13.2 Electromagnetic Waves & Transmission Line Fields

13.2 Electromagnetic Waves & Transmission Line Fields

High-voltage power transmission lines, high-frequency communication cables, and industrial bus ducts behave not as lumped parameters, but as distributed field systems. In the PRC REE Licensure Examination, candidates are tested on uniform plane wave propagation, intrinsic wave impedance, Poynting power flow, transmission line equations, characteristic impedance $Z_0$, reflection coefficients, standing wave ratios ($VSWR$), and surge wave behavior.

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1. Wave Equations & Propagation Parameters

Applying the curl operator to Maxwell's equations in a linear, isotropic, homogeneous medium with conductivity $\sigma$, permeability $\mu$, and permittivity $\epsilon$ yields the vector wave equation for time-harmonic electric fields:

$$\nabla^2 \mathbf{E}_s - \gamma^2 \mathbf{E}_s = 0$$

where $\mathbf{E}_s$ is the phasor electric field intensity, and $\gamma$ is the Complex Propagation Constant:

$$\gamma = \alpha + j\beta = \sqrt{j\omega\mu (\sigma + j\omega\epsilon)} \quad (\text{m}^{-1})$$

Attenuation Constant ($\alpha$) & Phase Constant ($\beta$)

• Attenuation Constant ($\alpha$): Measures wave amplitude reduction per unit distance due to dielectric and conduction losses, expressed in Nepers per meter (Np/m) or decibels per meter ($\text{dB/m}$, where $1\ \text{Np} = 8.686\ \text{dB}$).
• Phase Constant ($\beta$): Measures phase shift per unit distance, expressed in radians per meter (rad/m): $$\beta = \frac{2\pi}{\lambda} = \frac{\omega}{v_p}$$

Intrinsic Impedance ($\eta$) & Phase Velocity ($v_p$)

The ratio of the transverse electric field to transverse magnetic field for a forward-propagating wave is the Intrinsic Impedance $\eta$:

$$\eta = \frac{E_x}{H_y} = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\epsilon}} \quad (\Omega)$$

In Free Space ($\sigma = 0, \mu_r = 1, \epsilon_r = 1$):

$$\eta_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} = \sqrt{\frac{4\pi \times 10^{-7}}{8.854 \times 10^{-12}}} \approx 120\pi \approx 377\ \Omega$$ $$v_p = c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8\ \text{m/s}$$

Low-Loss Dielectrics vs. Good Conductors

The ratio $\frac{\sigma}{\omega \epsilon}$ is known as the loss tangent ($\tan\delta$):

1. Good Dielectrics ($\frac{\sigma}{\omega \epsilon} \ll 1$):

   $$\alpha \approx \frac{\sigma}{2} \sqrt{\frac{\mu}{\epsilon}} \quad (\text{Np/m}), \quad \beta \approx \omega \sqrt{\mu \epsilon} \quad (\text{rad/m}), \quad \eta \approx \sqrt{\frac{\mu}{\epsilon}}$$

2. Good Conductors ($\frac{\sigma}{\omega \epsilon} \gg 1$):

   $$\alpha = \beta = \sqrt{\pi f \mu \sigma} \quad (\text{m}^{-1}), \quad \eta = (1 + j)\sqrt{\frac{\pi f \mu}{\sigma}}$$

   The depth at which the wave amplitude decays to $1/e$ ($36.8\%$) of its surface value is the Skin Depth ($\delta$):

   $$\delta = \frac{1}{\alpha} = \frac{1}{\sqrt{\pi f \mu \sigma}} \quad (\text{m})$$

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2. Poynting Vector & Electromagnetic Power Flow

Electromagnetic fields transport energy through space. The instantaneous rate of energy flow per unit area is defined by the Poynting Vector $\mathbf{S}$:

$$\mathbf{S} = \mathbf{E} \times \mathbf{H} \quad (\text{W/m}^2)$$

For time-harmonic fields, the Time-Average Poynting Vector $\mathbf{P}_{\text{avg}}$ represents true real power density:

$$\mathbf{P}_{\text{avg}} = \frac{1}{2} \text{Re} \left\{ \mathbf{E}_s \times \mathbf{H}_s^* \right\} = \frac{|E_0|^2}{2 |\eta|} e^{-2\alpha z} \cos(\theta_n) \mathbf{a}_z \quad (\text{W/m}^2)$$

where $\theta_n$ is the phase angle of the intrinsic impedance $\eta$. Total power crossing a surface $S$ is obtained by integrating over the area:

$$P_{\text{total}} = \iint_S \mathbf{P}_{\text{avg}} \cdot d\mathbf{A} \quad (\text{Watts})$$

