to obtain I3. Like the substitution method, Gauss-Jordan grows rapidly as the system size increases. The process tends to be formulaic though, and thus easier to handle. 401
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Determinants Determinants Determinants revolve around the concept of a matrix which itself is little more than an ordered collection of coefficients and/or constants. It is imperative that the unknowns be in the same order in each equation (i.e., I1 ascending to IX left to right) A simple coefficient matrix for the original 2x2 example is: 20 8 8 4 The resultant value (properly referred to as the determinant) for a 2x2 matrix such as this may be solved using Sarrus’ Rule: Simply multiply the two values along the upper right-lower left diagonal and then subtract that product from the product of the two values found along on the upper left-lower right diagonal. In this example that’s: 20*4 − 8*8 = 16 A solution involves dividing one determinant by another determinant (Cramer's Rule). That is, each matrix is solved for its resultant value and then these two values are divided to determine the final answer. One of these matrices will be the coefficient matrix just discussed. This will be placed in the denominator. The numerator matrix is a modified version of the basic coefficient matrix. It is created by replacing one column of coefficients with the constant values from the original system of equations. For example, the numerator matrix used to find I1 would replace the first column (the I1 coefficients 20 and 8) with the constants 10 and 2: 10 8 2 4 The resultant value is 40 − 16 or 24. Similarly, the numerator matrix for I2 would replace the I2 coefficients in the second column (8 and 4) with the constants 10 and 2: 20 10 8 2 The resultant value is 40 − 80 or −40. To find any particular unknown, simply divide the modified matrix by the basic coefficient matrix. 10 8 2 4 I1 = -------- 20 8 8 4 24 I1 = ----- 16 402
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I1 = 1.5 In like fashion I2 is found: 20 10 8 2 I2 = -------- 20 8 8 4 −40 I2 = ----- 16 I2 = −2.5 Sarrus’ Rule may also be used with a 3x3 matrix. This is achieved by extending the matrix. Fourth and fifth columns are added to the right of the 3x3 matrix by simply making copies of the first two columns. This creates three right to left diagonals with three values each and three left to right diagonals with three values each. The three values along each diagonal are multiplied together. The three right to left products are then subtracted from the sum of the three left to right products yielding a single resultant value (the determinant). To create the modified numerator matrix, replace the coefficient column of interest with the constant terms and then replicate columns one and two. For example, given these three equations: 10 = 20I1 + 8I2 + 3I3 2 = 8I1 + 4I2 + 5I3 7 = 3I1 + 5I2 + 6I3 The basic coefficient matrix (i.e., denominator) is: 20 8 3 8 4 5 3 5 6 The extended matrix is: 20 8 3 20 8 8 4 5 8 4 3 5 6 3 5 The result is: 20*4*6 + 8*5*3 + 3*8*5 − 3*4*3 − 20*5*5 − 8*8*6 Sarrus’ Rule does not work beyond 3x3. 403
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Expansion by Minors Expansion by Minors Expansion by Minors is another method that may be used to generate a determinant solution. This involves breaking the matrix into a series of smaller matrices (minors) that are combined using row-column coefficients. The position of these coefficients will also indicate whether the determinant of any particular sub-matrix is added or subtracted to the total. The first step is to establish a single row or column from which to derive the coefficients. This can be any horizontal row or vertical column (no diagonals). Each element of the chosen row or column determines the associated minor (essentially, that which is left over). Consider the 3x3 used previously: 20 8 3 8 4 5 3 5 6 Choosing the top row yields coefficients of 20, 8 and 3. For each of these, blot out its row and column and see what is left. This leaves three 2x2 matrices, one for each coefficient. Multiply each coefficient by the determinant of its 2x2 matrix. To determine whether this result is added or subtracted to the others, the sign may be found using the following map for the coefficients: + − + − etc. − + − + etc. + − + − etc. The origin in the upper left is positive and the signs continually alternate across from it and down from it. The result using the top row for the coefficients is found thus (the 2x2 matrices are bold red for clarity): 4 5 8 5 8 4 20 * 5 6 − 8 * 3 6 + 3 * 3 5 If the second column was used instead (8, 4, 5), the result is found like so: 8 5 20 3 20 3 −8 * 3 6 + 4 * 3 6 − 5 * 8 5 In closing, whichever method is used, always look for null coefficient terms (that is, places in the equations and matrices where the coefficients are zero). Smart use of these can considerably simplify the computations as there are few mathematical operations easier than multiplying by zero. For example, if a particular row of a matrix contains a few zeros, that would be a good candidate for the coefficient row when using expansion by minors because some 2x2 minors need not be computed (they will just be multiplied by the zero coefficient). 404
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Appendix C Appendix C Equation Proofs Equation Proofs RMS Equivalents for Non-Sines For any waveform, the root-mean-square equivalence factor is computed by normalizing the peak value to unity, squaring the waveform, finding the mean value of that intermediate result, and then taking the square root of the mean. This value can never be more than unity. The RMS value of a square wave is equal to its peak value. This can be proven by observation. Normalize the square wave's peak value to unity. Its negative peak will be −1. Squaring these values results in the constant value 1 at any time (assuming the rise and fall times are negligible). Obviously, the mean of this is 1 as is its square root. Thus the RMS equivalence factor is unity. For a rectangular pulse that is always positive (i.e., traversing from zero volts to some positive peak and back to zero), the RMS value is equal to its peak value times the square root of its duty cycle. As there is no negative portion to this waveform, we need only examine the positive portion. First, its normalized amplitude is always unity, as is its squared value. The mean is simply the time for this positive pulse divided by the period of the wave. By definition, that's equivalent to the duty cycle (i.e., the percentage of time the pulse is positive out of a full cycle). Thus, the RMS equivalence factor is the square root of the duty cycle. For a triangle wave, the RMS value is equal to its peak value times one over the square root of three. This can be proven by first noting that a triangle wave has quarter wave symmetry. Consequently, we need examine only the first quarter of a cycle because the other three will produce identical numerical results. To ease computation, normalize the amplitude and the time for the first quarter cycle to unity. The result is a straight line that starts at the origin and reaches an amplitude of 1 when time also reaches 1. Written as a function of time, the expression for such a voltage is: v(t)= t Squaring this gives us t2. To find the mean, we integrate this function: mean =∫ 0 t t 2dt mean = 1 3 t 3|t=0 t=1 mean = 1 3 Finally, taking the square root results in a factor of one over the square root of three. QED. 405
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Maximum Power Transfer: Finding the Maximizing Value of P = R/(R2+2R+1) While the algebraic and graphing technique explored in Chapter 5 leads to a proper answer, it is incomplete. A more rigorous treatment using differential calculus follows. We have already determined that the reactive portion of the load must have the same magnitude but opposite sign of the internal reactance in order to maximize load current, and therefore maximize load power. Thus, we need only examine the resistive portion which is described by the equation P = R/(R2+2R+1). This function exhibits a single peak and thus we may find the corresponding value by taking the first derivative of the function, setting it to zero (i.e., find the point where the slope goes to zero), and solving for R. Chain rule can be used to solve this. Chain rule states: dy du du dx dy dx One way to apply the chain rule is first to rewrite the main equation to remove the numerator R. This effectively removes the issues of having a quotient or product. We divide through by R and arrive at: R R P 1 2 1 or in a somewhat more convenient form: 1 1) 2 ( R R P Using the chain rule the derivative of this is: ) 1( ) 2 ( 2 2 1 R R R dR dP or in “prettier” form: 2 1 2 ) 2 ( 1 R R R dR dP Multiply through by R2 2 2 2 )1 2 ( 1 R R R dR dP Which, for a really anal retentive sort of completeness, can be rewritten as: 4 2 )1 ( 1 R R dR dP For dP/dR to be zero, R must equal 1. In other words, it must match the internal resistance. QED. 406
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Finding k0 : Determining f1 and f2 in Resonant Circuits The coefficient k0 was defined such that: f 1 = f 0 k 0 and f 2 = f 0×k 0 We start with the definition of circuit Q based on bandwidth and resonant frequency, and expand, solving in terms of f2. Qcircuit = f 0 BW = f 0 f 2 −f 1 f 2 = f 0 Qcircuit +f 1 (EQ 1) The resonant frequency is equal to the geometric mean of f1 and f2. f 0 = √f 1 f 2 f 1 = f 0 2 f 2 (EQ 2) Substituting EQ 2 into EQ 1 yields: f 2 = f 0 Qcircuit + f 0 2 f 2 (EQ 3) We normalize f0, taking it as unity. This means that f2 is now equivalent to k0. Rewriting EQ 3 yields: k 0 = 1 Q circuit + 1 k 0 0 = k 0 2 − 1 Q circuit k0 −1 We can solve this using the quadratic formula where a = 1, b = 1/Qcircuit and c = −1. −b±√b 2 −4ac 2a Substituting and simplifying results in the equation for k0 : k 0 = 1 2Qcircuit +√ 1 4Qcircuit 2 +1 407
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Appendix D Appendix D Answers to Selected Odd-Numbered Problems Answers to Selected Odd-Numbered Problems 1 Fundamentals 1 Fundamentals 1. 10, 7.07, 0, 1 kHz, 1 ms, 0° 3. 20, −3, 50 Hz, 20 ms, 0° 5. 10, 7.07, 0, 100 Hz, 10 ms, 45° 7. 1, 10, 400 Hz, 2.5 ms, −45° 9. 200 μs, 10 μs 11. 36° 13. 14.1445°, 11.2−63.4°, 102169°, 5k53.1° 15. 7.07 + j7.07, j0.4, −4.5 + j7.79, 70.7 − j70.7 17. 15 + j30, j4, −20−j4, −70 + j250 19. −34.5k + j36k, −725 + j95, 2.39 −j0.709, −2.71 −j0.457 21. 1000°, 10−115°, 0.5145°, 0.25−45° 23. 2.7180°, 4.91−92.7°, 0.076123°, 5444.5° 25. −j15.9 k, −j318, −j15.9, −j0.398, −j15.9E−3 Ω 27. −j318 M, −j6.77 M, −j144.7 k, −j96.5 Ω 29. j6.28, j314, j6.28 k, j251 k, j6.28 M Ω 31. j62.8, j3.14 k, j62.8E−3, j2.51 Ω 33. 35. 1.67 @ 3 kHz, 1 @ 5 kHz, 0.714 @ 7 kHz, 0.555 @ 9 kHz, 0.455 @ 11 kHz 408
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37. 284 nF, 482 pF, 339 pF, 132.6 nF, 212 nF 39. 892 nH, 525 μH, 748 μH, 1.91 μH, 1.19 μH 41. a 43. b 409
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2 Series RLC Circuits 2 Series RLC Circuits 1. 2k − j1.94 k Ω 3. 270 + j125.7 Ω 5. 1 k − j1.278 k Ω 7. 300 − j400 9. 1.447 μF 11. v(t)=0.1sin2π1000t (i and v are in phase) 13. i is 241 mA p-p and lags by 90° 15. v is 16.6 V p-p and leads by 90° 17. 1 k − j318 Ω 19. i = 953E−617.6° amps, vR = 953E−317.6° volts, vC = 303E−3−72.4° volts, delay = 25 μs 21. 1 k + j3.14 kΩ 23. 1 k + j942 Ω 25. 2 k − j33.4 Ω 27. i = 0.4999E−3.957° amps, vR = 0.998.957° volts, vC = 16.7E−3−89° volts, delay = 167 μs 29. i = 21.2E−345° amps, vR = 4.2445° volts, vC = 4.24−45° volts 31. vS = 60.9−23.2° volts, vR = 560° volts, vC = 24−90° volts 33. i = 329E−670.8° amps, vR = 329E−370.8° volts, vC = 1.05−19.2° volts, vL = 103E−3160.8° volts 35. i = 48.7E−3−13° amps, vR = 9.745−13° volts, vC = 3.88−103° volts, vL = 6.12577° volts 37. i = 493E−39.52° amps, vB = 58.241.6° volts, vAC = 63−29° volts 39. i = 1E−30° amps, vR = 10° volts, vC = 200E−3−90° volts, vL = 200E−390° volts 410
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41. i = 2.24E−363.4° amps, vB = 8.94−26.6° volts, vC = 2.24153.4° volts, vAC = 12−4.8° volts 43. i = 35.36E−3−45° amps, vC = 2.12−135° volts, vR = 1.414−45° volts, vL = 3.5445° volts 45. vR = 100° volts, vC = 5.3−90° volts, vL = 7.0790° volts 47. vR = 2.40° volts, vL = 85.4E−390° volts, vC = 424E−390° volts 49. vR = 1100° volts, vC = 88−90° volts, vL = 22090° volts 51. vAC = 6087.8° volts, vB = 20.1−83.3° volts, vC = 20−90° volts 53. vR = 0.32970.8° volts, vL = 0.1034160.8° volts, vC = 1.048−19.2° volts 55. L = 79.6 μH, C = 7.24 nF 57. vR = 7.23−130° volts, vL = 3.62−40.1° volts, vC = 1.45139.9° volts 59. f = 15.594 kHz 61. f = 3.185 kHz 411
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3 Parallel RLC Circuits 3 Parallel RLC Circuits 1. 73.2−12.5° (71.4 −j15.8) 3. 99−8.04° (98 −j13.8) 5. 182.952.4° (111.5 + j145) 7. f = 45.2 MHz 9. 11. is = 0.1200010.18°, iR = 0.120°, iC = 377E−690° 13. is = 9.43E−3−58°, iR = 5E−30°, iC = 2E−390°, iL = 10E−3−90° 15. is = 47.4E−316.3°, iR = 45.5E−30°, iC = 20E−390°, iL = 6.667E−3−90° (all peak) 17. iR = 19.99E−3−1.73°, iC = 603E−688.27° 19. vR = vL = 627E−387° volts 21. vs = 2.82−28.1° volts 23. i2.2k = 5.06E−3−68.2°, i4.7k = 2.37E−3−68.2°, iC = 55.7E−321.8°, iL = 37.1E−3−158.2° 25. iC = 670E−6150°, iL = 536E−6−30°, 27. iR = 1.67E−3−5°, iC = 835E−685°, 412
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iR = 2.23E−360° iL = 1.044E−3−95° 29. 2.84 μF 31. 7.7 nF 33. 34.5 nF 35. 65.7 nF 37. 390 nF 39. 507 nF 41. 128 mH 4 Series-Parallel RLC Circuits 4 Series-Parallel RLC Circuits 1. Z10k ≈ 1250°, Z1M ≈ 125−0.8°, Z100M ≈ 68.8−24.2° Ω 3. Z300 ≈ 0.7590°, Z30k ≈ 77.589.4°, Z3M ≈ 9075.4° Ω 5. Z = 18051.6° Ω 7. is = 9.64E−3−85.6°, iR = 740.7E−60°, iC = iL = 9.62E−3−90°, 9. vR = vC = 3.45−80.5°, vL = 7.2827.9° 11. i50 = 31.4E−690°, i91 = i20 = 12.6E−689.9°, is ≈ 44E−690° 13. vb = 9.19−50°, vab = 15.826.5° 15. ij4k = 5E−3−90°, i2.7k = 7.41E−30°, i3.9k = i−j1k = 4.97E−314.4°, is = 12.8E−3−17.1° 17. vb = 1200.7°, vab = 20−175.6° 19. i15k = iL = 887E−6−4°, i12k = 1.12E−33.17° 21. va = 4.814.1°, vb = 6.6214.1° 23. va = 571E−3−26.5°, vb = 537E−3−6.68° 25. va = 19.85127.2°, vb = 23.8127.9° 27. ic = 12.83E−3136.9° 29. va = 7.3261.5°, vb = 7.175.6° 31. vab = −133.30° 33. 217 nF 35. 3.24 μF 37. 5.9 μF 39. 19.1 mH 413
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5 Analysis Theorems and Techniques 5 Analysis Theorems and Techniques 1. vb = 2.993.5° 3. i82 = 13.3E−359.8° 5. vb = 1.08165° 7. i2.2k = 1.3E−349.2° 9. vb = 7.8498.9°, vcd = 10.2−44.2° 11. iS1 = 337E−6−73.6°, iS2 = 401E−6152° 13. vab = 14.3−25.4° 15. vab = 972E−3−166° 17. va = 1.31−174°, vb = 2.58−154° 19. iC = 103E−3101°, iL = 369E−3171° 21. vbc = 19.65−102.5° 23. iE = 14.8E−3177° 25. vb = 10.748.4° 27. iL = 286.7E−340.6° 29. vab = 7.18−115° 31. ZTH = 910°, ETH = 10° 33. ZTH = ZN = j714.3, ETH = 8.5710°, IN = 12E−3−90° 35. ZN = 514.531°, IN = 24E−3−45° 37. vb = 5.69−71.5° 39. ZN = 156417.1°, IN = 19.2E−3−17.1°, ZL = 1495 − j461 41. ZTH = ZN = 64.418.8°, ETH = 113.618.8°, IN = 1.7650°, Combo needed = 61 − j20.7, P = 52.9 W 43. All three pair = 3.33 k + j3.33 k 45. All three pair = 1.33 k − j1 k 47. Za = Zxy = 3.667 k + j3.667 k, Zb = Zxz = 5.5 k + j5.5 k, Zc = Zyz = 11 k + j11 k 49. Za = Zxy = 1.83 k − j1.83 k, Zb = Zxz = 2.75 k − j2.75 k, Zc = Zyz = 5.5 k − j5.5 k 51. vbc = 4.554565.4° 53. iS = 11.8E−3101°, in parallel with a series combination of 1 kΩ and 79.6 nF 55. iS = 1.54E−3−22.7°, in parallel with a series combination of 600 Ω and 2 mH 57. eS = 1.98E3−49.3°, in series with 4.3 kΩ and a capacitive reactance of −j5 k 59. eS = 96.319°, in parallel with a series combination of 9.1 kΩ and 5 mH 414
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61. 63. 65. −j43.8 Ω 415
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6 Nodal and Mesh Analysis 6 Nodal and Mesh Analysis 1. vc = 34.437.7° 3. i43 = 33.2E−344.4° 5. vc = 180−106° 7. i4Ω = 772E−347.1° 9. vac = 18.35−80.8° 11. i22 = 3.07−153° 13. vc = 6.09−3.68° 15. i3.3k = 9.86E−318.8° 17. vc = 1838.9° 19. vbc = 1.0959.8° 21. vba = 964E−3127.3° 23. vcb = 15.2149° 25. Loop ordering is left to right 20° = (4.9 k)i1 − (2.2 k)i2 −30° = − (2.2 k)i1 + (2.2 k − j106)i2 vb = 2.99−3.5° 27. i75 = 8.22E−3−138.5° 29. Loop ordering is left to right 200° = (6.6 k + j4 k)i1 − (2.7 k + j4 k)i2 −50° = − (2.7 k + j4 k)i1 + (4.5 k + j3 k)i2 vb = 7.8498.9° 31. ij200 = 4.36E−3−28.5° 33. vcd = 218E−3−153° 35. i−j200 = 49.5E−3−44.8° (up) 37. vc = 14.4133.3° 39. ij300 = 11.7E−3138° (up) 41. vbc = 4.54565.4° 43. iR3 = 9.81E−3175° 45. vb = 9.24−8.7° 47. i2.2k = 1.51E−3−2.04° 49. vab = 14.3−25.4° 51. i330 = 7.76E−314.1° 53. vbc = 58.9175° 55. vb = 2.6842.3° 57. i5k = 49.9E−6−29.9° 59. vc = 106.790.6° 61. i1k = 17.8E−321.8° 63. va = 9.776°, vb = 0.273−12.3° 65. i1k = 5.24E−332.9° 67. vab = 90.6−112° 69. i1k = 99.95E−60° 71. vd = 16.670.191° 416
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7 AC Power 7 AC Power 1. S = 6.25 mVA, P = 4.88 mW, Q = 3.9 mVAR (ind), PF = 0.781 3. S = 64.4 VA, P = 57.7 W, Q = 28.8 VAR (ind), PF = 0.894 5. S = 315 VA, P = 162 W, Q = 270 VAR (ind), PF = 0.515 7. S = 751 VA, P = 720 W, Q = 216 VAR (cap), PF = 0.958 9. i = 4.286 A, P = 90.9 W 11. P = 6.4 kW, Q = 4.8 kVAR, L = 2.86 mH 13. S = P = 180 W, Q = 0 VAR, i = 1.5 A 15. S = P = 1600 W, Q = 0 VAR 17. S = 1523 VA, P = 1400 W, Q = 600 VAR, PF = 0.919, i = 12.7 A 19. S = 1811 VA, P = 1800 W, Q = 200 VAR, PF = 0.994, i = 15.09 A 21. 89.5% 23. 0.829 25. S = 683 VA, P = 478 W, Q = 488 VAR, i = 5.69 A 27. S = 2403 VA, P = 2332 W, PF = 0.971, i = 20 A 29. S = 3154 VA, P = 2943 W, Q = 1135 VAR, PF = 0.933 31. 141 μF 33. 102 mH 35. 17.6 mH 37. 260 μF 417
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8 Resonance 8 Resonance 1. BW = 14.67 kHz, f1 = 432.7 kHz, f2 = 447.3 kHz 3. Qcoil = 50, Rcoil = 1.88 Ω 5. 15 kΩ || j600 Ω 7. Rp = 20.36 kΩ, Lp = 75 μH 9. f0 = 247 kHz, Q = 10.2 11. f0 = 5.03 kHz, Qsys = 8.35, BW = 602 Hz 13. f0 = 10.07 kHz, Qsys = 4.22, BW = 2.39 kHz, vC = 4.22 15. f0 = 10.07 kHz, Qsys = 3.51, BW = 2.87 kHz, vC = 3.51 17. f0 = 10.07 kHz, Qsys = 4.22, BW = 2.39 Hz, vR= 1.5, vC = 6.32 19. f0 = 136.8 kHz, Qsys = 11, BW = 12.4 kHz, iL = 424E−3−84.8°, iC = 425.5E−390° 21. f0 = 360.4 kHz, Qsys = 24.1, BW = 14.9 kHz, iR = 3.19E−30°, iC = 169.8E−390°, iL = 169.8E−3−88.7° 23. f0 = 142.4 kHz, Qsys = 14.35, BW = 9.92 kHz, iR = 5E−30°, iC = 111.9E−390°, iL = 111.8E−3−88.6° 25. f0 = 225 kHz, Qsys = 25.3, BW = 8.9 kHz, iR = 149E−60°, iC = 12.6E−390°, iL = 12.6E−3−88.4°, vR = 1.787 27. 135 μH 29. 6.94 kΩ 31. Qsys = 50, however, none of the inductors exhibit Qcoil ≥ 50 at 1 MHz. Therefore there are no viable candidates because Qsys can be no larger than Qcoil. 9 Polyphase Power 9 Polyphase Power 1. vLINE = 1200° V (with 120° and 240°, other phases not shown from here on), iLINE = 20.8 A, iLOAD = 12 A, PLOAD = 4320 W 3. vLINE = 208 V, iLINE = iLOAD = 10.4 A, PLOAD = 6490 W 5. vLINE = 120 V, vLOAD = 69.3 V, iLINE = 13.86 A, iGEN-PHASE = 8 A, PTOTAL = 2880 W 7. vLINE = vLOAD = 208 V, iLINE = 6 A, PLOAD = 2160 W 9. vLINE = 208 V, iLINE = 33.4 A, SLOAD = 12 kVA, PLOAD = 11.1 kW 11. vLINE = 120 V, iLINE = iLOAD = 11.14 A, SLOAD = 4 kVA, PLOAD = 3.72 kW 13. vLINE = vLOAD = 398 V, iLINE = 13.8 A, iLOAD = 7.97 A, SLOAD = 9.52 kVA, PLOAD = 7.61 kW 15. vLINE = 120 V, iLINE = iLOAD = 11. A, SLOAD = 4 kVA, PLOAD = 3.72 kW 17. vLINE = 120 V, iLINE = 2.747 A, SLOAD = 571 VA, PLOAD = 566 W 19. 91.5 μF 418
