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What is an Open Textbook? Open textbooks are complete textbooks that have been funded, published, and licensed to be freely used, adapted, and distributed. As a particular type of Open Educational Resource (OER), this open textbook is intended to provide authoritative, accurate, and comprehensive subject content at no ... | Electromagnetics_Vol1_Page_6_Chunk1301 |
Contents Preface xii 1 Preliminary Concepts 1 1.1 What is Electromagnetics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Fundamentals of Waves . . . . . . . . . . . . . . . . . . .... | Electromagnetics_Vol1_Page_7_Chunk1302 |
. . . . . . . 27 vi | Electromagnetics_Vol1_Page_7_Chunk1303 |
CONTENTS vii 3 Transmission Lines 30 3.1 Introduction to Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Types of Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Transmission Lines as Two-Port Devices . . . . . . . . . . . . .... | Electromagnetics_Vol1_Page_8_Chunk1304 |
. . . . . . . . . . . . . . . 56 3.19 Quarter-Wavelength Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.20 Power Flow on Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.21 Impedance Matching: General Considerations . . . . . . . . . . . . . .... | Electromagnetics_Vol1_Page_8_Chunk1305 |
viii CONTENTS 4.3 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | Electromagnetics_Vol1_Page_9_Chunk1306 |
. . . . . . . 105 5.9 Independence of Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.10 Kirchoff’s Voltage Law for Electrostatics: Integral Form . . . . . . . . . . . . . . . . . . . . . 108 5.11 Kirchoff’s Voltage Law for Electrostatics: Differential Form . . . . . . . . . .... | Electromagnetics_Vol1_Page_9_Chunk1307 |
CONTENTS ix 5.19 Charge and Electric Field for a Perfectly Conducting Region . . . . . . . . . . . . . . . . . . . 122 5.20 Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.21 Dielectric Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | Electromagnetics_Vol1_Page_10_Chunk1308 |
. . . . . . . . . . . . . . . 149 7.5 Magnetic Field of an Infinitely-Long Straight Current-Bearing Wire . . . . . . . . . . . . . . . 150 7.6 Magnetic Field Inside a Straight Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.7 Magnetic Field of a Toroidal Coil . . . . . . . . . . . . . . . . . ... | Electromagnetics_Vol1_Page_10_Chunk1309 |
x CONTENTS 7.12 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.13 Inductance of a Straight Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.14 Inductance of a Coaxial Structure . . . . . . . . . . . . . . . . . . . . . . . .... | Electromagnetics_Vol1_Page_11_Chunk1310 |
. . . . . . . . . . . . . . . . . . . . 198 9.4 Uniform Plane Waves: Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 9.5 Uniform Plane Waves: Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 9.6 Wave Polarization . . . . . . . . . . . . . . . . . . . . .... | Electromagnetics_Vol1_Page_11_Chunk1311 |
CONTENTS xi A.3 Conductivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 B Mathematical Formulas 217 B.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 B.2 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . ... | Electromagnetics_Vol1_Page_12_Chunk1312 |
Preface About This Book [m0146] Goals for this book. This book is intended to serve as a primary textbook for a one-semester introductory course in undergraduate engineering electromagnetics, including the following topics: electric and magnetic fields; electromagnetic properties of materials; electromagnetic waves; and... | Electromagnetics_Vol1_Page_13_Chunk1313 |
xiii assembled (“remixed”) to create new and different versions of the book. The text “[m0146]” that you see at the beginning of this section uniquely identifies the module within the larger set of modules provided by the project. This identification is provided because different remixes of this book may exist, each cons... | Electromagnetics_Vol1_Page_14_Chunk1314 |
xiv PREFACE About the Open Electromagnetics Project [m0148] The Open Electromagnetics Project was established at Virginia Tech in 2017 with the goal of creating no-cost openly-licensed textbooks for courses in undergraduate engineering electromagnetics. While a number of very fine traditional textbooks are available on ... | Electromagnetics_Vol1_Page_15_Chunk1315 |
