Maximum Power Transfer Theorem
For a source V_s with source impedance Z_s = R_s + jX_s: Maximum power delivered to load Z_L when: Z_L = Z_s* = R_s − jX_s P_max = |V_s|² / (4·R_s) [maximum available power] η_max = 100% when Z_L = Z_s* η = (1 − |Γ|²) × 100% for any Z_L mismatch
Three Types of RF Impedance Match
| Match Type | Objective | When Used |
|---|---|---|
| Conjugate match | Z_L = Z_s* (max power) | PA output, receiver LNA output |
| Real match (50 Ω) | Z = Z₀ = 50 Ω (system standard) | Cascade integration, test ports |
| Noise match | Z_s = Z_noise_opt (min NF) | LNA input for minimum noise figure |
Conjugate Match ≠ 50 Ω Match
Confusion often arises: "matching" can mean different things. A 50 Ω system uses "real matching" at every interface for consistent cascade behavior. But for maximum power extraction from a transistor (e.g., a GaN PA output), the "conjugate match" transforms the transistor's output impedance (typically 3–15 Ω) to 50 Ω — this IS also called "matching" even though the transformation is significant.
PA output impedance: Z_out = 5 + j3 Ω Conjugate match load: Z_L = 5 − j3 Ω (still very different from 50 Ω!) Matching network transforms: 50 Ω (cable) → 5 − j3 Ω (PA load)
Bode-Fano Bandwidth Limit
You cannot simultaneously achieve arbitrarily low reflection AND arbitrarily wide bandwidth. For a parallel RC load (capacitive antenna, device input):
∫₀^∞ ln(1/|Γ(ω)|) dω ≤ π / (R·C)
This means: high Q (narrow BW) → very low |Γ| possible
wideband matching → higher minimum reflection
Practical consequence: matching a 5 Ω PA output over 500 MHz BW
is fundamentally harder than matching over 50 MHz BW.
RF View Auto Matching: RF View synthesizes conjugate matching networks from measured S22 (output impedance). The matched S11 or S22 response shows how closely the Bode-Fano limit allows achieving perfect match over the specified bandwidth.