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10 Decibels and Bode Plots 10 Decibels and Bode Plots 1) A) 10 dB B) 19 dB C) 26.99 dB D) 0 dB E) −6.99 dB F) −15.23 dB 3) 33 dB 5) G = 501, Pout = 12.53 W 7) A) 1.06 B) 1 C) 199.5 D) 3.43 E) 0.398 F) 0.188 9) A = 8.57 A' = 18.66 dB 11) A) 0 dBW B) 13.6 dBW C) 8.13 dBW D) −7 dBW E) −26.4 dBW F)30.8 dBW G) −43.5 dBW H) −65.2 dBW I) −172.5 dBW 13) A) 150 dBf B) 163.6 dBf C) 158.1 dBf D) 143 dBf E) 123.6 dBf F) 180.8 dBf G) 106.5 dBf H) 84.8 dBf I) −22.5 dBf 15) G'total = 28 dB, G = 631 17) For P'in = 4 dBm: output stage 1 = 6 dBm, stage 2 = 0 dBm, stage 3 = 15 dBm. For P'in = −34 dBm: output stage 1 = − 24 dBW, stage 2 = −30 dBW, stage 3 = −15 dBW. 19) a. 200 mW 21) V'out = 21 dBV, 21 dBV = 11.2 V (final output) For stage 1: 4 dBV = 1.58 V For stage 2: 9 dBV = 2.82 V 23) a. 15 V 25) At 50 kHz: −0.022 dB, −4.09 degrees At 700 kHz: −3 dB, −45 degrees At 10 MHz: −23.1 dB, −86 degrees Tr = 500 μsec 27) The amplitude portion does not change. Phases are: At 30 kHz −188.5 degrees, at 200 kHz −225 degrees, at 1 MHz −258.7 degrees 419
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29) 31) At 4 kHz: 78.7 degrees, at 20 Hz: 45 degrees, at 100 Hz: 11.3 degrees 33) 35) Net gain at 20 kHz = 35.87 dB Net phase at 20 kHz = 51.7 degrees At 100 kHz: phase = −5.1 degrees, A'v = 40 dB At 800 kHz: phase = −70.5 degrees, A'v = 30.5 dB 37) 420 f A'v 200 kHz 18 dB f θ 0° -180° -270° 2 MHz 20 kHz f A'v 20 Hz 32 dB f θ 0° 90° 200 Hz 2 Hz 45° 20 Hz f A'v 250 kHz 300 kHz 30 kHz
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39) 41) Each lag network rolls off at 20 dB/decade for a 60 dB/decade total (i.e., above 1.2 MHz). 43) 0.775 V 45) 360 W 47) 71.5 dBV 49) Greater than 30 Hz. 421 f A'v 750 kHz 36 dB 1.2 MHz 100 kHz
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Appendix E Appendix E A Closing Observation A Closing Observation This title is the twelfth in a series of free open educational resources, now at five texts and seven laboratory manuals. People occasionally ask why these titles have not gone through the typical route of using a traditional publisher (indeed, the Operational Amplifiers & Linear Integrated Circuits text went that route originally and then reverted to OER for its third edition). Surely, money is to be made, and after all, this is the culture that invented the modern use of the word “monetize”. In a culture that seeks to monetize everything, creating something of value and then giving it away is a subversive act. Some people do not see the incessant search for profit as necessarily a societal good. 422
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DC Electrical Circuit Analysis DC Electrical Circuit Analysis A Practical Approach A Practical Approach James M. Fiore James M. Fiore
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DC Electrical Circuit Analysis DC Electrical Circuit Analysis A Practical Approach A Practical Approach by James M. Fiore Version 1.0.11, 11 January 2024 3
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This DC Electrical Circuit Analysis, by James M. Fiore is copyrighted under the terms of a Creative Commons license: This work is freely redistributable for non-commercial use, share-alike with attribution Device data sheets and other product information are copyright by their respective owners and have been obtained through publicly accessible manufacturer's web sites. Published by James M. Fiore via dissidents ISBN13: 978-1654515478 For more information or feedback, contact: James Fiore, Professor Electrical Engineering Technology Mohawk Valley Community College 1101 Sherman Drive Utica, NY 13501 jfiore@mvcc.edu oer@jimfiore.org For the latest revisions, related titles, and links to low cost print versions, go to: www.mvcc.edu/jfiore or my mirror sites www.dissidents.com and www.jimfiore.org YouTube Channel: Electronics with Professor Fiore Cover art, Chapman's Contribution, by the author 4
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Preface Preface Welcome to DC Electrical Circuit Analysis, an open educational resource (OER). The goal of this text is to introduce the theory and practical application of analysis of DC electrical circuits. It is offered free of charge under a Creative Commons non-commercial, share-alike with attribution license. For your convenience, along with the free pdf and odt files, print copies are available at a very modest charge. Check my web sites for links. This text is based on the earlier Workbook for DC Electrical Circuits, which it replaces. The original expository text has been greatly expanded and includes many examples along with computer simulations. For the convenience of those who used the Workbook, many of the problem sets are the same, with some re-ordering depending on the chapter. The text begins with coverage of scientific and engineering notation along with the metric system. Also included is a discussion of the scientific method, the underpinning of our entire system of investigation and technology. From there, basic concepts and quantities are introduced such as charge, current, energy, power and voltage. Subsequent chapters introduce resistance, series circuits, parallel circuits and series-parallel circuits. The text continues with chapters covering network theorems, more advanced techniques such as nodal and mesh analysis, and finally finishes with introductions to capacitors, inductors and magnetic circuits. The companion AC Electrical Circuit Analysis text picks up after this point. Each chapter begins with a set of learning objectives and concludes with practice exercises that are generally divided into four major types: analysis, design, challenge and simulation. Many SPICE-based circuit simulators are available, both free and commercial, that can be used with this text. The answers to most odd-numbered exercises can be found in the Appendix. A table of standard resistor sizes is also in the Appendix, which is useful for real-world design problems. Finally, the Appendix includes a section reviewing simultaneous equation solutions. If you have any questions regarding this text, or are interested in contributing to the project, do not hesitate to contact me. This text is part of a series of OER titles in the areas of electricity, electronics, audio and computer programming. It includes three other textbooks covering semiconductor devices, operational amplifiers, and embedded programming using the C language with the Arduino platform. There is a text covering AC electrical circuits similar to this one, and also seven laboratory manuals; one for each of the five texts plus individual titles covering computer programming using the Python language, and the science of sound. The most recent versions of all of my OER texts and manuals may be found at my MVCC web site as well as my mirror site: www.dissidents.com This text was created using several free and open software applications including Open Office, Dia, SciDAVis, and XnView. Special thanks to the following individuals for their efforts in reviewing and proofreading the DC and AC Electrical Circuit Analysis texts: Glenn Ballard, John Markham, João Nuno Carvalho, Mark Steffka and Jim Noon. 5
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For Those Without and Those Within “All we can do...is to run over several instances, and examine carefully the principle, which binds the different thoughts to each other, never stopping till we render the principle as general as possible.” — David Hume, An Enquiry Concerning Human Understanding (1748) 6
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Table of Contents Table of Contents Chapter 1: Fundamentals . . . . . . . 10 1.0 Chapter Objectives . . . . . . . . 10 1.1 Introduction . . . . . . . . . 10 1.2 Significant Digits and Resolution . . . . . . 11 1.3 Scientific and Engineering Notation . . . . . . 14 1.4 The Metric System . . . . . . . . 16 1.5 The Scientific Method. . . . . . . . 18 1.6 Critical Thinking . . . . . . . . 21 1.7 RoHS . . . . . . . . . 24 Summary . . . . . . . . . 25 Exercises . . . . . . . . . 26 Chapter 2: Basic Quantities . . . . . . 30 2.0 Chapter Objectives . . . . . . . . 30 2.1 Introduction . . . . . . . . . 30 2.2 An Atomic Model . . . . . . . . 30 2.3 Charge and Current . . . . . . . . 34 2.4 Energy and Voltage . . . . . . . . 36 2.5 Power and Efficiency . . . . . . . . 39 2.6 Energy Cost and Battery Life . . . . . . . 42 2.7 Resistance and Conductance . . . . . . . 47 2.8 Instrumentation and Laboratory . . . . . . 63 Summary . . . . . . . . . 67 Exercises . . . . . . . . . 68 Chapter 3: Series Resistive Circuits . . . . . 74 3.0 Chapter Objectives . . . . . . . . 74 3.1 Introduction . . . . . . . . . 74 3.2 Conventional Current Flow and Electron Flow . . . . . 74 3.3 The Series Connection . . . . . . . 75 3.4 Combining Series Components . . . . . . 77 3.5 Ohm's Law . . . . . . . . . 78 3.6 Kirchhoff's Voltage Law . . . . . . . 82 3.7 Series Analysis . . . . . . . . 83 3.8 Potentiometers and Rheostats . . . . . . 92 Summary . . . . . . . . . 96 Exercises . . . . . . . . . 97 7
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Chapter 4: Parallel Resistive Circuits . . . . . 110 4.0 Chapter Objectives . . . . . . . . 110 4.1 Introduction . . . . . . . . . 110 4.2 The Parallel Connection . . . . . . . 111 4.3 Combining Parallel Components . . . . . . 111 4.4 Kirchhoff's Current Law . . . . . . . 115 4.5 Parallel Analysis . . . . . . . . 117 4.6 Current Limiting: Fuses and Circuit Breakers . . . . . 125 Summary . . . . . . . . . 127 Exercises . . . . . . . . . 128 Chapter 5: Series-Parallel Resistive Circuits . . . . 138 5.0 Chapter Objectives . . . . . . . . 138 5.1 Introduction . . . . . . . . . 138 5.2 Series-Parallel Connections . . . . . . . 139 5.3 Simplifying Series-Parallel Components . . . . . 139 5.4 Series-Parallel Analysis . . . . . . . 143 Summary . . . . . . . . . 155 Exercises . . . . . . . . . 157 Chapter 6: Analysis Theorems and Techniques . . . . 170 6.0 Chapter Objectives . . . . . . . . 170 6.1 Introduction . . . . . . . . . 170 6.2 Source Conversions . . . . . . . . 171 6.3 Superposition Theorem . . . . . . . 177 6.4 Thévenin's Theorem . . . . . . . . 181 6.5 Norton's Theorem . . . . . . . . 187 6.6 Maximum Power Transfer Theorem . . . . . . 188 6.7 Delta-Y Conversions . . . . . . . . 191 Summary . . . . . . . . . 197 Exercises . . . . . . . . . 199 Chapter 7: Nodal & Mesh Analysis, Dependent Sources . . 214 7.0 Chapter Objectives . . . . . . . . 214 7.1 Introduction . . . . . . . . . 214 7.2 Nodal Analysis . . . . . . . . 215 7.3 Mesh Analysis . . . . . . . . 229 7.4 Dependent Sources . . . . . . . . 238 Summary . . . . . . . . . 243 Exercises . . . . . . . . . 244 8
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Chapter 8: Capacitors . . . . . . . 260 8.0 Chapter Objectives . . . . . . . . 260 8.1 Introduction . . . . . . . . . 260 8.2 Capacitance and Capacitors . . . . . . . 261 8.3 Initial and Steady-State Analysis of RC Circuits . . . . . 272 8.4 Transient Response of RC Circuits . . . . . . 274 Summary . . . . . . . . . 285 Exercises . . . . . . . . . 287 Chapter 9: Inductors . . . . . . . 294 9.0 Chapter Objectives . . . . . . . . 294 9.1 Introduction . . . . . . . . . 294 9.2 Inductance and Inductors . . . . . . . 295 9.3 Initial and Steady-State Analysis of RL Circuits . . . . . 303 9.4 Initial and Steady-State Analysis of RLC Circuits . . . . . 306 9.5 Transient Response of RL Circuits . . . . . . 307 Summary . . . . . . . . . 317 Exercises . . . . . . . . . 318 Chapter 10: Magnetic Circuits and Transformers . . . 324 10.0 Chapter Objectives . . . . . . . . 324 10.1 Introduction . . . . . . . . 324 10.2 Electromagnetic Induction. . . . . . . . 325 10.3 Magnetic Circuits . . . . . . . . 329 10.4 Transformers . . . . . . . . 342 Summary . . . . . . . . . 348 Exercises . . . . . . . . . 350 Appendices A: Standard Component Sizes . . . . . . . 355 B: Methods of Solution of Linear Simultaneous Equations . . . 356 C: Equation Proofs . . . . . . . . 361 D: Answers to Selected Odd-Numbered Problems . . . . 363 E: Base Units . . . . . . . . 373 F: Appropriate Commentary . . . . . . . 374 9
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1 1 Fundamentals Fundamentals 1.0 Chapter Learning Objectives 1.0 Chapter Learning Objectives After completing this chapter, you should be able to: • Describe significant digits and resolution. • Express and compute numeric values using scientific and engineering notation. • Describe the metric system and detail its advantages. • Define the scientific method. • Give examples of cognitive biases and logical fallacies that affect critical thinking. • Describe the RoHS directive. 1.1 Introduction 1.1 Introduction A Little Background and Perspective A Little Background and Perspective This text focuses on the analysis of DC (direct current) electrical circuits. It assumes no prior knowledge of electrical quantities, systems or circuit theory. As with any new endeavor, it is important to define the terminology and tools to be used at the outset. We shall be examining the basic electrical quantities, their relationships, proper terminology, and a variety of analysis techniques and theorems that have broad application in the field. In this regard, our analytical “tools” are the appropriate mathematics and standards, and the scientific method, which are detailed in this chapter. The definition of specific electrical quantities and their relationships begins in Chapter Two. Various analysis techniques and theorems are detailed in subsequent chapters. The initial research into electricity occurred in the late eighteenth and early nineteenth centuries by individuals such as Alessandro Volta, André-Marie Ampère, Michael Faraday and Georg Ohm. This work was expanded later in the nineteenth century by Gustav Kirchhoff, James Clerk Maxwell, Léon Charles Thévenin and others. The late 1800s and early 1900s saw the practical application of electrical theory to solve practical problems (i.e., the field of electrical engineering). Perhaps the two names most associated with this period are Thomas Edison and Nikola Tesla. Numerous other individuals made contributions as well, eventually leading to the age of electronics, fully coming into its own by the mid-twentieth century with the introduction of solid state devices such as the bipolar junction transistor. The pace of these developments has been quite rapid. For example, a little over a century ago the average person did not have ready access to something as simple as a modern flashlight. To put this into perspective, if we were to scale all of human history from the emergence of modern homo sapiens to today into a single year; radio, television, digital computers and all the rest would show up only within the last couple of hours before midnight on the last day of the year. 10
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At this point we need to distinguish between electricity and electronics. The term electricity tends to refer to the general relationships between electrical quantities such as voltage and current. In practical use, an electrical system tends to refer to a system where electrical energy is used directly to perform some manner of physical work. Examples include commercial and residential wiring systems for lighting, heating and the like. Electrical power generation and transmission would normally fall into this category, such as the high voltage transmission line seen in Figure 1.1. In contrast, electronic systems, or simply electronics, tends to refer to systems where electrical signals are used to represent, store and/or manipulate some kind of information. This runs the gamut from radio and television to computing devices, cell phones, non-acoustic musical instruments, etc. Some of these applications may be obvious, where the individual interacts directly with the device such as a cell phone or tablet. On the other hand, the human interaction may be minimal such as with the engine management system of a modern car. A good example of a modern device packed with electronics is the DSLR, or Digital Single Lens Reflex camera. These devices include numerous electronic sensors and actuators to adjust to ambient light, for automatic focus, and the like. An example is shown in Figure 1.2. The usage and scale of power transmission lines versus digital cameras may seem to be wildly separated but, ultimately, they are both made possible by the application of the basic laws of electrical circuit theory. Another good example of the shear scale of difference is to look at a single electrical device such as a DC motor. Figure 1.3 shows a simple DC hobby motor, the kind of device that might be used for a motorized toy. This device runs on 12 volts, as available from a battery, and draws only a few hundredths of an amp of current (to put this in perspective, the typical old-fashioned incandescent light bulb draws between one half and one amp of current from a 120 volt socket). Compare this to the industrial DC motor seen in Figure 1.4. This is a 4000 horsepower motor used to reduce eight inch thick copper slabs down to ½ inch thickness. It draws 4680 amps maximum at 700 volts. It clearly dwarfs the technician standing nearby. Now that we have a little background and perspective, it's time to look at mathematics, specifically, how we represent and manipulate values that may be very, very large or very, very small, all while keeping appropriate accuracy. 1.2 Significant Digits and Resolution 1.2 Significant Digits and Resolution A key element of any measurement or derived value is the resulting resolution. Resolution refers to the finest change or variation that can be discerned by a measurement system. For digital measurement systems, this is typically the last or 11 Figure 1.1 345,000 volt transmission lines. Figure 1.2 A digital SLR camera. Figure 1.3 A DC hobby motor.
