How do you solve the simultaneous conjugate match equations?

For an unconditionally stable device (K > 1, |Delta| < 1), the simultaneous conjugate match source and load reflection coefficients are: Gamma_MS = (B1 - sqrt(B1^2 - 4|C1|^2)) / (2*C1) and Gamma_ML = (B2 - sqrt(B2^2 - 4|C2|^2)) / (2*C2), where B1 = 1 + |S11|^2 - |S22|^2 - |Delta|^2, B2 = 1 + |S22|^2 - |S11|^2 - |Delta|^2, C1 = S11 - Delta*conj(S22), C2 = S22 - Delta*conj(S11), and Delta = S11*S22 - S12*S21. The minus sign in the formula gives |Gamma| 1. The resulting maximum available gain is MAG = |S21/S12| * (K - sqrt(K^2 - 1)). These equations yield unique solutions that provide the maximum possible gain with conjugate match at both ports simultaneously.

What happens when the device is conditionally stable (K < 1)?

When K 1 or |Gamma_ML| > 1, meaning passive matching networks cannot achieve simultaneous conjugate match. The maximum stable gain (MSG) replaces MAG: MSG = |S21/S12|, which is the gain at the boundary of stability (K = 1). Design options: (1) Add stabilizing elements (series/shunt resistors, feedback networks) to increase K > 1 at all frequencies, then apply bilateral matching for MAG. Cost: reduced gain and noise figure. (2) Use constant-gain circles on the Smith chart to select source and load impedances within the stable region that provide acceptable gain without risking oscillation. (3) Design for a specific gain less than MSG using bilateral gain circles, ensuring the chosen impedances remain within the stable region defined by the stability circles.

How does bilateral matching differ from unilateral matching?

Unilateral matching assumes S12 = 0, so Gamma_in = S11 and Gamma_out = S22 are independent of load and source. The matching is simply Gamma_S = conj(S11) and Gamma_L = conj(S22), designed independently. The unilateral transducer gain is GTU = |S21|^2 / ((1-|S11|^2)(1-|S22|^2)) when both ports are conjugate matched. Bilateral matching accounts for non-zero S12, which couples input and output. The simultaneous conjugate match requires solving coupled equations where Gamma_MS and Gamma_ML depend on each other through Delta and the B/C parameters. The bilateral gain is MAG = |S21/S12| * (K - sqrt(K^2 - 1)), which can differ significantly from GTU. The gain error from using unilateral approximation when the device is actually bilateral is bounded by: -20*log10(1+U) <= G_error <= 20*log10(1+U), where U is the unilateral figure of merit. For U = 0.1: error <= 0.83 dB. For U = 0.5: error <= 3.5 dB.

Amplifier / Matching

Bilateral Matching

Q: What happens when the device is conditionally stable (K < 1)?

When K 1 or |Gamma_ML| > 1, meaning passive matching networks cannot achieve simultaneous conjugate match. The maximum stable gain (MSG) replaces MAG: MSG = |S21/S12|, which is the gain at the boundary of stability (K = 1). Design options: (1) Add stabilizing elements (series/shunt resistors, feedback networks) to increase K > 1 at all frequencies, then apply bilateral matching for MAG. Cost: reduced gain and noise figure. (2) Use constant-gain circles on the Smith chart to select source and load impedances within the stable region that provide acceptable gain without risking oscillation. (3) Design for a specific gain less than MSG using bilateral gain circles, ensuring the chosen impedances remain within the stable region defined by the stability circles.

/bye-LAT-er-ul MATCH-ing/

Simultaneous conjugate matching for non-unilateral two-ports (S₁₂ ≠ 0). Input/output matching coupled through S₁₂ feedback: Γ_in depends on Γ_L and vice versa. Achieves maximum available gain MAG = |S₂₁/S₁₂| × (K − √(K²−1)) when K > 1. Conditionally stable devices (K < 1) limited to MSG = |S₂₁/S₁₂|.

Gain: MAG (K>1) / MSG (K<1)

Match: Simultaneous conjugate

Coupling: Via S₁₂ feedback

Understanding Bilateral Matching

When S₁₂ is not negligible, the input impedance of a two-port depends on the load, and the output impedance depends on the source. This coupling means input and output matching networks cannot be designed independently. The simultaneous conjugate match solves coupled equations to find Γ_MS and Γ_ML that achieve conjugate match at both ports simultaneously, yielding the maximum available gain (MAG).

If the device is only conditionally stable (K < 1), the conjugate match solutions are unrealizable (require active matching), and the designer must either stabilize the device first or design for a gain below MSG using constant-gain circles within the stable region.

Simultaneous Conjugate Match

Source Match:
Γ_MS = (B₁ − √(B₁² − 4|C₁|²)) / (2C₁)
B₁ = 1 + |S₁₁|² − |S₂₂|² − |Δ|²
C₁ = S₁₁ − Δ × S₂₂*

Load Match:
Γ_ML = (B₂ − √(B₂² − 4|C₂|²)) / (2C₂)

Maximum Available Gain:
MAG = |S₂₁/S₁₂| × (K − √(K² − 1))
Valid only when K > 1

Bilateral vs. Unilateral Matching

Aspect	Unilateral (S₁₂=0)	Bilateral (S₁₂≠0)
Input match	Γ_S = S₁₁*	Γ_MS = f(S₁₁,S₁₂,S₂₁,S₂₂)
Port coupling	Independent	Coupled through Δ
Max gain	G_TU = \|S₂₁\|²/((1−\|S₁₁\|²)(1−\|S₂₂\|²))	MAG = \|S₂₁/S₁₂\|(K−√(K²−1))
Design method	Two independent Smith charts	Simultaneous equations

Conditional Stability Options

Approach	K After	Gain Impact	NF Impact
Series resistor	>1	Reduced 1–3 dB	Increased
Shunt resistor	>1	Reduced 0.5–2 dB	Moderate
Negative feedback	>1	Reduced, flattened	Moderate
Gain circle design	<1 (accepted)	Below MSG	Best (no added R)

Common Questions

Frequently Asked Questions

Simultaneous conjugate match?

Γ_MS and Γ_ML solved from coupled B/C equations. Minus sign gives |Γ| < 1 when K > 1. MAG = |S₂₁/S₁₂|(K−√(K²−1)). Unique solutions for both source and load.

K < 1 handling?

Conjugate match unrealizable (|Γ| > 1). Options: stabilize with resistors (lose gain/NF), or design within stable region using constant-gain circles at gain < MSG = |S₂₁/S₁₂|.

Unilateral approximation error?

Error bounded by ±20 log(1+U) dB. U = 0.1: ≤0.83 dB. U = 0.5: ≤3.5 dB. Check U first; if >0.3, full bilateral matching required for accurate design.

Related Terms

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