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6.9. RECTANGULAR WAVEGUIDE: TE MODES 113 expression for eEz must account for all non-trivial modes. Summarizing: eEz = ∞ X m=1 ∞ X n=1 eE(m,n) z (6.186) where eE(m,n) z ≜E(m,n) 0 sin mπ a x sin nπ b y e−jk(m,n) z z (6.187) where E(m,n) 0 is an arbitrary constant (consolidating the constants B and D), and, since k... | Electromagnetics_Vol2_Page_126_Chunk2301 |
114 CHAPTER 6. WAVEGUIDES Equation 6.189 is a partial differential equation. This equation, combined with boundary conditions imposed by the perfectly-conducting plates, is sufficient to determine a unique solution. This solution is most easily determined in Cartesian coordinates, as we shall now demonstrate. First we e... | Electromagnetics_Vol2_Page_127_Chunk2302 |
6.9. RECTANGULAR WAVEGUIDE: TE MODES 115 Substituting this expression into Equation 6.206, we obtain: Y ∂2 ∂x2 X + X ∂2 ∂y2 Y + k2 ρXY = 0 (6.208) Next dividing through by XY , we obtain: 1 X ∂2 ∂x2 X + 1 Y ∂2 ∂y2 Y + k2 ρ = 0 (6.209) Note that the first term depends only on x, the second term depends only on y, and the... | Electromagnetics_Vol2_Page_128_Chunk2303 |
116 CHAPTER 6. WAVEGUIDES Each positive integer value of m and n leads to a valid expression for eHz known as a mode. Summarizing: eHz = ∞ X m=0 ∞ X n=0 eH(m,n) z (6.238) where eH(m,n) z ≜H(m,n) 0 cos mπ a x cos nπ b y e−jk(m,n) z z (6.239) where H(m,n) 0 is an arbitrary constant (consolidating the constants A an... | Electromagnetics_Vol2_Page_129_Chunk2304 |
6.10. RECTANGULAR WAVEGUIDE: PROPAGATION CHARACTERISTICS 117 6.10 Rectangular Waveguide: Propagation Characteristics [m0224] In this section, we consider the propagation characteristics of TE and TM modes in rectangular waveguides. Because these modes exhibit the same phase dependence on z, findings of this section appl... | Electromagnetics_Vol2_Page_130_Chunk2305 |
118 CHAPTER 6. WAVEGUIDES Example 6.4. Cutoff frequencies for WR-90. WR-90 is a popular implementation of rectangular waveguide. WR-90 is air-filled with dimensions a = 22.86 mm and b = 10.16 mm. Determine cutoff frequencies and, in particular, the lowest frequency at which WR-90 can be used. Solution. Since WR-90 is ai... | Electromagnetics_Vol2_Page_131_Chunk2306 |
6.10. RECTANGULAR WAVEGUIDE: PROPAGATION CHARACTERISTICS 119 The speed of a signal within a rectangular waveg- uide is given by the group velocity of the as- sociated mode (Equation 6.259). This speed is less than the speed of propagation in unbounded media having the same permittivity and perme- ability. Speed depends... | Electromagnetics_Vol2_Page_132_Chunk2307 |
120 CHAPTER 6. WAVEGUIDES Image Credits Fig. 6.1: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File:Geometry for analysis of fields in parallel plate waveguide.svg, CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/), modified. Fig. 6.2: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File... | Electromagnetics_Vol2_Page_133_Chunk2308 |
Chapter 7 Transmission Lines Redux 7.1 Parallel Wire Transmission Line [m0188] A parallel wire transmission line consists of wires separated by a dielectric spacer. Figure 7.1 shows a common implementation, commonly known as “twin lead.” The wires in twin lead line are held in place by a mechanical spacer comprised of ... | Electromagnetics_Vol2_Page_134_Chunk2309 |
122 CHAPTER 7. TRANSMISSION LINES REDUX c⃝S. Lally CC BY SA 4.0 Figure 7.3: Structure of the electric and magnetic fields for a cross-section of parallel wire line. In this case, the wave is propagating away from the viewer. The associated field structure is transverse electromagnetic (TEM) and is therefore completely de... | Electromagnetics_Vol2_Page_135_Chunk2310 |
7.2. MICROSTRIP LINE REDUX 123 fields, and that the line is suspended in air so that ǫr ≈1, the phase velocity vp for a parallel wire line is approximately that of any electromagnetic wave in free space; i.e., c. In practical twin-lead, the effect of a plastic jacket/spacer material is to reduce the phase velocity by a ... | Electromagnetics_Vol2_Page_136_Chunk2311 |
124 CHAPTER 7. TRANSMISSION LINES REDUX Figure 7.5: Approximate structure of the electric and magnetic fields within microstrip line, assuming TM0 operation. The fields outside the line are possibly sig- nificant, complicated, and not shown. In this case, the wave is propagating away from the viewer. all but the TM0 mode,... | Electromagnetics_Vol2_Page_137_Chunk2312 |