Chapter 1 Preliminary Concepts 1.1 What is Electromagnetics? [m0037] The topic of this book is applied engineering electromagnetics. This topic is often described as “the theory of electromagnetic fields and waves,” which is both true and misleading. The truth is that electric fields, magnetic fields, their sources, waves... | Electromagnetics_Vol1_Page_16_Chunk1316 |
2 CHAPTER 1. PRELIMINARY CONCEPTS electromagnetic theory that applies when these considerations are not important. Many instances of this “electromagnetics as generalization” vs. “lumped-element theory as special case” dichotomy appear in the study of electromagnetics. There is more to the topic, however. There are man... | Electromagnetics_Vol1_Page_17_Chunk1317 |
1.2. ELECTROMAGNETIC SPECTRUM 3 • Non-contact sensors • Photonics • Printed circuit board stackup and layout • Radar • Radio wave propagation • Radio frequency electronics • Signal integrity • Transformers • Waveguides In summary: Applied engineering electromagnetics is the study of those aspects of electrical engineer... | Electromagnetics_Vol1_Page_18_Chunk1318 |
4 CHAPTER 1. PRELIMINARY CONCEPTS Regime Frequency Range Wavelength Range γ-Ray > 3 × 1019 Hz < 0.01 nm X-Ray 3 × 1016 Hz – 3 × 1019 Hz 10–0.01 nm Ultraviolet (UV) 2.5 × 1015 – 3 × 1016 Hz 120–10 nm Optical 4.3 × 1014 – 2.5 × 1015 Hz 700–120 nm Infrared (IR) 300 GHz – 4.3 × 1014 Hz 1 mm – 700 nm Radio 3 kHz – 300 GHz 1... | Electromagnetics_Vol1_Page_19_Chunk1319 |
1.3. FUNDAMENTALS OF WAVES 5 Band Frequencies Wavelengths Typical Applications EHF 30-300 GHz 10–1 mm 60 GHz WLAN, Point-to-point data links SHF 3–30 GHz 10–1 cm Terrestrial & Satellite data links, Radar UHF 300–3000 MHz 1–0.1 m TV broadcasting, Cellular, WLAN VHF 30–300 MHz 10–1 m FM & TV broadcasting, LMR HF 3–30 MHz... | Electromagnetics_Vol1_Page_20_Chunk1320 |
6 CHAPTER 1. PRELIMINARY CONCEPTS p(x,t) x p(x,t) x c⃝Y. Qin CC BY 4.0 Figure 1.2: The differential pressure p(x, t) (top) a short time after the clap and (bottom) a slightly longer time after the clap. pressure to be continuous over space. So instead, we see a rounded pulse representing the rapid build-up and similarl... | Electromagnetics_Vol1_Page_21_Chunk1321 |
1.3. FUNDAMENTALS OF WAVES 7 Note that Am and ψ are not determined by the wave equation, but instead are properties of the source. Specifically, Am is determined by how hard we blow, and ψ is determined by the time at which we began to blow and the location of the trumpet. For simplicity, let us assume that we begin to ... | Electromagnetics_Vol1_Page_22_Chunk1322 |
8 CHAPTER 1. PRELIMINARY CONCEPTS Phase velocity vp = λf is the speed at which a point of constant phase in a sinusoidal waveform travels. Recall that in Equation 1.2 we declared that βx is subtracted from the argument of the sinusoidal function. To understand why, let’s change the sign of βx and see if it still satisfi... | Electromagnetics_Vol1_Page_23_Chunk1323 |
1.4. GUIDED AND UNGUIDED WAVES 9 1.4 Guided and Unguided Waves [m0040] Broadly speaking, waves may be either guided or unguided. Unguided waves include those that are radiated by antennas, as well as those that are unintentionally radiated. Once initiated, these waves propagate in an uncontrolled manner until they are ... | Electromagnetics_Vol1_Page_24_Chunk1324 |
10 CHAPTER 1. PRELIMINARY CONCEPTS Ame j magnitude Im{C} Re{C} Am phase ( | Electromagnetics_Vol1_Page_25_Chunk1325 |
1.5. PHASORS 11 dependence: C = Amejψ (1.10) This does not normally cause any confusion since the definition of a phasor requires that values of C and ψ are those that apply at whatever frequency is represented by the suppressed sinusoidal dependence ejωt. Table 1.4 shows mathematical representations of the same phasors... | Electromagnetics_Vol1_Page_26_Chunk1326 |
12 CHAPTER 1. PRELIMINARY CONCEPTS A(t) C Am cos (ωt) Am Am cos (ωt + ψ) Amejψ Am sin (ωt) = Am cos | Electromagnetics_Vol1_Page_27_Chunk1327 |
1.6. UNITS 13 Summarizing: Phasor analysis does not limit us to sinusoidal waveforms. Phasor analysis is not only applica- ble to sinusoids and signals that are sufficiently narrowband, but is also applicable to signals of arbitrary bandwidth via Fourier analysis. Additional Reading: • “Phasor” on Wikipedia. • “Fourier ... | Electromagnetics_Vol1_Page_28_Chunk1328 |
