Constant-Gain Circle
Understanding Constant-Gain Circles
In microwave amplifier design, achieving the absolute maximum theoretical gain (Maximum Available Gain, or MAG) from a transistor requires applying a simultaneous conjugate match to both the input and output ports. However, designing for maximum gain is almost never the actual goal. Maximum gain often results in a terrible Noise Figure, poor linearity, or forces the impedance dangerously close to the unstable regions of the Smith Chart. To intelligently trade off gain for these other parameters, engineers utilize Constant-Gain Circles.
A Constant-Gain Circle is a locus of points plotted on a Smith Chart. Every single complex impedance (resistance and reactance) that lies precisely on the circumference of that circle will force the amplifier to produce the exact same amount of gain (e.g., exactly 12 dB). By drawing multiple concentric circles (e.g., 15 dB, 14 dB, 13 dB) on the input and output Smith Charts, the designer can visually map the entire performance landscape of the transistor.
Visual Design Flow
The true power of Constant-Gain Circles is unlocked when they are overlaid with Constant-Noise Circles (which map the Noise Figure) and Stability Circles (which map the forbidden oscillation zones). By looking at all three circles simultaneously, an engineer can select a specific source impedance that sits on the 0.8 dB Noise Circle, lies safely away from the Stability Circle, and rests exactly on the 13 dB Gain Circle. This visual synthesis is the foundational technique of all classical RF Low Noise Amplifier (LNA) design.
Center: Ci = [ gi × Sii* ] / [ 1 + gi × (|Sii|2) ]
Radius: ri = √(1 - gi) × (1 - |Sii|2) / [ 1 + gi × (|Sii|2) ]
Where gi is the normalized gain parameter for the specific target gain. As the target gain approaches the Maximum Available Gain (MAG), the radius shrinks until the circle becomes a single dot (the conjugate match point).
Comparison
| Design Circle | What it Maps | Goal | Constraint |
|---|---|---|---|
| Constant-Gain | Operating Power Gain | Hit target spec (e.g., 15dB) | Shrinks as target gain increases |
| Constant-Noise | Noise Figure (LNA focus) | Minimize NF (e.g., < 1dB) | Requires impedance mismatch from optimum gain |
| Stability | Oscillation boundaries | Stay out of forbidden zone | Absolute hardest limit; dictates survival |
Frequently Asked Questions
What does the 'Unilateral Assumption' mean when plotting gain circles?
The Unilateral Assumption assumes that the transistor's reverse isolation (S12) is exactly zero—meaning no signal leaks backward from the output to the input. If S12=0, the input and output gain circles are completely independent. In reality, S12 > 0, which means changing the output match slightly shifts the input gain circles. Modern CAD software (like Keysight ADS) calculates Bilateral (true) gain circles continuously.
Why do Constant-Gain circles get smaller as the gain target increases?
There is only one exact complex impedance on the entire Smith Chart that yields the absolute Maximum Available Gain (MAG). That point is a single dot. As you accept lower and lower gain (e.g., dropping from 15 dB to 12 dB), there are vastly more mathematical combinations of resistance and reactance that will yield that lower gain, so the circle grows larger.
How do I physically build the impedance selected from the gain circle?
Once you drop a pin on the Smith Chart at the intersection of your desired Gain and Noise circles, you read the complex impedance (e.g., 25 + j15 ohms). You then design a passive matching network using series transmission lines, open stubs, or discrete inductors and capacitors to transform the 50-ohm antenna feed exactly to that 25 + j15 ohm point.