Transmission Line Equations

The behaviour of voltage and current on a transmission line is governed by a pair of coupled partial differential equations known as the telegrapher's equations. These equations arise directly from applying Kirchhoff's voltage and current laws to the distributed RLGC model of the line. Their solutions reveal that signals on transmission lines exist as travelling waves — not simple instantaneous responses — and this wave nature is responsible for phenomena such as reflections, standing waves, and impedance transformation.

Core Concept Explanation

Consider an infinitesimal section of line of length Δz. Applying KVL around the top loop: V(z) - RΔz·I(z) - LΔz·∂I/∂t - V(z+Δz) = 0. Dividing by Δz and taking Δz → 0 gives the first telegrapher's equation: -∂V/∂z = RI + L·∂I/∂t. Similarly, KCL at the middle node gives the second: -∂I/∂z = GV + C·∂V/∂t. These are coupled first-order PDEs in V and I.

Differentiating the first equation with respect to z and substituting the second eliminates I, yielding the wave equation for voltage: ∂²V/∂z² = (R + jωL)(G + jωC)V = γ²V, where γ is the complex propagation constant. An identical equation holds for I. This is the core result — voltage and current on a transmission line obey the wave equation, confirming their wave nature.

The general solution of the wave equation has two travelling wave components: V(z) = V⁺e^(-γz) + V⁻e^(+γz). The term V⁺e^(-γz) is the forward-travelling wave (in +z direction) and V⁻e^(+γz) is the backward-travelling reflected wave. Each wave decays exponentially with distance due to the factor e^(-αz) while oscillating at angular frequency ω due to e^(jβz).

The characteristic impedance Z₀ relates the voltage and current amplitudes of each individual travelling wave: for the forward wave, V⁺/I⁺ = Z₀ = sqrt((R + jωL)/(G + jωC)), and for the backward wave, V⁻/I⁻ = -Z₀. The total current is I(z) = (V⁺e^(-γz) - V⁻e^(+γz))/Z₀.

Mathematical Expressions

For phasor (sinusoidal steady-state) analysis at frequency ω, the telegrapher's equations become ordinary differential equations: dV/dz = -(R + jωL)I and dI/dz = -(G + jωC)V. Their solution gives:

V(z) = V⁺e^(-γz) + V⁻e^(+γz). I(z) = (1/Z₀)[V⁺e^(-γz) - V⁻e^(+γz)]. Propagation constant: γ = α + jβ = sqrt((R + jωL)(G + jωC)). Characteristic impedance: Z₀ = sqrt((R + jωL)/(G + jωC)). Phase velocity: v_p = ω/β. Wavelength on line: λ = 2π/β. For lossless line (R = G = 0): γ = jβ = jω sqrt(LC), Z₀ = sqrt(L/C), v_p = 1/sqrt(LC).

The distortionless condition is a special case where R/L = G/C. Under this condition, γ = sqrt(RG) + jω sqrt(LC), meaning α = sqrt(RG) is frequency-independent and β = ω sqrt(LC) is linear in ω — no dispersion. This is how telegraphic cables were designed historically and is still relevant to broadband cable design.

Practical Understanding

In practice, most RF and microwave transmission lines (coaxial cables, microstrip, stripline) are designed to be approximately lossless at their operating frequencies, with R << ωL and G << ωC. Under this condition, Z₀ ≈ sqrt(L/C) (real-valued, resistive) and γ ≈ jβ (purely imaginary, no attenuation). Standard cable impedances of 50 Ω and 75 Ω arise from geometric optimisation of coaxial lines for minimum loss and maximum power handling respectively.

At lower frequencies, such as audio or power transmission, R and G are not negligible, and the full complex expressions for γ and Z₀ must be used. Long-distance power lines exhibit significant R, and their wave nature becomes relevant at continental scales where line lengths exceed a significant fraction of the 50 Hz wavelength (6000 km at 50 Hz).

The input impedance of a finite-length line terminated in load ZL is given by Zin = Z₀·(ZL + Z₀·tanh(γl))/(Z₀ + ZL·tanh(γl)). For lossless lines, tanh(γl) = j·tan(βl), which oscillates with line length — this is the mechanism behind quarter-wave transformers and half-wave line sections.