14 CHAPTER 1. PRELIMINARY CONCEPTS Unit Abbreviation Quantifies: ampere A electric current coulomb C electric charge farad F capacitance henry H inductance hertz Hz frequency joule J energy meter m distance newton N force ohm Ω resistance second s time tesla T magnetic flux density volt V electric potential watt W power ... | Electromagnetics_Vol1_Page_29_Chunk1329 |
1.7. NOTATION 15 1.7 Notation [m0005] The list below describes notation used in this book. • Vectors: Boldface is used to indicate a vector; e.g., the electric field intensity vector will typically appear as E. Quantities not in boldface are scalars. When writing by hand, it is common to write “E” or “−→ E ” in lieu of ... | Electromagnetics_Vol1_Page_30_Chunk1330 |
16 CHAPTER 1. PRELIMINARY CONCEPTS Image Credits Fig. 1.1 c⃝V. Blacus, https://commons.wikimedia.org/wiki/File:Electromagnetic-Spectrum.svg, CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Fig. 1.2: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:M0074 fClap.svg, CC BY 4.0 (https://creativecommons.org... | Electromagnetics_Vol1_Page_31_Chunk1331 |
Chapter 2 Electric and Magnetic Fields 2.1 What is a Field? [m0001] A field is the continuum of values of a quantity as a function of position and time. The quantity that the field describes may be a scalar or a vector, and the scalar part may be either real- or complex-valued. In electromagnetics, the electric field inte... | Electromagnetics_Vol1_Page_32_Chunk1332 |
18 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS F c⃝M. Goldammer CC BY SA 4.0 Figure 2.2: A map of the force that would be expe- rienced by a second particle having a positive charge. Here, the magnitude and direction of the force is indi- cated by the size and direction of the arrow. In that scenario, we could make a map i... | Electromagnetics_Vol1_Page_33_Chunk1333 |
2.2. ELECTRIC FIELD INTENSITY 19 A B + - 9V 1mm c⃝Y. Qin CC BY 3.0 Figure 2.3: A simple circuit used to describe the con- cept of electric field intensity. In this example, E at point C is 9000 V/m directed from B toward A. E points in the direction in which electric poten- tial is most rapidly decreasing, and the magni... | Electromagnetics_Vol1_Page_34_Chunk1334 |
20 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS 2.3 Permittivity [m0008] Permittivity describes the effect of material in determining the electric field in response to electric charge. For example, one can observe from laboratory experiments that a particle having charge q gives rise to the electric field E = ˆR q 1 4πR2 1 ǫ ... | Electromagnetics_Vol1_Page_35_Chunk1335 |
2.4. ELECTRIC FLUX DENSITY 21 2.4 Electric Flux Density [m0011] Electric flux density, assigned the symbol D, is an alternative to electric field intensity (E) as a way to quantify an electric field. This alternative description offers some actionable insight, as we shall point out at the end of this section. First, what ... | Electromagnetics_Vol1_Page_36_Chunk1336 |
22 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS Additional Reading: • “Flux” on Wikipedia. 2.5 Magnetic Flux Density [m0003] Magnetic flux density is a vector field which we identify using the symbol B and which has SI units of tesla (T). Before offering a formal definition, it is useful to consider the broader concept of the ... | Electromagnetics_Vol1_Page_37_Chunk1337 |
2.5. MAGNETIC FLUX DENSITY 23 c⃝Y. Qin CC BY 4.0 Figure 2.5: Evidence that current can also create a magnetic field. now interested in quantifying its behavior. To begin, let us consider the effect of a magnetic field on a electrically-charged particle. First, imagine a region of free space with no electric or magnetic fi... | Electromagnetics_Vol1_Page_38_Chunk1338 |
24 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS and aptly-named electromagnetic force. The electromagnetic force also gives rises to the electric field, and it is only limited intuition, grounded in classical physics, that leads us to perceive the electric and magnetic fields as distinct phenomena. For our present purposes – ... | Electromagnetics_Vol1_Page_39_Chunk1339 |
2.6. PERMEABILITY 25 This is true in a sense even for field lines which seem to form straight lines (for example, those along the axis of the bar magnet and the coil in Figures 2.7 and 2.8), since a field line that travels to infinity in one direction reemerges from infinity in the opposite direction. Additional Reading: •... | Electromagnetics_Vol1_Page_40_Chunk1340 |
