How does impedance mismatch cause amplitude and phase ripple in a signal chain?

Impedance mismatch at any interface in an RF signal chain creates reflections that interfere with the forward-traveling signal, causing periodic amplitude and phase variations across frequency (ripple): (1) Mechanism: when two components with different impedances are connected, a portion of the signal reflects at the interface. The reflected signal travels back to the previous interface, where it partially reflects again. This creates a multiply-reflected wave that adds to the forward signal. The amplitude of the combined signal varies sinusoidally with frequency because the phase of the reflected signal changes with frequency (as the electrical length of the path changes). Maximum signal: occurs when the reflected wave adds constructively with the forward wave. Minimum signal: occurs when the reflected wave adds destructively. (2) Ripple amplitude: for two mismatched interfaces separated by a transmission line of length L: peak-to-peak ripple in dB: Ripple = 20 × log10((1 + Gamma1 × Gamma2) / (1 - Gamma1 × Gamma2)). Where Gamma1 and Gamma2 are the reflection coefficients at each interface. For VSWR = 1.5 at both interfaces (Gamma = 0.2): ripple = 20 × log10(1.04/0.96) = 0.35 dB. For VSWR = 2.0 at both interfaces (Gamma = 0.33): ripple = 20 × log10(1.11/0.89) = 1.9 dB. (3) Ripple period: the ripple repeats every frequency interval of: delta_f = c / (2 × L × sqrt(epsilon_eff)), where L is the physical distance between the two interfaces. For L = 100 mm on a microstrip (epsilon_eff = 2.8): delta_f = 3e8 / (2 × 0.1 × 1.67) = 898 MHz. A ripple period of ~900 MHz means the gain/loss oscillates approximately once per GHz. (4) Phase ripple: the phase of the transmitted signal also ripples with the same period as the amplitude: the group delay varies across the band. The peak-to-peak group delay variation: delta_tau ≈ 2 × L × sqrt(epsilon_eff) / c × 2 × Gamma1 × Gamma2 / (1 - (Gamma1 × Gamma2)²). Phase ripple is critical for digitally modulated signals (it causes EVM degradation).