Constant-Noise Circle
Mapping Noise Figure onto the Smith Chart
The noise figure of a microwave transistor is not a single number; it varies with the impedance the source presents to the device input. The complete behavior is captured by four real noise parameters: the minimum noise figure Fmin, the magnitude and phase of the optimum source reflection coefficient Γopt (the source termination that achieves Fmin), and the equivalent noise resistance Rn, which sets how rapidly the noise figure degrades as the source moves away from Γopt. Vendors publish these parameters versus frequency in the device noise-parameter or S2P file. Plotting every source Γs that yields a chosen noise figure produces a circle, and a family of these circles forms a contour map of noise figure across the entire input-matching plane.
Because Rn governs the spacing of the circles, it tells the designer how forgiving a part is. A device with Rn of 5 to 15 ohms (common for a GaAs pHEMT) yields widely spaced circles, so a few tenths of a dB above Fmin buys a large region of acceptable source impedances. A device with Rn above 40 ohms packs the circles tightly, meaning even a small source mismatch pushes noise figure up quickly and the input match must be held precisely. This is why Rn is as important as Fmin when selecting a transistor for a sensitive receiver front end.
In practice the constant-noise contours are rarely used alone. The source impedance that minimizes noise figure almost never coincides with the impedance that delivers a conjugate input match or maximum gain, so Γopt and the gain-optimum point sit at different places on the chart. The designer overlays the two circle families and chooses a compromise: a typical 12 GHz LNA might accept a noise figure 0.3 dB above Fmin to recover roughly 1.5 dB of associated gain and improve input return loss.
Noise Circle Geometry
F = Fmin + (4Rn / Z0) × |Γs − Γopt|2 / [(1 − |Γs|2) × |1 + Γopt|2]
Noise figure parameter (for a target F):
N = (F − Fmin) × |1 + Γopt|2 / (4Rn / Z0)
Circle center and radius:
CF = Γopt / (1 + N) RF = √(N(N + 1 − |Γopt|2)) / (1 + N)
Where F and Fmin are linear (not dB), Z0 ≈ 50 Ω. At F = Fmin, N = 0, CF = Γopt, and RF = 0, so the circle is the single point Γopt.
Worked Noise-Circle Example
| Target NF (dB) | F (linear) | N parameter | Center |CF| | Radius RF | Design note |
|---|---|---|---|---|---|
| 0.6 (Fmin) | 1.148 | 0.000 | 0.50 | 0.00 | Single point at Γopt |
| 0.8 | 1.202 | 0.057 | 0.47 | 0.20 | Tight, near Γopt |
| 1.0 | 1.259 | 0.116 | 0.45 | 0.28 | Practical LNA target |
| 1.5 | 1.413 | 0.276 | 0.39 | 0.42 | Room for gain match |
| 2.0 | 1.585 | 0.457 | 0.34 | 0.51 | Loose; wide source region |
Values assume Fmin = 0.6 dB, Γopt = 0.50∠65°, and Rn = 20 Ω (Z0 = 50 Ω), representative of a low-noise GaAs pHEMT near 12 GHz. As the target noise figure rises, N grows, the circle radius expands, and its center slides inward away from Γopt toward the chart origin.
Frequently Asked Questions
How do you compute the center and radius of a constant-noise circle?
Form N = (F − Fmin)|1 + Γopt|2 / (4Rn/Z0), with F and Fmin in linear units. The center is CF = Γopt/(1 + N), lying on the line from the chart origin toward Γopt, and the radius is RF = √(N(N + 1 − |Γopt|2))/(1 + N). When F = Fmin, N = 0 and the circle collapses to the point Γopt; larger targets expand the circle toward the chart edge.
Why do constant-noise and constant-gain circles rarely share a center?
Noise circles center toward Γopt, the source that minimizes noise figure, while available-gain circles center toward the source giving a conjugate input match. For most transistors these optima differ by 30 to 60° on the chart because the impedance that minimizes noise is not the one that maximizes gain. The designer picks a source on a compromise contour, often accepting 0.2 to 0.5 dB above Fmin to recover 1 to 2 dB of gain.
What four noise parameters do you need to draw the circles?
Fmin, the magnitude and angle of Γopt (two numbers), and the equivalent normalized noise resistance Rn. They come from the vendor S2P or noise file versus frequency. A low Rn (5 to 15 Ω) spaces the circles widely so the design tolerates source mismatch; a high Rn (above 40 Ω) packs them tightly so noise figure degrades quickly away from Γopt.