26 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS materials, and may exhibit values of µr as large as ∼106. A commonly-encountered category of magnetic materials is ferromagnetic material, of which the best-known example is iron. Additional Reading: • “Permeability (electromagnetism)” on Wikipedia. • Section 7.16 (“Magnetic M... | Electromagnetics_Vol1_Page_41_Chunk1341 |
2.8. ELECTROMAGNETIC PROPERTIES OF MATERIALS 27 boundary conditions on H constrain the component of the magnetic field which is tangent to the boundary separating two otherwise-homogeneous regions. If one ignores the characteristics of the magnetic field represented by H and instead considers only B, then only the perpen... | Electromagnetics_Vol1_Page_42_Chunk1342 |
28 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS • Isotropy. A material that is isotropic behaves in precisely the same way regardless of how it is oriented with respect to sources and fields occupying the same space. A counter-example is quartz, whose atoms are arranged in a uniformly-spaced crystalline lattice. As a result,... | Electromagnetics_Vol1_Page_43_Chunk1343 |
2.8. ELECTROMAGNETIC PROPERTIES OF MATERIALS 29 Image Credits Fig. 2.1: c⃝M. Goldammer, https://commons.wikimedia.org/wiki/File:M0002 fTwoChargedParticles.svg, CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 2.2: c⃝M. Goldammer, https://commons.wikimedia.org/wiki/File:M0002 fForceMap.svg, CC BY SA ... | Electromagnetics_Vol1_Page_44_Chunk1344 |
Chapter 3 Transmission Lines 3.1 Introduction to Transmission Lines [m0028] A transmission line is a structure intended to transport electromagnetic signals or power. A rudimentary transmission line is simply a pair of wires with one wire serving as a datum (i.e., a reference; e.g., “ground”) and the other wire bearing... | Electromagnetics_Vol1_Page_45_Chunk1345 |
3.2. TYPES OF TRANSMISSION LINES 31 0.001 · 360◦= 0.36◦over the length of the transmission line, which is about 0.72◦for a round trip. So, to a good approximation, the entire transmission line is at the same electrical potential and thus transparent to the source and destination. However, if l is increased to 3 m, or i... | Electromagnetics_Vol1_Page_46_Chunk1346 |
32 CHAPTER 3. TRANSMISSION LINES g ound plane dielectric slab metallic trace c⃝SpinningSpark CC BY SA 3.0 (modified) Figure 3.3: Structure of a microstrip transmission line. Figure 3.4: Structure of the electric and magnetic fields within coaxial line. In this case, the wave is propagating away from the viewer. Figure 3... | Electromagnetics_Vol1_Page_47_Chunk1347 |
3.3. TRANSMISSION LINES AS TWO-PORT DEVICES 33 c⃝BigRiz CC BY SA 3.0 Unported Figure 3.7: Strands of optical fiber. 3.3 Transmission Lines as Two-Port Devices [m0077] Figure 3.8 shows common ways to represent transmission lines in circuit diagrams. In each case, the source is represented using a Th´evenin equivalent cir... | Electromagnetics_Vol1_Page_48_Chunk1348 |
34 CHAPTER 3. TRANSMISSION LINES Additional Reading: • “Th´evenin’s theorem” on Wikipedia. 3.4 Lumped-Element Model [m0029] It is possible to ascertain the relevant behaviors of a transmission line using elementary circuit theory applied to a differential-length lumped-element model of the transmission line. The concep... | Electromagnetics_Vol1_Page_49_Chunk1349 |
3.5. TELEGRAPHER’S EQUATIONS 35 z z z z z Figure 3.9: Interpretation of a transmission line as a cascade of discrete series-connected two-ports. R'Δz L'Δz G'Δz C'Δz c⃝Omegatron CC BY SA 3.0 Unported (modified) Figure 3.10: Lumped-element equivalent circuit model for each of the two-ports in Figure 3.9. cross-sectional ... | Electromagnetics_Vol1_Page_50_Chunk1350 |
36 CHAPTER 3. TRANSMISSION LINES Applying Kirchoff’s current law at the right port, we obtain: i(z, t)−(G′∆z) v(z+∆z, t)−(C′∆z) ∂ ∂tv(z+∆z, t) −i(z + ∆z, t) = 0 (3.4) Moving terms referring to potential to the right side of the equation and then dividing through by ∆z, we obtain −i(z + ∆z, t) −i(z, t) ∆z = G′ v(z + ∆z,... | Electromagnetics_Vol1_Page_51_Chunk1351 |
3.6. WAVE EQUATION FOR A TEM TRANSMISSION LINE 37 3.6 Wave Equation for a TEM Transmission Line [m0027] Consider a TEM transmission line aligned along the z axis. The phasor form of the Telegrapher’s Equations (Section 3.5) relate the potential phasor eV (z) and the current phasor eI(z) to each other and to the lumped-... | Electromagnetics_Vol1_Page_52_Chunk1352 |