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lowest level digit displayed. For example, a bathroom scale may show weights in whole pounds. Thus, one pound would be the resolution of the measurement. Even if the scale was otherwise perfectly accurate, we could not be assured of a person's weight to within better than one pound using this scale as there is no way of indicating fractions of a pound. Related to resolution is a value's number of significant digits. Significant digits can be thought of as representing potential percentage accuracy in measurement or computation. Continuing with the bathroom scale example, consider what happens when weighing a 156 pound adult versus a small child of 23 pounds. As the scale only resolves to one pound, that presents us with an uncertainty of one pound out of 156 for the adult, but a much larger uncertainty of one pound out of 23 for the child. The 156 pound measurement has three significant digits (i.e., units, tens and hundreds) while the 23 pound measurement has but two significant digits (units and tens). In general, leading and trailing zeroes are not considered significant. For example, the value 173.58 has five significant digits while the value 0.00143 has only three significant digits (the “143” portion) as does 0.000000143. Similarly, if we compute the value 63/3.0, we arrive at 21, with two significant digits. If your calculator shows 21.0 or 21.00, those extra trailing zeroes do not increase accuracy and are not considered significant. An exception to this rule is when measuring values in the laboratory. If a high resolution voltmeter indicates a value of, say, 120.0 volts, those last two zeroes are considered significant in that they reflect the resolution of the measurement (i.e, the meter is capable of reading down to tenths of volts). When performing calculations, the results will generally be no more accurate than the accuracy of the initial measurements. Consequently, it is senseless to divide two measured values obtained with three significant digits and report the result with ten significant digits, even if that's what shows up on the calculator. The same would be true for multiplication. For these sorts of calculations, we can't expect the result to be any better than the “weakest link” in terms of resolution and resulting significant digits. For example, consider the value 3.5 divided by 2.3. Both of these values have two significant digits. Using a standard calculator, we find an answer of 1.52173913. The result has nine significant digits implying much greater accuracy and resolution than we started with, and thus is misleading. To two significant digits, the answer would be rounded to 1.5. On a long chain of calculations it may be advisable to round the intermediate results to a further digit and then round the final answer as indicated previously. This will help to mitigate accumulated errors. When it comes to addition and subtraction, the larger value will tend to indicate the number of significant digits available, particularly in the laboratory. This is due to the resolution of the measurement (that is, its finest digit). For example, suppose we have to add two distances. We drive a car from a parking lot to its exit, a distance we record with a tape measure as being 51.17 feet. We then drive from the exit to the next city, which the car's odometer records as being 60.0 miles. Is it fair to say the 12 Figure 1.4 An industrial DC motor.
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total distance is 60 miles plus 51.17 feet? No, it is not. Why? Because we cannot expect the car's odometer to be accurate to within 0.01 feet, like the tape measure. In fact, the odometer is reading out a value with one-tenth mile resolution. One-tenth of a mile is 528 feet, or more than ten times the entire measurement given by the tape. We would simply ignore the tape measurement because it is smaller than the resolution of the odometer. The proper result is 60.0 miles. Now in contrast, if the tape measure had indicated 552.7 feet, or slightly more than 0.1 miles, we could safely say the total distance was 60.1 miles, rounding the result to finest resolution digit of the more coarsely measured value. Of course, this example is a bit contrived but it is designed to show the limitations of measurement devices. It is also complicated by the fact that we are using using USA customary unit (miles, feet), complete with their odd conversion factors. Fortunately, virtually all current work in science and technology uses the much simpler metric system, which we will be examining shortly. In summary, • When multiplying and dividing, the value with the least number of significant digits indicates the number of significant digits in the result. • When adding and subtracting, the value with the least resolution (i.e., the most coarse) will set the limit of resolution in the result. Example 1.1 Determine the number of significant digits in the following values. A. 12.6 B. 0.0034 C. 43.001 D. 5400 Answers: A. All three digits are significant. B. Two. Only the 3 and 4 count because leading zeroes are not significant. C. Five. Embedded zeroes are significant. D. Two. Trailing zeroes are not generally considered significant. 13
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Example 1.2 Perform the following computations, leaving the answer with the appropriate number of significant digits. A. 55 ∙ 10.1 B. 2312.5 / 16.2 C. 1756.2 + 345.1 D. 750.2 − 0.004 Answers: A. 555.5 which rounds to 560 (55 has only two significant digits). B. 142.7469136... which rounds to 143 (16.2 has three significant digits). C. 2101.3 (both values have resolution to tenths place). D. 750.2 (do not extend the answer beyond the more coarse resolution of tenths place). 1.3 Scientific and Engineering Notation 1.3 Scientific and Engineering Notation Scientific and engineering notations are ways to express numbers without a lot of trailing or leading zeroes. They also simplify calculations. Before going any further, it would be a good idea to obtain a proper scientific calculator, such as the ones shown in Figure 1.5. The simplest can be had for $10 to $20, while more powerful graphing types will run ten times as much (although good buys can often be found on the used market). Some of the brands to consider are Casio, Sharp and Texas Instruments. If you plan to continue study into AC electrical circuits, it would be wise to make sure the calculator can solve simultaneous equations with complex coefficients. The idea behind scientific notation is to represent the value in two parts: a precision portion, or mantissa; and the magnitude, a power of ten called the exponent. Thus, 360 can be written as 3.6 times 100, or 3.6 ∙ 102, where 3.6 is the mantissa and 2 is the exponent. Similarly, 0.00275 can be written as the value 2.75 times 0.001, or 2.75 ∙ 10−3. As the base of the exponent is always 10, a more compact form replaces “∙ 10” with “E”. Thus, these two values can be written as 3.6E2 and 2.75E−3. When adding or subtracting values in this form, the first step is to make sure that all of the values have the same exponent. Then, the precision portions are simply added together. Thus, 3.6E2 + 1.1E2 is equal to 4.7E2. Similarly, 3.6E2 + 5E1 is converted as 3.6E2 + 0.5E2 yielding 4.1E2. Alternately, it can be converted as 36E1 + 5E1 yielding 41E1 (the same answer, or 410 in ordinary form). Where this notation is 14 Figure 1.5 Scientific calculators.
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particularly useful is when multiplying or dividing. For multiplying, multiply the mantissas and add the exponents. For dividing, divide the mantissas and subtract the exponents. For example, multiply 20000 by 360000. This is equivalent to 2E4 times 3.6E5. The result is 7.2E9 (i.e., 7200000000). Similarly, dividing the value 0.006 by 50000 yields 6E−3 divided by 5E4, or 1.2E−7 (or in ordinary form 0.00000012). Notice the cumbersome and error-prone quantities of trailing and leading zeroes in these examples when using ordinary form. With scientific notation, we no longer have to deal with them. Engineering notation is the same as scientific notation with the caveat that the exponent must by a multiple of 3. Thus, 390000 would be written as either 390E3 or possibly as 0.39E6. Each multiple of 3 has a name and abbreviating letter to make the written representation even more compact. You are probably already familiar with many of them. For example, 103 or 1000, is kilo which is abbreviated as k. Also, 106, or one million, is Mega, and abbreviated M (note capital letter). Exponent Name Abbreviation 12 Tera T 9 Giga G 6 Mega M 3 kilo k −3 milli m (note lower case) −6 micro μ (Greek letter mu) −9 nano n −12 pico p A partial list of exponents, their names, and abbreviations is shown in Figure 1.6. There are standards for both larger and smaller exponents, but this table is something that should be committed to memory as they are in wide use. Example 1.3 Convert the following into engineering notation using appropriate prefixes. A. 2100 grams B. 0.005 meters C. 32,000,000 bits per second (bps) D. 0.0000741 seconds 15 Figure 1.6 Common engineering notation prefixes and abbreviations.
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Answers: A. 2.1 kilograms (2.1 kg) B. 5 millimeters (5 mm) C. 32 Megabits per second (32 Mbps) D. 74.1 microseconds (74.1 μs) Notice that the final answers are much more compact and less error prone. After some initial effort, certain common shortcuts will become second nature. For example (and keeping it generic), micro times kilo will yield milli (i.e., millionths times thousands yields thousandths). Similarly, the reciprocal of kilo is milli and the reciprocal of micro is Mega. Further, milli divided by kilo yields micro, Mega divided by kilo yields kilo, and so on. Gaining facility with these kinds of shortcuts will allow you to make estimates in your head very quickly, a much desired skill. 1.4 The Metric System 1.4 The Metric System In order to make consistent measurements of physical phenomena, some system of measurement needs to be standardized. While almost any scheme can be made to work, some systems are more logical and easier to deal with than others. The current standard in the world of science and technology is the metric system, or more properly, the International System of Units (abbreviated SI, from the French Système International). This standard has found wide adoption across the globe and is the system used by roughly 95% of the human population. This is in contrast to the system of USA customary units which is based on a prior system of units originating in England. It should be noted that the USA, comprising some 5% of the global population, is the only country of reasonable economic power left that has not adopted the metric system for general use. This is only true in the consumer realm though as virtually all scientists, engineers and technicians in the USA routinely use the metric system1. Even many products built and sold in the USA use metric components internally; it's just not noticed by the consumer. Why has the metric system seen such wide adoption compared to USA customary units? The simple answer is because it is an easier system to learn and manipulate. First, a main strength of the metric system is that everything is based on powers of ten. Second, customary units often consist of many different variations on a single parameter while the metric system uses just a single parameter. For example, consider the measurement of length. For human-size lengths we might measure something in inches or feet. If it's a bit larger we might use yards. If it's even farther, we would use miles. In the metric system there is just one unit for length and that's 1 It is also true of consumers who buy soda and similar beverages that are routinely sold in metric two or three liter bottles. Go figure. 16
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the meter (roughly 39.37 inches). If we want to talk about particularly small or large values we would simply add our engineering notation prefixes to come up with millimeters or kilometers for everyday conversion (although if a value happened to be particularly large or small we would say something like 2.3E9 meters). But customary units complicate the system even more because the various versions of a parameter are usually not based on a power of ten. For example, there are 12 inches in a foot and 5280 feet in a mile. This is further complicated by the fact that there is more than one kind of mile (the common statute mile is 5280 feet but the nautical mile is roughly 6076 feet). Similarly, if we measure the weight of something we could use ounces, pounds and tons (more than one kind), and again, the conversion factors are not convenient (16 ounces to the pound, 2000 pounds to the common ton). These odd conversion factors unnecessarily complicate matters. For example, suppose we have a length measurement of 2 feet, 8 ½ inches and we need to multiply this by a factor of 3. Multiplying each part by three yields 6 feet, 25 ½ inches, but that needs to be simplified to 8 feet, 1 ½ inches. This extra step can be avoided if we base everything on powers of ten. For individuals only familiar with USA customary units, the ease of the metric system can be illustrated with currency. USA currency is, in effect, metric: 100 cents make up one dollar. Simply imagine how difficult it would be if, instead of cents, a dollar consisted of 13 flarkneks and 85 dollars made up a skroon. Further, imagine you decide to buy a new computer that costs 21 skroon, 10 dollars and 12 flarkneks. Now compute the total with a 6.5% local tax rate. Will 22 skroon be enough? What's the change? Ouch! It's not impossible but it's extra error prone work. That's precisely the issue with feet, miles, ounces, pounds, pints, gallons and so on. In the metric system typically we use the MKS variation: meters for length or distance, kilograms for mass and seconds for time. Further, we typically use liters for volume (liquid measure) and Celsius (AKA Centigrade) for temperature, although it is sometimes more convenient to use the absolute temperature scale of kelvin. Fortunately, the thermal energy indicated by a one degree Celsius change is identical to that for kelvin. The two schemes only differ in their zero reference point (absolute zero for kelvin and the point at which water freezes for Celsius). As virtually all of the calculations presented in this text are metric, there is little need to concern ourselves with conversions back and forth between the two systems. Indeed, if the USA were to finally make the switch to metric, no one would ever need to concern themselves with conversions, with the possible exception of historians, and perhaps people trying to recreate old family recipes and the like. Sometimes, careless people tend to think of units of measure as an afterthought. Don't get caught in this trap. Failure to pay attention to proper units can have catastrophic results. One example is the Mars Climate Orbiter. In 1999 this mission 17
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to study the planet Mars suffered a spectacular failure. A subcontractor had used USA customary units for its software which then fed values to other systems that were expecting metric/SI units (as specified in the system contract). The result was the destruction of the orbiter as it attempted orbital insertion. The cost of the mission was nearly $330 million, completely wasted. A table of commonly used quantities, their SI units and typical equivalents is shown in Figure 1.7. Further details on units can be found in Appendix E. Quantity (abbreviation) Unit (abbreviation) Approximate Equivalent Length (l) meter (m) 1 m ≈ 39.37 inches Mass (m) kilogram (kg) 1 kg ≈ 2.2 pounds (on Earth) Time (t) second (s) NA Force (F) newton (N) 1 N ≈ 0.225 pound-force Energy (w) joule (J) 3.6E6 J ≈ 1 kWh Power (P) watt (W) 746 W ≈ 1 horsepower Temperature (T) kelvin (K) or Celsius (°C) Fahrenheit = 32 + 9/5 ∙ °C kelvin = °C + 273.15 Electric Charge (Q) coulomb (C) NA Current (I) amp (A) NA Voltage (V or E) volt (V) NA 1.5 The Scientific Method 1.5 The Scientific Method The scientific method is a means of uncovering and explaining physical phenomena. It relies on observation and logical reasoning to achieve insight into the actions and relations of the phenomena, and a means of relating it to similar items. The method starts with observations and measurements. Based on these and background knowledge such as mathematics, physics, the behavior of similar phenomena, and so on, a tentative explanation, or hypothesis, is derived. The hypothesis is then tested using experimental or field data. The precise nature of the tests depend on the area of study, but the idea will always be the same: namely to see if the predictions of the hypothesis are borne out under alternate applicable conditions or data sets. A proper hypothesis must be testable and falsifiable. That is, the hypothesis must be able to be proven wrong by subsequent observation. If it is not, then the hypothesis is better classified as philosophy. For example, Newtonian gravitation could be proven false by letting go of a suspended hammer and watching it remain motionless or fall upwards, rather than fall down toward the Earth. Similarly, Evolution could be proven false by the discovery of “fossil rabbits in the Precambrian”, to quote famous 18 Figure 1.7 Common quantities, SI units and equivalents. Note: As a general rule, when units are abbreviated they are capitalized. When they are spelled out they are treated as common nouns and therefore not capitalized, even when named in honor of someone (an exception being some temperature scales).
