Plotting Constant Noise Figure Circles for an LNA
Each noise circle has center Γ_opt/(1+N) and radius sqrt(N² + N(1−|Γ_opt|²))/(1+N), where N sets how far F rises above Fmin. With |Γ_opt| = 0.5 and N = 0.5, the circle is centered at 0.333 along Γ_opt and has radius 0.527.
Engineer’s note: I won't plot noise circles until I know the noise parameters came from the same bias and frequency as the S-parameters behind the gain circles. Mix two bias points and you get a chart that looks perfectly reasonable and means nothing.
For practitioners
An LNA reaches its minimum noise figure for only one source reflection coefficient, Γ_opt. In a real design you rarely land exactly on it. Gain, stability and input match all pull the source termination somewhere else. Constant noise figure circles show the cost of that move. Each circle on the source (Γ_S) Smith chart marks every source termination that gives the same noise factor. This page walks through the math step by step and works one numeric example.
The noise factor equation behind the circles
A linear two-port is described for noise by four numbers: Fmin, Rn, and the complex Γ_opt (magnitude and angle). The noise factor for any source reflection coefficient Γ_S is:
The second term is never negative. It is zero only when Γ_S = Γ_opt, so F = Fmin only at that one point. Rn/Z0 sets how fast F rises as Γ_S moves away from Γ_opt. A device with a large Rn punishes any mismatch from Γ_opt. A device with a small Rn tolerates it.
Two unit rules apply before you calculate anything:
- F is the linear noise factor, not the noise figure in dB. Convert with F = 10^(NF/10). For example, 3 dB gives F ≈ 1.995.
- Rn is divided by Z0, usually 50 Ω. If your data already gives the normalized value rn = Rn/Z0, do not divide by Z0 a second time.
From the equation to a circle
Fix a target noise factor F_i and gather the constant terms into one number, the noise circle parameter:
Put this into the noise factor equation and rearrange. You get a condition on Γ_S alone:
Expanding and completing the square shows that the points meeting this condition form a circle in the Γ_S plane. Its center and radius are:
Three properties follow directly from these formulas:
- The center lies on the Γ_opt line. C_Fi is Γ_opt scaled by the real number 1/(1+N_i), so it has the same angle as Γ_opt and a smaller magnitude.
- N_i = 0 gives a point. When F_i = Fmin, the radius is 0 and the circle is the point Γ_opt.
- Larger N_i gives a bigger circle closer to the origin. As N_i grows, the center moves toward the chart center and the radius approaches 1. A loose noise spec therefore covers most of the chart.
The circles are nested, and every one of them contains Γ_opt. They are not concentric.
Worked example: |Γ_opt| = 0.5 at 0°
Take Γ_opt = 0.5 at an angle of 0°, so it sits on the real axis. Then |1+Γ_opt|² = 1.5² = 2.25 and 1 − |Γ_opt|² = 0.75.
Circle A, N = 0.5:
- Center: 0.5/1.5 = 0.333 on the real axis
- Radius: sqrt(0.25 + 0.375)/1.5 = sqrt(0.625)/1.5 = 0.527
Circle B, N = 2:
- Center: 0.5/3 = 0.167 on the real axis
- Radius: sqrt(4 + 1.5)/3 = sqrt(5.5)/3 = 0.782
To find the noise factor each circle stands for, solve the N_i definition for F_i:
F_i − Fmin = N_i · (4 Rn/Z0)/|1+Γ_opt|² = N_i · (4 Rn/Z0)/2.25
Circle A gives F − Fmin ≈ 0.889 · Rn/Z0. Circle B gives F − Fmin ≈ 3.556 · Rn/Z0. Because N is linear in F − Fmin, circle B allows four times the noise-factor rise of circle A. Insert your device's Fmin and Rn to get the actual values. Convert F back to dB with NF = 10 log10(F).
Check one point. Circle A crosses the real axis on the right at 0.333 + 0.527 = 0.860. Put Γ_S = 0.860 into the condition: |0.860 − 0.5|² / (1 − 0.860²) = 0.1296/0.2604 ≈ 0.50. The left crossing is at 0.333 − 0.527 = −0.194, which gives 0.481/0.962 ≈ 0.50. Both points agree with N = 0.5, so the circle is correct.
Both circles stay inside the unit circle. Circle A spans −0.194 to 0.860 on the real axis, and circle B spans −0.615 to 0.949. Γ_opt = 0.5 lies inside both.
