Pi-Network Matching
A Pi network matches two resistances with a chosen loaded Q — a freedom the L-network doesn't give you. It behaves as two back-to-back L-sections meeting at a virtual resistance Rv smaller than both ports. The low-pass form (two shunt capacitors, one series inductor) also suppresses harmonics; the high-pass form (two shunt inductors, one series capacitor) blocks DC. For a combined Pi/T tool see Pi/T Network Matcher.
Equations & Parameters ▸
\(R_v = \dfrac{R_1}{Q^2+1}, \qquad Q_2 = \sqrt{\dfrac{R_2}{R_v}-1}\)
Low-pass: \(C_1=\dfrac{Q}{\omega R_1},\ \ L=\dfrac{R_v(Q+Q_2)}{\omega},\ \ C_2=\dfrac{Q_2}{\omega R_2}\)
High-pass: \(L_1=\dfrac{R_1}{\omega Q},\ \ C=\dfrac{1}{\omega R_v(Q+Q_2)},\ \ L_2=\dfrac{R_2}{\omega Q_2}\)
Low-pass: \(C_1=\dfrac{Q}{\omega R_1},\ \ L=\dfrac{R_v(Q+Q_2)}{\omega},\ \ C_2=\dfrac{Q_2}{\omega R_2}\)
High-pass: \(L_1=\dfrac{R_1}{\omega Q},\ \ C=\dfrac{1}{\omega R_v(Q+Q_2)},\ \ L_2=\dfrac{R_2}{\omega Q_2}\)
| f | Matching frequency (MHz). |
| R₁, R₂ | Source and load resistances (Ω). Q is referenced to the R₁ (source) port. |
| Q | Loaded Q. When R₁ > R₂ it must exceed √(R₁/R₂ − 1); higher Q → narrower bandwidth, stronger harmonic rejection. |
| Rv | Virtual resistance at the junction of the two L-sections. |
References: F. E. Terman, Radio Engineers' Handbook, McGraw-Hill, 1943. · C. Bowick, RF Circuit Design, 2nd ed., Newnes, 2008.
Inputs
MHz
Match frequencyΩ
Q referenced hereΩ
Load resistance> √(R1/R2 − 1)
LP or HP
Results
Elements
Shunt @ R₁—
Series—
Shunt @ R₂—
Performance
Loaded Q—
Bandwidth (−3 dB)—
Virtual Rv—
Diagram