# Background

In the process of trying to fix an LNA input impedance match, we discovered we need to understand more about Smith charts! The question is the at the bottom of this post, here's some background:

We are trying to fix the input match on an LNA. As the input match is currently designed it gets an VSWR of 1.7 . This is pretty good but we thought, maybe, ignoring parasitics, we can make it better as follows:

1. Measure S11 using a VNA (.s1p) and optimize an arbitrary ideal network that provides a good match. We call this ideal network "Z_Match":

1. We then place "Z_Match" in series with the component models (.s2p) of the existing matching circuit that was measured with the VNA. The ignoring parasitics, theoretically the RF behavior of this combined network is what we need to get a better match. We call this the "adjusted old match":

1. We can then use the "adjusted old match" to optimize a "new match" with component models that has the same behavior as the "adjusted old match". As you can see the "new match" has the same footprint as the old match but with different components so we can swap them out and hope for a better match (C1, C2, L3, L1 are optimized indexes into MDF data files):

Thus, when

$${S11}_{adjustedoldmatch}-{S11}_{newmatch} \approx 0$$

then the "New Match" network is nearly as good as it would be if the old network where augmented with the ideal matching network.

# Results

So we went to work and ended up with a great optimization. Here's the Smith chart plotting S11:

As you can see above, the "new match" and the "adjusted old match" are right on top of each other, and the ideal match is right in the middle where we want it.

Out of curiosity, I wondered what the smith chart of Z11 looked like, and was surprised to see that the Z11 behavior does not overlap, in fact it appears to be exactly 90 degrees apart:

# Questions:

1. Why does the Z-parameter Smith chart show "new match" and "adj old match" at 90 degrees apart when the S-parameter Smith chart is nicely overlapped? What is the difference between a smith sharp plotting S11 compared to a smith chart plotting Z11?

2. If not S11, then which parameters should I optimize to be equal between the "adjusted old match" and "new match"?

3. Does S21 matter In this case since we are only trying to optimize the input and not concerned about phase shift through the network?

Why does the Z-parameter Smith chart show "new match" and "adj old match" at 90 degrees apart when the S-parameter Smith chart is nicely overlapped?

$$\Z_{11}\$$ isn't only related to $$\S_{11}\$$. The formula is (stolen from Wikipedia)

$$Z_{11} = {((1 + S_{11}) (1 - S_{22}) + S_{12} S_{21}) \over \Delta_S} Z_0$$

where $$\\Delta_S = (1 - S_{11}) (1 - S_{22}) - S_{12} S_{21}\$$.

If $$\S_{11}\$$ of two networks are nearly equal, but the other S-parameters are different, there's no reason to expect $$\Z_{11}\$$ to match closely.

What is the difference between a smith sharp plotting S11 compared to a smith chart plotting Z11?

There is no point in plotting $$\Z_{11}\$$ on a Smith chart. The point of the Smith chart is to graphically relate the reflection coefficient ($$\S_{11}\$$) to the input impedance of a network.

Imagine superimposing a set of polar coordinates on top of the Smith chart. If you plot $$\S_{11}\$$ on those polar coordinates (which is exactly what your software has done, despite not displaying the polar axes) then the lines of the Smith chart allow you to read off the input impedance for each $$\S_{11}\$$ reading.

Since the same transformation doesn't appply to $$\Z_{11}\$$ the Smith chart doesn't tell you anything useful about $$\Z_{11}\$$.

If not S11, then which parameters should I optimize to be equal between the "adjusted old match" and "new match"?

It depends on your goal. If you want to maximize return loss, optimize $$\S_{11}\$$ of the system. If you want to maximize power delivered to port 2, then optimize that.

Does S21 matter In this case since we are only trying to optimize the input and not concerned about phase shift through the network?

If you know your network is lossless, and you don't care about phase shift, then you probably don't need to worry about $$\S_{21}\$$. Minimizing $$\S_{11}\$$ will maximize $$\|S_{21}|\$$.

If your network is lossy, then $$\S_{21}\$$ contains information about that loss.