7.2. MICROSTRIP LINE REDUX 125 c⃝C. Wang CC BY SA 4.0 Figure 7.6: View from the side of a microstrip line, used to determine L′. where H is the magnitude of H. Next, recall that: L ≜Φ I (7.15) So, we may determine L if we are able to obtain an expression for I in terms of H. This can be done as follows. First, note tha... | Electromagnetics_Vol2_Page_138_Chunk2313 |
126 CHAPTER 7. TRANSMISSION LINES REDUX Figure 7.8: Noting the similarity of the fields in nar- row microstrip line to those in a parallel wire line. Figure 7.9: Modeling the fields in narrow microstrip line as those of a parallel wire line, now introducing the ground plane. Figure 7.8. Note that the fields above the diel... | Electromagnetics_Vol2_Page_139_Chunk2314 |
7.2. MICROSTRIP LINE REDUX 127 Figure 7.10: Z0 for FR4 as a function of h/W, as determined by the “wide” and “narrow” approxima- tions, along with the Wheeler 1977 formula. Note that the vertical and horizontal axes of this plot are in log scale. the wide and narrow expressions (blue and green curves, respectively) for... | Electromagnetics_Vol2_Page_140_Chunk2315 |
128 CHAPTER 7. TRANSMISSION LINES REDUX method; e.g., using the Wheeler 1977 formula or from measurements. FR4 circuit board construction is so common that the result from the previous example deserves to be highlighted: In FR4 printed circuit board construction (sub- strate thickness 1.575 mm, relative permittivity ≈4... | Electromagnetics_Vol2_Page_141_Chunk2316 |
7.3. ATTENUATION IN COAXIAL CABLE 129 • “Printed circuit board” on Wikipedia. • “Stripline” on Wikipedia. • “Single-ended signaling” on Wikipedia. • Sec. 8.7 (“Differential Circuits”) in S.W. Ellingson, Radio Systems Engineering, Cambridge Univ. Press, 2016. • H.A. Wheeler, “Transmission Line Properties of a Strip on a... | Electromagnetics_Vol2_Page_142_Chunk2317 |
130 CHAPTER 7. TRANSMISSION LINES REDUX inner conductor, so: R′ ic ≈ 1 (2πa · δic) σic for δic ≪a (7.30) This expression is only valid for δic ≪a because otherwise the cross-sectional area through which the current flows is not well-modeled as a thin ring near the surface of the conductor. Similarly, we find the resistan... | Electromagnetics_Vol2_Page_143_Chunk2318 |
7.3. ATTENUATION IN COAXIAL CABLE 131 where Z0 is the characteristic impedance Z0 ≈η0 2π 1 √ǫr ln b a (low loss) (7.46) and where KR is a unitless constant to be determined. The justification for Equation 7.45 is as follows: First, αR must increase monotonically with increasing R′. Second, R′ must be divided by an imped... | Electromagnetics_Vol2_Page_144_Chunk2319 |
132 CHAPTER 7. TRANSMISSION LINES REDUX Here we see that αR is minimized by minimizing ǫr/σic. It’s not surprising to see that we should maximize σic. However, it’s a little surprising that we should minimize ǫr. Furthermore, this is in contrast to αG, which is minimized by maximizing ǫr. Clearly there is a tradeoff to... | Electromagnetics_Vol2_Page_145_Chunk2320 |
7.4. POWER HANDLING CAPABILITY OF COAXIAL CABLE 133 and ∂ ∂a ln b a = ∂ ∂a [ln (b) −ln (a)] = −∂ ∂a ln (a) = −1 a (7.59) So: ∂ ∂a a2 ln b a = [2a] ln b a + a2 −1 a = 2a ln b a −a (7.60) This result is substituted for a2 ln(b/a) in Equation 7.51 to obtain Equation 7.52. 7.4 Power Handling Capabili... | Electromagnetics_Vol2_Page_146_Chunk2321 |
134 CHAPTER 7. TRANSMISSION LINES REDUX The electric field intensity is given by: E = −∇V (7.67) Again we have ∂V/∂φ = ∂V/∂z = 0, so E = −ˆρ ∂ ∂ρV (7.68) = −ˆρ ∂ ∂ρ −V0 ln (b/a) ln ρ + V0 ln (b) ln (b/a) (7.69) = +ˆρ V0 ρ ln (b/a) (7.70) Note that the maximum electric field intensity in the spacer occurs at ρ = a; i.... | Electromagnetics_Vol2_Page_147_Chunk2322 |
7.5. WHY 50 OHMS? 135 Using the chain rule, we find: ∂ ∂a 1/a + C/b ln (b/a) = ∂ ∂a 1 a + C b ln−1 b a + 1 a + C b ∂ ∂a ln−1 b a (7.79) Note ∂ ∂a 1 a + C b = −1 a2 (7.80) To handle the quantity in the second set of square brackets, first define v = ln u, where u = b/a. Then: ∂ ∂av−1 = ∂ ∂v v−1 ... | Electromagnetics_Vol2_Page_148_Chunk2323 |
136 CHAPTER 7. TRANSMISSION LINES REDUX capability is also important, and is addressed in Section 7.4. In that section, we find the power handling capability of coaxial cable is optimized when the ratio of radii of the outer to inner conductors b/a is about 1.65. For the air-filled cables typically used in high-power app... | Electromagnetics_Vol2_Page_149_Chunk2324 |
7.5. WHY 50 OHMS? 137 Image Credits Fig. 7.1: c⃝SpinningSpark, Inductiveload, https://commons.wikimedia.org/wiki/File:Twin-lead cable dimension.svg, CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Minor modifications. Fig. 7.2: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File:Parallel wire t... | Electromagnetics_Vol2_Page_150_Chunk2325 |
