Basic impedance transformation constraints

I'm trying to optimize an impedance transformer for the output of a narrowband RFPA. Let's presume I have an antenna impedance ZA and the output impedance of my amplifier is Zamp. I want my impedance transformer to present the optimum load impedance Zol to the amplifier. Zol, is not a conjugate match of Zamp; it's something different in order to maximize efficiency as opposed to power transfer.

Obviously there are tons of possible ways to implement this impedance transformation, but what if we also care about the impedance seen by the antenna port looking back towards Zamp through the transformation network (we'll call this Zs)? Are there any theoretical constraints on what that impedance Zs can be given ZA, Zamp, and Zol? Assuming a lossless, reciprocal transformation network, but with no constraints on the construction of the network.

Thanks in advance.

I'm trying to optimize an impedance transformer for the output of a narrowband RFPA. Let's presume I have an antenna impedance ZA and the output impedance of my amplifier is Zamp. I want my impedance transformer to present the optimum load impedance Zol to the amplifier. Zol, is not a conjugate match of Zamp; it's something different in order to maximize efficiency as opposed to power transfer.

Obviously there are tons of possible ways to implement this impedance transformation, but what if we also care about the impedance seen by the antenna port looking back towards Zamp through the transformation network (we'll call this Zs)? Are there any theoretical constraints on what that impedance Zs can be given ZA, Zamp, and Zol? Assuming a lossless, reciprocal transformation network, but with no constraints on the construction of the network.

Thanks in advance.

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Hi Mtwieg
For this aim there are plenty of ways . one of the most popular method is using a single or double stage stub . if your cable between aerial and PA is too long , you must use a 2nd stage stub matching .
the other method is using some filters , which in high powers is the most reasonable way .

Anyway , have a look through the Secrets of RF circuit design by Joseph J carr and RF circuit design by chris bowick . these two books are dealing with all of the art of matching .


Best Wishes
Goldsmith

To clarify, I'm not working at frequencies high enough for stubs or lines to be useful (<100MHz, small size constraints). I'll likely be working with lumped elements only.

And I'm not really concerned with practical implementations at this point, more with pure theory. Most resources I've found only deal with impedance transformation on one side of a network, not both ends.

So you want to impedance match a radio with antenna for maximum efficiency and have a transformer somewhere along the line. Impedance matching can be done on either or both sides of transformer. It is only a question of what is the most practical alternative.
Assume components have been added resulting in conjugate match between antenna and transformer. Then is it automatically a corresponding conjugate match also between transformer and radio. It is total bidirectional result as long as matching is done with inductors, caps and coil-transformers . If you care for impedance seen by antenna, measured result will show same conjugate match in both direction also near the antenna.

maximize efficiency as opposed to power transfer.

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Makes no difference, if you now required impedance value that you want to present at the radio-interface is it the value you should use for conjugate matching. Simplest can a impedance matching network be designed by measuring impedance near radio interface and calculate a correcting network that results in a conjugate match for the impedance that results in optimal efficiency.

When calculating optimal system efficiency can not a TX-circuit efficiency be isolated from antenna efficiency. If TX impedance is offset from impedance resulting in lowest reflection factor, do it also cost dropped antenna efficiency due to not optimal antenna impedance matching. Lees antenna efficiency must be compensated by increased TX power to reach same amount of TRP, if TRP is a measure of system efficiency.

Assuming a lossless, reciprocal transformation network,

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Have not seen any lossless transformation IRL, and if it is a question about system efficiency, is it often an important factor.

If you consider only "theoretical" constraints, bandwidth is essential. It's also the principle constraint of electrical small antennas, as you probably know.

For practical matching networks, component tolerances should be considered besides losses.

E Kafeman said:

Makes no difference, if you now required impedance value that you want to present at the radio-interface is it the value you should use for conjugate matching.

When calculating optimal system efficiency can not a TX-circuit efficiency be isolated from antenna efficiency. If TX impedance is offset from impedance resulting in lowest reflection factor, do it also cost dropped antenna efficiency due to not optimal antenna impedance matching. Lees antenna efficiency must be compensated by increased TX power to reach same amount of TRP, if TRP is a measure of system efficiency.

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Believe me, I know for certain that a conjugate match is not optimal for me. My switchmode PA gets best efficiency when my reflection coefficient is between 0.5 and 0.75. And I also desire a large reflection coefficient presented to the antenna port. Sounds strange, but my application is not typical.

FvM said:

If you consider only "theoretical" constraints, bandwidth is essential. It's also the principle constraint of electrical small antennas, as you probably know.

For practical matching networks, component tolerances should be considered besides losses.

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My bandwidth requirements are quite narrow, about a thousandth of my carrier frequency. The matching network itself doesn't have to have any specific frequency response (the antenna itself is quite narrowband). And of course I'm not expecting perfect tolerances or zero losses, but as of now I'm only asking what's achievable at a single frequency with ideal components, so that I know if my desired Zs is possible to achieve or not in theory. If theory says it is possible, then I should move on to consider practical matters. If not then I'd be wasting my time, that's why I'm keeping the question simple.

Are there any theoretical constraints on what that impedance Zs can be given ZA, Zamp, and Zol?

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You can calculate mutual dependency of the s parameters of a lossless, reciprocal two-port network based on the general properties of s parameters. There are no other constraints for an "ideal" impedance matching network as far as I see.