38 CHAPTER 3. TRANSMISSION LINES 3.7 Characteristic Impedance [m0052] Characteristic impedance is the ratio of voltage to current for a wave that is propagating in single direction on a transmission line. This is an important parameter in the analysis and design of circuits and systems using transmission lines. In this... | Electromagnetics_Vol1_Page_53_Chunk1353 |
3.8. WAVE PROPAGATION ON A TEM TRANSMISSION LINE 39 The characteristic impedance Z0 (Ω) is the ratio of potential to current in a wave traveling in a single direction along the transmission line. Take care to note that Z0 is not the ratio of eV (z) to eI(z) in general; rather, Z0 relates only the potential and current ... | Electromagnetics_Vol1_Page_54_Chunk1354 |
40 CHAPTER 3. TRANSMISSION LINES z v+(z,t=0) Figure 3.12: The potential v+(z, t) of the wave travel- ing in the +z direction at t = 0 for ψ = 0. or equivalently in the time domain: v+(z, t) = Re n eV +(z) ejωto = Re V + 0 e−γzejωt = V + 0 e−αz cos (ωt −βz + ψ) (3.44) where ψ is the phase of V + 0 . Figure 3.... | Electromagnetics_Vol1_Page_55_Chunk1355 |
3.9. LOSSLESS AND LOW-LOSS TRANSMISSION LINES 41 Similarly, we find that the current eI−(z) associated with eV −(z) for the wave traveling in the −z direction is eI−(z) = −V − 0 Z0 e−γz (3.50) The negative sign appearing in the above expression emerges as a result of the sign conventions used for potential and current i... | Electromagnetics_Vol1_Page_56_Chunk1356 |
42 CHAPTER 3. TRANSMISSION LINES Of course if the line is strictly lossless (i.e., R′ = G′ = 0) then these are not approximations, but rather the exact expressions. In practice, these approximations are quite commonly used, since practical transmission lines typically meet the conditions expressed in Inequalities 3.54 ... | Electromagnetics_Vol1_Page_57_Chunk1357 |
3.10. COAXIAL LINE 43 Figure 3.14: Structure of the electric and magnetic fields within coaxial line. In this case, the wave is propagating away from the viewer. Section 7.14, the inductance per unit length is L′ = µ0 2π ln b a (3.62) The loss conductance G′ depends on the conductance σs of the spacer material, and ... | Electromagnetics_Vol1_Page_58_Chunk1358 |
44 CHAPTER 3. TRANSMISSION LINES C′ ≈67.7 pF/m. The conductivity of polyethylene is σs ∼= 5.9 × 10−5 S/m, yielding G′ ≈200 µS/m. Typical resistance per unit length R′ is on the order of 0.1 Ω/m near DC, increasing approximately in proportion to the square root of frequency. From the above values, we find that RG-59 sati... | Electromagnetics_Vol1_Page_59_Chunk1359 |
3.11. MICROSTRIP LINE 45 conductor geometry is asymmetric and the one conductor – namely, the ground plane – also normally serves as ground for the source and load. The spacer material is typically a low-loss dielectric material having permeability approximately equal to that of free space (µ ≈µ0) and relative permitti... | Electromagnetics_Vol1_Page_60_Chunk1360 |
46 CHAPTER 3. TRANSMISSION LINES Using this concept, we obtain λ = 2π β = 2π β0√ǫr,eff = λ0 √ǫr,eff (3.73) where λ0 is the free-space wavelength c/f. Similarly the phase velocity vp, can be estimated using the relationship vp = ω β = c √ǫr,eff (3.74) i.e., the phase velocity in microstrip is slower than c by a factor o... | Electromagnetics_Vol1_Page_61_Chunk1361 |
3.12. VOLTAGE REFLECTION COEFFICIENT 47 3.12 Voltage Reflection Coefficient [m0084] We now consider the scenario shown in Figure 3.18. Here a wave arriving from the left along a lossless transmission line having characteristic impedance Z0 arrives at a termination located at z = 0. The impedance looking into the terminat... | Electromagnetics_Vol1_Page_62_Chunk1362 |
48 CHAPTER 3. TRANSMISSION LINES If the terminating impedance is equal to the char- acteristic impedance of the transmission line, then there is no reflection. If, on the other hand, ZL ̸= Z0, then |Γ| > 0, V − 0 = ΓV + 0 , and a leftward-traveling reflected wave exists. Since ZL may be real-, imaginary-, or complex-valu... | Electromagnetics_Vol1_Page_63_Chunk1363 |
