1.0 TRANSMISSION LINE FUNDAMENTALS
1.1 Transmission Line Parameters & Primary Constants
Transmission lines are modeled using distributed parameters per unit length:
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R (Ω/m): Resistance due to conductor loss.
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L (H/m): Inductance due to magnetic field energy storage.
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G (S/m): Conductance representing dielectric leakage loss.
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C (F/m): Capacitance due to electric field energy storage.
These are called primary constants. They determine the secondary constants: propagation constant (γ), attenuation constant (α), phase constant (β), characteristic impedance (Z₀), and velocity (v).
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Common Exam Question: "Define R, L, G, C and give their units."
Remember: R and G cause power loss; L and C store energy.
1.2 Transmission Line Equations & Wave Propagation
Consider an infinitesimal length dx of a transmission line. Applying Kirchhoff's laws:
Voltage equation:
$$\displaystyle V(x) - V(x+dx) = (R + j\omega L)dx \cdot I(x) $$
$$\frac{\partial V}{\partial x} = -(R + j\omega L)I \quad (1)$$
Current equation:
$$\displaystyle I(x) - I(x+dx) = (G + j\omega C)dx \cdot V(x) $$
$$\frac{\partial I}{\partial x} = -(G + j\omega C)V \quad (2)$$
Differentiating (1) w.r.t. x and substituting (2) yields the wave equation:
$$\frac{\partial^2 V}{\partial x^2} = \gamma^2 V$$
where the propagation constant is:
$$\gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)}$$
Similarly for current. The general solutions are:
$$V(x) = V^+ e^{-\gamma x} + V^- e^{\gamma x}$$
$$I(x) = \frac{V^+}{Z_0} e^{-\gamma x} - \frac{V^-}{Z_0} e^{\gamma x}$$
Characteristic impedance (Z₀):
$$Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} \quad \boxed{Z_0 = \frac{V^+}{I^+} = \frac{V^-}{-I^-}}$$
For a lossless line (R=0, G=0):
$$\gamma = j\omega\sqrt{LC}, \quad Z_0 = \sqrt{\frac{L}{C}}, \quad v = \frac{1}{\sqrt{LC}}$$
Velocity of propagation:
$$v = \frac{\omega}{\beta} = \frac{1}{\sqrt{LC}} \text{ (lossless)}$$
Wavelength:
$$\lambda = \frac{2\pi}{\beta} = \frac{v}{f}$$
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Very High Priority: Derivation of transmission line equations is frequently asked.
Key Insight: γ describes wave attenuation (α) and phase shift (β); Z₀ is the ratio of voltage to current for a traveling wave.
1.3 Line Types & Special Cases
| Case | Condition | Propagation Constant (γ) | Characteristic Impedance (Z₀) | Input Impedance (Z_in) |
|---|---|---|---|---|
| Lossless | R=0, G=0 | $j\omega\sqrt{LC}$ | $\sqrt{L/C}$ | $$\displaystyle Z_{in} = jZ_0 \tan(\beta l) $$ (general) |
| Distortionless | R/L = G/C | $$\displaystyle \alpha = R\sqrt{C/L} $$, $$\displaystyle \beta = \omega\sqrt{LC} $$ | $\sqrt{L/C}$ | Same as lossless but with α |
| Quarter-wave | $$\displaystyle l = \lambda/4 $$ | — | — | $$\displaystyle \boxed{Z_{in} = \frac{Z_0^2}{Z_L}} $$ |
| Half-wave | $$\displaystyle l = \lambda/2 $$ | — | — | $$\displaystyle \boxed{Z_{in} = Z_L}} $$ |
| Short-circuited | $$\displaystyle Z_L = 0 $$ | — | — | $$\displaystyle Z_{in} = jZ_0 \tan(\beta l) $$ |
| Open-circuited | $$\displaystyle Z_L = \infty $$ | — | — | $$\displaystyle Z_{in} = -jZ_0 \cot(\beta l) $$ |
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High Priority: Quarter-wave line inverts impedance; half-wave repeats it.
Short/Open lines act as reactances: $$\displaystyle Z_{in} \propto \tan(\beta l) $$ or $\cot(\beta l)$.