Drawing them in practice
The workflow is four steps:
- Read Fmin, Rn and Γ_opt for the frequency and bias point you care about.
- Pick a target F_i in linear units and compute N_i.
- Compute C_Fi and r_Fi.
- Draw the circle on the Γ_S Smith chart. The chart's outer rim has radius 1, so r_Fi is in the same units as |Γ|.
Repeat for a family of targets, for example Fmin plus a few small steps. Overlay the available-gain circles and the input stability circle on the same chart. The source termination you choose should sit inside the noise circle you can accept, at the best gain you can get there, and in the stable region. If you need help placing Γ_S points, see Smith Chart Basics: How to Read Impedance and Reflection.
What the circles tell you about the device
The spacing between circles describes the device. If Rn/Z0 is small, a given F_i produces a large N_i and a large circle. You can then move Γ_S toward the gain match with little noise penalty. If Rn/Z0 is large, the circles are tight around Γ_opt, and only a precise match gets you close to Fmin. The magnitude of Γ_opt matters too. A large |Γ_opt| puts the optimum near the chart edge, where matching networks are harder to build and more sensitive to component tolerance.
In plain words · for students and beginners
Noise figure circles in plain words
An everyday analogy
Think of a hill with a single lowest point at the bottom of a bowl. That lowest point is Γ_opt, where noise is smallest (Fmin). A noise circle is like a contour line on a map: every spot on the line is at the same height, meaning the same noise. A higher contour line makes a bigger loop. The analogy breaks down in one way. On this chart, the loops are not centered on the lowest point. Each bigger loop shifts toward the middle of the chart.
The same example, in simple steps
The Smith chart here is a map of every possible "source," meaning what the amplifier sees looking back toward the antenna. Γ (gamma) is a point on that map.
- Read the device data. Fmin is the lowest noise possible. Rn is how quickly noise rises when you miss the best point. Γ_opt is the best point. In our example, Γ_opt = 0.5, on the horizontal axis.
- Pick how much extra noise you accept and turn it into N. N is a single number for "how far above the minimum." A bigger N means more noise is allowed. We use N = 0.5 and N = 2.
- Compute the center and radius. For N = 0.5, the center is 0.5 ÷ 1.5 = 0.333 and the radius is 0.527. For N = 2, the center is 0.5 ÷ 3 = 0.167 and the radius is 0.782.
- Draw the circle. Any source on the N = 0.5 circle gives the same noise. Sources inside it are quieter, and sources outside it are noisier.
If N = 0, you accept no extra noise at all. The radius becomes 0, and the "circle" is just the single point Γ_opt.
Mistakes beginners make
- Using dB in the formula. The formula needs the plain ratio F, not dB. Convert first: 3 dB is F ≈ 1.995.
- Putting the center on Γ_opt. The center is Γ_opt ÷ (1 + N), which is always closer to the middle of the chart.
- Dividing Rn by 50 Ω twice. If the data already gives Rn as a normalized number (already divided by Z0, the reference impedance), use it as is.
- Ignoring gain and stability. The quietest point may give poor gain, or it may make the amplifier oscillate. Draw those circles on the same chart too.
Remember
- F = Fmin only at Γ_S = Γ_opt. Every other source adds noise.
- Center = Γ_opt/(1+N), radius = sqrt(N² + N(1−|Γ_opt|²))/(1+N).
- For |Γ_opt| = 0.5 and N = 0.5, the center is 0.333 and the radius is 0.527.
Related pages on RF View
Frequently asked questions
How do I calculate the center and radius of a noise figure circle?
First compute N_i = (F_i − Fmin)/(4 Rn/Z0) · |1+Γ_opt|², using linear noise factors. The center is Γ_opt/(1+N_i) and the radius is sqrt(N_i² + N_i(1−|Γ_opt|²))/(1+N_i).
Why are noise circles not concentric?
Every center lies on the line from the chart origin to Γ_opt, at Γ_opt/(1+N_i). As N_i grows, the center moves toward the origin and the radius grows, so the circles nest around Γ_opt without sharing a center.
Can I use noise figure in dB directly in the noise circle formula?
No. The formula uses the linear noise factor. Convert with F = 10^(NF/10) first, so 3 dB becomes F ≈ 1.995.
What does a noise circle of zero radius mean?
When F_i equals Fmin, N_i = 0 and the radius is zero. The circle shrinks to the single point Γ_opt, the only source reflection coefficient that gives the minimum noise figure.