10.4 Transmission Line Parameters, SWR & Smith Chart

10.4 Transmission Line Parameters, SWR & Smith Chart

Board Examination Focus: Characteristic impedance ($Z_0 = \sqrt{L/C}$), velocity factor ($VF = 1/\sqrt{\epsilon_r}$), reflection coefficient ($\Gamma = (Z_L - Z_0)/(Z_L + Z_0)$), Voltage Standing Wave Ratio ($\text{VSWR} = (1+|\Gamma|)/(1-|\Gamma|)$), quarter-wave transformer matching ($Z_0' = \sqrt{Z_0 Z_L}$), and Smith Chart impedance transformations are heavily tested EST topics.

Transmission lines are specialized electrical conductors designed to carry radio frequency (RF) energy from a transmitter source to an antenna load, or from an antenna to a receiver front-end, with minimum attenuation and distortion. Unlike low-frequency power lines where physical length is negligible relative to wavelength, transmission lines must be modeled using distributed parameters per unit length.

Distributed Transmission Line Parameters

At high frequencies, wire resistance, loop inductance, dielectric insulation conductance, and inter-conductor capacitance are distributed continuously along the line:

• $R$: Distributed resistance of conductors ($\Omega/\text{m}$)
• $L$: Distributed inductance of conductors ($\text{H/m}$)
• $G$: Distributed dielectric conductance of insulation ($\text{S/m}$)
• $C$: Distributed capacitance between conductors ($\text{F/m}$)

             R (ohms/m)    L (H/m)
   Input o----/\/\/-------UUUUU------+-------------------o Output
                                      |
                                    [G] Conductance (S/m)
                                      |
                                     ---
                                     --- Capacitance (C) (F/m)
                                      |
   Return o---------------------------+-------------------o Return

Characteristic Impedance ($Z_0$)

The characteristic impedance ($Z_0$) is the input impedance of an infinitely long transmission line. Applying telegrapher's equations yields:

$Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}$

For low-loss or high-frequency RF lines where $R \ll \omega L$ and $G \ll \omega C$, the line approaches the lossless approximation:

$Z_0 = \sqrt{\frac{L}{C}}$

Physical dimensions determine $Z_0$ for common cable types:

1. Coaxial Cable: $Z_0 = \frac{138}{\sqrt{\epsilon_r}} \log_{10}\left(\frac{D}{d}\right)$ Where $D$ is inner diameter of outer shield, $d$ is outer diameter of inner conductor, and $\epsilon_r$ is dielectric constant. Standard commercial coax values are $50\ \Omega$ (RF power transmission) and $75\ \Omega$ (TV/cable distribution, low loss).

2. Parallel Two-Wire Line (Twin-Lead): $Z_0 = \frac{276}{\sqrt{\epsilon_r}} \log_{10}\left(\frac{2 D}{d}\right)$ Where $D$ is center-to-center conductor spacing. Standard twin-lead value is $300\ \Omega$.

Velocity Factor ($VF$) and Wavelength

The velocity of propagation ($v_p$) along a line surrounded by dielectric $\epsilon_r$ is less than speed of light in vacuum ($c = 3 \times 10^8\text{ m/s}$):

$v_p = \frac{1}{\sqrt{L C}} = \frac{c}{\sqrt{\epsilon_r}} \implies VF = \frac{v_p}{c} = \frac{1}{\sqrt{\epsilon_r}}$

Wavelength along the transmission line ($\lambda_{\text{line}}$) is:

$\lambda_{\text{line}} = \frac{v_p}{f} = \frac{c \times VF}{f} = \lambda_{\text{free space}} \times VF$

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Reflection Coefficient and Standing Wave Ratio (VSWR)

When a transmission line terminated in load impedance $Z_L$ is not equal to characteristic impedance $Z_0$ ($Z_L \neq Z_0$), a portion of incident voltage wave ($V_i$) is reflected back toward the source ($V_r$).

Voltage Reflection Coefficient ($\Gamma$)

The complex reflection coefficient ($\Gamma$) at the load is defined as:

$\Gamma = \frac{V_r}{V_i} = \frac{Z_L - Z_0}{Z_L + Z_0} = |\Gamma| e^{j\theta}$

• Matched Load ($Z_L = Z_0$): $\Gamma = 0$. All power absorbed by load; no reflection.
• Short Circuit ($Z_L = 0$): $\Gamma = \frac{0 - Z_0}{0 + Z_0} = -1 = 1 \angle 180^\circ$. Total reflection with phase reversal.
• Open Circuit ($Z_L = \infty$): $\Gamma = \frac{\infty - Z_0}{\infty + Z_0} = +1 = 1 \angle 0^\circ$. Total reflection in phase.
• Purely Reactive Load ($Z_L = \pm j X$): $|\Gamma| = 1.0$. Total reflection with phase shift.