1.4 Power Flow & Losses
For a lossless line ($$\displaystyle \alpha=0 $$) with matched load ($$\displaystyle Z_L = Z_0 $$), average power is:
$$P_{avg} = \frac{|V|^2}{2Z_0} = \frac{|I|^2 Z_0}{2}$$
With mismatch, incident power $$\displaystyle P_i $$, reflected power $$\displaystyle P_r $$, and forward power $$\displaystyle P_f $$:
$$P_r = |\Gamma|^2 P_i, \quad P_f = (1 - |\Gamma|^2) P_i$$
Reflection coefficient (Γ):
$$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} \quad \boxed{|\Gamma| \leq 1 \text{ for passive } Z_L}$$
Voltage Standing Wave Ratio (VSWR):
$$S = \frac{1 + |\Gamma|}{1 - |\Gamma|} \quad \boxed{S \geq 1}$$
Inverse: $$\displaystyle |\Gamma| = \frac{S-1}{S+1} $$
Reflection loss (mismatch loss):
$$L_r = -10 \log_{10}(1 - |\Gamma|^2) \text{ dB}$$
Insertion loss (when source and load are matched to Z₀) equals reflection loss.
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Very High Priority: VSWR and Γ are interconvertible. Always check if load is passive (|Γ| ≤ 1).
Common Pitfall: Insertion loss ≠ reflection loss if source/load not matched.
1.5 Line Analysis & Measurement
Smith Chart
Construction:
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Based on normalized impedance $$\displaystyle z = Z / Z_0 = r + jx $$.
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Constant resistance circles: $$\displaystyle r = \text{constant} $$, centers at $(r/(r+1), 0)$.
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Constant reactance arcs: $$\displaystyle x = \text{constant} $$, arcs intersecting real axis.
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Wavelength scale: clockwise rotation corresponds to moving toward load.
Applications:
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Impedance matching: Using lumped elements or stubs.
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Finding Γ and VSWR: Distance from center to point gives |Γ|; intersection with real axis gives VSWR.
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Admittance transformation: Rotate 180° or use admittance chart.
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Determining electrical length: Move along constant |Γ| circle.
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Very High Priority: Smith Chart is essential for impedance matching problems. Practice reading normalized impedances and moving toward load/generator.
Determination of Primary & Secondary Constants
Given open-circuit impedance $$\displaystyle Z_{oc} $$ and short-circuit impedance $$\displaystyle Z_{sc} $$:
Characteristic impedance:
$$Z_0 = \sqrt{Z_{oc} Z_{sc}} \quad \boxed{Z_0 = \sqrt{Z_{oc} Z_{sc}}}$$
Propagation constant:
$$\gamma = \cosh^{-1}\left(\frac{Z_{oc}}{Z_0}\right) = \cosh^{-1}\left(\frac{Z_{sc}}{Z_0}\right)$$
Then $$\displaystyle \alpha = \text{Re}(\gamma) $$, $$\displaystyle \beta = \text{Im}(\gamma) $$.
Primary constants from $\gamma$ and $$\displaystyle Z_0 $$:
$$R + j\omega L = \gamma Z_0, \quad G + j\omega C = \frac{\gamma}{Z_0}$$
If given attenuation A (dB/km) and phase constant β (rad/km):
$$\alpha = \frac{A \ln(10)}{20} \text{ (Np/km)}$$
Then $$\displaystyle \gamma = \alpha + j\beta $$, and compute R, L, G, C as above.
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Common Exam Problem: "Find R, L, G, C from Z_oc, Z_sc, and frequency." Use the boxed formulas step-by-step.
Summary of High-Priority Formulas (Unit 1)
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Transmission line equations:
$$\displaystyle \frac{\partial V}{\partial x} = -(R+j\omega L)I $$, $$\displaystyle \frac{\partial I}{\partial x} = -(G+j\omega C)V $$
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Propagation constant:
$$\displaystyle \gamma = \sqrt{(R+j\omega L)(G+j\omega C)} = \alpha + j\beta $$
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Characteristic impedance:
$$\displaystyle Z_0 = \sqrt{\frac{R+j\omega L}{G+j\omega C}} $$
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Quarter-wave input impedance:
$$\displaystyle Z_{in} = Z_0^2 / Z_L $$
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Reflection coefficient:
$$\displaystyle \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$
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VSWR:
$$\displaystyle S = \frac{1+|\Gamma|}{1-|\Gamma|} $$
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From Z_oc and Z_sc:
$$\displaystyle Z_0 = \sqrt{Z_{oc} Z_{sc}} $$, $$\displaystyle \gamma = \cosh^{-1}(Z_{oc}/Z_0) $$
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Exam Strategy: For derivation questions, start with the infinitesimal model and apply KVL/KCL. For calculation problems, first identify given/required constants and use the appropriate formula chain.