Transducer, Available and Operating Power Gain Compared
G_T = P_L/P_avs, G_P = P_L/P_in and G_A = P_avn/P_avs. With passive terminations on a stable two-port, G_T never exceeds G_P or G_A, and all three become equal under simultaneous conjugate match.
Engineer’s note: Your 50 Ω S21 is G_T, and that's the number that belongs in the cascade budget. G_P and G_A are never lower with passive terminations on a stable device, which is exactly why they look so nice on a slide.
For practitioners
Four powers in one circuit
A two-port sits between a source and a load. You can count power at four places. Each gain definition takes a ratio of two of them.
- P_avs is the power available from the source. The source delivers this much only into a conjugate-matched load.
- P_in is the power that actually enters the input port of the two-port.
- P_avn is the power available at the output of the network. A load equal to Γ_out* would absorb this much.
- P_L is the power actually delivered to the load.
The source has reflection coefficient Γ_S and the load has Γ_L. If you look into the input port with the load connected, you see Γ_in. If you look back into the output port with the source connected, you see Γ_out.
The figure brackets the three ratios:
- Transducer gain, G_T = P_L / P_avs, runs from the source to the load.
- Operating power gain, G_P = P_L / P_in, runs from the input plane to the load.
- Available power gain, G_A = P_avn / P_avs, runs from the source to the output plane.
The definitions show what each gain depends on. P_in is already "past" the source. So G_P does not depend on the source impedance. P_avn is the power the output could give, whatever load is connected. So G_A does not depend on the load. G_T has a real source on one end and a real load on the other, so it depends on both. It is the gain you see in a system with a power-calibrated source and a terminated output.
The S-parameter expressions
The input and output reflection coefficients come first:
Transducer gain:
Operating power gain:
Available power gain:
Check what each formula contains. G_P has no Γ_S anywhere, because Γ_in depends only on the S-parameters and Γ_L. G_A has no Γ_L, because Γ_out depends only on the S-parameters and Γ_S. That matches the definitions. When Γ_S = Γ_L = 0, as in a 50 Ω measurement on a 50 Ω VNA, G_T reduces to |S21|². This is why S21 in dB is the transducer gain of the device in a matched 50 Ω system. For more on the parameters themselves, see S11, S21, S12 and S22 Explained for a Two-Port.
Why G_T is never the largest
Take passive terminations, |Γ_S| ≤ 1 and |Γ_L| ≤ 1, on a two-port with |Γ_in| < 1 and |Γ_out| < 1. An unconditionally stable device meets this condition (see Amplifier Stability: How to Read the Rollett K Factor). Divide the formulas:
Each ratio is a mismatch factor. The first is the fraction of P_avs that enters the input. The second is the fraction of P_avn that reaches the load. The first ratio equals 1 only when Γ_S = Γ_in*. The second equals 1 only when Γ_L = Γ_out*. So:
- G_T = G_P when the input is conjugately matched, Γ_S = Γ_in*.
- G_T = G_A when the output is conjugately matched, Γ_L = Γ_out*.
- G_T = G_P = G_A when both ports are conjugately matched at the same time.
In decibels, G_T is G_P minus the input mismatch loss, or G_A minus the output mismatch loss. The same mismatch-loss idea appears in VSWR Explained: Reflection Coefficient and Mismatch Loss.
Worked example
Use a unilateral two-port (S12 = 0) with real S-parameters: S11 = 0.5, S21 = 4, S22 = 0.4. With S12 = 0, Γ_in = S11 and Γ_out = S22 for any terminations. Both have magnitudes below 1, so the conditions above hold. |S21|² = 16.
Case 1: 50 Ω source and load, Γ_S = Γ_L = 0.
- G_T = |S21|² = 16, or 12.04 dB.
- G_P = 16 / (1 − 0.25) = 21.33, or 13.29 dB.
- G_A = 16 / (1 − 0.16) = 19.05, or 12.80 dB.
G_T/G_P = 0.75, an input mismatch loss of 1.25 dB. G_T/G_A = 0.84, an output mismatch loss of 0.76 dB. G_T is the smallest, as expected.
