Amplifier Design Circles
Small-signal amplifier design is done graphically on the Smith chart. From the transistor's S-parameters this tool draws, in the source (ΓS) plane: the input stability circle (which source impedances risk oscillation), constant available-gain circles (loci of equal gain), and constant noise-figure circles (loci of equal NF). Picking ΓS then trades gain against noise while staying in the stable region — the classic LNA design compromise. Rollett's K, the μ-factor, and MAG/MSG are reported too.
Equations & Parameters ▸
\(\Delta=S_{11}S_{22}-S_{12}S_{21},\quad K=\dfrac{1-|S_{11}|^2-|S_{22}|^2+|\Delta|^2}{2|S_{12}S_{21}|}\)
input stability: \(C_S=\dfrac{(S_{11}-\Delta S_{22}^*)^*}{|S_{11}|^2-|\Delta|^2},\ R_S=\Big|\dfrac{S_{12}S_{21}}{|S_{11}|^2-|\Delta|^2}\Big|\)
avail. gain \(g_a=G_a/|S_{21}|^2\): \(C_a=\dfrac{g_a(S_{11}-\Delta S_{22}^*)^*}{1+g_a(|S_{11}|^2-|\Delta|^2)}\)
NF \(N=\dfrac{(F-F_{min})|1+\Gamma_{opt}|^2}{4r_n}\): \(C_F=\dfrac{\Gamma_{opt}}{1+N},\ R_F=\dfrac{\sqrt{N(N+1-|\Gamma_{opt}|^2)}}{1+N}\)
input stability: \(C_S=\dfrac{(S_{11}-\Delta S_{22}^*)^*}{|S_{11}|^2-|\Delta|^2},\ R_S=\Big|\dfrac{S_{12}S_{21}}{|S_{11}|^2-|\Delta|^2}\Big|\)
avail. gain \(g_a=G_a/|S_{21}|^2\): \(C_a=\dfrac{g_a(S_{11}-\Delta S_{22}^*)^*}{1+g_a(|S_{11}|^2-|\Delta|^2)}\)
NF \(N=\dfrac{(F-F_{min})|1+\Gamma_{opt}|^2}{4r_n}\): \(C_F=\dfrac{\Gamma_{opt}}{1+N},\ R_F=\dfrac{\sqrt{N(N+1-|\Gamma_{opt}|^2)}}{1+N}\)
| Sij | Transistor S-parameters as magnitude ∠ angle (degrees). |
| Ga | Target available gain (dB) for the gain circle. |
| Fmin, Γopt, rn | Minimum noise figure (dB), optimum source reflection, normalised noise resistance. Optional — for the NF circle. |
| NF | Target noise figure (dB) for the NF circle. |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., Ch. 12 (amplifier design, stability & gain circles).
S-parameters (mag ∠ deg)
∠
Input ∠
Forward ∠
Reverse ∠
OutputGain & noise targets
dB
Gain circledB
NF circle ∠
Opt. sourceResults
Stability
Rollett K—
|Δ|—
μ-factor—
Verdict—
MAG / MSG—
Circles (ΓS plane: centre ∠, radius)
Input stability—
Output stability (ΓL)—
Available gain—
Noise figure—
Source-plane Smith chart