Available Gain
Understanding Available Gain
In two-port network theory, three gain definitions are commonly used: transducer gain (GT), available gain (GA), and operating gain (GP). Each fixes different variables and is suited to a different design approach. Available gain fixes the source impedance as the independent variable and assumes the output is always conjugate-matched. This makes GA a function of ΓS alone, which is precisely what a noise-optimized design needs: the designer picks ΓS to minimize noise figure (setting ΓS = Γopt), reads the resulting available gain, and then conjugate-matches the output to deliver that gain to the load.
Available gain circles are the graphical design tool that makes this practical. Plotted on the Smith chart in the ΓS plane, each circle represents all source impedances that yield a specific available gain value. When overlaid with the constant noise figure circles (also in the ΓS plane), the designer can visually identify the optimal compromise between noise figure and gain, choosing a source impedance that achieves an acceptable noise figure while maintaining sufficient gain.
Available Gain Formulas
GA = (|S21|2 × (1 - |ΓS|2)) / (|1 - S11ΓS|2 × (1 - |Γout|2))
Where output reflection coefficient:
Γout = S22 + (S12S21ΓS) / (1 - S11ΓS)
Maximum Available Gain (K > 1, unconditionally stable):
MAG = |S21/S12| × (K - √(K2 - 1))
Maximum Stable Gain (K < 1, potentially unstable):
MSG = |S21/S12|
Gain Definitions Comparison
| Gain Type | Symbol | Fixed Variable | Design Use | When to Use |
|---|---|---|---|---|
| Transducer Gain | GT | Both ΓS and ΓL | Complete design verification | Final check after both matches designed |
| Available Gain | GA | ΓS (output conjugate-matched) | Noise-optimized LNA design | When source match is set for noise, output matched for gain |
| Operating Gain | GP | ΓL (input conjugate-matched) | Power amplifier design | When load match is set for power, input matched for gain |
| Unilateral Gain | GTU | S12 = 0 assumed | Simplified first-pass design | When |S12| is negligibly small |
Frequently Asked Questions
What is available gain and how is it different from transducer gain?
Available gain (G_A) is the ratio of power available from the two-port output to power available from the source, assuming the output is conjugate-matched. It depends only on the source impedance and device S-parameters. Transducer gain (G_T) depends on both source and load impedances and represents the power actually delivered to the load versus the power available from the source. When the output is conjugate-matched, G_A equals G_T. The key difference is that G_A is independent of load impedance, making it the natural parameter for LNA design where the source impedance is fixed at the optimum noise match and the output matching is designed afterward.
How are available gain circles used in LNA design?
Available gain circles are constant-gain contours on the Smith chart in the source reflection coefficient plane. Each circle shows all source impedances producing a specific available gain. The designer overlays noise figure circles and available gain circles on the same chart, then chooses the source impedance where the desired noise figure circle intersects an acceptable gain circle. This gives the optimal source match that achieves the target noise figure with known gain. The output matching network is then designed to conjugate-match the resulting output impedance, ensuring all available gain reaches the load.
What is the maximum available gain of a transistor?
Maximum Available Gain (MAG) is the highest transducer gain when both ports are simultaneously conjugate-matched, defined only when the device is unconditionally stable (K greater than 1). MAG equals |S21/S12| times (K minus the square root of K squared minus 1). When potentially unstable (K less than 1), the Maximum Stable Gain (MSG) equals |S21/S12| and represents the gain at the stability boundary. For a typical GaAs pHEMT at 10 GHz, MAG might be 15 dB with 0.5 dB minimum noise figure, while the available gain at the optimum noise match might be 13 dB, reflecting a 2 dB gain tradeoff for minimum noise.