Telegrapher's Equations: Voltage and Current Waves

Why Transmission Lines Need Wave Analysis

From lumped circuits to distributed parameters

Every circuits course starts with Ohm's law: V = IR, applied instantaneously and uniformly across a component. That assumption works as long as the physical size of your circuit is much smaller than the wavelength of the signals passing through it. A 10 cm trace carrying a 1 kHz signal is electrically invisible — the wavelength is 300 km. The same trace carrying a 10 GHz signal is a different story: the wavelength in free space is 3 cm, and in a typical PCB dielectric it shrinks further to around 1.5–2 cm. The trace is now longer than a wavelength, and voltage at one end can be meaningfully different — in both magnitude and phase — from voltage at the other end at the same instant.

This is the core reason lumped-element thinking fails at RF: a lumped model assumes zero propagation delay across a component. A transmission line is what you get when you refuse to make that assumption and instead treat the conductor pair as a continuum of infinitesimal circuit elements, each with its own delay.

When circuit models break down at RF frequencies

A useful rule of thumb: once a conductor's physical length exceeds about 1/10 of the signal wavelength, lumped analysis starts giving visibly wrong answers — phase shifts, standing waves, and frequency-dependent impedance that a simple R-L-C model cannot explain. Below that threshold, treat it as a node; above it, you need distributed analysis. This is why a 1 MHz audio cable is "just a wire" but the same cable carrying a 2.4 GHz Wi-Fi signal is a transmission line with its own impedance, loss, and reflection behavior.

Physical motivation: inductance and capacitance per unit length

The physical reason distributed behavior exists is that every conductor pair has series inductance (from the magnetic field linking the current loop) and shunt capacitance (from the electric field between the conductors), both distributed continuously along the length. You cannot localize these to one point — a microstrip trace has inductance and capacitance in every millimeter of its length, not lumped at its ends. This distributed L and C is what converts a conductor pair into a medium capable of supporting a traveling wave rather than an instantaneously-updating node.

Deriving the Telegrapher's Equations

Starting from Maxwell's equations and Ampere-Maxwell law

The telegrapher's equations are not an independent postulate — they are Maxwell's equations applied to the specific geometry of two parallel conductors, with the fields reduced to equivalent circuit quantities (voltage and current) through line integrals. The Ampere-Maxwell Law: ∇×H = J + ∂D/∂t ties the magnetic field circulating around a conductor to both the conduction current and the displacement current (∂D/∂t). Applied around a conductor cross-section, it produces the inductive voltage drop and leakage current terms in the line equations. Faraday's law, applied to the loop between the two conductors, produces the relationship between the rate of change of magnetic flux and the voltage gradient along the line. The telegrapher's equations are the one-dimensional, circuit-equivalent distillation of this field behavior.

Series impedance (R, L) and shunt admittance (G, C) per unit length

Model an infinitesimal length dz of line as a series impedance R·dz + jωL·dz and a shunt admittance G·dz + jωC·dz, where:

• R = series resistance per unit length (Ω/m), from conductor loss
• L = series inductance per unit length (H/m), from the magnetic field
• G = shunt conductance per unit length (S/m), from dielectric leakage
• C = shunt capacitance per unit length (F/m), from the electric field between conductors

These four distributed parameters fully characterize an ideal transmission line.

Coupled differential equations for V(z,t) and I(z,t)

Applying Kirchhoff's voltage and current laws across the dz segment and taking the limit as dz → 0 gives the telegrapher's equations:

∂V(z,t)/∂z = -R·I(z,t) - L·∂I(z,t)/∂t
∂I(z,t)/∂z = -G·V(z,t) - C·∂V(z,t)/∂t

These are coupled, first-order partial differential equations: the spatial derivative of voltage depends on current (and its rate of change), and vice versa. That coupling is the mathematical signature of wave behavior — energy sloshing between the electric field (voltage, capacitance) and the magnetic field (current, inductance) as it propagates.