The relations are
S21 = S12 (reciprocal)
S11² + S21² = S22² + S12² = 1 (lossless)

Right, but I have trouble relating those rules back to port impedances. Does this also imply that |S11| is always equal to |S22|, and therefore the reflection coefficients at both ports must have the same magnitude? That seems intuitive, but I can think of examples that violate that rule. For example if Zs=1 and ZL=1+j, and I decide to conjugate match, my matching network can just be a series capacitor with impedance -j. So on the source side Zin will be 1 while on the load side Zout will be 1-j. So my reflection coefficient at the input will be (1-1)/(1+1)=0, while at the output it will be [(1-j)-(1+j)]/[(1-j)+(1+j)]=-j. Where's the mistake?

edit: ah apparently the equation Γ=(ZL-Zs)/(ZL+Zs) is not correct in general. Using Γ=(ZL-Zs*)/(ZL+Zs) seems to be consistent with |S11|=|S22|. But the vast majority of sources I've seen use the first equation, I guess presumably because they assume Zs is real...

So if that's all fine, then is my constraint just |Γin|=|Γout|, and therefore |(Zol-Zamp*)/(Zol+Zamp)|=|(Zs-ZA*)/(Zs+ZA)| ??

ah apparently the equation Γ=(ZL-Zs)/(ZL+Zs) is not correct in general. Using Γ=(ZL-Zs*)/(ZL+Zs) seems to be consistent with |S11|=|S22|.

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I can't follow the considerations. Γ=(ZL-Zs)/(ZL+Zs) is correct with no loss of generality.

Yes I think you're right, I found the other formula in papers on reflection of "power waves" instead of voltages. So if Γ=(ZL-Zs)/(ZL+Zs) is correct, then what's wrong with my hypothetical example above? I'm making the following assumptions:
1. Γ=(ZL-Zs)/(ZL+Zs)
2. For a two port lossless reciprocal network, |S11|=|S22|
3. Reflection coefficient at port 1 is equal to S11, and at port 2 is S22.

I'm not completely sure about your calculation. The problem might be that you are terminating both ports with Z ≠ Z0 but still assume, that S11 and S22 are representing the input impedance. But you must put S21 and S12 in the calculation to determine the reflection factor at one port if the other port isn't terminated with Z0.

FvM said:

I'm not completely sure about your calculation. The problem might be that you are terminating both ports with Z ≠ Z0 but still assume, that S11 and S22 are representing the input impedance.

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In my application I'm not dealing with cable or transmission lines, at least not at the moment, so I don't really have a defined Z0 (except as an arbitrary reference impedance I can choose, I suppose). I was under the impression that reflection coefficient and S11/S22 were, by definition, equal. And if that formula for reflection coefficient is correct in general, then that means that S11 and S22 should also be defined by ZL and ZS at any node in the network, right?

But you must put S21 and S12 in the calculation to determine the reflection factor at one port if the other port isn't terminated with Z0.

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What exactly is Z0, in this case where all my impedances are different?

Z0 is the reference impedance for the S-parameters, sometimes also called ZL. In a modfied S-parameter scheme, you can probably work with different Z0 values for both ports although I'm not totally clear about all implications. But as far as I understand, Z0 must be real.

So in the context of a two port network DUT, is ZL/Z0 assumed to be looking out of the network into the VNA (or whatever), or is it looking into the ports of the DUT?

And if analysis insists on Z0/ZL being real, then I suppose I can lump its complex part into the other side of the "port" and do analysis like that, then shift the complex impedance back afterwards? Seems absurd though, I thought S parameters and reflection coefficients were generalized to any combination of complex impedances.

Did you try to design a Smith diagram with complex or imaginary reference impedance? I'm not sure if it's possible at all, but at least the metric will be weird. Most known properties of S-parameters become invalid, e.g. |S11| <= 1 for a passive network.

The definition of S-parameters assumes that the ports are terminated with Z0 for the measurement. In so far you "see" Z0 when looking into the VNA. Based on this parameters you can also calculate the behaviour with arbitrary source and load impedances.

It is possible to use imaginary values as Smith chart center reference, but I doubt you like it.

http://www.3dsmithchart.com/

Do not know if it is to any help in this case, but as I understand it do this kind of work cause me very minor problem.
In average each week do I some kind of impedance matching, often related to a cellphone under development. This kind of matching involves wideband/multiband matching between two complex curves.
For active side do I have curves for RX impedance for best s/n and TX curves for best efficiency and for antenna side do I have free space impedance and impedance for handheld position.
The both impedance curves for the antenna are weighted together to one single curve and the both curves at active side are weighted in a similar way.
From this can a wide band matching circuit be calculated.
A problem is that both ends are complex and irregular impedances which not exist more then as calculations as they are weighted from other curves. As the curves not exist must VNA measurement be mathematically compensated with aid of an external software, which also calculates optimal matching network, based on real or ideal components. When implementing by software proposed network, do the software verify each component value, so a complete matching job is often done within minutes from the first measurement.
Result is better matching with fewer components as it no longer require a (assumed) 50 Ohm reference halfway along impedance matching network, which previously was the normal way but as cellphones not any longer have any external antenna port with a fixed impedance, is a such reference not needed.

Okay I think I might have found my solution. Basically I used Z parameters under the assumption of reciprocity and no losses.

So given the following setup with Zs, ZL, and Zin defined:


I found that Zout=Z22-(Z11-Zin)(Z22+ZL)/(Z11+ZS)
Where Z22=(Z11*real(ZL)-j*imag(Zin*ZL))/real(Zin)
And Z11 can be any purely imaginary number. I could combine these equations but I'm pretty sure the result wouldn't be any easier to look at.

The results all make sense, and I also happen that my power reflection coefficient on both the input and output always has the same magnitude (basically what I suggested at the bottom of post #8). The normal reflection coefficients, however, don't seem to obey and constraints once I start plugging in complex inputs (in fact, the reflection coefficients sometimes have magnitudes greater than unity, despite the system still being stable and lossless). I presume that I could also get the solutions directly from the assumption that power reflection magnitude is equal at both ports, but so far I haven't been able to get a closed form solution from this approach.