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3. Distributed Transmission Line Parameters & Telegrapher's Equations

A uniform transmission line is modeled using per-unit-length distributed parameters:

• $R$: Series resistance per unit length ($\Omega/\text{m}$)
• $L$: Series inductance per unit length ($\text{H/m}$)
• $G$: Shunt conductance per unit length ($\text{S/m}$)
• $C$: Shunt capacitance per unit length ($\text{F/m}$)

Telegrapher's Equations

Applying Kirchhoff's laws to an incremental line section $dz$ yields the time-domain Telegrapher's Equations:

$$\frac{\partial v(z,t)}{\partial z} = -R i(z,t) - L \frac{\partial i(z,t)}{\partial t}$$ $$\frac{\partial i(z,t)}{\partial z} = -G v(z,t) - C \frac{\partial v(z,t)}{\partial t}$$

In the frequency domain (phasor form):

$$\frac{d^2 V_s(z)}{dz^2} = \gamma^2 V_s(z), \quad \frac{d^2 I_s(z)}{dz^2} = \gamma^2 I_s(z)$$

where the line propagation constant is $\gamma = \sqrt{(R + j\omega L)(G + j\omega C)} = \alpha + j\beta$.

Characteristic Impedance (Z_0)

The ratio of forward voltage wave to forward current wave is the Characteristic Impedance $Z_0$:

$$Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} \quad (\Omega)$$

• Lossless Line ($R = 0, G = 0$):

  $$Z_0 = \sqrt{\frac{L}{C}} \quad (\text{Purely real resistance}), \quad \alpha = 0, \quad \beta = \omega \sqrt{LC}$$

• Distortionless Line (Condition: $\frac{R}{L} = \frac{G}{C}$):

  $$Z_0 = \sqrt{\frac{L}{C}}, \quad \alpha = \sqrt{RG}, \quad \beta = \omega \sqrt{LC}$$

  A distortionless line transmits signals without amplitude distortion across frequency components because $\alpha$ is independent of frequency $\omega$.

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4. Line Reflections, VSWR & Impedance Matching

When a transmission line with characteristic impedance $Z_0$ is terminated by a load impedance $Z_L \neq Z_0$, a portion of the incident wave reflects back toward the source.

Voltage Reflection Coefficient ($\Gamma_L$)

The ratio of the reflected voltage wave $V_0^-$ to the incident voltage wave $V_0^+$ at the load is:

$$\Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0} = |\Gamma_L| e^{j\theta_r}$$

Key Load Termination Cases:

1. Matched Load ($Z_L = Z_0$): $\Gamma_L = 0$ (No reflected wave; maximum power transfer).
2. Open-Circuit Load ($Z_L = \infty$): $\Gamma_L = +1$ (Total voltage reflection in-phase; $V_{\text{load}} = 2 V_0^+$).
3. Short-Circuit Load ($Z_L = 0$): $\Gamma_L = -1$ (Total voltage reflection $180^\circ$ out-of-phase; $V_{\text{load}} = 0$).
4. Purely Reactive Load ($Z_L = jX$): $|\Gamma_L| = 1$ (Total power reflection).

Voltage Standing Wave Ratio (VSWR)

The superposition of incident and reflected waves creates a standing wave pattern. The ratio of peak voltage to minimum voltage along the line is:

$$VSWR = \frac{V_{\text{max}}}{V_{\text{min}}} = \frac{1 + |\Gamma_L|}{1 - |\Gamma_L|}$$

• Bounds: $1 \le VSWR \le \infty$.
• $VSWR = 1.0$ indicates a perfectly matched line ($|\Gamma_L| = 0$).

