Transmission-Line Wave Visualizer

EENG530 · Passive RF & Microwave Devices — bounce-diagram animation of V(x,t) and I(x,t) on a lossless line
~ V0 cos(2πt)·u(t) Zs x = 0 x = ℓ ZL
Voltage V(x,t) y ± V
Power P(x,t) y ± mW
Current I(x,t) y ± mA
Bounce (lattice) diagram
Voltage at the two ends vs time t ≤  y ± V
Theory — what exactly is being animated?

A lossless line of characteristic impedance Z₀ and one-way transit time T = ℓ/v is driven at x = 0 by a Thévenin source (V₀, Zs) switched on at t = 0 and terminated at x = ℓ by ZL. Voltage and current are superpositions of a forward and a backward travelling wave:

V(x,t) = V⁺(t − x/v) + V⁻(t + x/v),    I(x,t) = [V⁺(t − x/v) − V⁻(t + x/v)] / Z₀

Each end reflects with its reflection coefficient, and the initial launch is set by the voltage divider between Zs and Z₀ (the line first "looks like" a resistor of value Z₀):

ΓL = (ZL − Z₀)/(ZL + Z₀),   ΓS = (Zs − Z₀)/(Zs + Z₀),   V⁺₀ = V₀ Z₀/(Zs + Z₀)

The n-th forward bounce leaves the source at t = 2nT with amplitude V⁺₀(ΓSΓL)ⁿ; the n-th backward bounce leaves the load at t = (2n+1)T with amplitude V⁺₀ΓL(ΓSΓL)ⁿ. The animation sums this (causally gated) series — that is the entire simulation.

Time-harmonic mode. Each bounce is a phasor; the geometric series sums to the exact steady-state standing wave shown as the dotted envelope:

V(x) = V⁺₀ · [e−jβx + ΓL e−jβ(2ℓ−x)] / [1 − ΓSΓL e−j2βℓ],    Zin = Z₀ (1+ΓLe−j2βℓ)/(1−ΓLe−j2βℓ)

Step / pulse modes. With resistive ends the reflections are frequency-flat, so the drawn waveforms are exact. Under step drive the load voltage builds up as the classic staircase toward V∞ = V₀RL/(Rs+RL); a rect or Gaussian pulse instead returns as a train of successively scaled echoes decaying to zero.

Caveat (reactive terminations, harmonic mode): reflecting each switched-on wavefront by the phasor Γ(jω) is the standard bounce-diagram idealization — a real L or C also produces a decaying natural response at each wavefront arrival (time constant L/(R+Z₀) for an inductive load, (R+Z₀)C for a capacitive one — a fraction of a period when |X| is comparable to Z₀), which is omitted here. The steady-state standing wave the animation settles into is exact regardless.

Power. The power plot shows the instantaneous flow P(x,t) = V(x,t)·I(x,t) = [ (V⁺)² − (V⁻)² ]/Z₀ (positive toward the load). In harmonic steady state it oscillates at 2ω about the time average ⟨P⟩(x) = ½ Re{V(x) I*(x)}, which on a lossless line is the same at every x — including across junctions — and equals the power delivered to the load.

Cascaded sections. Each junction between Z0,k and Z0,k+1 partially reflects (ρk = (Z0,k+1−Z0,k)/(Z0,k+1+Z0,k)) and transmits (1+ρk). With more than one section the app switches from the bounce series to an exact frequency-domain solve — an ABCD chain per frequency, inverse-transformed to time — so nothing is truncated; the only numerical artifact is the band-limiting of ideal sharp edges. The single-line bounce amplitudes and the Zin formula above apply to one section; for cascades Zin is computed by chaining the section transformations.

Normalization: harmonic mode uses source period = 1, λ = 1, v = 1 (so T = ℓ/λ periods and all sections share λ); pulse modes use v = 1 with the total transit equal to the summed section lengths. Amplitudes assume V₀ = 1 V; currents are in mA for Z₀ in Ω.