Time and frequency domain formulations

For sinusoidal steady-state analysis, switch to phasors V(z) and I(z) with time dependence e^(jωt) implied:

dV(z)/dz = -(R + jωL)I(z)
dI(z)/dz = -(G + jωC)V(z)

Differentiating once more and substituting decouples them into two second-order wave equations:

d²V(z)/dz² = γ²V(z)
d²I(z)/dz² = γ²I(z)

where γ = √((R + jωL)(G + jωC)) is the complex propagation constant.

Wave Solutions: Forward and Backward Traveling Waves

General solution as superposition of V⁺ and V⁻

The general solution to the voltage wave equation is:

V(z) = V⁺e^(-γz) + V⁻e^(γz)

The first term travels in +z, the second in −z. Any voltage pattern on a transmission line, however complex, is a superposition of these two traveling components — this is the single most important conceptual shift from lumped-circuit thinking: voltage isn't a shared node value, it's two counter-propagating waves whose sum you measure at any point.

Relationship between voltage and current waves

Current follows the same form, but scaled by the characteristic impedance:

I(z) = (V⁺/Z0)e^(-γz) - (V⁻/Z0)e^(γz)

The minus sign on the reverse wave matters — it reflects that power flow direction reverses for the backward component.

Wave velocity and propagation constant

Writing γ = α + jβ, α is the attenuation constant (Np/m) and β is the phase constant (rad/m). Phase velocity is vp = ω/β, and for a low-loss line this approaches 1/√(LC), the same form as the speed of light in a medium scaled by permittivity and permeability.

Characteristic impedance from the wave equation

Dividing the voltage and current phasor equations yields:

Z0 = √((R + jωL)/(G + jωC))

This is the impedance seen by a single traveling wave — not a static resistance, but the ratio fixed by the line's distributed geometry. In the lossless limit this reduces to the well-known Characteristic Impedance: Z0 = √(L/C) (Lossless) relation, which is why a 50 Ω coax cable is 50 Ω regardless of length — it's a ratio set by L and C per meter, not a lumped resistor you could measure with an ohmmeter (a DC ohmmeter reads near-zero resistance on a coax's center conductor, not 50 Ω).

Lossless vs. Lossy Transmission Lines

Lossless case: purely imaginary propagation constant

Set R = G = 0. Then γ = jω√(LC) is purely imaginary — α = 0, no attenuation, and Z0 = √(L/C) is purely real. Waves propagate without amplitude decay, only phase accumulation. This is the idealization used for short, high-quality lines (good coax, short microstrip runs) where loss is a second-order correction.

Impact of R and G losses on wave attenuation

Real lines have R > 0 (finite conductor resistivity) and G > 0 (imperfect dielectric). Both contribute to α, causing the traveling wave amplitude to decay as e^(-αz). R dominates at lower frequencies; G (through dielectric loss tangent) dominates at higher frequencies for many practical dielectrics.

Skin effect in conductors and dielectric loss

R itself is frequency-dependent: skin effect confines current to a thin layer near the conductor surface, with skin depth shrinking as 1/√f, so R rises roughly as √f at high frequency. Dielectric loss, characterized by the material's loss tangent, causes G to rise roughly linearly with f. Consult the dielectric manufacturer's datasheet for loss tangent vs. frequency — it's rarely constant across a wide band.

Low-loss approximation for practical lines

When R << ωL and G << ωC (true for well-designed RF lines well above their resistive corner frequency), α ≈ (R/2)√(C/L) + (G/2)√(L/C), and Z0 stays approximately real and close to the lossless value. This approximation is what lets engineers use the lossless Z0 formula for practical matching work while still budgeting insertion loss separately.

Boundary Conditions and Terminations

Open, short, and matched load reflections

A traveling wave only reflects when it meets a discontinuity — a load impedance ZL that differs from Z0. A matched load (ZL = Z0) absorbs the wave entirely, mimicking an infinite line. An open circuit (ZL = ∞) reflects the voltage wave in phase; a short circuit (ZL = 0) reflects it inverted.