Chapter 8 Optical Fiber 8.1 Optical Fiber: Method of Operation [m0178] In its simplest form, optical fiber consists of concentric regions of dielectric material as shown in Figure 8.1. A cross-section through the fiber reveals a circular region of transparent dielectric material through which light propagates. This is su... | Electromagnetics_Vol2_Page_151_Chunk2326 |
8.1. OPTICAL FIBER: METHOD OF OPERATION 139 c⃝S. Lally CC BY-SA 4.0 Figure 8.2: Total internal reflection in optical fiber. the fiber, and is reflected onward. Otherwise, power is lost into the cladding. Example 8.1. Critical angle for optical fiber. Typical values of nf and nc for an optical fiber are 1.52 and 1.49, respect... | Electromagnetics_Vol2_Page_152_Chunk2327 |
140 CHAPTER 8. OPTICAL FIBER 8.2 Acceptance Angle [m0192] In this section, we consider the problem of injecting light into a fiber optic cable. The problem is illustrated in Figure 8.3. In this figure, we see light incident from a medium having index of refraction n0, with angle of incidence θi. The light is transmitted ... | Electromagnetics_Vol2_Page_153_Chunk2328 |
8.3. DISPERSION IN OPTICAL FIBER 141 Example 8.2. Acceptance angle. Typical values of nf and nc for an optical fiber are 1.52 and 1.49, respectively. What are the numerical aperture and the acceptance angle? Solution. Using Equation 8.17 and presuming n0 = 1, we find NA ∼= 0.30. Since sin θa = NA, we find θa = 17.5◦. Ligh... | Electromagnetics_Vol2_Page_154_Chunk2329 |
142 CHAPTER 8. OPTICAL FIBER c⃝S. Lally CC BY-SA 4.0 Figure 8.6: A digital signal that might be applied to the input of a fiber optic cable. the cable. Figure 8.5(b) represents the continuum of possibilities between the extreme cases of (a) and (c), with associated propagation times greater than that of case (a) but les... | Electromagnetics_Vol2_Page_155_Chunk2330 |
8.3. DISPERSION IN OPTICAL FIBER 143 overlap problem in greater detail. To avoid overlap: τmax −τmin + ton < T (8.18) Let us define the quantity τ ≜τmax −τmin. This is sometimes referred to as the delay spread.1 Thus, we obtain the following requirement for overlap-free transmission: τ < T −ton (8.19) Note that the dela... | Electromagnetics_Vol2_Page_156_Chunk2331 |
144 CHAPTER 8. OPTICAL FIBER Image Credits Fig. 8.1: c⃝S. Lally, https://commons.wikimedia.org/wiki/File:Figure 8.1-01.svg, CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Modified by author. Fig. 8.2: c⃝Offaperry (S. Lally), https://commons.wikimedia.org/wiki/File:Internal Reflection in Optical Fiber.svg... | Electromagnetics_Vol2_Page_157_Chunk2332 |
Chapter 9 Radiation 9.1 Radiation from a Current Moment [m0194] In this section, we begin to address the following problem: Given a distribution of impressed current density J(r), what is the resulting electric field intensity E(r)? One route to an answer is via Maxwell’s equations. Viewing Maxwell’s equations as a syst... | Electromagnetics_Vol2_Page_158_Chunk2333 |
146 CHAPTER 9. RADIATION in precisely this form. Nevertheless, the current moment turns out to be generally useful as a “building block” from which practical distributions of current can be constructed, via the principle of superposition. Radiation from current distributions constructed in this manner is calculated sim... | Electromagnetics_Vol2_Page_159_Chunk2334 |
exercise for the student. | Electromagnetics_Vol2_Page_159_Chunk2335 |
9.2. MAGNETIC VECTOR POTENTIAL 147 Note that the expression we have obtained for the radiated electric field is approximate (hence the “≈”). This is due in part to our presumption of a simple spherical wave, which may only be valid at distances far from the source. But how far? An educated guess would be distances much ... | Electromagnetics_Vol2_Page_160_Chunk2336 |
148 CHAPTER 9. RADIATION The magnetic vector potential eA is defined by the following relationship: eB ≜∇× eA (9.12) where eB = µ eH is the magnetic flux density. The magnetic field appears in three of Maxwell’s equations. For Equation 9.12 to be a reasonable definition, ∇× eA must yield reasonable results when substituted... | Electromagnetics_Vol2_Page_161_Chunk2337 |
9.2. MAGNETIC VECTOR POTENTIAL 149 equation. We established earlier that eV can be essentially any scalar field – from a mathematical perspective, we are free to choose. Invoking this freedom, we now require eV to satisfy the following expression: ∇· eA + jωµǫeV = 0 (9.27) Clearly this is advantageous in the sense that ... | Electromagnetics_Vol2_Page_162_Chunk2338 |