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biologist J. B. S. Haldane (Haldane was being somewhat snippy, but in general, he meant anything that would be clearly out of the expected time-line, in this case a small mammal predating even the most simple creatures with backbones). A hypothesis is tested by repeated cycles of prediction, observation and measurement, and also subject to peer review, that is, the critique of others in the field of study who may point out inconsistencies in the explanations or logic, errors in experimental design, and so on. This cycle continues until the weight of data and scientific opinion elevates the hypothesis to a theory. One might say that a theory is a hypothesis that has withstood the rigors of the scientific method. This cycle was well expressed by the Marquise du Châtelet2. She explained hypotheses as “probable propositions”. While it would only take one observation to falsify a hypothesis, several would be required to vindicate it: “each non-contradictory result would add to the probability of the hypothesis and ultimately…we would arrive at a point where its ‘certitude’ and even its ‘truth’, was so probable that we could not refuse our assent”. It is important to note that the scientific usage of the word theory is entirely different from its popular usage, which is perhaps closer to “hunch” or “seat-of-the-pants guess”. Also, a scientific theory is not true in the same sense as a fact. Facts come in three main varieties: direct observations, indirect observations and those that may be logically deduced. A direct observation is something that you have measured yourself, such as noting the time it takes for a ball to reach the ground when released from a given height. An indirect observation is something that may be inferred from other known quantities or proper historical data, such as “George Washington was the first president of the United States of America”. An example of the third variety would be “If x is an even integer larger than four and smaller than eight, then x must be six”. At first glance, it may seem that facts are highest on the pecking order, but scientific theories are much more useful than facts because isolated facts have very little predictive capacity. It is the predictive and explanatory ability of theories that make them so powerful. Consider the following. Suppose you hold out a stone at arm’s length and let go. It drops to the ground. That’s a fact. You saw it happen. Unfortunately, by itself, it doesn’t tell you very much. Suppose you repeat this several times and each time the stone drops in precisely the same way as it did initially. This is beginning to get useful because you’re noticing a pattern, and patterns can be predictive. Now, suppose you pick up stones of differing sizes, say 100 grams, 200 grams, half a kilogram and a kilogram, and drop each of them in turn. You observe that they each hit the ground in the same amount of time. Further, you drop them from different heights and you notice that the higher up they are, the longer it takes for them to hit the ground, but they all take the same amount of time to reach the ground. 2 Du Châtelet was that most rare of 18th century women: a mathematician and physicist. She translated Newton’s Principia Mathematica into French and was a companion of Voltaire. Unfortunately, after an affair with the poet Jean François de Saint-Lambert in her early 40’s, she became pregnant and died six days after giving birth. 19
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You might now formulate a hypothesis: namely that the mass of a stone doesn’t have an effect on how fast it falls from a given height and that height and fall time are directly related. Your hypothesis is predictive. Although you used only four sizes of stones and a few heights, your broadened hypothesis should apply to any stone dropped from any height. So now you (and a bunch friends) starting picking up random pairs of stones and drop them from random heights, and sure enough, you see the same effect again and again. If you do this enough and it is continually verified without exception, you might even make a “law of falling stones”, particularly if you were able to quantify the times and heights through careful measurement and reduce the relation to a nice formula. It is useful because you can now predict what will happen with any stone dropped from any height. But this law is rather limited. It only applies to stones because you may have noticed that stones drop much faster than pieces of cork. While you might then proceed to make a “law of falling cork”, that would unnecessarily complicate things. Instead, you could take a step back and try to figure out why stones and cork both fall, but not at the same rate. Eventually, you might discover that the difference has to do with air friction and you can now create a law governing falling bodies in a frictionless environment. That’s even more useful than the original “law of falling stones”. But even this new and improved “law of falling bodies” doesn’t offer a lot of insight into what is really going on in the larger scheme of things. Through repeated observations and experiments this could be extended to cover not just falling bodies on the Earth, but the interactions between any bodies, including falling stones and cork on the moon, or the interaction between the Earth and the Sun, the Sun and the other planets, the Sun and other stars, and so on. What you’ll have arrived at is a full-blown theory of gravitation (Newtonian gravitation). Now that is an extremely useful tool. It helps us design airplanes, get satellites into orbit, even get people to the moon and back safely. Thus a theory is a “best estimate so far”, a model to explain some observable aspect of the universe. It is understood that as our view of what is observable widens and our knowledge extends, so too a given theory may be refined. The Newtonian gravitation model was sufficient to describe the movements of the planets around the sun and is still used to plan the flight of spacecraft. In the early 1900’s, however, Einstein’s Theory of Relativity refined Newtonian Gravitation to include more extreme (relativistic) effects. It is important to note that we did not “throw out” Newton’s equations; rather, we now have a better understanding of the set of conditions under which they apply. While this trek towards more and more refinement is not truth in and of itself, to paraphrase the late Harvard paleontologist, Stephen Jay Gould, to ignore the weight of scientific data behind an established theory would be perverse. 20
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1.6 Critical Thinking: Avoiding Being Fooled 1.6 Critical Thinking: Avoiding Being Fooled As humans, we need to recognize that we are fallible. No matter how good our intentions, we make mistakes and can be fooled. The first step toward reducing and ultimately eliminating these sources of error is to understand them. We will lump these into two broad categories: cognitive bias and logical fallacies. Understanding these will enhance our ability to think critically and avoid being fooled (or fooling ourselves). Cognitive Bias Cognitive Bias A cognitive bias is an inclination toward looking at a situation in an unbalanced or unfair manner. Generally, people are not aware of these biases. One example is confirmation bias (AKA, confirmation of expected outcomes). That is, we expect (or hope) to see a certain result and thus we tend to overvalue evidence that confirms it while discounting evidence that contradicts it. One way to avoid this is through the use of a double-blind test. Suppose we wish to test a new drug to see if it is effective and safe. As we may have invested a lot of time and money developing the drug, it is only natural that we want it to work, and this may skew our analysis (unintentionally, of course). What we do is have a third party create two sets of pills; one is the drug under test and the other is a placebo (it looks like the other pill but does nothing). These sets are identified using codes known only to the third party. The sets are then given to the researchers who, in turn, give them to the patients. The important thing is that neither the patients nor the researchers know which pills are which. When the trial has run its course, the researchers (us) analyze the data to determine if any set of pills was successful. Only after the analysis is completed does the third party tell the researchers which set was real and which set was the placebo. Another cognitive bias is the Dunning-Kruger effect, named after the two social psychologists who studied it. This states that the knowledge needed to determine if someone is competent in a certain field is competence in that same field. Thus, individuals who have low competence are not in a position to accurately evaluate their own level of competence. Consequently, these individuals often over estimate their competence. This is known as illusory superiority. To put it crudely, these individuals are too ignorant of the subject to understand just how ignorant they are. Among the highly competent, two other effects may be seen. First, the advanced individual may be keenly aware of any shortcomings or gaps in their competence and may undervalue their level as a result. Second, they may assume that their level of competence is typical, and that most people are therefore “at their level”. It is useful to remember that in our complex and interdependent society, no one can be an expert at everything, or even at most things. Instead, it is likely that we are all largely ignorant of a majority of subjects and/or incompetent at a variety of skills. 21
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Logical Fallacies Logical Fallacies Logical fallacies represent faulty reasoning. They are “thinking traps” that people sometimes fall into. Familiarity with them will help reduce their occurrence. There are dozens of logical fallacies but we shall only investigate a representative few. To help explain the process, we'll begin with a syllogism. This is, in essence, a simple argument. It starts with a major premise (a generalization) which is followed by a minor premise (a more specific statement). From these, we derive a conclusion. For example: All humans breathe air. (major premise) Alice is a human. (minor premise) Therefore, Alice breathes air. (conclusion) Errors can occur when either premise is false or when the conclusion does not follow (the latter being referred to as a non-sequitur). For example: All fish live in water. Lobsters live in water. Therefore, lobsters are fish. The problem with this is the linkage between the major and minor premises. Saying “all fish live in water” does not preclude something else (like a lobster or a sea snake or kelp) from living in water. Compare the prior example to this version: All fish live in water. Trout are fish. Therefore, trout live in water. While these examples may seem obvious, there are trickier versions. For example: I am made of nothing but atoms. Atoms are not alive. Therefore, I am not alive. Nope. Doesn't work. This error is called the fallacy of composition. Basically, it says that what is true of the parts must be true of the whole, and vice versa. It ignores the concept of emergent properties (consider the behavior of a single bird to a flock, or a single fish to a school). The fallacy of composition can be illustrated without using a syllogism. Suppose you are in a crowded movie theater. If you stand up, you will have a better view of the screen. In contrast, it is not true that if everyone stands up, everyone will have a better view. In fact, everyone will most likely have a worse view. If one person 22
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stands, they are unique. That unique character is lost when everyone stands. Turning to a different fallacy, the Latin phrase post hoc ergo propter hoc can be translated as “Before this, therefore because of this”. This fallacy is sometimes referred to as the post hoc fallacy or the causation fallacy. It is an error regarding causality; the assumption being that because event A occurred before event B, then event A must have caused event B. On the surface, it seems logical enough. For example, you might see a lightning strike and then hear a clap of thunder. It seems reasonable to assume that the lightning caused the thunder (generally speaking, that is the case). On the other hand, you might wake up some morning when it's dark outside. Shortly thereafter, the sun rises. Obviously, your waking did not cause the sun to rise. Another error involves proportional contribution to an outcome. Relative size is mistakenly seen as a determiner. That is, the error assumes that only large contributors have any sway in the outcome. Basically, this fallacy proposes that if something makes up only a small percentage of the total, then its effect must be minimal. This is easily proven wrong. As an example, the atmosphere of the Earth is comprised largely of nitrogen (78%) and oxygen (21%) along with a number of trace gases such as argon, carbon dioxide and so forth. If the atmosphere was suddenly altered so that it included just 0.1% hydrogen sulfide, every human likely would be dead after their first breath of this new combination. Along with size, there is a similar issue regarding linearity of effect. A linear function is one that can be plotted as a straight line. More to the point, if we have a linear system, then doubling an input to that system doubles its effect. To wit, if you order two pieces of pie for dessert, it will cost twice as much as one piece. The reality is that many systems do not behave in a linear fashion. Systems or relationships can be logarithmic, square law, cubic, or follow some other characteristic. For instance, the braking distance of a car does not vary linearly with its speed, it varies in accordance with the square of its speed. Therefore, if you're traveling twice as fast, it doesn't take twice as far to come to a stop, it takes four (two squared) times farther to stop. Remember this the next time you're speeding down a highway. The next two items sometimes appear in arguments for or against a proposition. They are the excluded middle and the ad hominem. The excluded middle presents a false set of choices. Essentially, it falsely reduces the set of possible outcomes and then proceeds to disprove all but one of them. By process of elimination, the remaining outcome should be true. For example, someone might complain that a particular politician would only support a particular bill if said politician was either stupid or a communist (or a fascist, take your pick). They then show that the politician is not stupid, so by their logic the politician must be a communist. Of course, there are any number of other possible scenarios that have been excluded; for instance, the politician might have taken a hefty bribe to vote for the bill or the analysis of the bill by the complainer might be faulty. 23
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Finally, ad hominem is a Latin term meaning “to the person”. The ad hominem attempts to disprove a point by arguing against the person making a claim, not the claim itself. For example, suppose Doug makes a claim in favor of a new theory of gravity. Fran's counterargument is that Doug is an evil person because he likes to spray paint foul words on other people's pet cats, and therefore he can't be trusted. The reality is that, in spite of his proclivity for penning feline profanity, Doug's ideas regarding gravity might be spot on. Those ideas need to be addressed directly. 1.7 RoHS 1.7 RoHS Electrical components are not without a downside. Some devices potentially may contain hazardous substances such as toxic metals. Even if the devices are constructed and used with care, these substances can still create life cycle issues; that is, when their useful life is over, simply “throwing them away” can create environmental contamination. In the early 2000s the European Union ratified the RoHS Directive. This stands for the Restriction of Hazardous Substances, and controls the use of several materials used in the construction of electrical and electronic systems. These materials include lead, mercury, cadmium, hexavalent chromium (used in chrome plating), and other specific chemicals such as certain flame retardants and plasticizers. The RoHS Directive specifies the amount of these materials allowed in any homogeneous material that makes up the device or product. Homogeneous materials, are, in essence, the smallest parts that could be separated mechanically. Examples would be items such as a machine screw, the toggle lever of a switch, or the magnet of a loudspeaker. Each of these items must individually meet the limits set by the directive. Most of these substances have limits of 1000 ppm (parts per million) although cadmium (used in rechargeable Ni-Cd batteries and certain light detectors) has a limit of 100 ppm. The RoHS Directive has been updated periodically. The third variation (RoHS 3) became effective in the summer of 2019. Related to RoHS is WEEE, which stands for Waste from Electrical and Electronic Equipment. This controls the recovery and recycling of these devices. Generally, products sold in the EU market must meet the WEEE requirements. Several countries outside of the EU have adopted similar requirements including Japan, China, South Korea and Turkey. The USA has not adopted RoHS per se, however, several US states have adopted “RoHS-like” requirements, including California, Colorado and New York. With increasing environmental consciousness, it is likely that more countries will adopt similar measures. To help electronic and electrical system designers make their products meet the RoHS requirements, electrical component manufacturers often use some manner of “RoHS compliant” logo or statement on their data sheets. There is no standard for 24 Figure 1.8 RoHS compliance label.
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these logos, however, a finished product will bear the standard RoHS compliance label, as shown in Figure 1.8. 1.8 Summary 1.8 Summary An electrical system is one that refers to the direct use of electrical energy in terms of power generation, transmission or application. In contrast, an electronic system is one that tends to use electrical signals as a means to represent, store and manipulate information. When measuring physical quantities and in computations, care must be taken to maintain accuracy. Of particular importance are the concepts of resolution and significant digits. Resolution refers to the finest digit place that may be discerned (e.g., tenths or hundredths place). Generally, accuracy increases with the number of significant digits used to represent a value. Leading and trailing zeroes are not considered significant, the exception being trailing zeroes as reported by a measurement device such as a digital multimeter. In computations, results can be no more accurate or have greater resolution than the original values being manipulated. Thus, if the division of two values, each with three significant digits, yields a calculated result with much more than three significant digits, those extra digits only lead to a false sense of accuracy and resolution. As a general rule, the results of multiplication and division will have as many usable significant digits as the least precise of the original values. For addition and subtraction, the resolution of the result can be no greater than that of the original value with the least resolution. Scientific and engineering notations are ways of representing values in a compact and less error prone form. They consist of two parts: a mantissa consisting of the significant digits and an exponent, or power of ten, for scaling. Engineering notation further stipulates that the exponent must be a multiple of three, and these multiples use prefix names such as kilo and milli for further simplification. When multiplying or dividing values, the mantissas are multiplied (or divided) while the exponents are added (or subtracted). For addition and subtraction the values are adjusted so that they have the same exponent and the mantissas are then added or subtracted as needed. The metric system (or SI) is a measurement system based on powers of ten. Conversions between units of the same type are no longer necessary which simplifies computations. It is a global standard that is ubiquitous in the field of science, engineering and technology. The scientific method is a technique used to uncover the reality behind the physical world. It begins with observations of phenomena which then lead to a tentative hypothesis. The hypothesis is tested experimentally with the results either confirming or denying the hypothesis. The process is repeated creating a feedback 25
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loop. If the hypothesis is repeatedly verified and not rejected, it may be elevated to a scientific theory. It is important during this process to be aware of logical fallacies and cognitive biases that can lead to false interpretations of the experimental results. Finally, RoHS, or the Restriction of Hazardous Substances, is a European Union directive aimed at reducing environmental contamination by restricting the use of certain substances such as lead and mercury. RoHS has a direct impact on the construction and disposal of electrical and electronic products. Other countries and states have similar directives and restrictions in place to help protect the environment. Review Questions Review Questions 1. Describe the differences between scientific notation and engineering notation. 2. List the terms and abbreviations (i.e, words and symbols) for engineering notation from 10−12 through 1012. 3. Give at least one advantage of the metric system over the customary/Imperial system of measurement. 4. Outline the process of the scientific method. 5. Explain the differences between a scientific theory, a hypothesis and a fact. 6. Describe at least three examples of cognitive biases and logical fallacies. 7. What is RoHS? 1.9 Exercises 1.9 Exercises Analysis Analysis 1. Round the following to four significant digits: a) 14.5423 b) 30056 c) 76.90032 d) 0.00084754 2. Round the following to three significant digits: a) 354.005 b) 9100.46 c) 1.0054 d) 0.000052753 3. Round the following to five significant digits: a) 5.100237 b) 1020.765 c) 1.00540 d) 0.00004578053 4. Round the following to four significant digits: a) 5.100237 b) 1020.765 c) 1.00540 d) 0.00004578053 5. Convert the following to scientific notation: a) 23.61 b) 12000 c) 7632 d) 0.00509 6. Convert the following to scientific notation: a) 4253 b) 640000000 c) 2.03 d) 0.00000658 26
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7. Convert the following to scientific notation: a) 41.56 b) 954000 c) 84.035 d) 0.0001632 8. Convert the following to scientific notation: a) 11200 b) 30000000 c) 325.2 d) 0.00002504 9. Convert the following to engineering notation: a) 12000 b) 470 c) 6.5 d) 0.00198 10. Convert the following to engineering notation: a) 3500 b) 17.9 c) 5601000 d) 0.0000355 11. Convert the following to engineering notation: a) 33.2 b) 313.6 c) 43000 d) 0.000076 12. Convert the following to engineering notation: a) 0.23 b) 76.95 c) 45500 d) 0.00890 For problems 13 through 18 express the results using engineering notation without rounding or truncating. 13. Compute the following: a) 1.2E2 times 3.0E4 b) 5.4 times 3.1E3 c) 6.01E3 times 2.0E−1 d) 5.3E9 times 4.1E−5 14. Compute the following: a) 7.1E1 times 4.0E2 b) 2.32 times 5.6E3 c) 6.01E3 times 3.0E−2 d) 5.2E5 times 8.2E−7 15. Compute the following: a) 8.2E1 / 4E2 b) 2.42 / 2.0E3 c) 6.09E3 / 3.0E−2 d) 9.6E5 / 3.1E−7 16. Compute the following: a) 4.6E2 / 2.0E2 b) 2.35 / 4.0E6 c) 5.15E4 / 3.0E−3 d) 6.8E−2 / 5.0E−4 17. Compute the following: a) 8.2E1 + 4E2 b) 2.42 + 2.0E1 c) 6.09E3 − 3.0E−2 d) 9.6E5 − 5.1E6 18. Compute the following: a) 4.2E3 + 9E2 b) 3.53 + 4.2E−1 c) 1.05E−2 − 5.0E−2 d) 9.4E5 − 3.2E4 For problems 18 through 22 express the results using engineering notation with proper rounding to reflect the resulting resolution. 19. Compute the following: a) 16.2 − 0.4 b) 4356 + 378 c) 0.012 − 0.005 20. Compute the following: a) 4.5 ∙ 43.1 b) 1201 / 23.6 c) 890.1 ∙ 0.172 21. Compute the following: a) 51E3 ∙ 62.7E2 b) 6.733E2 / 1.01E−3 c) 20.12E6 / 65.6E9 22. Compute the following: a) 17.9E3 − 10.4E6 b) 81E3 + 12E3 c) 76.0E3 + 1465 27
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For problems 23 through 26 determine the result of the computation in SI base units (meters, kilograms, seconds) with appropriate engineering notation units (kilo, milli, etc.). 23. Determine the result of 20 meters times 0.10 kilograms in kg·m. 24. Determine the result of 34 kilometers divided by 10 millimeters. 25. Determine the resulting velocity in m/s if a distance of 3.6 meters is covered in the space of 1.2 milliseconds. 26. Determine the result of 24 μs divided by 3.0 km. 28
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Notes Notes ♫♫ ♫♫ 29
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2 2 Basic Quantities Basic Quantities 2.0 Chapter Learning Objectives 2.0 Chapter Learning Objectives After completing this chapter, you should be able to: • Describe a basic, functional atomic model. • Describe fundamental quantities and relations including charge, current, energy, voltage, power, resistance and conductance; and perform basic computations using them. • Compute efficiency, energy cost, battery life and DMM accuracy. • Utilize the resistor color code. 2.1 Introduction 2.1 Introduction In this chapter we shall examine the major quantities of interest in electrical systems and describe their relationships. These include voltage, current, power, energy and resistance. We begin with basic definitions and proceed to example computations. We shall go beyond the theoretical and investigate practical aspects as found in electrical laboratories. This includes measurement devices and power sources, as well as various resistive devices. We shall examine batteries from a functional perspective, and perform computations regarding efficiency and cost of energy. We will also introduce some of the device symbols used to create electrical circuit schematics. This material presents the foundation upon which we will build circuit analysis techniques in subsequent chapters. Treat it accordingly. 2.2 An Atomic Model 2.2 An Atomic Model In order to understand electrical circuit behavior we must first define what it is that we are chasing. Ultimately, electrical circuits are all about the movement of electric charge. What do we mean by charge? To answer that question, we need to come up with a usable model of an atom, in other words, a functional description of its internal structure. To be sure, atoms are far smaller than most people imagine. There are far more atoms in a gallon of water than there are gallons of water in all of the oceans, lakes and rivers of planet Earth combined. It would be pointless to ask what an atom might “look like” because its components are all smaller than the shortest wavelengths of light that humans can see. That would be a little like asking a human what a dog whistle sounds like, when in fact it's at a pitch that's higher than humans can hear. Ultimately though, all we really need is a model to explain its observed behavior. Perhaps the most prolific model in the popular imagination is the planetary model shown in Figure 2.1, so named because it is reminiscent of simple models of our solar system. In this model, the core, or nucleus, is drawn at the center and contains positively charged protons and non-charged neutrons. Revolving around this 30