Given:
Lossless transmission line:
  L = 250 nH/m, C = 100 pF/m
  Line length: l = 0.3 m
  Frequency: f = 1 GHz
  Load: ZL = 100 Ω (resistive)

Why this formula applies:
Lossless line → α = 0, γ = jβ = jω√(LC)
Need β, Z₀, and then Zin

Formula:
β = ω√(LC)
Z₀ = √(L/C)
Zin = Z₀ · (ZL + jZ₀·tan(βl)) / (Z₀ + jZL·tan(βl))

Substitution:
ω = 2π × 10⁹ = 6.283 × 10⁹ rad/s
Z₀ = √(250×10⁻⁹ / 100×10⁻¹²) = √2500 = 50 Ω
β = 6.283×10⁹ × √(250×10⁻⁹ × 100×10⁻¹²)
  = 6.283×10⁹ × √(2.5×10⁻¹⁷)
  = 6.283×10⁹ × 1.581×10⁻⁸.5

Calculation:
√(LC) = √(25×10⁻¹⁸) = 5×10⁻⁹ s/m
β = 6.283×10⁹ × 5×10⁻⁹ = 31.42 rad/m
βl = 31.42 × 0.3 = 9.425 rad
9.425 rad = 3π - 0.283 rad → tan(9.425) = tan(-0.283) ≈ -0.293

Zin = 50 × (100 + j50×(-0.293)) / (50 + j100×(-0.293))
     = 50 × (100 - j14.65) / (50 - j29.3)
|numerator| ≈ 50 × 101.07 / 57.97

Final Answer:
Z₀ = 50 Ω, β = 31.42 rad/m
Zin ≈ 50 × (100 - j14.65)/(50 - j29.3) ≈ 87.1 + j34.2 Ω
(Input impedance is complex even for resistive load — line transforms impedance)

Exam Tip: For a lossless line of length λ/4 (quarter-wave transformer), Zin = Z₀²/ZL — this is one of the most GATE-tested results. For a half-wave line (l = λ/2), Zin = ZL regardless of Z₀. Also remember: an open-circuited quarter-wave section looks like a short circuit at its input, and a short-circuited quarter-wave section looks like an open circuit.

Mechanism — Forward and Backward Waves

• The telegrapher's equations are two coupled PDEs obtained by applying KVL and KCL to the distributed RLGC model. They reduce to the wave equation when decoupled.
• Solution V(z) = V⁺e^(-γz) + V⁻e^(+γz) shows two travelling wave components. V⁺ travels in +z direction; V⁻ travels in -z direction (reflection from load).
• Characteristic impedance Z₀ = V⁺/I⁺ = -V⁻/I⁻ = sqrt((R+jωL)/(G+jωC)) — purely resistive for lossless line.
• Propagation constant γ = α + jβ: α gives amplitude decay per unit length (Np/m); β gives phase change per unit length (rad/m).
• Quarter-wave transformer: Zin = Z₀²/ZL — used for impedance matching between two different impedance levels.

Quick Revision

• Telegrapher's equations: -∂V/∂z = RI + L∂I/∂t and -∂I/∂z = GV + C∂V/∂t (from KVL, KCL on distributed model).
• Propagation constant: γ = sqrt((R+jωL)(G+jωC)) = α + jβ; for lossless: γ = jβ = jω sqrt(LC).
• Characteristic impedance: Z₀ = sqrt((R+jωL)/(G+jωC)); for lossless: Z₀ = sqrt(L/C).
• General solution: V(z) = V⁺e^(-γz) + V⁻e^(+γz); I(z) = (V⁺e^(-γz) - V⁻e^(+γz))/Z₀.
• λ/4 line: Zin = Z₀²/ZL. λ/2 line: Zin = ZL. Short-circuited λ/4: Zin = ∞. Open-circuited λ/4: Zin = 0.
• Distortionless condition R/L = G/C → frequency-independent α, linear β (no dispersion).
• Common trap: Z₀ is the impedance of each individual wave, NOT the impedance seen at the input — input impedance depends on length and load.