150 CHAPTER 9. RADIATION 9.3 Solution of the Wave Equation for Magnetic Vector Potential [m0196] The magnetic vector potential eA due to a current density eJ is given by the following wave equation: ∇2 eA −γ2 eA = −µeJ (9.29) where γ is the propagation constant, defined in the usual manner7 γ2 ≜−ω2µǫ (9.30) Equation 9.2... | Electromagnetics_Vol2_Page_163_Chunk2339 |
9.3. SOLUTION OF THE WAVE EQUATION FOR MAGNETIC VECTOR POTENTIAL 151 Now consider the factor e±αr/r, which determines the dependence of magnitude on distance r. If we choose the negative sign in the exponent, this factor decays exponentially with increasing distance from the origin, ultimately reaching zero at r →∞. Th... | Electromagnetics_Vol2_Page_164_Chunk2340 |
152 CHAPTER 9. RADIATION c⃝C. Wang CC BY-SA 4.0 Figure 9.4: A filament of current lying along the path C. This current distribution may be interpreted as a collection of current moments lying along C. magnetic vector potential: eA(r) ≈ N X n=1 ∆eA(r; rn) (9.43) ≈µ 4π N X n=1 ˆl(rn) eI(rn) e−γ|r−rn| |r −rn| ∆l (9.44) Now... | Electromagnetics_Vol2_Page_165_Chunk2341 |
9.4. RADIATION FROM A HERTZIAN DIPOLE 153 The solution for the magnetic vector potential due to a ˆz-directed Hertzian dipole located at the origin was presented in Section 9.3. In the present scenario, it is: eA(r) = ˆz µ eI ∆l e−γr 4πr (9.49) where the propagation constant γ = α + jβ as usual. Assuming lossless media... | Electromagnetics_Vol2_Page_166_Chunk2342 |
154 CHAPTER 9. RADIATION • Notice the factor β∆l has units of radians; that is, it is electrical length. This tells us that the magnitude of the radiated field depends on the electrical length of the current moment. • The factor e−jβr/r indicates that this is a spherical wave; that is, surfaces of constant phase corresp... | Electromagnetics_Vol2_Page_167_Chunk2343 |
9.5. RADIATION FROM AN ELECTRICALLY-SHORT DIPOLE 155 9.5 Radiation from an Electrically-Short Dipole [m0198] The simplest distribution of radiating current that is encountered in common practice is the electrically-short dipole (ESD). This current distribution is shown in Figure 9.6. The two characteristics that define ... | Electromagnetics_Vol2_Page_168_Chunk2344 |
156 CHAPTER 9. RADIATION length of the dipole as well as being short relative to a wavelength, we may approximate the current over each segment as approximately constant. In other words, we may interpret each of these segments as being, to a good approximation, a Hertzian dipole. The advantage of this approach is that ... | Electromagnetics_Vol2_Page_169_Chunk2345 |
9.5. RADIATION FROM AN ELECTRICALLY-SHORT DIPOLE 157 Equation 9.75 that exhibits varying phase is e−jβ|r−ˆzz′|. Using Equation 9.76, we find e−jβ|r−ˆzz′| ≈e−jβre+jβˆr·ˆzz′ (9.78) The worst case in terms of phase variation within the integral is for field points along the z axis. For these points, ˆr · ˆz = ±1 and subsequ... | Electromagnetics_Vol2_Page_170_Chunk2346 |
158 CHAPTER 9. RADIATION c⃝S. Lally CC BY-SA 4.0 Figure 9.9: Magnitude of the radiated field in any plane of constant φ. c⃝S. Lally CC BY-SA 4.0 Figure 9.10: Orientation of the electric and magnetic fields in any plane of constant φ. c⃝S. Lally CC BY-SA 4.0 Figure 9.11: Magnitude of the radiated field in any plane of cons... | Electromagnetics_Vol2_Page_171_Chunk2347 |
9.6. FAR-FIELD RADIATION FROM A THIN STRAIGHT FILAMENT OF CURRENT 159 9.6 Far-Field Radiation from a Thin Straight Filament of Current [m0199] A simple distribution of radiating current that is encountered in common practice is the thin straight current filament, shown in Figure 9.13. The defining characteristic of this ... | Electromagnetics_Vol2_Page_172_Chunk2348 |
160 CHAPTER 9. RADIATION shown that a ˆz-directed Hertzian dipole at the origin radiates the electric field eE(r) ≈ˆθjη eI (β∆l) 4π (sin θ) e−jβr r (9.84) where eI and ∆l may be interpreted as the current and length of the filament, respectively. In this expression, η is the wave impedance of medium in which the filament ... | Electromagnetics_Vol2_Page_173_Chunk2349 |
9.7. FAR-FIELD RADIATION FROM A HALF-WAVE DIPOLE 161 These simplifications are known collectively as a far field approximation, since they are valid only for distances “far” from the source. Applying these simplifications for magnitude and phase to Equation 9.90, we obtain: eE(r) ≈ˆθj ηβ 4π e−jβr r (sin θ) · Z +L/2 −L/2 e... | Electromagnetics_Vol2_Page_174_Chunk2350 |