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core are negatively charged electrons, each following a nice, regular, planar path much like a planet around the sun. Unfortunately, this model is not accurate to say the least, although its popularity can sometimes lead to wildly erroneous and darkly humorous conclusions. In spite of its inaccuracy, the components (nucleus, protons, neutrons and electrons) are perfectly valid. Before we come up with a more accurate and useful model, let's take a closer look at these items. Unlike people, all particles of a type are indistinguishable: every proton is identical to every other proton and every electron is identical to every other electron. The number of protons determine what the element is. The simplest element, hydrogen, has a single proton in its nucleus. In contrast, helium has two protons in its nucleus and copper has 29. The nucleus may also contain a number of neutrons as well (not necessarily equal to the number of protons). In any stable atom, the number of electrons will equal the number of protons. If they aren't the same, the atom is called an ion (technically, a cation has fewer electrons than protons and an anion is its opposite). With an external energy source, it is possible to separate an electron from an atom. This leaves us with an electron and an atom that now has a net positive charge. It is this ability to separate these particles, and thus separating charge, that gives rise to our concept of electricity including such items as current and voltage, as we shall see. Most of the mass of any given atom is from the protons and neutrons. Protons and neutrons have similar masses, about 1.67E−24 grams each. The mass of an electron is roughly 2000 times smaller. The radius of a proton is approximately 0.87E−15 meters and the mean distance to the nearest electron is about 5.3E−11 meters. Therefore an electron is about 60,000 times farther away from the proton than the size of said proton. To put this into perspective, that's roughly the same as the ratio between a golf ball and a sphere with a radius of 3/4ths of a mile or 1200 meters. This would be the case for a hydrogen atom as it consists of a single proton and electron. The magnitude of this ratio is not much different for other substances, including things like diamond and quartz that are very hard and solid. If you think about that for a moment, you realize that the idea of “solidity” is in some ways an illusion because the vast majority of what we call “something” is really just empty space (like a golf ball rattling around inside a sphere that's a mile and a half wide). In reality, the feeling of solidity when you grab something with your hand is just the result of the interaction of atomic forces between your hand and the object. Indeed, if we could somehow get rid of that “empty space”, we could fit a mountain inside of a car (although it would probably require a severely upgraded suspension). For our immediate work, perhaps the most important issue is that of charge, an essential part of the theory of electromagnetism. It was mentioned that protons are positively charged and electrons are negatively charged. What then is charge? Charge is not an obvious physical attribute like someone's height or eye color. It's closer to a behavioral characteristic. Saying a particle is charged is like saying someone has an attractive personality. The magnitude of the charge on an electron is the same as the magnitude of the charge on a proton. The only difference is the 31 Figure 2.1 Planetary atomic model: Popular but sadly incorrect. Image source (modified)
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polarity: protons are positively charged while electrons are negatively charged. An important thing to remember is that particles with the same polarity repel each other while opposites attract3. Charge is measured in coulombs, named after Charles-Augustin de Coulomb a French physicist of the eighteenth century. The charge on a single electron is tiny, a mere 1.602E−19 coulombs. Alternately, it takes 6.242E18 electrons to create a charge of one coulomb. More on this in a moment. One of the major issues with the planetary model is the idea that electrons whirl around the nucleus in stable, planet-like orbits. That's simply not true. First, the electron inhabits a region of 3D space and does not simply move through a plane. Second, due to the Heisenberg Uncertainty Principle, we can't precisely plot the position and trajectory of a given electron. The best we can do is make a plot of where the electron is likely to be. This is called a probability contour. To understand this concept, imagine that you could record the position of a specific electron relative to the nucleus at a specific time. A moment later you record its new position, a moment after that you record the next position, and so on for thousands of measurements. If you attempted to plot them all, you would wind up with a cloud of dots around the nucleus. This cloud is referred to as an orbital. You wouldn't know how the electron got from one position to the next but you would get a general idea of where it was likely to be. Ultimately, an orbital looks nothing like a planetary orbit. A good analogy to an electron probability contour is the mapping of the flight of migratory birds. For example, a certain group of Canada geese might migrate from their summer range in the far north of Canada to their summer abode in the southern USA via s section of the Atlantic Flyway, perhaps through central New York State, as shown in purple, in Figure 2.2. The flyway tells you where these birds are likely to be found while migrating. If they are taking the route through central New York, at some point it is likely that any particular goose in that flock will be found near the cities of Utica, Syracuse and so forth. It is much less likely that any particular goose from that flock will be found in Ohio or New Hampshire. Texas, of course, is right out. The flyway gives a good bit of data regarding where these birds are likely to be but in no way can it predict with any accuracy precisely where any individual bird will be on a specific date, nor precisely where it will end its migration. Returning from our avian diversion, we note that there are several potential orbitals. Due to quantum physics, only certain orbitals are allowed. The permissible electron energy levels are first grouped into shells, then subshells and finally orbitals. It is important to remember that orbitals indicate the electron energy level. That is, a higher orbital implies a higher energy level. Further, orbitals fill in first from lowest energy level to highest energy level. 3 The obvious question might be “Why doesn't the nucleus break apart?” The answer is that the sub-atomic strong nuclear force has a greater effect at atomic distances than electromagnetism, and thus binds the protons and neutrons together in the nucleus. 32 Figure 2.2 Electron behavior versus migratory birds. Figure 2.3 Electron probability contour for innermost orbital, 1s. Image source
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Shells are denoted by their principal quantum number, n; 1, 2, 3, etc. The higher the number, the more subshells it can contain. Subshells are designated by letters, the first four being s, p, d, and f. Shell 1 contains only subshell s while shell 2 contains subshell types s and p. Shell 3 contains subshell types s, p and d, and so on. These subshells may also have variations within them. There is one variation on s, three variations on p, five variations on d, etc. These variations are the orbitals and each orbital can hold a maximum of two electrons. Putting this all together, we find that the first shell can contain a maximum of two electrons: two in the single s subshell orbital (1s). The second shell can contain a maximum of eight electrons: two in the s subshell (2s) plus two in each of the three p subshell orbitals (2p). In like manner the third shell can contain a maximum of 18 electrons: two in 3s, six in 3p and two in each of the the five d subshell orbitals (3d). You can condense this into a simple formula: Number of electrons per shell = 2n2, where n is the shell number Figure 2.3 shows the electron probability contour of the innermost orbital, namely 1s (i.e., principle quantum number 1, subshell s). As you can see, it is spherical in shape. The nucleus is located at the center, obscured here. All s orbitals are similarly spherically shaped although the internals change. 1s is the lowest energy orbital. Orbitals are not limited to simple spherical shapes. Higher order orbitals can take on a variety of forms. Figure 2.4 shows a much more complex electron probability contour. The nucleus is situated in the center of the concentric rings. Obviously, this is nothing like the well-behaved elliptical orbits of planets around the sun (nor the paths of migratory birds, unless they are very confused). As visually interesting as these graphics are, they are cumbersome to work with. Consequently, a more functional graphic is called for. Such a device is the Bohr model, named after Danish physicist Niels Bohr. An example is shown in Figure 2.5. It is important to understand that the Bohr model is an energy description of the atom, not an attempt to mimic its physical appearance or structure. The nucleus is placed at the center. It is surrounded by concentric rings that represent the electron shells. The higher the number, the larger the ring and the greater the energy level. If an electron were to move from a higher level to a lower level, the energy difference is radiated out. This could be in the form of heat or light. This is a point worth remembering. For example, this transition is what makes light emitting diodes (LEDs) function. The inverse is also possible, namely that by absorbing energy, an electron can move into a higher orbital. This is an equally powerful concept. Using the Bohr model we can create diagrams to represent individual elements. For example, copper has an atomic number of 29 meaning that it has 29 protons and 29 electrons. The electron shell configuration is 2-8-18-1. That is, the first three shells are completely filled and there is a single electron in the fourth shell. The Bohr 33 Figure 2.4 A higher order electron probability contour. Image source Figure 2.5 Generic Bohr model. Image source
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model for copper would simply show four rings, the first three being filled and with a single electron in the fourth ring. This is shown in Figure 2.6. In this version, the individual electrons are drawn in each shell and the atomic number is indicated at the nucleus. Again, please do not imagine this representing individual electrons orbiting the nucleus in lanes. This is an energy level depiction. It is worth noting that a lone outer electron is only loosely bound to the nucleus thus making the movement of charge through said material relatively easy, given the application of some external energy source. Consequently, copper is a very good conductor. This is a characteristic shared by common metals: they are good conductors because they only have one or two loosely held electrons in their outer shell. Examples include silver, gold and aluminum (one outer electron for each, all of them being very good conductors); and iron, tin and nickel (two outer electrons for each, and this group being not quite as conductive as those in the first group). 2.3 Charge and Current 2.3 Charge and Current As already noted, charge is an attractive force. It is denoted by the letter Q and has units of coulombs. Electrons are negatively charged and protons are positively charged. All electrons and protons exhibit the same magnitude of charge, roughly 1.602E−19 coulombs. Thus, one coulomb is equivalent to the charge exhibited by approximately 1/1.602E−19, or 6.242E18 electrons. Further, opposite charges attract while like charges repel, similar to the poles of a magnet. It is possible to move charge from one point to another. The rate of charge movement over time is called current. It is denoted by the letter I and has units of amps (or amperes)4. One amp of current is defined as one coulomb per second. 1 amp ≡ 1 coulomb / 1 second (2.1) That is, one amp can be visualized as approximately 6.242E18 electrons passing 4 I stands for Intensity (of current), and was so named by André-Marie Ampère. 34 Figure 2.6 Bohr model of copper. +29 Figure 2.7 Defining current as charge flow through a wire.
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through a wire in a period of one second. Consider Figure 2.7. Here we have a wire with electrons flowing through it in the direction of the arrow. We cut this wire with an imaginary plane, leaving us with the highlighted disk. Now imagine that we could count the number of electrons passing through this disk over the course of one second. Because we know the charge possessed by one electron, we simply multiply the number of electrons by the charge for each to yield the total charge, and thus we arrive at the current. As a formula: I = Q / t (2.2) Where I is the current in amps, Q is the the charge in coulombs, t is the time in seconds. A common analogy for electric current is the flow of water through a pipe or river. Just as we can imagine water flow as “gallons or liters per minute”, we imagine electric current as “coulombs per second”. Example 2.1 In the course of one-half second, a certain battery delivers a charge of three coulombs. Determine the resulting current. I= Q t I= 3C 0.5s I= 6 A Example 2.2 A device delivers a current of 25 mA. Determine the charge transferred in two seconds along with the equivalent total number of electrons. I = Q t Q = I×t Q = 25 mA×2s from Definition 2.1, amp-seconds is coulombs Q = 50 mC As one coulomb is equivalent to 6.242E18 electrons, just multiply to find the total number of electrons transferred. 35
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Total electrons = Q×number of electrons per coulomb Total electrons = 50 mC×6.242E18 Total electrons = 3.121E17electrons In sum, the larger the charge transferred within a given time period, the greater the current. Modern electrical and electronic systems might deal with currents of under a picoamp or, at the other extreme, thousands of amps. That is an astonishing range. It is roughly equivalent to a single drop of water dripping from a leaky faucet each second compared to 1000 times the flow of water over Niagara Falls. 2.4 Energy and Voltage 2.4 Energy and Voltage Energy is defined as the ability to do work. It is denoted by the letter W. The basic unit is the joule although other units are sometimes used (for example, the calorie or the kilowatt-hour, kWh). If we were to move a charge from one point to another (for example, separating an electron from an atom), we would have to expend energy to do so. This is illustrated in Figure 2.8. In this Figure, we would say that B has a higher electric potential than A. In other words, there is a potential difference between B and A. We refer to this change as voltage. It is denoted by the letter V (or sometimes E5) and has units of volts, in honor of Alessandro Volta. One volt is defined as one joule per coulomb. 1 volt ≡ 1 joule / 1 coulomb (2.3) As you might guess, the bigger the charge to be moved, the greater the energy required. Expressed as a formula, V = W / Q (2.4) Where V is the voltage in volts, W is the energy in joules, Q is the charge in coulombs. Unlike current, voltage always implies two points for measurement because it involves a difference. Often, one of the points is a common reference, such as earth ground or a circuit ground (i.e., chassis ground). Sometimes people will refer to a point in a circuit as having a certain voltage, as in “point X is 12 volts”. Although common in use, this is somewhat sloppy and not strictly correct. It is important to always remember that this value is relative to some reference point. As a general rule, a voltage will be denoted using the two points as subscripts, for example, VAB, that is, the voltage at point A relative to point B. If only a single subscript is used, as 5 E is used for voltage sources such as batteries. It is short for EMF, or electromotive force. 36 Figure 2.8 Defining voltage as work to move charge. A B Energy
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in VA, then the second, or reference, point is assumed to be the system common or ground. In this case, we're referring to the voltage at point A relative to the system common point. Finally, by definition, VAB = VA − VB, as they have the same reference. Example 2.3 100 joules are expended to move a 20 coulomb charge from point A to point B. Determine the resulting voltage. V BA = W Q V BA = 100J 20C V BA = 5 V Note that it is possible for a voltage to be negative. This simply means that the potential at the point of interest is less than that of the reference point. In Example 2.3 we discovered that point B is five volts above point A. We could just as easily say that point A is five volts below point B, or VAB = −5 V. Further, we can state the difference in terms of the individual ground-referenced voltages, or VBA = VB − VA. Static Electricity and ESD Static Electricity and ESD While it is obviously true that higher voltages imply higher associated energies (charge being held constant), it is not true that a particularly high voltage is necessarily lethal. This is because a very high voltage can be achieved by moving a small charge with only modest energy input. A good example of this is static electricity, so called because it is not associated with a moving current. Static electricity is commonly generated through the triboelectric effect which involves the transfer of electrons from one material to another via physical contact such as rubbing or scraping. If said materials are good electrical insulators, charges will remain on the materials and can build to very high levels, creating a large voltage. Many plastics, such as polystyrene and polyester, are good candidates. The effect can be noticed with certain fabrics, especially under low humidity. For example, removing a polyester fleece pullover or jacket can elicit a certain crackling sound. The sliding of the fleece builds up the charge and eventually the voltage will be become so large that it will arc through the air to surrounding objects which have a lower voltage. This happens quickly over many parts of the garment, each crackle being an individual arc. In fact, if tried in darkness, it is possible to see a cascade of small sparks. This is the same phenomenon that causes a spark when you touch a car after sliding off the seat on a cold and dry winter's day, or a small shock when you touch an object (or another person) after walking across a carpet in a dry library. 37
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Anyone who has opened a box filled with polystyrene packing peanuts can attest to the troublesome nature of the triboelectric effect, as the very light peanuts can easily adhere to other objects due to the electric charge generated through their displacement. No amount of manic brushing or throwing of the pieces will reduce the effect and may, in fact, make it worse. A simple solution in some instances is to spray a fine mist of water on the packing peanuts as the water will provide a conduction path, draining off the charges. Of course, this will not be appropriate in all situations, particularly in the one shown in Figure 2.9. In the prior examples, the static voltage may be on the order of a few thousand volts but the associated energy may be just a few microjoules. In spite of the high voltage, this is not enough to kill someone. On the other hand, the same voltage achieved with a much higher charge and energy could be lethal. Beyond its simple inconvenience and inadvertent feline entertainment capabilities, high static potential can damage sensitive electronic devices. Care must be taken to prevent the accidental buildup of damaging charges. In the electronics industry this is commonly referred to as ESD, or electrostatic discharge. Steps to reduce ESD include humidity control and the use of conductive devices such as resistive wrist straps for technicians to continually bleed off the charges, thus preventing the creation of a high static potential. The Height Analogy The Height Analogy Just as the flow of water can be seen as an analogy for electrical current, a serviceable analogy for voltage involves pressure or height. Indeed, sometimes voltage is referred to as “electrical pressure”. The height analogy ties together the concepts of voltage and energy. In this analogy, height corresponds to voltage and mass corresponds to charge. To begin, we note that there are two kinds of energy: kinetic energy, or energy of motion; and potential energy, or energy by virtue of position. Potential energy is the product of mass, gravity and height, or w = mgh. Keeping gravity equal, we see that the more mass something has or the higher up it is, the greater its potential energy. We might think of potential energy as the object's potential to inflict damage when released. To illustrate, we shall call upon 18th century Scottish philosopher and noted soccer fan6, David Hume. If a soccer ball is held stationary over Mr. Hume's head, as shown in Figure 2.10, the ball has energy by virtue of its position. Releasing the ball from the position shown will scarcely bother Mr. Hume as there is little energy associated with this position relative to the top of his noggin. In fact, he would be hard pressed to head the ball to another player. If, on the other hand, the ball were held 6 Hume, author of An Enquiry Concerning Human Understanding, died in 1776. The modern game of soccer was established in 1863. Let's not let that deter our own inquiry. 38 Figure 2.9 Triboelectric effect and static electricity: Cat fur meets polystyrene. Image source. Figure 2.10 Understanding voltage: David Hume does a header.
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considerably higher, its potential energy would be much greater. Therefore, the impact on Mr. Hume's head would be increased dramatically and he would have little difficulty heading the ball down field (i.e., the transformation of potential energy into kinetic energy), although the chances of him gaining a concussion are greatly increased. Thus, we note that the height of the ball gives us some insight into its potential energy, although these terms are not synonymous. That is to say, if, instead of a soccer ball, we had used a ping-pong ball, even when dropped from an extreme height, the chances of a concussion are non-existent7. On the other hand, if the ball had been replaced with one made of solid iron, a drop from even a modest height could render our most excellent philosopher seriously dead. So it is with voltage. If the charge associated with the voltage is small, even a relatively high voltage will not be lethal (as in the case of simple static electricity on clothing), however, if associated with a sufficiently large charge, a much lower voltage can be deadly. 2.5 Power and Efficiency 2.5 Power and Efficiency The terms power and energy are often used incorrectly as synonyms. Although related, they are not the same thing. As already mentioned, energy is the ability to do work. In contrast, power is the rate of energy usage. Power is denoted by the letter P and has units of watts, although other units are sometimes used such as the horsepower (1 horsepower ≈ 746 watts). One watt is defined as one joule of energy consumed per second. 1 watt ≡ 1 joule / 1 second (2.5) As a formula, P = W / t (2.6) Where P is the power in watts, W is the energy in joules, t is the time in seconds. To better understand the concept, consider for a moment a delicious peanut butter and banana sandwich. This sandwich contains a certain number of food calories, let's say 300 in total. A food calorie refers to a certain amount of energy that humans can extract from an item of food. That energy enables us to do some manner of work such as walking, swimming or just breathing. The sandwich can be seen as an energy storage medium, a battery for biological units called humans. The question 7 The construction of the referred sentence owes a certain debt to the writing style of Mr. Hume. 39
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is, what do we do with the energy, and more to the point, how fast do we use it? For example, that sandwich might be sufficient to allow someone to run a 5k (3.1 mile) road race in 17 minutes. In contrast, it also might be sufficient to allow that same person to watch television for three hours. It's the same amount of energy being used, it's just being used at a much faster rate in the former case. That rate is power. The 5k runner has a much higher power output than the TV watcher. Example 2.4 100 joules are consumed by a device in 0.1 seconds. Determine the power in watts and in horsepower. P = W t P = 100J 0.1s P = 1000W As one horsepower is approximately 746 watts, this is equivalent to Php = PW 746W/hp Php = 1000W 746W/hp Php = 1.34 hp Power can also be found by multiplying a current by the associated voltage. To begin, we note the definitions of current and voltage, Equations 2.2 and 2.4 respectively, and then combine them. I = Q t V = W Q I×V = Q t × W Q = W t From equation 2.6, we know that P = W/t, thus P = IV. This is known as power law. P = I×V (2.7) Where P is the power in watts, I is the current in amps, V is the voltage in volts. 40
