What Conjugate Matching Really Is — The Chart Reference and the Target Are Different Things

Many people remember that the center of the Smith chart is 50 Ω. That is half right and half assumption. Strictly, the center is the chart's normalizing impedance Z0, which is a different concept from the target of the match.

Textbook examples nearly always match to 50 Ω in a 50 Ω system, so the two coincide and the distinction disappears. Trouble starts in practice when they stop coinciding.

Two different "50 Ω"s

The two things to keep apart are:

• Z0 (chart reference) — the normalizing impedance used to draw the grid. All it decides is where the constant-resistance circle through "1" lies
• Zt (target impedance) — where Zin has to end up. Get there and the match is done

Z0 is a display setting; Zt is the design goal. Change Z0 from 50 to 75 and nothing about the circuit changes — only the grid is redrawn. Change Zt, and the element values you need change.

Mismatched cases are not rare

Z0 ≠ Zt comes up all the time:

• A 50 Ω measurement system and a 75 Ω device — video equipment, some antenna systems
• Power amplifier output matching — the load at which a transistor delivers maximum power or efficiency is not 50 Ω but some other complex impedance found by load-pull
• Interstage matching — matching one stage's output to the next stage's input. There is no reason either should be 50 Ω

The second is the important one. In PA design, "matching to 50 Ω" is actually the exception: you match to the optimum load the device wants. Yet measurement and display stay in a 50 Ω system, so you end up with a chart still normalized to 50 Ω, and a target sitting somewhere else.

An optimum load from load-pull is a value at one particular frequency, so if you are judging bandwidth, how you model the load also needs care.

When the target is complex

If Zt is real, things are simple. If it is complex, the definition of the reflection coefficient needs care. Writing it naively as

Γ = (Z − Zt) / (Z + Zt)

goes wrong when Zt is complex. Even a passive circuit can give |Γ| > 1, which breaks its meaning as a reflection coefficient.

The correct definition uses power waves:

Γt = (Z − Zt) / (Z + conj(Zt))

The point is that only the denominator takes the conjugate. Now Z = Zt gives exactly zero, and when Zt is real you are back to the familiar formula.

How this relates to conjugate matching

The condition for drawing maximum power from a source of impedance Zs is to make the load the complex conjugate of Zs:

Zin = conj(Zs)

That is conjugate matching. It makes sense if you picture the reactances canceling and only the resistances remaining.

Textbooks write the reflection coefficient referred to the source Zs as

Γ = (Z − conj(Zs)) / (Z + Zs)

It looks different from the formula above, but set the target to Zt = conj(Zs) and it is exactly the same formula. Substitute, and the denominator becomes Z + conj(conj(Zs)) = Z + Zs.

For Zs = 25 + j40, set Zt = 25 − j40
  → Γt = (Z − Zt)/(Z + conj(Zt))
     matches the textbook (Z − conj(Zs))/(Z + Zs)

So the only difference is whether you specify the source impedance or the impedance you need to reach. During design, what you want to see is the latter — where Zin has to go — so specifying the target directly is the more natural choice.

A match that does not land in the center

When Z0 and Zt differ, a good match does not land in the center of the chart.

Take matching a 200 Ω load to 75 Ω. An L-section gives

Q = √(200/75 − 1) ≈ 1.2910

which fixes the element values, and Zin comes out at exactly 75 Ω. But viewed on a chart still normalized to 50 Ω,

|Γ| = (75 − 50) / (75 + 50) = 0.2

it lands 0.2 away from the center. That is not a design failure. It just means that, seen from a 50 Ω system, 75 Ω reflects.

Concluding "it is not in the center, so it is not matched" is the classic mistake when Z0 and Zt get mixed up. What matters is the distance from the target, not from the center.

Reading the readouts

When Zt is off center and you look at VSWR or return loss, you need to be aware of which one the value is referred to:

• |Γ| referred to the chart center (Z0) — the reflection seen by the measurement system. Meaningful for cables and instruments
• |Γt| referred to the target (Zt) — how well the design goal is met

For a PA output match, the latter is pass or fail. The former can never reach zero. Being able to see both at once avoids confusion, and I think design tools should be built that way.

Try it

The distinction sinks in once you can actually set the two separately.

Smith Match — Matching Network Designer

This tool keeps the chart reference Z0 and the target Zt independent. With the chart normalized to 50 Ω, you can set the target to 75 Ω or to 25 − j40 and design against it. The target is drawn as a marker away from the center, and how well the match is achieved is judged by the distance from that point.

The default target is deliberately set off center for exactly this reason — so that the difference between the two is clear from the very first screen.