Input Impedance at Distance l from Load

The input impedance $Z_{\text{in}}(l)$ looking into a lossless transmission line of length $l$ terminated by load $Z_L$ is:

$$Z_{\text{in}}(l) = Z_0 \left[ \frac{Z_L + j Z_0 \tan(\beta l)}{Z_0 + j Z_L \tan(\beta l)} \right]$$

• Quarter-Wave Line ($l = \lambda/4$, $\beta l = \pi/2$): $$Z_{\text{in}} = \frac{Z_0^2}{Z_L} \quad (\text{Quarter-wave impedance transformer})$$
• Half-Wave Line ($l = \lambda/2$, $\beta l = \pi$): $$Z_{\text{in}} = Z_L \quad (\text{Impedance repeats every half-wavelength})$$

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5. Transient Surge Propagation on Power Lines (Bounce Diagrams)

When a circuit breaker closes or lightning strikes an overhead transmission line, a high-voltage step wave $V_1$ travels along the line at the surge velocity $v = \frac{1}{\sqrt{LC}}$.

• Surge Impedance: $Z_c = \sqrt{\frac{L}{C}}$
• Sending-End Reflection Coefficient: $\Gamma_S = \frac{Z_S - Z_0}{Z_S + Z_0}$
• Receiving-End Reflection Coefficient: $\Gamma_R = \frac{Z_R - Z_0}{Z_R + Z_0}$

A Bewley Lattice Diagram (Bounce Diagram) tracks successive position-time reflections ($t_1 = l/v, 2t_1, 3t_1$) to compute transient voltage profiles at any line position.

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Solved Board Exam Numerical Problems

Problem 1: Intrinsic Impedance & Power Density of an EM Wave

Question: An electromagnetic wave operating at $f = 100\ \text{MHz}$ propagates through a lossless, non-magnetic medium ($\mu_r = 1.0$) with a relative permittivity $\epsilon_r = 9.0$. The electric field amplitude is $E_0 = 120\ \text{V/m}$. Calculate: (a) the phase velocity $v_p$, (b) the wavelength $\lambda$, (c) the intrinsic impedance $\eta$, and (d) the average power density $P_{\text{avg}}$.

Solution:

1. Calculate phase velocity $v_p$:

   $$v_p = \frac{c}{\sqrt{\epsilon_r}} = \frac{3.0 \times 10^8\ \text{m/s}}{\sqrt{9.0}} = 1.0 \times 10^8\ \text{m/s}$$

2. Calculate wavelength $\lambda$:

   $$\lambda = \frac{v_p}{f} = \frac{1.0 \times 10^8\ \text{m/s}}{100 \times 10^6\ \text{Hz}} = 1.0\ \text{meter}$$

3. Calculate intrinsic impedance $\eta$:

   $$\eta = \frac{\eta_0}{\sqrt{\epsilon_r}} = \frac{377\ \Omega}{\sqrt{9.0}} = \frac{377}{3} = 125.67\ \Omega$$

4. Calculate average power density $P_{\text{avg}}$:

   $$P_{\text{avg}} = \frac{E_0^2}{2\eta} = \frac{(120)^2}{2 \times 125.67} = \frac{14,400}{251.34} = 57.28\ \text{W/m}^2$$

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Problem 2: Transmission Line Reflection & VSWR Calculation

Question: A $50\ \Omega$ lossless transmission line is terminated by an antenna load impedance $Z_L = 75 + j50\ \Omega$. Determine: (a) the load reflection coefficient $\Gamma_L$ in polar form, (b) the reflection coefficient magnitude $|\Gamma_L|$, and (c) the Voltage Standing Wave Ratio ($VSWR$).

Solution:

1. Compute load reflection coefficient $\Gamma_L$:

   $$\Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0} = \frac{(75 + j50) - 50}{(75 + j50) + 50} = \frac{25 + j50}{125 + j50}$$

2. Convert numerator and denominator to polar coordinates:

   $$\text{Numerator: } 25 + j50 = \sqrt{25^2 + 50^2} \angle \arctan(50/25) = 55.90 \angle 63.43^\circ$$ $$\text{Denominator: } 125 + j50 = \sqrt{125^2 + 50^2} \angle \arctan(50/125) = 134.63 \angle 21.80^\circ$$ $$\Gamma_L = \frac{55.90 \angle 63.43^\circ}{134.63 \angle 21.80^\circ} = 0.4152 \angle 41.63^\circ$$

3. Extract magnitude $|\Gamma_L|$:

   $$|\Gamma_L| = 0.4152$$

4. Calculate $VSWR$:

   $$VSWR = \frac{1 + |\Gamma_L|}{1 - |\Gamma_L|} = \frac{1 + 0.4152}{1 - 0.4152} = \frac{1.4152}{0.5848} = 2.42$$