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Example 2.5 If a 9 volt battery delivers a current of 0.1 amps, determine the power delivered in watts. P = I×V P = 0.1amps × 9volts P = 0.9W Efficiency Efficiency Efficiency is the ratio of useful output power to applied power expressed as a percentage. It is denoted by the Greek letter η (eta) and is always less than 100%. Expressed as a formula, η = Pout Pin ×100% (2.8) Where η is the efficiency in percent, Pout is the output power, Pin is the input power. Generally speaking, the higher the efficiency, the better. This implies less waste. In other words, if a system is 30% efficient, then 70% of the input power is wasted, whereas if a system is 99% efficient, then only 1% of the input power is wasted. The concept is illustrated graphically in Figure 2.11. In most systems, waste power turns into heat which is not a desired commodity, and in fact often reduces the lifespan of electrical components. Example 2.6 If a device draws 200 watts of power and has a useful output of 120 watts, determine the efficiency. η = Pout Pin × 100 % η = 120W 200W × 100% η = 60% In this case, the device is wasting 40% of the input power, or 80 watts. 41 Figure 2.11 Basic concept of efficiency. Input Useful Output Waste
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Example 2.7 An audio amplifier has a maximum rated output of 100 watts to a loudspeaker. If it exhibits an efficiency of 70%, determine the input power required and the amount of power wasted. η = Pout Pin × 100% Pin = Pout η × 100% Pin = 100W 70 % × 100% Pin = 142.9watts As 142.9 watts are drawn by the amplifier and only 100 watts are delivered to the loudspeaker, then the difference, or 42.9 watts, is wasted power (most likely just making the amplifier hot). 2.6 Energy Cost and Battery Life 2.6 Energy Cost and Battery Life As we have seen, knowing the voltage and current demands of a given device allows us to determine its power rating and energy consumption. The next steps are to determine the cost of operating a device and, if it's battery powered, how long the device will last before needing new batteries. Computing Energy Cost Computing Energy Cost Once we know the power drawn by the device and for how long it will be used, the cost to operate a device can be determined given the per unit cost of energy. In spite of the fact that many people refer to their local electricity supplier as “the power company”, we do not buy “power”, per se. Rather, we are billed for energy. Although it would be possible to determine the cost per joule (or more practically, per megajoule), suppliers normally bill based on kilowatt-hours, or kWh, the product of the power and time8. This unit is used because most residential and commercial devices and appliances are rated in terms of power consumption in watts. Multiplying the power consumption in watts by the length of time the device is used in hours yields a watt-hour value. This is then scaled by a factor of one thousand to arrive at kWh. For example, a 1500 watt toaster oven used for 30 minutes (i.e., 0.5 hours) yields 750 watt-hours, or 0.75 kWh. Finally, knowing the per kWh cost, a simple multiplication yields the cost of electricity. Thus, if the utility charges 10 8 Remember, power is the rate of energy usage per unit time, and thus multiplying power by time yields energy. One kWh is approximately equal to 3.6 megajoules. 42
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cents per kWh, then the cost to run that toaster oven is 7.5 cents. If it is used for a full hour then it costs 15 cents, and so on. To put usage in perspective, a typical home in the USA or Canada uses about 900 kWh per month while households in many countries in Europe might use one half to one quarter of that amount. Global electricity generation is around 25 million gigawatt hours per year. Typical electricity rates in the USA are between ten and twenty cents per kWh, depending on geographic region and sector (e.g., residential or commercial). Example 2.8 A 100 watt incandescent light bulb is left on for 24 hours. If the cost of electricity is 15 cents per kWh, determine the cost to run the light. Cost = P× t × rate Cost = 100W × 24 hours × 0.15 $/kWh Cost = $0.36 At this point, it should be obvious that the more efficient a device is, the less expensive it will be to run. In fact, it is quite possible that in the long run, a device that is more expensive than a similar, though less efficient, device may be considerably less expensive to use over the course of its lifespan. A good example is a comparison between an ordinary incandescent light bub and an LED light. LED lighting can be an order of magnitude more efficient than incandescent lighting. Indeed, incandescent bulbs may convert less than five percent of their input into usable light output, the remaining 95 percent just turning into heat. While they are less expensive to purchase initially, their operating cost is much higher. A proper way to compare lights is to examine their light output in lumens, not their power consumption. For example, a 60 watt incandescent light produces about 800 lumens of illumination. That same level of illumination can be obtained with an LED drawing just 9 watts. Further, typically LED lights last ten to twenty times longer than incandescent bulbs. This will be illustrated in the next example. Example 2.9 A certain 14 watt LED light produces the same illumination as a 75 watt incandescent light bulb. Assume the LED costs $11 and has an expected life of 15,000 hours. The incandescent costs 50 cents each and has an expected life of 1000 hours. If the cost of electricity is 12 cents per kWh, determine the cost to run each version for 15,000 hours. 43
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First, it should be noted that 15 incandescent bulbs will be needed. At 50 cents each, that's $7.50 for the bulbs. The cost to run them is, Cost = P× t × rate Cost = 75W × 15000 hours × 0.12 $/kWh Cost = $135.00 The total is $ 142.50. Now for the single LED required: Cost = P× t × rate Cost = 14W × 15000 hours × 0.12 $/kWh Cost = $25.20 The total is $36.20, a considerable savings, not to mention other positive factors including only having to change the light once instead of fifteen times; a considerable reduction in wasted energy, thus lowering demand and environmental impact; and finally, a reduction in burned out light bulbs for a further lowering of environmental impact (waste stream reduction). Batteries Batteries A battery is a device used to store electrical energy, generally in the form of a chemical cell. Ideally, it presents a constant voltage, its current varying according to what it drives. In reality, as the battery is used, its voltage will begin to decrease. Eventually, the energy stored in the battery will be exhausted and its voltage will drop to zero. The storage capacity of a battery is measured in amp-hours, Ah (or milliamp-hours, mAh, for smaller batteries). All other factors being equal, the battery with the higher amp-hour rating will last longer before being depleted. This is true for the same size of battery using different compositions (e.g., zinc-carbon versus alkaline), as well as batteries of different physical sizes but having the same voltage. For example, all AAA, AA, C and D cell batteries have a nominal voltage of 1.5 volts. If they are all of the same type, such as alkaline, then the immediate practical difference is that the larger the physical size, the greater its energy storage, and thus the longer it will last. Over a modest range of currents, the expected lifespan of a battery can be computed based on its amp-hour rating and the current drawn from it. Battery life ≈ amp-hour rating / current draw (2.9) This equation is best used as a rough guide. If the current draw is considerably higher than the current at which the battery was tested, the predicted lifespan will be overly optimistic. On the other hand, if the current draw is considerably lower, it is likely that the battery will last longer than predicted. 44
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The maximum current output of a battery is likewise limited. If such were not the case, we might expect even very small batteries to produce phenomenally large currents for very shorts periods of time. This is not the case. A graph of discharge curves at various load currents is shown in Figure 2.12. Notice how the battery voltage begins at the rated 1.5 volts and then begins to fall. After a certain point, the rate of decrease accelerates and “drops off of a cliff”. The actual service life will also be application dependent in that some devices can tolerate a lower voltage than others. For example, an old-fashioned flashlight will still work with largely depleted batteries, it simply won't be that bright. If we were to consider 75% of rated voltage as the useful lower limit (a little over 1.1 volts), we see that at a 50 mA draw, the battery will last around 18 hours, achieving about 900 mAh. At 100 mA, it will last around 7 hours, yielding 700 mAh. If we used 1.0 volts as our lower usable limit, we would arrive at 1.05 Ah and 910 mAh, respectively. Example 2.10 A certain battery is rated at 10 Ah. Approximately how long will it last with a 0.5 amp draw? Lifespan ≈Ah I Lifespan ≈10 Ah 500mA Lifespan ≈20 hours 45 Figure 2.12 Battery discharge curves for various load currents at room temperature. Courtesy of Duracell
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Remember, 20 hours is only an approximation. For a more accurate rendering, consider the following example. Example 2.11 Using the graph of Figure 2.12, determine the expected lifespan with a 100 milliamp draw for a lower voltage limit of 1.2 volts. Also determine the effective amp-hour rating at this point. The 50 mA curve (purple) passes through 1.2 volts at approximately 5 hours. This is the expected lifespan. The corresponding amp-hour rating is Ah ≈I×t Ah ≈50 mA×5hours Ah ≈250mAh Batteries also tend to show decreased performance with reductions in temperature. This is shown in Figure 2.13. For this particular battery, the capacity at freezing (0°C) is roughly half of what it is at room temperature (21°C). Peak current capacity may also be reduced with temperature. This is true of many kinds of batteries, including large 12 volt automotive batteries. This is a major reason why cars are often much more difficult to start on very cold days; their current capacity is reduced, thereby reducing the output of the starter motor. 46 Figure 2.13 Battery discharge curves for different temperatures. Courtesy of Duracell
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A table of typical amp hour ratings for common battery sizes is shown in Figure 2.14. These values are appropriate for good quality alkaline batteries. Rechargeable NiMH (nickel-metal hydride) would be around the same. Remember, these are just approximations. 2.7 Resistance and Conductance 2.7 Resistance and Conductance We have seen that both current and voltage deal with the movement of charge. Consequently, in any electrical system, voltage and current are interrelated. Let us consider the most simple case. This would involve a single voltage source, such as a battery, driving a single, homogeneous item such as a length of wire or a block of a given material. The physical characteristics of this item will dictate how much current will flow. In general terms, Effect = Cause Oppostion In this case the cause is the voltage source and the effect is the resulting current. The opposition is the characteristic of the item in question, in other words, its ability to restrict current flow. We call this characteristic resistance. In other words, resistance is a measure of how difficult it is to establish a flow of current (i.e., “resistance to current flow”) under a given set of circumstances. It is denoted by the letter R and has units of ohms, in honor of Georg Ohm, a researcher from the early 1800s. The unit is denoted by the capital Greek letter omega, Ω. Sometimes it is more convenient to use the inverse view of this phenomenon, and instead of referring to how difficult it is to establish a current, we would be interested in how easy it is to establish a current. This is called conductance and it is the reciprocal of resistance. Conductance is denoted by the letter G and has units of siemens, named after Werner von Siemens. The unit is abbreviated as S. R = 1 G G = 1 R (2.10) 47 Battery Size Capacity (mAh) AAA 1000 AA 2500 C 5000 D 10000 9 Volt 500 Figure 2.14 Typical battery capacities.
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In this simple scenario, resistance is a function of the material the current is passing through along with its shape. This is illustrated in Figure 2.15 where the arrow shows the direction of current flow. An obvious question is “What is this block made of?” It should come as no surprise that the material chosen has a great impact on the current. We have already seen that metals such as copper are good conductors of electricity. Other materials, such as certain plastics and ceramics, are not. These materials are referred to as insulators. The measure of a material's inherent and general ability to restrict current flow is called resistivity. Resistivity is denoted by the Greek letter ρ (rho). All other factors being equal, the higher the resistivity, the greater will be the resistance. Further, the greater the length of the material that the current must pass through, the greater the resistance. Finally, as the surface area of the cross section (i.e, the face) grows, the resistance decreases. Expressed as a formula R =ρl A (2.11) Where R is the resistance in ohms, ρ is the resistivity, l is the length of the material, A is area of the face (h times w). Resistivity is often specified in ohm-centimeters with the length and area similarly specified in centimeters and square centimeters, respectively. A table of resistivity values for a variety of materials is shown in Figure 2.16. Note that resistivity is not necessarily constant across temperature. Indeed, this change can be exploited as a means of measuring temperature. In other applications, we might need it to be as stable as possible across temperature. This need led to the creation of the alloys Constantan and Manganin in the late 1800s which exhibit very high stability across temperature. From this table we can see that silver has lower resistivity than copper which in turn is lower than gold. This means that if we made identically sized wires of these three materials, the silver version would have the least resistance and gold the highest. Why then, do audio and video cables often feature gold plating? Certainly, it isn't due to lower resistivity and enhanced conductivity. The reason is that gold is a noble metal, meaning that it does not tarnish. In contrast, the surface of both silver and copper will oxidize, creating a patina (the dark “stain” noticeable on old silver and copper implements). The oxide will create a high resistance layer and reduce the integrity of the connection. It is important to note that Formula 2.11 does not include a term for the mass of the material. It is only concerned with the shape of the item. This is an important distinction. If we were to alter the mass but keep the ratio of the length versus the 48 Figure 2.15 Defining resistance. w l h
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area the same, the resulting resistance would be unchanged. In general, the increase in mass by itself does not necessarily alter the resistance but it may have an impact on the power handling capability of the device. In contrast, if we take that that original mass and reshape it, or just apply the current to a difference face, such that the effective surface area and length change, then the resulting resistance will also change. This is illustrated in Figure 2.17. Here we have taken the item shown in Figure 2.15 and directed the current flow from top to bottom rather than through the side. In this orientation, the surface area is much increased and the length through which the current must travel is greatly reduced. 49 Figure 2.16 Resistivity values for various materials. Material Resistivity ρ (Ω·cm) at 20 °C Temperature coefficient (K−1) Silver 1.59×10−6 0.0038 Copper 1.68×10−6 0.00404 Gold 2.44×10−6 0.0034 Aluminium 2.65×10−6 0.0039 Zinc 5.90×10−6 0.0037 Nickel 6.99×10−6 0.006 Iron 9.7×10−6 0.005 Platinum 1.06×10−5 0.00392 Tin 1.09×10−5 0.0045 Titanium 4.20×10−5 0.0038 Manganin 4.82×10−5 0.000002 Constantan 4.90×10−5 0.000008 Stainless steel 6.90×10−5 0.00094 Nichrome 1.10×10−4 0.0004 Carbon (amorphous) 5×10−4 to 8×10−3 −0.0005 Silicon 6.4×104 −0.075 Glass 1×1013 to 1×1017 Carbon (diamond) 1×1014 Hard rubber 1×1015 Air 1011 to 1017 PET 1×1023 Teflon 1×1025 to 1×1027 Figure 2.17 Resistance will change due to to altering area and length with mass unchanged.
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Consequently, the effective resistance in this orientation will be considerably less than that seen in the original. Example 2.12 A certain material has a resistivity of 0.2 ohm-centimeters. Determine the resistance of a piece that is 0.3 cm wide, 0.5 cm high and 4 cm long. A =h×w A =0.5cm×0.3cm A =0.15 cm2 R = ρl A R =0.2Ωcm×4cm 0.15cm 2 R =5.333Ω For our next example, let's consider a spool of wire. In many cases we treat wire ideally, as if it has no resistance. While this can be a good approximation in many instances, especially with relatively short runs of wire, such is not always the case. Example 2.13 A certain gauge of copper wire has a diameter of 0.6 mm. Determine the resistance if the spool is 200 meters long. The table of Figure 2.16 indicates that the resistivity of copper is 1.68E−6 ohm-centimeters. The trick here is that we must keep the units consistent. As there are 100 centimeters to the meter, the length is 200 meters times 100, or 20,000 centimeters (i.e., 20E3). Given that there are 10 millimeters to the centimeter, the diameter must be decreased by a factor of ten, yielding a diameter of 0.06 cm and thus a radius of 0.03 cm. A = π r2 A = π(0.03cm) 2 A =2.83E-3cm 2 R = ρl A R =1.68E-6Ωcm×20E3cm 2.83E-3cm 2 R =11.9Ω 50
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The amount of resistance seen in Example 2.13 would be considered excessive if the item to be connected is something as simple as a loudspeaker, which typically would be around 8 Ω. And while no one would likely need 200 meters of cable to connect a loudspeaker in their home, that sort of distance would be unremarkable in a large stadium or airport terminal. Don't forget, we'd need wire to and from the loudspeaker, achieving a total separation of 100 meters at most. Various thicknesses of cables and wires are used for a wide variety of purposes. To make this easier, wire thicknesses have been standardized. The most common standard in North America is the American Wire Gauge, or AWG. This is a non- metric specification with origins in the mid nineteenth century. The larger the gauge number, the smaller the wire diameter, and the less current it safely can carry. To put the gauge numbers in perspective, typical small home appliances use 16 or 18 gauge wire, basic home wiring uses 12 gauge (with a 20 amp circuit breaker), and hook-up wire used in an electrical circuits or electronics laboratory solderless breadboard is commonly 22 or 24 gauge. For general purpose wiring, copper is by far the most common metal used because it is highly conductive and relatively inexpensive. Some other metals are used in special cases, for instance, aluminum is used for long distance power transmission lines. 51 AWG Diameter Resistance/length (in) (mm) (Ω/km) (Ω/1000ft) 0000 (4/0) 0.4600 11.684 0.1608 0.04901 00 (2/0) 0.3648 9.266 0.2557 0.07793 0 (1/0) 0.3249 8.251 0.3224 0.09827 2 0.2576 6.544 0.5127 0.1563 4 0.2043 5.189 0.8152 0.2485 6 0.1620 4.115 1.296 0.3951 8 0.1285 3.264 2.061 0.6282 10 0.1019 2.588 3.277 0.9989 12 0.0808 2.053 5.211 1.588 14 0.0641 1.628 8.286 2.525 16 0.0508 1.291 13.17 4.016 18 0.0403 1.024 20.95 6.385 20 0.0320 0.812 33.31 10.15 22 0.0253 0.644 52.96 16.14 24 0.0201 0.511 84.22 25.67 26 0.0159 0.405 133.9 40.81 32 0.00795 0.202 538.3 164.1 40 0.00314 0.0799 3441 1049 Figure 2.18 AWG wire data for copper conductors.
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A table of gauge sizes and associated parameters is shown in Figure 2.18. This table assumes copper is being used for the wire. Note that as the diameter of the wire decreases, the amount of resistance for a particular length increases, as expected from Equation 2.11. Using this table we can perform a quick crosscheck of Example 2.13. The wire diameter used in that example was 0.6 mm which is just a little smaller than AWG 22 as listed in the table. Further, #22 wire is listed as having a resistance of approximately 53 Ω per km. For 200 meters, as used in the example problem, this works out to 10.6 Ω. As #22 wire is slightly larger in diameter, we expect it to show slightly less resistance than the calculated value of 11.9 Ω, which it does. While it may not be immediately apparent, gauge numbers proceed in a logarithmic fashion based on diameter. Stepping up one gauge number (e.g., from #10 to #11) decreases the diameter by a factor of approximately 0.89. As resistance is inversely proportional to the square of the diameter (i.e., the area), the resistance increases by over 25%. Even numbered sizes are particularly common in use and a jump to the next higher even gauge number (e.g., #18 to #20) produces a resistance increase of nearly 60%. Resistors and the Resistor Color Code Resistors and the Resistor Color Code Resistors are devices used to control the currents and voltages in a circuit. They are available in many shapes and sizes, and are normally designed to maintain stable ohmic values in spite of environmental changes such as temperature and humidity. A sample of different resistor styles is shown in Figure 2.19. As a general rule, the larger the resistor, the more power it can handle. At the back of the figure is a large ceramic power resistor using a rectangular aluminum housing. This device is rated for 200 watts of dissipation. Immediately in front and to the right of it are several smaller ceramic power resistors with ratings in the 5 to 20 watt range. Along the left side is a set of carbon composition and carbon film resistors ranging from 1 watt down to one-tenth watt in dissipation. Toward the center is a multi-lead chip resistor that contains several resistors in one package. With few exceptions, all of these items are classified as “through-hole” components, that is, their leads are designed to go through pre-drilled holes in a printed circuit board. These are also the most commonly used components in an electrical lab as the leads fit into solderless breadboards and the components are a convenient size. 52
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With the increasing desire to shrink components, modern production designs use surface mounting techniques in place of through-hole construction. At the bottom-center is a small dot which is, in fact, a surface mount resistor capable of ½ watt of power dissipation. A close-up is shown in Figure 2.20. Obviously, these devices are too small to be practical to use without special equipment. Resistors are linear bilateral devices. Being linear, their current-voltage relation can be drawn as a straight line. Bilateral means that the polarity of voltage or direction of current will not matter. In other words, unlike a battery, these devices cannot be inserted into a circuit backwards because either orientation works the same way. If the horizontal axis is voltage and the vertical is current, then the slope of the line yields the conductance. Thus, the steeper the line, the lower its resistance. This is illustrated in Figure 2.21. Not all electrical components are linear and bilateral. A good example is the semiconductor diode, a commonly used device in electronic systems. The current-voltage plot of a diode is shown in Figure 2.22 for comparison. Note that the plot is not a simple straight line, thus it is not linear. Further, the first and third quadrant responses are wildly different, indicating that polarity is of utmost importance. Clearly, it matters which way these kinds of devices are inserted into a circuit. 53 Figure 2.19 A variety of different resistors. Figure 2.20 Close up of a surface mount resistor. Figure 2.21 Current-voltage plot of two resistors. V I -I -V
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Resistors are available in standardized ohmic values and at standardized power ratings (see Appendix A). Along with their resistance value, resistors also have a specified tolerance. This specifies an allowable range of variation of the stated value. For example, a 220 ohm resistor may have a tolerance of 10%. This means that the actual value of any given specimen from a box of these resistors may be off of the nameplate or nominal value by 10% or 22 ohms. Thus, any particular resistor might be as high as 242 ohms or as low as 198 ohms. General purpose resistors use a color code to denote their value and tolerance. Typically, this will involve four color stripes: two for the precision/mantissa, one for the power of ten, and the fourth to indicate the tolerance. For higher precision, a five stripe version may be used with the first three denoting the precision/ mantissa. Alternately, high precision resistors may have their nominal value printed directly on them. Refer to Figure 2.23 for an example of the basic variety. The first two bands, here yellow and violet, indicate the precision or leading digits. The third band, here orange, indicates the power of ten or “number of zeroes” to add. The fourth band, silver in this example, indicates the tolerance. Note that the fourth band is spaced away from the others to avoid accidentally reversing the order. The color code is shown in Figure 2.24. To assist in remembering the sequence, a number of mnemonic aids have been used, the first letter of each word starting with the same letter as the corresponding color. One example is the “Picnic Basket Mnemonic” which is: Black Bears Robbed Our Yummy Goodies Beating Various Gray Wolves. Another possibility is to note that the middle section follows a rainbow with black and white at the extreme ends. Color Numeral Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Violet 7 Gray 8 White 9 For small resistor values If the multiplier band is gold then divide by10; if silver, divide by 100. 54 Figure 2.22 Current-voltage plot of a diode: neither linear nor bilateral. V I -I -V Figure 2.23 Example of basic four stripe resistor color code. Figure 2.24 Resistor color code.
