Simultaneous Conjugate Match
For maximum gain from a bilateral transistor (S₁₂ ≠ 0), the source and load must be conjugately matched to the device's input and output at the same time — but each depends on the other through S₁₂. This tool solves the coupled equations for ΓS and ΓL, gives the resulting source/load impedances and the maximum transducer gain, and checks the solution by confirming Γin = ΓS*. It requires an unconditionally stable device (K > 1).
Equations & Parameters ▸
\(B_1=1+|S_{11}|^2-|S_{22}|^2-|\Delta|^2,\quad C_1=S_{11}-\Delta S_{22}^*\)
\(\Gamma_S=\dfrac{B_1\pm\sqrt{B_1^2-4|C_1|^2}}{2C_1}\quad(\text{and similarly }\Gamma_L\text{ with }B_2,C_2)\)
\(G_{T,max}=\dfrac{|S_{21}|}{|S_{12}|}\big(K-\sqrt{K^2-1}\big)\)
\(\Gamma_S=\dfrac{B_1\pm\sqrt{B_1^2-4|C_1|^2}}{2C_1}\quad(\text{and similarly }\Gamma_L\text{ with }B_2,C_2)\)
\(G_{T,max}=\dfrac{|S_{21}|}{|S_{12}|}\big(K-\sqrt{K^2-1}\big)\)
| Sij | Transistor S-parameters as magnitude ∠ angle (degrees). |
| Z0 | Reference impedance for the source/load impedances (Ω). |
| ΓS, ΓL | Source and load reflection coefficients for the simultaneous conjugate match. |
| GT,max | Maximum transducer gain (= MAG). Exists only when K > 1. |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., §12.3 (conjugate matching & maximum gain).
S-parameters (mag ∠ deg)
∠
Input ∠
Forward ∠
Reverse ∠
OutputΩ
ReferenceResults
Source side
ΓS—
Source impedance ZS—
Check Γin vs ΓS*—
Load side
ΓL—
Load impedance ZL—
Max transducer gain—
Stability
Rollett K—
|Δ|—
Diagram