162 CHAPTER 9. RADIATION of current may be calculated using the method described in Section 9.6, in particular: eE(r) ≈ˆθj η 2 e−jβr r (sin θ) · " 1 λ Z +L/2 −L/2 eI(z′)e+jβz′ cos θdz′ # (9.99) which is valid for field points r far from the dipole; i.e., for r ≫L and r ≫λ. For the HWD, the quantity in square brackets is... | Electromagnetics_Vol2_Page_175_Chunk2351 |
9.8. RADIATION FROM SURFACE AND VOLUME DISTRIBUTIONS OF CURRENT 163 any distribution of current that is constrained to flow along a single path through space, as along an infinitesimally-thin wire. In this section, we derive an expression for the radiation from current that is constrained to flow along a surface and from ... | Electromagnetics_Vol2_Page_176_Chunk2352 |
164 CHAPTER 9. RADIATION Image Credits Fig. 9.1: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File:A z-directed current moment located at the origin.svg, CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 9.2: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File:A z-directed curre... | Electromagnetics_Vol2_Page_177_Chunk2353 |
9.8. RADIATION FROM SURFACE AND VOLUME DISTRIBUTIONS OF CURRENT 165 Fig. 9.15: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File:Parallel ray approximation for an esd.svg, CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 9.16: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File... | Electromagnetics_Vol2_Page_178_Chunk2354 |
Chapter 10 Antennas 10.1 How Antennas Radiate [m0201] An antenna is a transducer; that is, a device which converts signals in one form into another form. In the case of an antenna, these two forms are (1) conductor-bound voltage and current signals and (2) electromagnetic waves. Traditional passive antennas are capable... | Electromagnetics_Vol2_Page_179_Chunk2355 |
10.1. HOW ANTENNAS RADIATE 167 Hertzian dipoles are identical in magnitude but opposite in sign. Therefore, the radiated field from any such pair of Hertzian dipoles is approximately zero at distances sufficiently far from the transmission line. Continuing to sum all such pairs of Hertzian dipoles, the radiated field rema... | Electromagnetics_Vol2_Page_180_Chunk2356 |
168 CHAPTER 10. ANTENNAS 10.2 Power Radiated by an Electrically-Short Dipole [m0207] In this section, we determine the total power radiated by an electrically-short dipole (ESD) antenna in response to a sinusoidally-varying current applied to the antenna terminals. This result is both useful on its own and necessary as... | Electromagnetics_Vol2_Page_181_Chunk2357 |
10.3. POWER DISSIPATED BY AN ELECTRICALLY-SHORT DIPOLE 169 that is, the length of the antenna expressed in radians, where 2π radians is one wavelength. Thus, we see that the power radiated by the antenna increases as the square of electrical length. Example 10.1. Power radiated by an ESD. A dipole is 10 cm in length an... | Electromagnetics_Vol2_Page_182_Chunk2358 |
170 CHAPTER 10. ANTENNAS calculated using Equation 4.17 (Section 4.2, “Impedance of a Wire”): Rseg ≈1 2 r µf πσ · ∆l a (10.12) where µ is permeability, f is frequency, σ is conductivity, and ∆l is the length of the segment. Substitution into Equation 10.11 yields: Pseg(zn) ≈1 4a r µf πσ eI(zn) 2 ∆l (10.13) Now the tota... | Electromagnetics_Vol2_Page_183_Chunk2359 |
10.4. REACTANCE OF THE ELECTRICALLY-SHORT DIPOLE 171 find that the loss resistance Rloss ≈9.49 mΩ. Subsequently, the power dissipated within this antenna is Ploss = 1 2 |I0|2 Rloss ≈47.5 µW (10.21) We conclude this section with one additional caveat: Whereas this section focuses on the limited conductivity of wire, othe... | Electromagnetics_Vol2_Page_184_Chunk2360 |
172 CHAPTER 10. ANTENNAS What is the impedance of an open circuit? One might be tempted to say “infinite,” since the current is zero independently of the voltage. However, we must now be careful to properly recognize the real and imaginary components of this infinite number, and signs of these components. In fact, the im... | Electromagnetics_Vol2_Page_185_Chunk2361 |
10.5. EQUIVALENT CIRCUIT MODEL FOR TRANSMISSION; RADIATION EFFICIENCY 173 10.5 Equivalent Circuit Model for Transmission; Radiation Efficiency [m0202] A radio transmitter consists of a source which generates the electrical signal intended for transmission, and an antenna which converts this signal into a propagating ele... | Electromagnetics_Vol2_Page_186_Chunk2362 |
