Unilateral Figure of Merit: When Can You Ignore S12?
The true gain lies between 1/(1+U)² and 1/(1−U)² times the unilateral gain G_TU. For U = 0.1 that is −0.83 dB to +0.92 dB, and for U = 0.05 it is about −0.42 dB to +0.45 dB.
Engineer’s note: I'd do the bilateral match the moment the U window eats most of the stage's gain tolerance. Use the worst U across the band, since 1.75 dB at U = 0.1 is fine for a loose gain block and too wide for a tight cascade budget.
For practitioners
Why the unilateral assumption is tempting
Designing an amplifier match with all four S-parameters is a coupled problem. The input reflection depends on the load through S12 and S21, and the output reflection depends on the source the same way. Change the output match and the input match moves.
Set S12 = 0 and the coupling goes away. The input match depends only on S11, and the output match depends only on S22. Each side can be designed on its own Smith chart, and the transducer gain splits into three separate factors:
This is the unilateral transducer gain. It is exact only when S12 = 0. No real transistor has zero reverse transmission, so the question is how large an error you accept when you use G_TU anyway. The unilateral figure of merit answers that question with one number.
Computing U from S-parameters
The unilateral figure of merit uses only the magnitudes of the four S-parameters:
The behaviour follows directly from the formula:
- U = 0 when S12 = 0. A truly unilateral device has no error.
- U grows linearly with |S12|. More reverse feedback means more error.
- U grows linearly with |S21|. A high-gain device turns the same S12 into a stronger feedback loop.
- U grows with |S11| and |S22|, and fast as either approaches 1, because the denominator terms (1−|S11|²) and (1−|S22|²) go to zero. A strongly reflective device bounces the fed-back signal back and forth.
All four quantities come straight from a measured or modelled S-parameter set at the frequency of interest. U changes with frequency, so compute it at each frequency point in your band, not only at the centre. For a refresher on what each term means physically, see S11, S21, S12 and S22 Explained for a Two-Port.
From U to a gain-error band
Terminate the device at the unilateral conjugate-match values Γ_S = S11* and Γ_L = S22*. G_TU then reaches its maximum:
With S12 ≠ 0, the true transducer gain G_T at those same terminations differs from G_TU. For U < 1 the ratio is bounded:
Either bound is reached when the feedback term has the worst-case phase. Which way the error goes depends on the phases of the S-parameters, and U uses magnitudes only. So U gives you a band, not a correction. In dB the band is:
- lower bound: −20·log10(1 + U)
- upper bound: −20·log10(1 − U)
The band is not symmetric. The upper bound grows faster than the lower bound, because 1 − U approaches zero as U approaches 1.
The plot shows both curves starting at 0 dB at U = 0 and spreading apart as U grows. The shaded region between them is the possible G_T/G_TU error. Two reference points are worth keeping in mind:
- U = 0.05: about −0.42 dB to +0.45 dB
- U = 0.1: 1/1.21 = 0.826 (about −0.83 dB) to 1/0.81 = 1.235 (about +0.92 dB)
Worked example
Take a device at one frequency with these magnitudes (an illustrative set chosen for round numbers):
- |S11| = 0.5
- |S22| = 0.5
- |S21| = 4.5 (20·log10(4.5) ≈ 13.06 dB)
- |S12| = 0.05 (≈ −26.0 dB)
Numerator: 0.5 × 0.05 × 4.5 × 0.5 = 0.05625. Denominator: (1 − 0.25)(1 − 0.25) = 0.75 × 0.75 = 0.5625. U = 0.05625 / 0.5625 = 0.1.
Maximum unilateral gain with Γ_S = S11* and Γ_L = S22*:
G_TU,max = (1/0.75) × 20.25 × (1/0.75) = 1.333 × 20.25 × 1.333 = 36.0, which is 10·log10(36) ≈ 15.56 dB.
With U = 0.1 the true gain at these terminations lies between 15.56 − 0.83 ≈ 14.73 dB and 15.56 + 0.92 ≈ 16.48 dB. That is a 1.75 dB wide window.
Now halve the reverse transmission to |S12| = 0.025 (≈ −32.0 dB) with the other terms unchanged. U falls to 0.05, and the window shrinks to about 15.14 dB to 16.01 dB, under 0.9 dB wide. Halving U roughly halves the band.