3.13. STANDING WAVES 49 value of |Γ| is limited to the range 0 to 1. To see this, note: Γ = ZL −Z0 ZL + Z0 = ZL/Z0 −1 ZL/Z0 + 1 (3.88) Note that the smallest possible value of |Γ| occurs when the numerator is zero; i.e., when ZL = Z0. Therefore, the smallest value of |Γ| is zero. The largest possible value of |Γ| occur... | Electromagnetics_Vol1_Page_64_Chunk1364 |
50 CHAPTER 3. TRANSMISSION LINES z V + 0 2 V + 0 V(z) / z=0 (a) Potential. I(z) ! " z=0 z I + 0 2 I + 0 (b) Current. Figure 3.19: Standing wave associated with an open- circuit termination at z = 0 (incident wave arrives from left). eI(z) = |V + 0 | Z0 p 2 −2 cos (2βz + φ) (3.98) where φ = 0 for an open circuit and φ =... | Electromagnetics_Vol1_Page_65_Chunk1365 |
3.14. STANDING WAVE RATIO 51 3.14 Standing Wave Ratio [m0081] Precise matching of transmission lines to terminations is often not practical or possible. Whenever a significant mismatch exists, a standing wave (Section 3.13) is apparent. The quality of the match is commonly expressed in terms of the standing wave ratio (... | Electromagnetics_Vol1_Page_66_Chunk1366 |
52 CHAPTER 3. TRANSMISSION LINES • SWR = 2.0 corresponds to |Γ| = 1/3. • SWR = 3.0 corresponds to |Γ| = 1/2. 3.15 Input Impedance of a Terminated Lossless Transmission Line [m0087] Consider Figure 3.22, which shows a lossless transmission line being driven from the left and which is terminated by an impedance ZL on the... | Electromagnetics_Vol1_Page_67_Chunk1367 |
3.15. INPUT IMPEDANCE OF A TERMINATED LOSSLESS TRANSMISSION LINE 53 e−jβl: Zin(l) = Z0 1 + Γe−j2βl 1 −Γe−j2βl (3.109) Recall that Γ in the above expression is: Γ = ZL −Z0 ZL + Z0 (3.110) Summarizing: Equation 3.109 is the input impedance of a lossless transmission line having characteristic impedance Z0 and which is te... | Electromagnetics_Vol1_Page_68_Chunk1368 |
54 CHAPTER 3. TRANSMISSION LINES 3.16 Input Impedance for Open- and Short-Circuit Terminations [m0088] Let us now consider the input impedance of a transmission line that is terminated in an open- or short-circuit. Such a transmission line is sometimes referred to as a stub. First, why consider such a thing? From a “lu... | Electromagnetics_Vol1_Page_69_Chunk1369 |
3.17. APPLICATIONS OF OPEN- AND SHORT-CIRCUITED TRANSMISSION LINE STUBS 55 Following the same procedure detailed above for the short-circuit case, we find Zin(l) = −jZ0 cot βl (3.119) Figure 3.23(b) shows the result for open-circuit termination. As expected, Zin →∞for l = 0, and the same λ/2 periodicity is observed. Wha... | Electromagnetics_Vol1_Page_70_Chunk1370 |
56 CHAPTER 3. TRANSMISSION LINES bipolar transistors in common-emitter configuration, it is often useful to introduce a little inductance between the emitter and ground. This is known as “inductive degeneration,” “emitter induction,” or sometimes by other names. It can be difficult to find suitable inductors, especially f... | Electromagnetics_Vol1_Page_71_Chunk1371 |
3.19. QUARTER-WAVELENGTH TRANSMISSION LINE 57 be possible to unambiguously determine βl. Although we shall not present the method here, it is possible to resolve this ambiguity by making multiple measurements over a range of frequencies. Once βl is determined, it is simple to determine l given β, β given l, and then vp... | Electromagnetics_Vol1_Page_72_Chunk1372 |
58 CHAPTER 3. TRANSMISSION LINES l= /4 Figure 3.24: Impedance-matching using a quarter- wavelength transmission line. transmission line of length λ/4 is sometimes referred to as a quarter-wave inverter or simply as a impedance inverter. Quarter-wave lines play a very important role in RF engineering. As impedance inver... | Electromagnetics_Vol1_Page_73_Chunk1373 |
3.19. QUARTER-WAVELENGTH TRANSMISSION LINE 59 L D/4 (completely real-valued) Figure 3.25: Impedance-matching a complex-valued load impedance using quarter-wavelength transmis- sion line. patch antenna, use characteristic impedance Z01 = 50 Ω. Determine the lengths l1 and l2 of the two segments of transmission line, and... | Electromagnetics_Vol1_Page_74_Chunk1374 |
60 CHAPTER 3. TRANSMISSION LINES RF out (moderate Z) DC power in (low E) Figure 3.26: Use of an inductor to decouple the DC input power from the RF output signal at the output of a common-emitter RF amplifier. will flow predominantly in the direction of the power supply as opposed to following the desired path, which exh... | Electromagnetics_Vol1_Page_75_Chunk1375 |