174 CHAPTER 10. ANTENNAS antenna is PA = 1 2Re n eVAeI∗ A o (10.26) where we have assumed peak (as opposed to root mean squared) units for voltage and current. Since eVA = ZAeIA, we have: PA = 1 2Re n (Rrad + Rloss + jXA) eIAeI∗ A o (10.27) which reduces to: PA = 1 2 eIA 2 Rrad + 1 2 eIA 2 Rloss (10.28) As expected, th... | Electromagnetics_Vol2_Page_187_Chunk2363 |
10.6. IMPEDANCE OF THE ELECTRICALLY-SHORT DIPOLE 175 Using Equations 10.28–10.30, we see that this efficiency can be expressed as follows: erad = Rrad Rrad + Rloss (10.34) Once again, the equivalent circuit formalism proves useful. Example 10.4. Impedance of an antenna. The total power radiated by an antenna is 60 mW wh... | Electromagnetics_Vol2_Page_188_Chunk2364 |
176 CHAPTER 10. ANTENNAS Assuming free-space conditions, η ∼= 376.7 Ω, which is ≈120π Ω. Subsequently, Rrad ≈20π2 L λ 2 (10.43) This remarkably simple expression indicates that the radiation resistance of an ESD is very small (since L ≪λ for an ESD), but increases as the square of the length. At this point, a warning... | Electromagnetics_Vol2_Page_189_Chunk2365 |
10.7. DIRECTIVITY AND GAIN 177 techniques. Additional Reading: • R.C. Johnson (Ed.), Antenna Systems Handbook (Ch. 4), McGraw-Hill, 1993. 10.7 Directivity and Gain [m0203] A transmitting antenna does not radiate power uniformly in all directions. Inevitably more power is radiated in some directions than others. Directi... | Electromagnetics_Vol2_Page_190_Chunk2366 |
178 CHAPTER 10. ANTENNAS proportional to r−2. This is a key point: Directivity is a convenient way to characterize an antenna because it does not change with distance from the antenna. In general, directivity is a function of direction. However, one is often not concerned about all directions, but rather only the direc... | Electromagnetics_Vol2_Page_191_Chunk2367 |
10.8. RADIATION PATTERN 179 The receive case. To conclude this section, we make one additional point about directivity, which applies equally to gain. The preceding discussion has presumed an antenna which is radiating; i.e., transmitting. Directivity can also be defined for the receive case, in which it quantifies the e... | Electromagnetics_Vol2_Page_192_Chunk2368 |
180 CHAPTER 10. ANTENNAS “co-polarized” or simply “co-pol,” and the ˆφ-polarization of the transmitted field as “cross-pol.” At this point, the reader may wonder what purpose is served by defining cross polarization, since the definition given above seems to suggest that cross-pol should always be zero. In common engineer... | Electromagnetics_Vol2_Page_193_Chunk2369 |
10.8. RADIATION PATTERN 181 c⃝S. Lally CC BY-SA 4.0 Figure 10.6: H-plane co-pol pattern for the ˆz-oriented ESD. In the unnormalized pattern scaling, the radius of the circle is the maximum value of Equation 10.58. denominator is the maximum value of the electric field at distance r. A normalized pattern is scaled to a ... | Electromagnetics_Vol2_Page_194_Chunk2370 |
182 CHAPTER 10. ANTENNAS c⃝S. Lally CC BY-SA 4.0 Figure 10.8: Half-power beamwidth (HPBW). The main lobe is bounded on each side by a null, where the magnitude reaches a local minimum, perhaps zero. Many antennas also exhibit a lobe in the opposite direction, known as a backlobe. (Many other antennas exhibit a null in ... | Electromagnetics_Vol2_Page_195_Chunk2371 |
10.9. EQUIVALENT CIRCUIT MODEL FOR RECEPTION 183 10.9 Equivalent Circuit Model for Reception [m0206] In this section, we begin to address antennas as devices that convert incident electromagnetic waves into potentials and currents in a circuit. It is convenient to represent this process in the form of a Th´evenin equiv... | Electromagnetics_Vol2_Page_196_Chunk2372 |
184 CHAPTER 10. ANTENNAS le is uniquely defined to be the factor that converts this component into eVOC. Summarizing: The vector effective length le = ˆlle is defined as follows: ˆl is the real-valued unit vector cor- responding to the polarization of the electric field that would be transmitted from the antenna in the fa... | Electromagnetics_Vol2_Page_197_Chunk2373 |
10.9. EQUIVALENT CIRCUIT MODEL FOR RECEPTION 185 The output impedance ZA of the equivalent cir- cuit for an antenna in the receive case is equal to the input impedance of the same antenna in the transmit case. This remarkable fact is a consequence of the reciprocity property of antenna systems, and greatly simplifies th... | Electromagnetics_Vol2_Page_198_Chunk2374 |
186 CHAPTER 10. ANTENNAS 10.10 Reciprocity [m0214] The term “reciprocity” refers to a class of theorems that relate the inputs and outputs of a linear system to those of an identical system in which the inputs and outputs are swapped. The importance of reciprocity in electromagnetics is that it simplifies problems that ... | Electromagnetics_Vol2_Page_199_Chunk2375 |
