Bode–Fano Matching Limit
The Bode–Fano criterion sets a hard ceiling on how well a reactive load can ever be matched: no passive network, however complex, can beat it. For a parallel-RC or series-RL load it says the integrated match ∫ln(1/|Γ|)dω is bounded — so wider bandwidth forces higher reflection, and vice versa. Use it to see the best possible match over a band before you spend effort on a network that can't get there.
Equations & Parameters ▸
parallel RC: \(\displaystyle\int_0^\infty\ln\dfrac{1}{|\Gamma|}\,d\omega\le\dfrac{\pi}{RC}\); series RL: \(\displaystyle\int_0^\infty\ln\dfrac{1}{|\Gamma|}\,d\omega\le\dfrac{\pi R}{L}\)
rectangular band: \(\Delta\omega\,\ln\dfrac{1}{\Gamma_m}\le K\;\Rightarrow\; \Delta\omega_{max}=\dfrac{K}{\ln(1/\Gamma_m)},\ \ \Gamma_{m,min}=e^{-K/\Delta\omega}\)
rectangular band: \(\Delta\omega\,\ln\dfrac{1}{\Gamma_m}\le K\;\Rightarrow\; \Delta\omega_{max}=\dfrac{K}{\ln(1/\Gamma_m)},\ \ \Gamma_{m,min}=e^{-K/\Delta\omega}\)
| Load | Parallel RC or series RL — the two canonical single-reactance loads. |
| R, C / L | Load resistance and its reactive element. |
| f0 | Band center frequency, for the fractional bandwidth. |
| Γm, BW | A target ripple gives the max bandwidth; a target bandwidth gives the best-possible ripple. |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., §5.9; H. W. Bode (1945) & R. M. Fano, J. Franklin Inst. (1950).
Inputs
Canonical
Ω
Load RpF
capacitanceGHz
For %BW→ max BW
%
→ min ΓmResults
For target ripple → bandwidth
Max fractional BW—
Max BW (absolute)—
For target BW → best match
Min |Γ| (best)—
Best VSWR / return loss—
Load
Bode–Fano constant K—
Load Q at f0—
Diagram