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The tolerance band colors are as follows: For basic parts silver is ±10% and gold is ±5%. If the fourth band is omitted, this indicates a tolerance of ±20%, although it is seldom used in modern designs. For precision parts some colors are reused but follow the color code numerals: Brown is ±1% tolerance and red is ±2% tolerance. Tighter tolerances are also available. Sometimes an extra band is added that indicates a reliability rating or temperature coefficient. In such cases, it is best to refer to the manufacturer's data sheets for details. We will not pursue these further. Example 2.14 Determine the nominal, maximum and minimum acceptable resistance values for the resistor pictured in Figure 2.17. The colors are yellow-violet-orange. This translates to 4, 7 and 3. The value is “47 with 3 zeroes”, or 47000 ohms. The silver fourth band indicates 10% tolerance. Thus, the resistor pictured is nominally 47 kΩ with ±10% variation around the nominal value being acceptable. The tolerance yields ±4.7 kΩ, so the acceptable range is from 42.3 kΩ to 51.7 kΩ. Example 2.15 A precision resistor has a color code of orange-blue-green-brown-brown. Determine its value and acceptable range. The final band indicates that this is a ±1% tolerance component. The first four colors translate to 3, 6, 5 and 1. The value is “365 with 1 zero”, and thus 3650 ohms, or 3.65 kΩ. The allowable range is 3.65 kΩ ± 36.5 Ω. Example 2.16 Determine the nominal value and tolerance of a resistor with the color code green-blue-gold-silver. The first two colors translate to 5 and 6. Gold in the third band means “multiply by 0.1”. The final band indicates that this is a ±10% tolerance component. Therefore, nominal value is 5.6 Ω. The allowable range is 5.6 Ω ± 0.56 Ω. 55
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An example of a resistor data sheet is shown in Figure 2.25. This data sheet is for a series of surface mount chip resistors. The available tolerance grades range from 0.5% to 20%. Also, for each variant there are two temperature coefficients available with the most stable being 100 ppm/°C (ppm is short for “parts per million”, thus 100 ppm is equivalent to 0.01%). Note that these devices are too small to use the resistor color code. Instead, a numeric code is printed directly on them that follows the rules for color bands, meaning that the final digit is the multiplier. Thus “150” is 15 Ω (1-5 with no zeroes). The exception is very small values where the letter “R” is used in place of a decimal point. Consequently, “2R4” is 2.4 Ω. A particularly important characteristic of most electronic devices is their power handling capacity. For this series there are several variations with power handling from 0.1 watts to 1.5 watts, and maximum working voltages up to 500 volts. Power handling is also a function of ambient temperature. In general, devices have a maximum internal temperature that they can reach before they are damaged. Control of heat tends to be a major issue in many system designs. If the ambient temperature increases, there is less “headroom” for the device's temperature rise, and thus the device will not be able to dissipate as much power. This is illustrated in the graph of Figure 2.26. 56 Figure 2.25 Example resistor data sheet. Courtesy of Stackpole Electronics
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Note how the power dissipation is constant at temperatures at and below 70 °C. This temperature is considered the maximum normal operating temperature for this device and the power dissipation at this temperature is the one quoted in the data sheet. If the device is operated in a warmer environment, the power dissipation is derated by the percentage given in the graph. For example, at 100 °C the dissipation is only about 65% of the nominal rating. Of particular importance, at 155 °C the device can no longer dissipate any power, and therefore this temperature serves as the absolute ceiling. After all, a resistor that cannot dissipate any power is a resistor that cannot have any current through it or voltage across it without burning up. It is essentially non-functional. Of course, just like a skillet on a stove, these devices do not heat up instantly, that is, they exhibit a thermal time constant. They take a certain amount of time to heat (and also to cool). For short periods, devices can handle considerably more power than their long term rating. The graph of Figure 2.27, shows the power dissipation for single pulses. In some cases, these resistors can handle powers of an order of magnitude greater. For example, consider the type RPC0402 (bottom-most line). From the data sheet of Figure 2.25 this device is specified as having a long term rating of 0.2 watts. In contrast, the graph of Figure 2.27 shows it is capable of withstanding a single pulse of greater than 20 watts for 100 microseconds, just under 10 watts for a millisecond pulse and roughly 2 watts for a one-tenth second pulse. 57 Figure 2.26 Resistor power derating graph. Courtesy of Stackpole Electronics
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Other Resistive Devices Other Resistive Devices Along with standard fixed resistors, there are several other kinds of resistive devices that have been designed to be sensitive to changes in their environment. Thus, they can serve as sensors because as their resistance changes, it impacts the flow of current and the resulting voltage. We will look at examples of this in upcoming work. Some of the environmental inputs include temperature, force and light levels. Force Sensing Resistor (FSR) The force sensing resistor consists of two layers of material with a nominal separation distance. It is presented as a flat membrane that is usually round or rectangular, and perhaps with an adhesive backing. An example of an FSR is shown in Figure 2.28. With no force applied, the device shows an extremely high resistance, well into the megohms. As force is applied to the surface, the two layers come into better contact which decreases the net resistance. This is borne out in the graph of Figure 2.29. The graph shows roughly a straight line response between resistance and force when plotted on a log-log scale. At the highest force levels, the resistance may drop to just a few hundred ohms. 58 Figure 2.27 Pulse power dissipation. Courtesy of Stackpole Electronics Figure 2.28 A force sensing resistor (FSR).
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Photoresistor As their name implies, photoresistors are sensitive to changes in light level. They are also called LDRs, short for Light Dependent Resistor. Different materials may be used in their construction, but the most common is cadmium sulfide, CdS. As a consequence, photoresistors are sometimes generically referred to as “CdS cells”. A photoresistor is shown in Figure 2.30 along with its corresponding resistance graph in Figure 2.31. In total darkness the device exhibits a very high resistance. As light levels increase, the resistance decreases. As with the FSR, we see an reverse relation between the resistance and the environmental factor: as the environmental input increases (force, light level), the resistance does the opposite and decreases. And once again, we see a straight line when plotted with log-log scales. Technically, we refer to this as a negative relation because the slope of the plot line is negative. 59 Figure 2.29 FSR response curve. Courtesy of Interlink Electronics Figure 2.30 A photoresitor, or LDR. Figure 2.31 Photoresitor response curve. Courtesy of Advanced Photonix
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To put the brightness of the light into common terms, 0.01 foot-candles (roughly 0.1 lux) is equivalent to a clear moon-lit evening. At this level, the photoresistor is showing over 1 megohm of resistance. In contrast, 100 foot-candles (roughly 1000 lux) is equivalent to an overcast day (for reference, direct sunlight is perhaps 100 times stronger). At this level, the photoresistor's value has dropped to about 1000 ohms. A point worth noting is that the light that the photoresistor “sees” is not necessarily the same as what a human sees. The sensitivity of the device at various wavelengths (i.e., colors) may differ starkly from human vision. In fact, differing materials exhibit different sensitivities at various wavelengths. Some of these may be wavelengths that the unaided human eye cannot see at all (infrared or ultraviolet, for example). An example of sensitivity curves is shown in Figure 2.32. Note the variations between the different materials, CdS being one of the three (left side). The peak sensitivities vary as do the precise shapes. In practice, this means that some of these units will be more or less sensitive to certain colors than other units. The CdS curve indicates peak sensitivity of about 540 nm, which corresponds to green. In comparison, the CdSe (cadmium selenide) cell exhibits a peak of just over 700 nm, which corresponds to red. At some wavelengths, the relative response of one material may be no more than 10% of the response of a different material. It is important to note that the use of cadmium, such as in CdS cells and the like, is severely restricted by the RoHS directive (see Chapter One). Thermistor A thermistor is a device whose resistance is a function of temperature. These devices are available in two basic types. Either PTC, for Positive Temperature Coefficient; or NTC, for Negative Temperature Coefficient. PTC devices show an increase in resistance as temperature increases and NTC devices show a decrease in resistance as temperatures rise. Ideally, these are linear relationships with the plots showing straight lines. The reality is that linearity can only be assumed across fairly 60 Figure 2.32 Photoresitor sensitivity curves. Courtesy of Token Electronics Figure 2.33 NTC thermistor.
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narrow ranges of temperature. For wider ranges, there will be noticeable deviation from a straight line as the curve is logarithmic in nature. A basic NTC thermistor is shown in Figure 2.33. Generic thermistor performance graphs are shown in Figure 2.34 with the idealized straight line response at top and the more realistic non-linear response below. Thermistors will be specified in terms of their room temperature resistance (usually taken to be 25 °C) along with their sensitivity which is denoted as beta (β). The larger the value of beta, the steeper the curve and the greater the sensitivity. The curves labeled Beta 2 show increased sensitivity, meaning that there will be a greater change in resistance for a certain change in temperature. The following equation can be used to determine the resistance of a thermistor at some other temperature of interest with greater accuracy than using a simple linear approximation. All that is needed is a reference temperature and corresponding resistance, the beta value and the new temperature of interest. Using a reference temperature of 25 °C: RT =R25e β( 1 T − 1 298.15) (2.12) Where RT is the resistance at the new kelvin temperature, T, R25 is the resistance at 25 °C, T is the temperature of interest in kelvin, β is the device beta. Note that the constant “298.15” in Equation 2.12 is equivalent to the reference temperature of 25 °C (0 K is −273.15 °C). Consequently, if a different reference point is used, simply insert the new reference temperature in its place and use the corresponding resistance in place of R25. Varistor Varistors are used as limiting devices, primarily to suppress unwanted voltage spikes in electronic equipment. The varistor is a unique device in that it has a highly non- linear current-voltage characteristic. This is shown in Figure 2.35. Remember, when plotted with the voltage on the horizontal axis, the slope of the line represents the conductance. Consequently, the varistor shows a region of near zero conductance or extremely high resistance (the horizontal section), and two sections that are nearly vertical, which indicate extremely high conductance or near zero resistance. This characteristic allows the varistor to act as a limiting device. Imagine that a lightning strike effects a local power line. This will create a sudden but short-lived spike in the voltage. A normal 120 volt wall outlet normally produces regular peaks of approximately 170 volts. A lightning strike might add several 61 Figure 2.34 NTC thermistor response curves: Ideal (top) and real. R (log) T (C) 25 25 R -25 0 50 75 Beta 1 Beta 2 R (log) T (C) 25 25 R -25 0 50 75 Beta 1 Beta 2
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hundred volts to this. The resulting voltage could be so high that it would damage electronic equipment attached to the outlet. To alleviate this problem, a varistor can be placed across the incoming voltage lines. The vertical break point voltages would be set for a value just over 170 volts, the normal maximum. Under typical circumstances the varistor would see a voltage in its horizontal region and thus behave as a very high resistance. It would drain virtually no current from the outlet, and consequently it would have no impact on the rest of the circuitry. On the other hand, if a large spike hits the line, the varistor will swing into the vertical region, show a much reduced resistance, and act as a shunting path for the spike's current. It will effectively clamp the voltage to some maximum rated value. Of course, the varistor has to absorb the energy presented by that spike, and important parameters of a varistor include the amount of energy that it can absorb (in joules) and its maximum current capacity, along with the maximum clamping voltage. Strain gauge The strain gauge is device used to measure mechanical strain on some item. Strain occurs when an item is under compression (its length gets shortened or squished) or when it's under tension (it gets elongated or stretched). Both of these things can happen simultaneously, for example when a bar is experiencing bending or torque (one side is under compression while the other is under tension). Engineering strain is defined as the change in length over the initial length. If an item experiences too much strain, it can become permanently deformed or fail (for example, the landing gear in an aircraft or the suspension components in a car). Simply put, strain gauges are used to measure this effect. A strain gauge is made typically of very thin metal foil in a specific pattern. See Figure 2.36 for example shapes. Typically, the shape is that of a series of fine lines connected in a back-and-forth pattern, the two ends terminating into larger pads for soldering on connecting leads. In operation, the strain gauge is glued to the material being investigated, for example, a metal bar that is part of a suspension system. As they are glued together, the strain gauge experiences the same deformation as the metal bar. Any deformation will create changes in the length of the strain gauge's foil wires as well as their frontal surface area. For example, under tension, the length increases while the surface area decreases (the surface area must decrease because the foil wire has finite mass). Recalling the basic resistance relation, Equation 2.11, both of these effects will cause the resistance to rise. The greater the strain under tension, the greater the rise in resistance. The opposite will occur under compression and the resistance will decrease. These changes in resistance are not large but they are sufficient to alter the associated voltage or current which can then be calibrated to determine the applied strain. 62 Figure 2.35 Current-voltage characteristic of a varistor. I V + + - - Figure 2.36 Strain guages. Courtesy of Zemic
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2.8 Instrumentation and Laboratory 2.8 Instrumentation and Laboratory At this point we need to shift gears and focus on a few practical aspects, the sort of things we might deal with in an electrical laboratory. It is one thing to discuss abstract concepts such as voltage and current, and quite another to deal with them in the real world. Sources Sources For starters, let's consider the idea of power sources, that is, devices that can produce reliable, stable and constant voltages and/or currents. As mentioned previously, a battery approximates an ideal voltage source. The problem, of course, is that batteries only last a finite period of time and their voltage begins to sag after being used for a while. Also, their voltage is fixed and not adjustable. An adjustable voltage source would be much more flexible in a laboratory. To address this problem, most electrical labs use a variable DC voltage supply in place of simple batteries. These sources typically are adjustable from 0 to 25 volts or higher and can produce one amp of current or more. An example is shown in Figure 2.37. This particular unit has two main outputs plus a third auxiliary output. All outputs are independent, meaning that the voltage of each can be set independent of the others. Further, the two main outputs offer programmable current limiting. This feature can be thought of as a programmable fuse or circuit breaker: it limits the amount of current a circuit can draw to some pre- selected level. It is worth noting that even if a source is rated for, say, two amps, that only refers to the maximum value that can be drawn; that is not the current that it would produce by default. Multiple output power supplies are usually wired in a “floating” form, meaning that the negative terminal is not tied back to earth ground. A separate connector is presented (usually green) that is tied back to earth ground if it is needed. Having floating supplies gives the user much more flexibility. For example, two floating supplies can be wired in sequence to create a single higher voltage. They can also be wired to present a positive voltage and a negative voltage. The unit shown in Figure 2.37 also features twin LED displays for voltage and current for each of the main outputs. It is worthwhile to point out that the color coding used for electronic equipment is not the same as that used for residential wiring in North America. The electronics standard is that the common or ground terminal is black while the “hot” or positive terminal is red. While there are a large number of makes and models of affordable laboratory DC voltage sources available, the same is not true of DC current sources. This is not usually a problem. As we shall see in upcoming work, it is possible to emulate a constant current source using a voltage source and some attached components. 63 Figure 2.37 A multi-output adjustable DC voltage supply.
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Schematic Symbols Schematic Symbols Clearly, it would not be practical to create circuit drawings (schematics) using pictures of the actual devices. Instead, we use simple schematic symbols to represent them. There are two widely used schemes: ANSI (American National Standards Institue) and IEC (International Electrotechnical Commission). For many electronic components the ANSI and IEC versions are the same, but there are notable differences. ANSI tends to dominate in North America (surprise!) while IEC tends to dominate elsewhere. Figure 2.38 shows the symbols for DC current and voltage sources. The long bar at the top of the voltage source is the positive terminal while the shorter bar at the bottom denotes the negative terminal. For the current source, the arrow points in the direction of conventional current flow. If the source is adjustable, it will often be shown with a diagonal arrow drawn through it. In some schematics, a DC voltage source will simply be drawn as a “connection dot” or node with the voltage labeled. It is assumed that the other end of the source is tied back to the system common or ground. Speaking of ground, there are three different symbols used for a circuit's common connection point. These are: earth ground, chassis ground and signal or digital ground. These are shown in Figure 2.39. Earth ground is used when the circuit common is tied back to true earth (e.g., to the third pin on an AC receptacle). Chassis ground is a more generic term which would represent a common reference point that did not go back to true earth (e.g., a portable device). Finally, digital or signal ground is used to distinguish between common points where they might be separated for noise or interference concerns (e.g., a sensitive analog signal common separate from a digital logic common). The symbols for resistors and other resistive devices vary considerably between ANSI and IEC versions. ANSI uses a zig-zag line while IEC uses a simple rectangle. These are illustrated in Figure 2.40. This text will focus on using the ANSI standard symbols for resistors. One final note about component values: As decimal points are easy to lose on photocopies or small display screens, engineering notation multipliers are sometimes placed in the position of a decimal point. For example, a 4.7 k Ω resistor may be listed as simply 4k7. 64 Figure 2.38 Voltage (left) and current source schematic symbols. Figure 2.39 Schematic symbols for ground. (L-R) earth, chassis, signal. Figure 2.40 Schematic symbols for resistive devices. (L-R) resistor (ANSI), resistor (IEC), photoresistor, thermistor.
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Measurement – The Digital Multimeter Measurement – The Digital Multimeter Perhaps the most handy measurement device in the electrical laboratory is the digital multimeter, or DMM for short. These handheld devices are used to measure voltage, current and resistance; and depending on model may have other measurement capabilities as well. A pair of typical units are shown in Figure 2.41. Beyond the measurement functions of the DMM, perhaps the single most important characteristic is its accuracy. It is important to remember that no measurement device is perfect and there is always some uncertainly between what the device indicates and the true value of the parameter being measured. As a DMM uses a digital display, it has both finite range and resolution. As you might guess, the more digits the meter can display, the greater its potential accuracy. DMMs are commonly described as having 3 ½ digits, 3 ¾ digits, 4 ½ digits, and so on for their displays. A fractional digit is simply a leading digit that cannot go up to nine. By common use, the term “½ digit” means that the leading digit can be no more than one. A ¾ digit specification typically means that the first digit can't be larger than three (it might also be four or even five as this terminology is not standardized)9. To clarify, a 3 ½ digit display is also referred to as a “2000 count” display. This is because there are 2000 possible values, from 0000 to 1999. Similarly, a 4 ¾ digit display is called a 40,000 count display because it has 40,000 possible values, from 00,000 to 39,000. Using a 2000 count display as an example, the next question to ask is where to place the decimal point. This display could be set up to read from 0 to 1999 volts. For convenience, we would call that a “2000 volt scale”, meaning that the maximum voltage that can be displayed is approximately 2000 volts. With an otherwise perfect meter, the error can be as large as 1 volt because we have no way of indicating fractions of volts. To solve this limitation we could add other scales by shifting the decimal point. For instance, we could make a nominal “20 volt scale” which would range up to 19.99 volts, giving us resolution down to 10 millivolts. We could go further and make a 2 volt scale with a maximum of 1.999 volts and 1 millivolt resolution as well as a 200 millivolt scale with a maximum of 199.9 millivolts yielding a resolution of one-tenth millivolt. We might need to measure very large as well as very small voltages, so the scale setting is user adjustable. It is important to use the lowest scale that can display the desired value otherwise a loss of resolution and accuracy will occur. For example, if we need to measure a voltage somewhere around five or six volts with this 2000 count meter, it should be set for the 20 volt scale. A lower setting, such as a 2 volt or 200 millivolt scale, will not be able to display the measurement and instead will indicate an overload condition (usually by flashing an abbreviated error message such as “Err” or “OL”). On the other hand, if the higher 2000 volt scale is used, the meter will only be able 9 You might rightly ask how it is that having just one numeral out of nine counts as “half” and three out of nine counts as “three fourths”. Well, that's marketing for you. 65 Figure 2.41 Digital Multimeters.