10.10. RECIPROCITY 187 having SI base units of V/m. The volume may consist of any combination of linear time-invariant matter; i.e., permittivity ǫ, permeability µ, and conductivity σ are constants that may vary arbitrarily with position but not with time. Here’s a key idea: We may interpret this scenario as a “two-por... | Electromagnetics_Vol2_Page_200_Chunk2376 |
188 CHAPTER 10. ANTENNAS Next, let us subtract Equation 10.82 from Equation 10.81. Again using the vector identity, the left side of the resulting equation is eE2 · ∇× eH1 −eH1 · ∇× eE2 = ∇· eH1 × eE2 (10.83) So we find: ∇· eH1 × eE2 = eE2 ·eJ1 +jωǫeE1 · eE2 +jωµ eH1 · eH2 (10.84) Finally, subtracting Eq... | Electromagnetics_Vol2_Page_201_Chunk2377 |
10.10. RECIPROCITY 189 c⃝C. Wang CC BY-SA 4.0 Figure 10.13: A two-port consisting of two dipole antennas. Reciprocity of two-ports consisting of antennas. Equation 10.90 allows us to establish the reciprocity of two-ports consisting of pairs of antennas. This is most easily demonstrated for pairs of thin dipole antenna... | Electromagnetics_Vol2_Page_202_Chunk2378 |
190 CHAPTER 10. ANTENNAS Applying the exact same procedure for port 2 (or by simply exchanging subscripts), we find: Z V2 eE1 · eJ2 dv = −eIt 2 eV r 2 (10.96) Now substituting these results into Equation 10.90, we find: eIt 1 eV r 1 = eIt 2 eV r 2 (10.97) At the beginning of this section, we stated that a two-port is rec... | Electromagnetics_Vol2_Page_203_Chunk2379 |
10.11. POTENTIAL INDUCED IN A DIPOLE 191 Fortunately, we can bypass this obstacle using the principle of reciprocity. In a reciprocity-based strategy, we establish a relationship between two scenarios that take place within the same electromagnetic system. The first scenario is shown in Figure 10.16. In this scenario, w... | Electromagnetics_Vol2_Page_204_Chunk2380 |
192 CHAPTER 10. ANTENNAS other antennas – it is simple to determine the open circuit potential. As explained earlier: eV r 2 = − Z gap eEgap · dl (10.101) For the Hertzian dipole, eEgap is simply the incident electric field, since there is negligible structure (in particular, a negligible amount of material) present to ... | Electromagnetics_Vol2_Page_205_Chunk2381 |
10.11. POTENTIAL INDUCED IN A DIPOLE 193 equal to the expression from Equation 10.108, yielding le ≈2π βeIt 1 " 1 λ Z +L/2 −L/2 eI(z′)e+jβz′ cos θdz′ # sin θ (10.112) Noting that β = 2π/λ, this simplifies to: le ≈ " 1 eIt 1 Z +L/2 −L/2 eI(z′)e+jβz′ cos θdz′ # sin θ (10.113) Thus, you can calculate le using the following... | Electromagnetics_Vol2_Page_206_Chunk2382 |
194 CHAPTER 10. ANTENNAS 10.12 Equivalent Circuit Model for Reception, Redux [m0216] Section 10.9 provides an informal derivation of an equivalent circuit model for a receiving antenna. This model is shown in Figure 10.18. The derivation of this model was informal and incomplete because the open-circuit potential Ei · ... | Electromagnetics_Vol2_Page_207_Chunk2383 |
10.12. EQUIVALENT CIRCUIT MODEL FOR RECEPTION, REDUX 195 can be identified that completely determine the relationship between the port potentials and currents. We also established in Section 10.10 that: eIt 1 eV r 1 = eIt 2 eV r 2 (10.124) Therefore, eV r 1 eIt 2 = eV r 2 eIt 1 (10.125) Referring to Equations 10.122 and... | Electromagnetics_Vol2_Page_208_Chunk2384 |
196 CHAPTER 10. ANTENNAS Summarizing: The Th´evenin equivalent circuit for an antenna in the presence of an incident electric field eEi is shown in Figure 10.18. The series impedance ZA in this model is equal to the impedance of the an- tenna in transmission. Mutual coupling. This concludes the derivation, but raises a ... | Electromagnetics_Vol2_Page_209_Chunk2385 |
10.13. EFFECTIVE APERTURE 197 sinusoidally-varying plane wave eEi co incident on the antenna. Further, let eEi co = Eiˆe (10.136) where ˆe is the reference direction of eEi co. The co-polarized power density incident on the antenna is: Si co = Ei 2 2η (10.137) where η is the wave impedance of the medium (e.g., ∼= 377 Ω... | Electromagnetics_Vol2_Page_210_Chunk2386 |
198 CHAPTER 10. ANTENNAS wire antennas. For the electrically-short dipole (ESD) of length L, le ≈(L/2) sin θ and Rrad ≈20π2 (L/λ)2. Thus, we find the effective aperture assuming free space (i.e., η = η0) is: Ae ≈0.119λ2 |sin θ|2 (lossless ESD) (10.149) Remarkably, the effective aperture of the ESD does not depend on its... | Electromagnetics_Vol2_Page_211_Chunk2387 |