Voltage Standing Wave Ratio (VSWR)

Interference between incident and reflected waves creates stationary voltage peaks ($V_{\text{max}}$) and troughs ($V_{\text{min}}$) along the line. The ratio is the Voltage Standing Wave Ratio (VSWR):

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

Conversely, magnitude of reflection coefficient is:

$|\Gamma| = \frac{\text{VSWR} - 1}{\text{VSWR} + 1}$

Power Relations and Return Loss

• Reflected Power Ratio: $P_r / P_i = |\Gamma|^2$
• Transmitted Power Ratio: $P_t / P_i = 1 - |\Gamma|^2$
• Return Loss (RL): $RL = -20 \log_{10}|\Gamma|\text{ (dB)}$

| Matching Condition | Reflection Coeff $|\Gamma|$ | VSWR | Return Loss (dB) | Reflected Power (%) | |---|---|---|---|---| | Ideal Match | $0.00$ | $1.0 : 1$ | $\infty\text{ dB}$ | $0.0%$ | | Good RF System | $0.05$ | $1.11 : 1$ | $26.0\text{ dB}$ | $0.25%$ | | Acceptable System | $0.10$ | $1.22 : 1$ | $20.0\text{ dB}$ | $1.0%$ | | Marginal System | $0.33$ | $2.00 : 1$ | $9.54\text{ dB}$ | $11.1%$ | | Severe Mismatch (Open/Short) | $1.00$ | $\infty : 1$ | $0.0\text{ dB}$ | $100.0%$ |

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Special Line Lengths & Impedance Transformations

Input impedance ($Z_{\text{in}}$) of a lossless line of length $l$ terminated in $Z_L$ is:

$Z_{\text{in}}(l) = Z_0 \left[ \frac{Z_L + j Z_0 \tan(\beta l)}{Z_0 + j Z_L \tan(\beta l)} \right] \quad \text{where } \beta = \frac{2\pi}{\lambda}$

Critical Quarter-Wave and Half-Wave Resonators

1. Quarter-Wave Transformer ($l = \lambda/4$): $\tan(\beta l) = \tan\left(\frac{2\pi}{\lambda} \frac{\lambda}{4}\right) = \tan\left(\frac{\pi}{2}\right) \to \infty \implies Z_{\text{in}} = \frac{Z_0^2}{Z_L}$

   • Converts short circuit ($Z_L = 0$) into open circuit ($Z_{\text{in}} = \infty$).
   • Used for impedance matching: To match a main line $Z_{01}$ to a load $Z_L$, insert a $\lambda/4$ section with characteristic impedance: $Z_0' = \sqrt{Z_{01} Z_L}$

2. Half-Wave Line ($l = \lambda/2$): $\tan(\beta l) = \tan(\pi) = 0 \implies Z_{\text{in}} = Z_L$

   • Repeats load impedance directly regardless of line characteristic impedance.

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The Smith Chart & Impedance Matching

Invented by Phillip H. Smith in 1939, the Smith Chart is a polar plot of complex voltage reflection coefficient $\Gamma = u + j v$ mapped onto normalized impedance planes ($z = Z / Z_0 = r + j x$).

                      +jx (Inductive Reactance Region)
                            r=0.5   r=1.0   r=2.0
                               |      |      |
        Short Circuit (0+j0) --+------+------+-- Open Circuit (inf+j-inf)
        (Leftmost point)       |  (1+j0) Main |  (Rightmost point)
                               |   Center    |
                            r=0.5   r=1.0   r=2.0
                      -jx (Capacitive Reactance Region)

Key Coordinates and Topography

• Prime Center ($1 + j0$): Perfect match ($z = 1 \implies Z = Z_0, \Gamma = 0, \text{VSWR} = 1.0$).
• Normalized Values: $r = R / Z_0$, $x = X / Z_0$.
• Constant Resistance Circles: Circles centered along horizontal real axis; all points share identical normalized resistance $r$.
• Constant Reactance Arcs: Curves radiating from right-hand open-circuit point; upper half is inductive ($+j x$), lower half is capacitive ($-j x$).
• VSWR Circle: Drawn centered at $(1+j0)$ with radius equal to distance to normalized load point. VSWR is read directly at the rightmost intersection with real axis.
• Wavelengths Toward Generator (WTG): Moving clockwise along perimeter outer scale moves away from load toward generator ($0.5\lambda$ full revolution).

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Step-by-Step Worked Example: Transmission Line & SWR Analysis

Problem: A $50\ \Omega$ coaxial transmission line ($Z_0 = 50\ \Omega$) is terminated by a load impedance $Z_L = 100 + j50\ \Omega$. The line carries an incident forward power of $100\text{ Watts}$. Calculate: (a) Complex reflection coefficient ($\Gamma$), (b) VSWR, (c) Reflected power ($P_r$), and (d) Characteristic impedance ($Z_0'$) of a quarter-wave matching section required to match a $200\ \Omega$ resistive antenna load to the $50\ \Omega$ main coax line.

Solution:

1. Calculate reflection coefficient ($\Gamma$): $\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} = \frac{(100 + j50) - 50}{(100 + j50) + 50} = \frac{50 + j50}{150 + j50}$ Convert numerator and denominator to polar form: $\text{Numerator } = \sqrt{50^2 + 50^2} \angle \arctan(50/50) = 70.71 \angle 45^\circ$ $\text{Denominator } = \sqrt{150^2 + 50^2} \angle \arctan(50/150) = 158.11 \angle 18.43^\circ$ $\Gamma = \frac{70.71}{158.11} \angle (45^\circ - 18.43^\circ) = 0.4472 \angle 26.57^\circ \implies |\Gamma| = 0.4472$

2. Calculate VSWR: $\text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|} = \frac{1 + 0.4472}{1 - 0.4472} = \frac{1.4472}{0.5528} \approx 2.618 \implies 2.62 : 1$

3. Calculate reflected power ($P_r$): $P_r = P_i |\Gamma|^2 = 100\text{ W} \times (0.4472)^2 = 100\text{ W} \times 0.200 = 20.0\text{ Watts}$

4. Calculate quarter-wave matching section impedance ($Z_0'$): $Z_0' = \sqrt{Z_{01} Z_L} = \sqrt{50\ \Omega \times 200\ \Omega} = \sqrt{10,000} = 100.0\ \Omega$