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to resolve the measurement to the nearest volt. Indeed, this is so important that some meters have auto-ranging, meaning that they will automatically choose the scale setting to give the best result. The accuracy specification for a DMM is in two parts. The first part is the percent deviation around the measured value. The second part is the added number of counts. The accuracy specification for a typical 3 ½ digit meter might be ± 2% of reading plus 3 counts. To determine the range of possible values for a reading, first determine the percentage of the reading and then add in the number of counts (i.e., one count represents the resolution of the meter on that particular scale). The resulting value represents the uncertainty of the reading. It can also be visualized as an “error envelope” surrounding the displayed value. Somewhere in that envelope is the true value measured. This is shown in the following examples. Example 2.17 A 3 ½ digit (2000 count) DMM has an accuracy specification of ± 2% of reading plus 3 counts. On its 20 volt scale it measures 5.01 volts. Determine the uncertainty in this measurement. On its 20 volt scale this meter can display up to 19.99 volts. Therefore its resolution, or single count value, is 0.01 volts. Thus, 3 counts is 0.03 volts. To this we add 2% of the reading of 5.01 volts, or 0.1002 volts. The total is 0.1302 volts or 130.2 millivolts. This represents the error envelope on either side of the reading. That is, the true value is within the range of 5.01 volts ± 130.2 millivolts, or somewhere between 4.8798 volts and 5.1402 volts. In other words, the ambiguity is about 130 millivolts out of about 5 volts. Example 2.18 A 4 ¾ digit (40,000 count) DMM has an accuracy specification of ± 0.1% of reading plus 8 counts. On its 4 volt scale it measures 3.0035 volts. Determine the uncertainty in this measurement. On its 4 volt scale this meter can display up to 3.9999 volts. Therefore its resolution is 0.0001 volts or 0.1 millivolts. 8 counts is 0.8 millivolts. To this we add 0.1% of the reading of 3.0035 volts, or 3.0035 millivolts. The total is 3.8035 millivolts. This represents the error envelope on either side of the reading and the true value is somewhere between 2.9996965 volts and 3.0073035 volts. The ambiguity here is just a few millivolts out of about 3 volts. Clearly, this meter's level of ambiguity is much reduced compared to the meter used in Example 2.17. 66
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2.9 Summary 2.9 Summary In this chapter we have examined the basic quantities that make up an electrical circuit. Charge, measured in coulombs, is a characteristic of subatomic particles; protons being positively charged and electrons being negatively charged. Like charges repel and opposite charges attract. The Bohr model of an atom shows that electrons are contained in energy shells and a single electron in the outermost shell is only loosely bound to an atom. With an applied external energy source these electrons (and thus their charge) can be moved from place to place. The rate of charge movement over time is called current and is measured in amps. Energy is defined as the ability to do work and is measured in joules. Voltage is defined as the energy required to move a charge divided by that charge, with units of volts. Power is the rate of energy usage over time and is measured in watts. Some materials allow the easy flow of current and are called conductors. In contrast, materials that inhibit the flow of electrical current are called insulators. The measure of this inhibition is called resistance and has units of ohms. Its reciprocal is called conductance and is measured in siemens. Resistors are devices designed to restrict the flow of current. They are available in a wide range of resistive values, power ratings, and physical sizes and configurations. Common resistors use a series of colored stripes, or color code, to signify their resistance value and tolerance. Efficiency is the ratio of useful power output to applied power input, normally expressed as a percentage. The higher a system's efficiency, the less energy it will use, the less it will cost to run, and generally the less heat it will generate. If the system is battery powered, higher efficiency leads to longer battery life. Battery capacity is measured in amp-hours. All other factors being equal, the higher the amp-hour rating, the longer the battery will last. The digital multimeter, or DMM, is a versatile tool for measuring voltage, current, resistance, and possibly other circuit or electrical device parameters. The accuracy of a DMM is dependent on its resolution which is in turn set by the number of digits used, or its total count. The count specification is added to a percent tolerance to arrive at a worst case error value for some specific reading. 67
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Review Questions Review Questions 1. Describe the Bohr atomic model. 2. How does charge relate to current and voltage? 3. What is the relationship between energy and power? 4. What is the relationship between resistance and conductance? 5. Define efficiency. How does it relate to operating cost? 6. Define count in terms of a digital multimeter accuracy specification. 7. Describe the resistor color code and its use. 2.10 Exercises 2.10 Exercises Analysis Analysis 1. A 2000 count digital multimeter has an accuracy specification of ±1%. If it reads a voltage of 34.2 volts, what is the possible range of the true voltage? 2. A 4000 count digital multimeter has an accuracy specification of ±0.5%. If it reads a current of 142.0 mA, what is the possible range of the true current? 3. Determine the expected voltage scales for a standard “three and a half” digit meter between 1 and 1000 volts. 4. Determine the expected current scales for a standard “three and a half” digit meter between 1 mA and 1 amp. 5. A 2000 count digital multimeter has an accuracy specification of ±2% and ±5 counts. If it reads a voltage of 7.00 volts, what is the possible range of the true voltage? 6. A 50000 count digital multimeter has an accuracy specification of ±0.1% and ±4 counts. If it reads a voltage of 7.00 volts, what is the possible range of the true voltage? 7. What is the charge in coulombs of a million million (1012) electrons? 8. What is the charge in coulombs of 1015 electrons? 9. How many electrons would be needed for a charge of 20 coulombs? 10. How many electrons would be needed for a charge of 1 microcoulomb? 11. If a charge of 2 coulombs passes through a wire in 5 seconds, what is the current? 12. If a charge of 300 millicoulombs passes through a wire in 0.1 seconds, what is the current? 68
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13. How much charge must be transferred in 0.1 seconds in order to achieve a current of 5 amps? 14. How much charge must be transferred in 20 seconds in order to achieve a current of 10 microamps? 15. Determine the resulting voltage if it takes 2 joules to move 10 coulombs of charge. 16. Determine the voltage if 15 joules is used to move 0.5 coulombs of charge. 17. How much energy is required to create a 10 volt potential difference with a 2 coulomb charge? 18. How much energy is required to create a 50 millivolt potential difference with a 0.1 coulomb charge? 19. What is the wattage equivalent of two horsepower (2 hp)? 20. What is the horsepower equivalent of 1000 watts? 21. If a device draws 2 amps of current from a 12 volt battery, determine the power delivered. 22. If a device draws 10 milliamps of current from a 1.5 volt battery, determine the power delivered. 23. A 2 hp motor draws 1800 watts from its source. Determine its efficiency. 24. A 5 hp motor draws 4.5 kw from its source. Determine its efficiency. 25. An audio power amplifier is rated for 500 watts of maximum output at an efficiency of 80%. Determine the amount of wasted power. 26. A compressor draws 10 amps of current from a 120 volt source. Its rated output is 1 hp. Determine the efficiency. 27. An application requires a battery to deliver 15 mA for at least 200 hours. Determine the required amp-hour rating. 28. Determine the required rating for a battery to deliver 0.8 A for at least 30 hours. 29. A certain 12 volt battery has a rating of 6 Ah. Determine the expected battery life using a 5 mA draw. 30. A certain AA battery has a rating of 800 mAh. Determine the expected battery life using a 20 mA draw. 31. Assume that a certain piece of material has a resistance of 80 ohms. Determine the new resistance if the length of the piece is doubled and no other parameters are changed. 69
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32. Assume that a certain piece of material has a resistance of 2 k ohms. Determine the new resistance if the width and height of the piece are doubled and no other parameters are changed. 33. Assume that a certain piece of material has a resistance of 4 ohms. Determine the new resistance if the resistivity is doubled and no other parameters are changed. 34. Assume that a certain piece of material has a resistance of 10 k ohms. Determine the new resistance if the length, width and height of the piece are all halved. 35. A certain material has a resistivity of 100 ohm-centimeters. Determine the resistance of a piece that is 1 cm wide, 0.5 cm high and 6 cm long. 36. A certain material has a resistivity of 2000 ohm-centimeters. Determine the resistance of a piece that is 2 mm wide, 4 mm high and 10 mm long. 37. A 40 ohm resistor has dimension of 0.4 cm wide by 0.2 cm high by 1 cm long. Determine the resistivity in ohm-centimeters. 38. A 5000 ohm resistor has dimension of 5 mm wide by 3 mm high by 6 mm long. Determine the resistivity in ohm-centimeters. 39. A resistor with the color code yellow-violet-red-silver has a measured value of 4806 ohms. Is this resistor within tolerance? As a percentage, how far is it from the nominal value? 40. A resistor with the color code orange-orange-yellow-gold has a measured value of 33.9 k ohms. Is this resistor within tolerance? As a percentage, how far is it from the nominal value? 41. A resistor with the color code brown-black-orange-silver has a measured value of 9980 ohms. Is this resistor within tolerance? As a percentage, how far is it from the nominal value? 42. A resistor with the color code green-blue-black-gold has a measured value of 50 ohms. Is this resistor within tolerance? As a percentage, how far is it from the nominal value? 43. Determine the value of the resistors pictured in Figure 2.42 (left-to- right: red-black-yellow, blue-gray-orange, red-violet-red-silver). 44. Determine the value of the resistors pictured in Figure 2.43 (left-to- right: red-red-orange-gold, brown-black-red-gold, yellow-violet- orange-silver). 45. Determine the value of the resistors pictured in Figure 2.44 (left-to- right: orange-orange-green-silver, white-brown-black-silver, orange- white-brown-gold). 70 Figure 2.42 a b c Figure 2.43 a b c
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46. Determine the value of the resistors pictured in Figure 2.45 (left-to-right: silver-orange-violet-yellow, green-blue-yellow-gold, gold-black-orange- yellow). 47. Determine the maximum and minimum allowed values of the resistors pictured in Figure 2-42. 48. Determine the maximum and minimum allowed values of the resistors pictured in Figure 2-43. 49. Determine the maximum and minimum allowed values of the resistors pictured in Figure 2-44. 50. Determine the maximum and minimum allowed values of the resistors pictured in Figure 2-45. Design Design 51. Determine the resistor color code for the following ohmic values using 10% tolerance: a) 56 Ω b) 33 kΩ c) 470 kΩ d) 1.2 kΩ e) 750 Ω 52. Determine the resistor color code for the following ohmic values using 5% tolerance: a) 47 Ω b) 22 kΩ c) 390 kΩ d) 2.2 kΩ e) 560 Ω Challenge Challenge 53. A radio transmitter is rated for 100 watts of maximum output at an efficiency of 90%. If it is fed from a 120 volt source, determine the current draw. 54. A certain 75 watt incandescent bulb produces 71 watts worth of heat and the remainder in the form of light. Determine its efficiency as a lighting device and its efficiency as a heating device. 55. Assume it takes an 1800 watt toaster 3 minutes to toast a bagel “just right”. If you toast a bagel every morning for a year and electricity costs 15 cents/kWh, how much will you have spent in that year to toast bagels? 71 Figure 2.44 a b c Figure 2.45 a b c
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56. Assume that you can buy standard 60 watt incandescent light bulbs for 50 cents each and that each has an expected life span of 1000 hours. In comparison, you can buy an LED light bulb that produces the same amount of light but only consumes 7 watts. The LED bulbs cost $5.50 each and have an expected life of 20,000 hours. Assuming electricity costs 14 cents/kWh, determine the total cost of running incandescent lights versus LEDs for 40,000 hours. 57. Given a 1.5 volt battery with a 500 mAh rating, how much current can it produce continuously for 25 hours?. 58. Given a 9 volt battery with a 100 mAh rating, determine the total energy storage in joules. 59. A 2000 count DMM with ±1% accuracy is used to measure the resistance of a spool of AWG 22 wire. If the measurement is 4.50 Ω, determine the length of the wire and the possible inaccuracy (in feet or meters). 60. A length of AWG 26 wire is attached to one end of a length of AWG 32 wire. The total length of the two wires is 200 meters. Determine the length and resistance of each piece if the total resistance is 75 Ω. 72
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Notes Notes ♫♫ ♫♫ 73
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3 3 Series Resistive Circuits Series Resistive Circuits 3.0 Chapter Learning Objectives 3.0 Chapter Learning Objectives After completing this chapter, you should be able to: • Describe the differences between conventional current flow and electron flow. • Identify series resistive circuits that include one or more voltage sources or a single current source. • Compute equivalent resistance, and component and node voltages for series resistive circuits. • Compute circulating current and component powers for series resistive circuits. • Utilize Ohm's law, Kirchhoff's voltage law (KVL) and the voltage divider rule (VDR) to aid in the analysis of series resistive circuits. • Identify and describe the usage of potentiometers and rheostats. • Utilize computer simulation tools to investigate and verify basic electric circuit quantities such as component voltages. 3.1 Introduction 3.1 Introduction With the requisite background information of the first two chapters now behind us, in this chapter we shall begin our analysis of electrical circuits. Here we will introduce the most simple configuration, the series loop. It can contain any number of resistors and voltage sources, or in place of the voltage sources, a single current source. We shall examine how to determine the current flow through each component, the voltage across each component and the power either dissipated or generated by each component. Other practical issues will also be examined, such as the effect of component tolerance on the circuit parameters. 3.2 Conventional Current Flow and Electron Flow 3.2 Conventional Current Flow and Electron Flow Before we dive into series circuits we need to consider an interesting question involving the direction of current flow. Does it flow from positive to negative or from negative to positive? For that matter, does it even make a difference as far as our analyses will be concerned? Benjamin Franklin (pictured in Figure 3.1) began experimenting with the phenomenon of electricity in 1746. In 1752 he performed his famous kite experiment proving that lightning is a form of electricity by capturing charge from storm clouds in a leyden jar (an early form of an electrical charge storage device)10. At this time the 10 It is worth noting that Franklin's kite was not struck by lightning. If it had been, he likely would have been killed. The hemp string that was used for the kite was sufficiently wet from rain that it was possible to transfer charge from the atmosphere to the leyden jar, and subsequently to a metal key which would emit a spark. 74
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modern concept of an atomic model with electrons and protons did not exist and electricity was conceived of as a sort of fluid. Franklin surmised that the “electrical flow” moved from positive to negative. This idea was accepted and became the conventional view. Today we call this idea conventional current flow. In this model, current flows from a more positive voltage to a less positive voltage. We know now that the electron is the charge carrier in metals and the electrons travel in the reverse direction. Essentially, Franklin guessed wrong. Electrons move from a lower potential to a higher potential. We call this model electron flow. For most work, engineers and technicians use conventional flow, although in some cases, such as the explanation of semiconductors, electron flow is easier to visualize for some people. In short, conventional flow exists for historical reasons, and it is the model used for most analyses, including this text. You might think the current direction would make a big difference in an analysis; after all, it certainly makes a big difference if you drive a car in the wrong direction. It turns out that both forms will achieve the desired results, we just have to be consistent with the usage. To better understand this, consider that the movement of a net negative charge in one direction can be thought of as a movement of a net positive charge in the other direction. That is, the movement of an electron creates a “hole” where it used to be and that hole is net positive. This is illustrated in Figure 3.2. Here we start at the top with a tube of identical marbles, all pushed to the right. In each step below it we move a marble to the left, mimicking the flow of electrons in a circuit. When we reach the bottom, each marble has been pushed left by one place. We can also arrive at the bottom drawing by simply taking the right-most marble in the top drawing and inserting it to the extreme left by jumping over the other three marbles. Here's the important bit: instead of imagining the marbles moving left, we can also think in terms of “negative space” and imagine the empty slot moving to the right. That's hole flow. The two views are functionally identical as they lead to the same outcome. 3.3 The Series Connection 3.3 The Series Connection The word circuit comes from the Latin root circ, meaning “ring” or “around”. An electrical circuit consists of at least one ring or loop through which current flows. For example, if we have a battery attached to a lamp as in Figure 3.3, the current exits the battery, flows through the lamp, and then returns to the other side of the battery creating a loop or completed circuit. Without a path back to the battery, current will not flow. Thus, if we cut one of the wires connecting the battery and lamp, there is no path for current and no current flows. This is referred to as an open circuit and is a common fault that occurs when electronic systems are dropped or struck forcefully. Obviously, this will tend to render the circuit unusable. In this example, the lamp will not illuminate. The opposite of the open circuit is the short 75 Figure 3.1 Benjamin Franklin: Technically incorrect but it doesn't really matter. Figure 3.2 Electron versus hole flow. Figure 3.3 Battery and lamp circuit.
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circuit. In a short circuit, an unintended alternate path for current flow exists and this also can create a malfunction. In the case of our battery and lamp, a short circuit can occur if a piece of wire or metal accidentally fell across the terminals of the lamp. The current would then have a high conductance (i.e., low resistance) path around the lamp. The vast majority of the current would take this low resistance path instead of the higher resistance path presented by the lamp. The result would be that the lamp would not light. In either the open or short case, the light does not function but there is an important difference: for the short circuit, excessive current will flow out of the battery because there is little to resist the flow of current. Thus, the battery will be drained very quickly. For the open, no current flows and thus the battery is not drained. Series connections are not limited to just two components. In general, a series connection is any connection of components configured such that the current through each component must be the same. This is illustrated in Figure 3.4. Inside each of the lettered boxes would be a component such as a resistor or a voltage source. Note that for each component, there is one entrance point and one exit point. No matter which box you pick, the current flowing through it must be the same as the current flowing into the next box or out of the preceding box, and it doesn't matter if you follow this path in a clockwise or counterclockwise fashion. Thus, this entire configuration is a series connection. The idea that current is consistent throughout should be self- evident. After all, the only way the current through, say, item B could be different from that flowing through item C or D is if some of it somehow “disappeared” along the way by magic. It is important to remember that consistent current is the hallmark feature defining a series connection: The current is the same everywhere in a series connection. (2.1) It is possible that only a portion of a circuit exhibits a series connection. Consider the more complex diagram presented in Figure 3.5. Some of these items are in series and some are not. For example, items A and B are in series with each other but not in series with the remaining items. Why? Because if we imagine the current flowing through A and then through B they must be the same, however, once beyond B, the current could split and flow down other branches: a portion entering C, a portion entering D and the remainder flowing into E. On the other hand, items C and F are in series with each other because whatever current is flowing through one of them must be flowing through the other. Thus, items A and B are in series with each other and items C and F are in series with each other, although all four are not in series as a group. It is possible that no two items in a circuit are strictly in series. We will see examples of this in upcoming chapters. Further, just because the currents through two items happen to be the same, that does not necessarily mean they are in series. Identical currents could be just a by-product of the component values chosen. For example, 76 Figure 3.4 A generic series configuration. A B C D E Figure 3.5 A more complex configuration. A B C D F E
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even if items D and E in Figure 3.5 have the same numeric value for current, we would not say that they are in series, anymore than we would say that any two people with the same last name would have to be siblings. 3.4 Combining Series Components 3.4 Combining Series Components Typically, a series connection will include multiple resistors. Referring back to the resistance equation presented in Chapter 2, Equation 2.11, we can see that resistors in series add. R =ρl A If we consider two identical resistors placed in series, one after the other as in Figure 3.6, the effective length would double while keeping the resistivity and area unchanged. The combined result would be a doubling of the resistance of just one of them. If we then generalize this to two arbitrary resistors of identical resistivity and area, the lengths would dictate the resistance of each, and the combined lengths would then reflect the resistance of the pair. We can generalize this further for N resistors. Thus we find that the equivalent resistance of a group of series resistors is their sum: RTotal = R1+R2+R3+...+RN (3.2) Consequently, as resistors in series add, total resistance may be found by summing the individual resistors. Example 3.1 A string of resistors is placed in series as shown in Figure 3.7. Their values are: 120 Ω, 390 Ω, 560 Ω and 470 Ω. Determine the equivalent series value. RT = R1+R2+R3+R4 RT = 120Ω+390Ω+560Ω+470Ω RT =1540Ω Multiple voltage sources in series may also be added, however, polarities must be considered as opposing voltages partially cancel each other (i.e., adding a negative). This concept is presented in the next example. 77 Figure 3.6 Resistors in series. A B Figure 3.7 Resistor string for Example 3.1.
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Example 3.2 Determine the equivalent series value of the voltage sources presented in Figure 3.8. If we use point b as our reference, by inspection the top of the 12 volt source is 12 volts above point b (reminder, the long bar denotes the positive terminal). Also, by inspection, the right side of the 3 volt source (point a) is negative with respect to its left side. As the left side of this source is connected to the positive terminal of the 12 volt source, then it too must be 12 volts above point b. As its right side is 3 volts less than this side, point a must be 3 volts less than 12 volts, or 9 volts above point b. Thus Vab = 9 volts. Chasing this further, if the 3 volt source had been flipped so that it had the same polarity as the 12 volt source (positive toward the right) then Vab = 15 volts. If the 12 volt source had been flipped (positive toward bottom) with the 3 volt source as drawn originally, then Vab = −15 volts. If both sources had been flipped then Vab = −9 volts. Finally, if point a had been taken as the reference then these four potentials would have the opposite polarity because, by definition, Vba = −Vab. In contrast to voltage sources, differing current sources are not placed in series as they would each attempt to establish a different series current, a practical impossibility. Refer to Figure 3.9 as a reminder! 3.5 Ohm's Law 3.5 Ohm's Law In Chapter 2, the concept of resistance was introduced using the following generality: Effect = Cause Oppostion To review, the cause is a voltage source, the opposition is the resistance, and the effect is the resulting current. If the item the current passes through is linear, such as the simple resistors depicted in Figure 3.10, then this relationship can be rewritten as: I = V R 78 Figure 3.8 Voltage sources in series. Figure 3.9 Placing current sources in series generally is evil. Figure 3.10 Current-voltage plots for simple resistors. V I -I -V
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