Deciding whether a unilateral design is good enough
Use U as a go/no-go check before choosing a design method:
- Compute U at every frequency point in the band.
- Take the worst (largest) U and convert it to the dB band.
- Compare the band with the gain tolerance you have for that stage.
If the band fits inside your tolerance, design the matches from S11 and S22 alone and accept the error. If it does not fit, use a bilateral design: simultaneous conjugate match or an operating-gain or available-gain approach that keeps S12 in the equations.
The bounds hold for the conjugate-match terminations Γ_S = S11* and Γ_L = S22*. If you deliberately mismatch for noise or bandwidth, the feedback interacts with different reflection coefficients, so treat the U band as a guide rather than a guarantee for those terminations.
In plain words · for students and beginners
Unilateral figure of merit in plain words
An everyday analogy
Think of a one-way street with a narrow side lane where a few cars drive the wrong way. The main traffic is the signal going forward through the amplifier (that is S21, the forward gain). The wrong-way cars are the small part of the signal that leaks backward from output to input (that is S12, the reverse transmission). If very few cars go the wrong way, you can plan the traffic as if the street were fully one-way. U tells you how many wrong-way cars there really are, and how much they can upset your plan.
The drawing shows this. The thick arrow is the main forward path. The thin arrow going back is the S12 feedback, which you may ignore when U is small. If S12 = 0, U = 0 and the plan is exact.
Where the analogy breaks down: in a real amplifier the backward signal does not only "slow things down". Depending on its phase (its timing compared with the forward signal), it can make the gain higher or lower. That is why U gives a range of possible error, not a single fixed error.
The same example, step by step
- Write down the sizes of the four S-parameters: |S11| = 0.5, |S22| = 0.5, |S21| = 4.5, |S12| = 0.05.
- Multiply all four: 0.5 × 0.05 × 4.5 × 0.5 = 0.05625.
- Work out (1 − 0.5²) twice and multiply: 0.75 × 0.75 = 0.5625.
- Divide: U = 0.05625 / 0.5625 = 0.1.
- Look up U = 0.1: the real gain can be about 0.83 dB lower or 0.92 dB higher than the simple estimate. (dB, or decibel, is a log scale for ratios: +3 dB is about double the power.)
- The simple estimate is 15.56 dB, so the real gain is somewhere from about 14.73 dB to 16.48 dB.
Mistakes beginners make
- Checking U at one frequency only. The S-parameters change with frequency, so U does too. Check the whole band.
- Thinking small U means a safe amplifier. U only measures gain error. Whether the amplifier oscillates (makes its own unwanted signal) is a separate check.
- Treating the error as a fixed correction. You do not know if the gain goes up or down, only how far it can go.
- Forgetting the error is uneven. The upward error is always a bit bigger than the downward one.
Remember
- U = |S11||S12||S21||S22| / ((1−|S11|²)(1−|S22|²)), and U = 0 when S12 = 0.
- The real gain lies between 1/(1+U)² and 1/(1−U)² times the unilateral estimate.
- U = 0.1 means about −0.83 dB to +0.92 dB; U = 0.05 means about −0.42 dB to +0.45 dB.
Related pages on RF View
Frequently asked questions
What is the unilateral figure of merit U?
U = |S11||S12||S21||S22| / ((1−|S11|²)(1−|S22|²)). It is a single number that shows how much the reverse transmission S12 can change the gain when you design as if S12 were zero.
How do I convert U into a gain error in dB?
For U < 1 and conjugate-match terminations Γ_S = S11* and Γ_L = S22*, the gain ratio G_T/G_TU lies between 1/(1+U)² and 1/(1−U)². In dB that is −20·log10(1+U) to −20·log10(1−U).
What value of U is small enough to ignore S12?
It depends on how much gain error your budget allows. U = 0.05 gives roughly −0.42 dB to +0.45 dB, and U = 0.1 gives −0.83 dB to +0.92 dB. If that band fits your tolerance, a unilateral design is good enough.
Does a small U mean the amplifier is stable?
No. U only bounds the gain error of the unilateral approximation. You still have to check stability separately, for example with the Rollett K factor.