10.13. EFFECTIVE APERTURE 199 per steradian of solid angle.5 The total power accessible to the antenna is one-half this amount, since an antenna is sensitive to only one polarization at a time, whereas the thermal radiation is equally distributed among any two orthogonal polarizations. PA is the remaining power, obtain... | Electromagnetics_Vol2_Page_212_Chunk2388 |
200 CHAPTER 10. ANTENNAS presented earlier, or using the reciprocity theorem developed in Section 10.10. The fact that effective aperture is easily calculated from transmit directivity is an enormously useful tool in antenna engineering. Without this tool, determination of effective aperture is limited to direct measur... | Electromagnetics_Vol2_Page_213_Chunk2389 |
10.14. FRIIS TRANSMISSION EQUATION 201 10.14 Friis Transmission Equation [m0219] A common task in radio systems applications is to determine the power delivered to a receiver due to a distant transmitter. The scenario is shown in Figure 10.22: A transmitter delivers power PT to an antenna which has gain GT in the direc... | Electromagnetics_Vol2_Page_214_Chunk2390 |
202 CHAPTER 10. ANTENNAS the path between antennas, and therefore this “spreading loss” increases with frequency. In fact, the reduction in power density due to spreading between any two distances R1 < R2 is: PT /4πR2 1 PT /4πR2 2 = R1 R2 2 (10.169) which is clearly independent of frequency. The path loss Lp, in cont... | Electromagnetics_Vol2_Page_215_Chunk2391 |
10.14. FRIIS TRANSMISSION EQUATION 203 Image Credits Fig. 10.1: c⃝Offaperry (S. Lally), https://commons.wikimedia.org/wiki/File:Standing Wave Creation.svg, CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 10.2: c⃝Offaperry (S. Lally), https://commons.wikimedia.org/wiki/File:Electrically-Short Dipole... | Electromagnetics_Vol2_Page_216_Chunk2392 |
204 CHAPTER 10. ANTENNAS Fig. 10.15: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File:Thin straight dipole respond to incident plane wave.svg, CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 10.16: c⃝Sevenchw (C. Wang), https://commons.wikimedia.org/wiki/File:Dipole of interest driven ... | Electromagnetics_Vol2_Page_217_Chunk2393 |
Appendix A Constitutive Parameters of Some Common Materials A.1 Permittivity of Some Common Materials [m0135] The values below are relative permittivity ǫr ≜ǫ/ǫ0 for a few materials that are commonly encountered in electrical engineering applications, and for which permittivity emerges as a consideration. Note that “re... | Electromagnetics_Vol2_Page_218_Chunk2394 |
206 APPENDIX A. CONSTITUTIVE PARAMETERS OF SOME COMMON MATERIALS lower end of the range. Other liquids typically exhibit ǫr in the range 10–90, with considerable variation as a function of temperature and frequency. Animal flesh and blood consists primarily of liquid matter and so also exhibits permittivity in this rang... | Electromagnetics_Vol2_Page_219_Chunk2395 |
A.3. CONDUCTIVITY OF SOME COMMON MATERIALS 207 Ferrites include a broad range of ceramic materials that are combined with iron and various combinations of other metals and are used as magnets and magnetic devices in various electrical systems. Common ferrites exhibit µr in the range 16–640. Additional Reading: • CRC Ha... | Electromagnetics_Vol2_Page_220_Chunk2396 |
208 APPENDIX A. CONSTITUTIVE PARAMETERS OF SOME COMMON MATERIALS Non-conductors. Most other materials that are not well-described as conductors or semiconductors and are dry exhibit σ < 10−12 S/m. Most materials that are considered to be insulators, including air and common dielectrics, exhibit σ < 10−15 S/m, often by ... | Electromagnetics_Vol2_Page_221_Chunk2397 |
Appendix B Mathematical Formulas B.1 Trigonometry [m0138] ejθ = cos θ + j sin θ (B.1) cos θ = 1 2 | Electromagnetics_Vol2_Page_222_Chunk2398 |
210 APPENDIX B. MATHEMATICAL FORMULAS Gradient in spherical coordinates: ∇f = ˆr∂f ∂r + ˆθ1 r ∂f ∂θ + ˆφ 1 r sin θ ∂f ∂φ (B.12) Divergence Divergence in Cartesian coordinates: ∇· A = ∂Ax ∂x + ∂Ay ∂y + ∂Az ∂z (B.13) Divergence in cylindrical coordinates: ∇· A = 1 ρ ∂ ∂ρ (ρAρ) + 1 ρ ∂Aφ ∂φ + ∂Az ∂z (B.14) Divergence in s... | Electromagnetics_Vol2_Page_223_Chunk2399 |
B.3. VECTOR IDENTITIES 211 B.3 Vector Identities [m0140] Algebraic Identities A · (B × C) = B · (C × A) = C · (A × B) (B.22) A × (B × C) = B (A · C) −C (A · B) (B.23) Identities Involving Differential Operators ∇· (∇× A) = 0 (B.24) ∇× (∇f) = 0 (B.25) ∇× (fA) = f (∇× A) + (∇f) × A (B.26) ∇· (A × B) = B · (∇× A) −A · (∇×... | Electromagnetics_Vol2_Page_224_Chunk2400 |
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