3.20. POWER FLOW ON TRANSMISSION LINES 61 i+(z, t) = V + 0 Z0 cos (ωt −βz + φ) (3.136) And so the associated time-average power is P + av(z) = 1 T Z T 0 v+(z, t) i+(z, t) dt = V + 0 2 Z0 · 1 T Z T 0 cos2 (ωt −βz + φ) dt (3.137) Employing a well-known trigonometric identity: cos2 θ = 1 2 + 1 2 cos 2θ (3.138) we may rewr... | Electromagnetics_Vol1_Page_76_Chunk1376 |
62 CHAPTER 3. TRANSMISSION LINES 96%, respectively. In either case (from Section 3.14): SWR = 1 + |Γ| 1 −|Γ| = 1.5 This is often acceptable, but may not be good enough in some particular applications. Suffice it to say that it is not necessarily required to use an impedance matching device to connect 50 Ω to 75 Ωdevices... | Electromagnetics_Vol1_Page_77_Chunk1377 |
3.22. SINGLE-REACTANCE MATCHING 63 employ discrete components and do not require knowledge of electromagnetics.8 To list just a few of these approaches: transformers, resistive (lossy) matching, single-reactance matching, and two-reactance (“L” network) matching. However, all of these have limitations. Perhaps the most... | Electromagnetics_Vol1_Page_78_Chunk1378 |
64 CHAPTER 3. TRANSMISSION LINES ZL Z1 Z0 I jXS Zin Figure 3.28: Single-reactance matching with a series reactance. Example 3.9. Single reactance in series. Design a match consisting of a transmission line in series with a single capacitor or inductor that matches a source impedance of 50Ωto a load impedance of 33.9 + ... | Electromagnetics_Vol1_Page_79_Chunk1379 |
3.22. SINGLE-REACTANCE MATCHING 65 YL J1 K 0 M jO p Pin Figure 3.29: Single-reactance matching with a parallel reactance. propagation constant β of the transmission line are independent variables and can be selected for convenience. In the present problem, we aim to solve the equation Re {Y1} = Re Y0 1 −Γe−j2βl 1 + Γ... | Electromagnetics_Vol1_Page_80_Chunk1380 |
66 CHAPTER 3. TRANSMISSION LINES must be true that − 1 2πfC ∼= −86.3 Ω (3.161) Thus, we find the parallel reactance is a capacitor of value C ∼= 1.2 pF. Comparing this result to the result from the series reactance method (Example 3.9), we see that the necessary length of transmission line is much shorter, which is norm... | Electromagnetics_Vol1_Page_81_Chunk1381 |
3.23. SINGLE-STUB MATCHING 67 fL h1 i 01 k1 l m n oin stub characteristics: q 02 r2 2 stub may be open- or short-circuited stub Figure 3.31: Single-stub matching. (such as a capacitor or inductor), which does not require that either of its terminals be tied to ground. This issue is avoided in the parallel-attached stub... | Electromagnetics_Vol1_Page_82_Chunk1382 |
68 CHAPTER 3. TRANSMISSION LINES yielding Y1 ∼= 0.0200 −j0.0116 mho for the input admittance after attaching the primary line. We now seek the shortest stub having an input admittance of ∼= +j0.0116 mho to cancel the imaginary part of Y1. For an open-circuited stub, we need Bp = +Y0 tan 2πl2/λ ∼= +j0.0116 mho (3.168) T... | Electromagnetics_Vol1_Page_83_Chunk1383 |
3.23. SINGLE-STUB MATCHING 69 Image Credits Fig. 3.1: Dmitry G, https://en.wikipedia.org/wiki/File:Mastech test leads.JPG, public domain. Fig. 3.2: c⃝Tkgd2007, https://commons.wikimedia.org/wiki/File:Coaxial cable cutaway.svg, CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/). Minor modifications from the origina... | Electromagnetics_Vol1_Page_84_Chunk1384 |
Chapter 4 Vector Analysis 4.1 Vector Arithmetic [m0006] A vector is a mathematical object that has both a scalar part (i.e., a magnitude and possibly a phase), as well as a direction. Many physical quantities are best described as vectors. For example, the rate of movement through space can be described as speed; i.e.,... | Electromagnetics_Vol1_Page_85_Chunk1385 |
4.1. VECTOR ARITHMETIC 71 | x z r1 r } c⃝K. Kikkeri CC BY SA 4.0 Figure 4.2: Position vectors. The vectors r1 and r1 are position-fixed and refer to particular locations. v ~ v c⃝K. Kikkeri CC BY SA 4.0 Figure 4.3: Two particles exhibiting the same veloc- ity. In this case, the velocity vector... | Electromagnetics_Vol1_Page_86_Chunk1386 |
72 CHAPTER 4. VECTOR ANALYSIS x z x z A c⃝K. Kikkeri CC BY SA 4.0 Figure 4.5: Components of a vector A in the Carte- sian coordinate system. vector as ˆa = A |A| = A q A2x + A2y + A2z = ˆxAx | Electromagnetics_Vol1_Page_87_Chunk1387 |