Reflection coefficient and transmission coefficient

The reflection coefficient at the load is:

Γ = (ZL - Z0)/(ZL + Z0)

Γ ranges from -1 (short) through 0 (matched) to +1 (open). The transmission coefficient into the load is T = 1 + Γ. This single complex number, measured versus frequency, is the basis of S-parameter characterization — and it's exactly what a VNA measures at each port.

Standing wave patterns in mismatched lines

When Γ ≠ 0, forward and backward waves interfere, producing a standing wave pattern with fixed voltage maxima and minima along the line. The voltage standing wave ratio, VSWR = (1+|Γ|)/(1-|Γ|), quantifies the severity — VSWR of 1.5:1 is often an acceptable design target, while 3:1 or worse typically signals a matching problem worth fixing before it causes amplifier instability or power loss.

Power flow and Poynting vector interpretation

Physically, the reflected wave carries real power back toward the source; net delivered power is proportional to |V⁺|² − |V⁻|², equivalent to integrating the Poynting vector over the line's cross-section. A mismatched load doesn't dissipate the reflected power — it returns it, which is why poor matching shows up as reduced delivered power even when no component is overtly failing.

Practical Applications in RF Circuit Design

Impedance matching and Smith chart foundations

The Smith chart is simply a graphical plot of Γ in the complex plane, mapped onto equivalent normalized impedance contours. Every impedance-matching technique — series/shunt stubs, L-networks, quarter-wave sections — is an exercise in manipulating Γ back toward zero at the frequency of interest.

Stub tuning and quarter-wave transformers

A quarter-wavelength line of impedance Zq transforms a load ZL to Zin = Zq²/ZL at its input — directly derivable from the traveling-wave solution evaluated at z = λ/4. Stub tuning adds a reactive shunt or series element, realized as an open- or short-circuited line segment, to cancel the imaginary part of an unwanted load impedance.

Measurement via network analyzers and VNA calibration

Because Γ is what's physically measurable at RF (direct voltage/current probing becomes unreliable above a few hundred MHz), nearly all RF characterization is done through reflection and transmission coefficients. A vector network analyzer measures these directly, but only after the systematic errors of cables, adapters, and connectors are removed — the subject of VNA Calibration: SOLT, TRL, SOLR, LRM Methods.

Link to S-parameters and two-port characterization

S-parameters generalize Γ and T to multi-port networks, describing how incident and reflected waves at each port relate to each other. They are one member of a family of two-port descriptions alongside Two-Port Network Parameters: Z, Y, H, ABCD, chosen for RF work specifically because they're defined in terms of traveling waves rather than open/short-circuit conditions that are hard to realize cleanly at high frequency.

Connecting to Physical Transmission Media

Coaxial cables, stripline, and microstrip implementations

The telegrapher's equations are geometry-agnostic — L, C, R, and G simply take different values depending on whether you're analyzing coax (concentric cylinders), stripline (a trace buried between two ground planes), or microstrip (a trace over a single ground plane with air/dielectric above). Each geometry has closed-form or numerically-solved expressions for L and C per unit length in standard RF textbooks and field-solver tools.

Waveguide modes and cutoff frequencies

Hollow waveguides don't support the simple TEM mode assumed above; they propagate TE or TM modes with a cutoff frequency below which no propagation occurs at all. The telegrapher's equation framework still applies conceptually but requires modified per-unit-length parameters derived from the relevant mode's field distribution.

Real-world loss models and frequency dependence

Datasheet-quoted insertion loss for coax or PCB laminate is almost never a single number — it's frequency-dependent through skin effect and dielectric loss tangent, as discussed above. For serious design work, pull the vendor's frequency-dependent loss curve rather than a single dB/m figure quoted at one frequency, since extrapolating a low-frequency loss number to a 10x higher frequency will understate real attenuation.