Case 2: input matched only, Γ_S = 0.5 = Γ_in, Γ_L = 0.*
G_T = 0.75 × 16 / |1 − 0.25|² = 12 / 0.5625 = 21.33. This equals G_P, which has not changed because G_P ignores the source.
Case 3: both ports matched, Γ_S = 0.5, Γ_L = 0.4.
G_T = 16 / (0.75 × 0.84) = 25.40, or 14.05 dB. G_P = 16 × 0.84 / (0.75 × 0.7056) = 25.40. G_A = 16 × 0.75 / (0.5625 × 0.84) = 25.40. All three are equal. Matching recovered 2.01 dB over the 50 Ω case. That is the sum of the two mismatch losses.
Which gain to use
- G_T is the system number. Use it for cascade budgets and to compare with measured S21 in a known environment.
- G_P fixes the load and ignores the source. It is the natural choice when you design the output network first, for example around a chosen PA load. Constant-G_P circles are drawn in the Γ_L plane.
- G_A fixes the source and ignores the load. It fits low-noise design, where Γ_S is set for noise figure. Constant-G_A circles are drawn in the Γ_S plane.
In plain words · for students and beginners
Three power gains in plain words
An everyday analogy
Think of water flowing from a tank through a pump into a bucket. The tank could deliver a certain maximum flow if the pipe fitted it perfectly. That is the available power from the source. Some water spills at the pump inlet, so less goes in. That is power in. The pump could push a certain maximum out. That is the power available at the output. Some spills at the bucket, so the bucket gets what is delivered.
Each gain is just "what you count at the end" divided by "what you count at the start":
- Transducer gain: delivered to the load ÷ available from the source. Both spills count.
- Operating gain: delivered to the load ÷ what entered the input. The inlet spill is ignored.
- Available gain: available at the output ÷ available from the source. The bucket spill is ignored.
Where the analogy breaks down: water does not reflect back up the pipe, but RF power does. A mismatch at one end can also change what the other end sees. A reflection coefficient (Γ, the fraction of a wave that bounces back) describes this.
The same example, step by step
The amplifier has S21 = 4. S21 is how much the voltage wave grows going through, so the power grows 4 × 4 = 16 times. The input reflects 0.5 and the output reflects 0.4.
- In a plain 50 Ω setup, the transducer gain is 16 (12.04 dB).
- At the input, 0.5 × 0.5 = 0.25 of the power bounces back. So 0.75 gets in. Operating gain ignores this loss: 16 ÷ 0.75 = 21.33.
- At the output, 0.4 × 0.4 = 0.16 bounces back. Available gain ignores this loss: 16 ÷ 0.84 = 19.05.
- Add matching circuits so nothing bounces at either end. All three gains become 25.40 (14.05 dB).
Mistakes beginners make
- Reading S21 from a 50 Ω measurement and calling it "the gain." It is the transducer gain in 50 Ω only.
- Expecting transducer gain to be the biggest. With normal (passive) terminations it is never larger than the other two.
- Matching the input, then the output, and assuming the input is still matched. If the amplifier sends some signal backwards (S12 not zero), each match changes the other.
Remember
- Top and bottom of the ratio decide the gain: P_L/P_avs, P_L/P_in, P_avn/P_avs.
- Operating gain ignores the source, and available gain ignores the load.
- Conjugate match at both ports makes all three equal.
Related pages on RF View
Frequently asked questions
What is the difference between transducer gain and operating power gain?
Transducer gain divides the power delivered to the load by the power available from the source, so it depends on both source and load. Operating power gain divides the load power by the power that actually enters the input port, so it does not depend on the source impedance.
Does available power gain depend on the load impedance?
No. Available power gain is the power available at the network output divided by the power available from the source, so it depends on the source and the two-port but not on the load.
When are transducer, available and operating power gain equal?
G_T equals G_P when Γ_S = Γ_in*, and G_T equals G_A when Γ_L = Γ_out*. All three are equal when the input and output are conjugately matched at the same time.
Can transducer gain be larger than available gain?
Not with passive terminations on a two-port where |Γ_in| < 1 and |Γ_out| < 1. Under those conditions the ratios G_T/G_P and G_T/G_A are mismatch factors that never exceed 1.