Characterizing output impedance of switchmode RF amps?

My academic research is centered around SMPA development, and one of the major issues I have to address is characterizing the output "impedance" of such topologies as class E or current-mode class D (basically if you put inject a voltage source between the amp output and its load, how is the RF current changed?). But from what I can tell, there is no way to define a single number for output impedance, since the amp is, from the load's perspective, a time-varying (depending on the phase of your injected signal and that phase of the drive signal to the amp) and non-linear (assuming the switch can only operate in one quadrant, as all do) system. So one should expect the effective output impedance to depend on the amplitude of both the injected signal, the amplitude of the signal from the amp, and the phase between the two.

So I'm wondering if there's any standard way of characterizing these systems? I've read up on large-signal S-parameters, but these seem directed at linear amps operating at or near compression, where these effects aren't very severe. From what I've researched so far, all the assumptions that typical schemes are based on break down when you have hard-switched transistors involved.

Any guidance would be appreciated.

The effects you are describing apply more or less to any amplifier in large signal operation. That's why load-pulling is a straightforward way to determine the output impedance for a given operation point.

Harmonic balancing should be a suitable analysis method, too.

Optimum Load Impedance would frequently be different than Output Impedance of a PA that works especially high power levels.
Switching mode systems can not have a fixed output impedance because the active device hardly works at 2 extreme points therefore I don't perosnally think that switching mode systems have very variable instantaneous output impedances.But I'm agree that this impedance plots a contour around a certain impedance point.

So how is output impedance defined and modeled in such cases? From what I know, load pulling just involves a (effectively) passive load with variable complex impedance, but I'm interested in the amp being connected to an actual complex variable source. I don't think a typical load pull test is the same thing.

I'm not trying to find an optimum load impedance, rather I'm trying to predict what happens to RF power directed at the amplifier output.

Depending on how I model my circuit, I get many interesting results. For example if I assume (incorrectly) that my transistors can operate bidirectionally (no body diodes) then my amp may appear as a passive transparent mixer, whose impedance may be close to zero at one phase and close to infinity when the phase is changed by 90 degrees. In a more accurate model, the effective impedance still varies substantially (by a factor of two or more) with phase, and makes some interesting patterns (looks like what you might see on a smith chart, except you're sweeping phase instead of frequency).

Model follows problem. The problem specification isn't yet completely clear.

Is the connected source same frequency as the PA output signal (phase difference stays constant)? In this case, it might be still modelled as a complex load, if we allow negative real impedance components also with net energy flow from load to PA. In so far, it could refer to the load-pulling setup, as one of several possible models.

During load pulling, the output impedance varies with connected load. This doesn't matter because you are only interested in a single point of this complex-valued function. If you vary a PA parameter, e.g. nominal output power, you get a different complex function and a different optimal point.

With your problem, you are apparently asking for the complete complex valued output impedance as function of load impedance.

Indeed. Currently I have a measurement setup where I can inject a known RF voltage between the amplifier output and the load, and measure the RF current. So I vary the injected voltage while observing the change in current, and dividing voltage by the change in current (relative to the case with no injection at all) gives me an effective complex impedance, which is a function of the bias voltage of the amp, the amplitude of the injected V, and the relative phase between the injected V and the carrier driving the amp. Is that reasonable?
Yes, the injected V and the amp are always at the same frequency for my tests, but phase offset can be anything. So far my simulations always show the impedance being on the right-half plane, but some of my measurements drift a bit to the left half, I suspect it's a an error I have to hunt down.
Could you be more specific what you mean by the "complex-valued function"? What are dependents and independents of this function in a LP test?
No, not really, I'm looking for complex output impedance as a function of amplifier bias, injected amplitude, and carrier phase. While varying these parameters the passive impedance seen by my amplifier is constant.

Should be a reasonable way to determine a differential impedance, which is valid at least for smaller external voltage respectively smaller load impedance variations.

That's good. What I mean is that output impedance generally can't be expected to be constant for a nonlinear device, e.g. a PA. This means, it can vary with the variation of any parameter. If it's almost constant, your amplifier seems to be more linear than expected.

In load pulling, you simply vary real and imaginary part of load impedance until you achieve maximum delivered power. You don't particularly care for possible variations of output impedance during load pull, it's sufficient to know that it's equal to the conjugate of the load impedance in the optimal point.

Presumed the output impedance is variable, you can record it against load impedance and get the said complex-valued function.

Okay, so after reading a quick overview of load pulling, I suppose my setup is comparable to an Open Loop Active Load Pull, but in my case I am also interested in cases where Γ>1 (the injected voltage is greater than the voltage produced by the DUT alone). That's where the effects of nonlinearity show themselves the most, according to my observations.

What I'm basically asking is how the results of a test with so many varying parameters is usually summarized. At the bottom of that page, it shows some example outputs, which are pretty hard to interpret (many sets of isocontours on a smith chart). My results largely have the same form (though I prefer to keep things on a complex cartesian plane). Is there any method beyond that of presenting the results in a more "compressed" way? Like, if you had to fit the results of a load pull test into half a page and present them to your boss in ten minutes, could it be done?

Thanks for the interesting link. I didn't know that the postulated equivalence of variable impedance load pulling and variable voltage injection ("open loop active load pull") has been implemented in a practical measurement setup.

The Γ>1 region, corresponding to negative real load impedance isn't interesting in usual amplifier tests, I think. But it can be of course covered by the active load pull setup, as long as the output transistor ratings aren't exceeded.

Provided that transistor saturation usually limits the PA output power, it seems plausible that connecting a negative load impedance will force the output transistor further into and beyond saturation and thus brings up heavy non-linearity.

I believe that the results are truely complex and possibly can't be compressed that much. I also assume that varying parameters like PA output power, supply voltage or bias will probably exponentiate the complexity. That's really different from the purpose of standard load-pulling that's asking just for a single point on the Smith chart.

I guess, you have to focus on the purpose of your research (which didn't become fully clear to me) and ask what's the essential information that should be extracted from all the measurements.

Okay, thanks for the input FvM. I sort of expected that there's not much way to compress the test results, but rather I'll just have to select a few distinct operating points with the most interesting results to demonstrate.

About the purpose, well I can't say too much before publishing, but basically we want to see what a bunch of SMPA will do when their outputs are coupled to each other, and output impedance is a key part of that.