4.1. VECTOR ARITHMETIC 73 from r1 to r2, the distance between these points, and the associated unit vector. Solution. The vector that points from r1 to r2 is R = r2 −r1 = (1 −2)ˆx + (−2 −3)ˆy + (3 −1)ˆz = −ˆx −5ˆy + 2ˆz (4.9) The distance between r1 and r2 is simply the magnitude of this vector: |R| = q (−1)2 + (−5)2 +... | Electromagnetics_Vol1_Page_88_Chunk1388 |
74 CHAPTER 4. VECTOR ANALYSIS A B ¥ c⃝K. Kikkeri CC BY SA 4.0 Figure 4.8: Calculation of the dot product. and any other dot product of basis vectors is zero. Thus, the whole mess simplifies to: A · A = A2 x + A2 y + A2 z (4.17) This is the square of the magnitude of A, so we have discovered that A · A = |A|2 = A2 (4.18)... | Electromagnetics_Vol1_Page_89_Chunk1389 |
4.1. VECTOR ARITHMETIC 75 n B A AB ¦ c⃝K. Kikkeri CC BY SA 4.0 Figure 4.9: The cross product A × B. x y z Figure 4.10: Cross products among basis vectors in the Cartesian system. The cross product of any two basis vectors is the third basis vector when the order of operands is counter-clockwise, as shown in the dia- gr... | Electromagnetics_Vol1_Page_90_Chunk1390 |
76 CHAPTER 4. VECTOR ANALYSIS 4.2 Cartesian Coordinates [m0004] The Cartesian coordinate system is introduced in Section 4.1. Concepts described in that section – i.e., the dot product and cross product – are described in terms of the Cartesian system. In this section, we identify some additional features of this syste... | Electromagnetics_Vol1_Page_91_Chunk1391 |
in | Electromagnetics_Vol1_Page_91_Chunk1392 |
4.3. CYLINDRICAL COORDINATES 77 some problems this sign becomes important. One example of a class of problems for which the sign of area is important is when the quantity of interest is a flux. If A were a flux density, then the integration over area that we just performed indicates the magnitude and direction of flux, an... | Electromagnetics_Vol1_Page_92_Chunk1393 |
78 CHAPTER 4. VECTOR ANALYSIS ¬ z Figure 4.12: Cross products among basis vectors in the cylindrical system. (See Figure 4.10 for instructions on the use of this diagram.) products of basis vectors are as follows: ˆρ × ˆφ = ˆz (4.47) ˆφ × ˆz = ˆρ (4.48) ˆz × ˆρ = ˆφ (4.49) A useful diagram that summarizes these relat... | Electromagnetics_Vol1_Page_93_Chunk1394 |
4.3. CYLINDRICAL COORDINATES 79 l ® x ¯ ° ± c⃝K. Kikkeri CC BY SA 4.0 Figure 4.13: Example in cylindrical coordinates: The circumference of a circle. The circumference of a circle of radius ρ is 2πρ. If only a fraction of the circumference is traversed, the associated arclength is the circumference scaled by φ/2π, wher... | Electromagnetics_Vol1_Page_94_Chunk1395 |
80 CHAPTER 4. VECTOR ANALYSIS y ¶ · ¸ ¹2 º1 dϕ dz 0 c⃝K. Kikkeri CC BY SA 4.0 Figure 4.15: Example in cylindrical coordinates: The area of the curved surface of a cylinder. Here we go. What is the integral of a vector field A = ˆρ over a cylindrical surface S concentric with the z axis having radius ρ0 and extending fro... | Electromagnetics_Vol1_Page_95_Chunk1396 |
4.4. SPHERICAL COORDINATES 81 4.4 Spherical Coordinates [m0097] The spherical coordinate system is defined with respect to the Cartesian system in Figure 4.16. The spherical system uses r, the distance measured from the origin;4 θ, the angle measured from the +z axis toward the z = 0 plane; and φ, the angle measured in ... | Electromagnetics_Vol1_Page_96_Chunk1397 |
82 CHAPTER 4. VECTOR ANALYSIS · ˆr ˆθ ˆφ ˆx sin θ cos φ cos θ cos φ −sin φ ˆy sin θ sin φ cos θ sin φ cos φ ˆz cos θ −sin θ 0 Table 4.2: Dot products between basis vectors in the spherical and Cartesian coordinate systems. coordinates is as follows: r = p x2 + y2 + z2 (4.75) θ = arccos (z/r) (4.76) φ = arctan (y, x) (4... | Electromagnetics_Vol1_Page_97_Chunk1398 |
4.4. SPHERICAL COORDINATES 83 expressed with the minimum number of varying coordinates in the spherical system. If we had attempted this problem in the Cartesian system, we would find that both z and either x or y (or all three) vary over C and in a relatively complex way. Integration Over Area. Now we ask the question,... | Electromagnetics_Vol1_Page_98_Chunk1399 |
84 CHAPTER 4. VECTOR ANALYSIS 4.5 Gradient [m0098] The gradient operator is an important and useful tool in electromagnetic theory. Here’s the main idea: The gradient of a scalar field is a vector that points in the direction in which the field is most rapidly increasing, with the scalar part equal to the rate of change.... | Electromagnetics_Vol1_Page_99_Chunk1400 |
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