Nonlinear Switch Models

Warning: this webpage gets a bit detailed in its description of nonlinear model issues and results.

One of my goals was to easily incorporate standard nonlinear switch models, such as SPICE transistor and diode models, in an averaged simulation.  Most of the early averaged models could use only linear switch models (resistive ideal diodes and switches). 
This goal has been achieved but the results were not as practically useful as I had hoped.  For further explanation, please review the following remarks.
This is not a trivial issue.  I'm aware of at least a few papers that employ a considerable amount of mathematics and explanation to incorporate nonlinear components in averaged models.  See, for example, "An Advanced PWM-Switch Model Including Semiconductor Device Nonlinearities" (Ammous, et al, IEEE Transactions on Power Devices, Sept 2003) or "SPICE modeling of switched DC-DC converters via generalized model of PWM switch" (Biolek, et al, Radioelektronica, 2007), or "Simple method of including conduction losses for average modeling of switched-inductor cells" (Davoudi, et al, Electronics Letters, 2006). 
None of these methods, however, is nearly as simple as mine, which only requires placing the nonlinear SPICE components in the "averaged" circuit SPICE model.
Another of my goals is to demonstrate realistic nonlinear switch effects in the simulation results.  This is actually a separate goal from simply being able to use nonlinear SPICE switch components in the model.  In particular, even though I didn't expect to achieve outstandingly accurate results from averaged models, I wished to have the simulation be realistic enough to at least act as a "flag" when nonlinear switch effects become significant. 
For example, if a MOSFET is run into its nonlinear (current-limiting) mode due to insufficient on-state gate voltage combined with switch current ripple, I would like the averaged simulation to show some clearly observable sign of this.  This goal is more challenging than simply being able to incorporate nonlinear SPICE components within the averaged model.

Why?  Because state-space-averaged models assume that that the circuit can be completely characterized, at every time point, by the values of the averaged "state variables".  Unfortunately, the real detailed waveforms which the averaged state variables attempt to represent, such as inductor current or output filter capacitor voltage, can actually vary considerably during a switching cycle.  The actual waveforms may include considerable excursions away from their "averaged" values (that is, ripple).  Even if nonlinear effects are included in calculating these "state variables", it only represents an averaged condition over an entire cycle, and can not evaluate the effects of current and voltage ripple during each cycle.
Yet I was a bit surprised (but shouldn't have been, in retrospect) that the incorporation of nonlinear switch models in the averaged simulation did not, by itself, guarantee that strong nonlinear switch effects would be flagged.  A demonstration will be provided below.
The basic problem is that, although you gain in execution speed and small-signal model generation, you really do give up a lot in going from a detailed transient model to an averaged model.  There's just no way around it.  In the detailed model you calculate the switches' characteristics at each timepoint during each switching cycle, including ripple currents and voltages.  For the averaged model, you only calculate the switch characteristics as if the currents and voltages were at a more-or-less constant average value during each cycle.  That is, there is no switching ripple in the averaged waveforms.  Therefore the effects of ripple excursions simply can not be evaluated in a conventional "averaged" model.
Nonlinear Switch Averaged Model Results

Now let's confirm that my averaged model actually does exhibit some of the effects of nonlinear switch models.
For this demonstration I've replaced my previous linear (resistive) ideal switch model with a SPICE subcircuit model for a MOSFET having a similar on-state resistance.  The transistor I chose was the IRLR024Z NMOS.  Its SPICE model was downloaded from the International Rectifier website.  Below is a plot of the nominal IV characteristics of this transistor from its data sheet.
                   IRLR024Z Id vs. Vds and Vgs Characteristics
I drove this transistor with one of two on-state gate voltages, either +4v or +5v. 

These values were NOT chosen to represent an acceptable design but, instead, to demonstrate nonlinear effects.
In addition, the boost rectifier diode's model consists of a standard exponential SPICE diode model with a (pretty unrealistically large) series resistance.  Again, this model was chosen not to be particularly realistic but, instead, to accentuate the nonlinearity of the diode.
First, let's demonstrate the diode current vs. voltage characteristics.  The following plot shows the averaged model's anode current vs. anode-to-cathode voltage for the rectifier when running the large-signal averaged model transient simulation.  Clearly, the diode model is nonlinear in the expected manner.
If you click on the plots they should expand to be a little more viewable.  Hit your browser back arrow to return to this page.
The results for the MOSFET nonlinearity demonstration are DEFINITELY more interesting.  I ran my detailed/averaged models at two gate drive voltages, +4v and +5v, under the usual circuit conditions.  The inductor current results for the Vgs = +5v simulation are shown below.  The agreement between the averaged and detailed models is not bad.
But now let's try the simulation with a gate drive of only +4v in the on state.  See the results below.
Again, the averaged results agree with the detailed simulation pretty well, EXCEPT at the higher inductor current (and thus switch current) levels.  At the higher duty cycle/inductor current levels the peak inductor current of the averaged model is significantly higher than that of the detailed model. 

The problem is that, with the low +4v gate drive, the IRLR024Z NMOS transistor is incapable of sourcing more than about 10 amps.  This is demonstrated by the detailed current results at Vgs = +4v above (also see the "IRLR024Z Id vs. Vds and Vgs Characteristics" plot above)
In the averaged simulation, the MOSFET switch averaged current (beige, in the plot just above) is assumed equal to the current averaged over a cycle, which, at its maximum, only comes near to 10 amps - but is just a little lower.  This doesn't quite reach the region of operation where the transistor current is severely limited by the low Vgs gate voltage.
Further evidence of this is shown below.
These curves plot the "detailed simulation" MOSFET drain current vs. drain voltage for both the Vgs = +4v and Vgs = +5v cases.  The vertical lines are an artifact caused by plotting the drain current during multiple states; on, off, and discontinuous states.  Here, the actual on state switch current is the upper extent of the plotted points.
Clearly, the MOSFET just barely goes into its current-limiting mode at about 10 amps when Vgs = +4v (red curves).  At Vgs = +5v (turquoise curves) it's still mostly linear (resistive) and OK.
If we plot the same drain currents for the averaged model, however, we see that the maximum AVERAGED drain current is still just barely less than 10 amps when Vgs = +4v and thus the drain current has not quite saturated (it's really close though!).  That's because the state-averaged switch current, which is used to calculate the averaged switch voltage drop, is less than its actual maximum value during a switching cycle.
The bottom line is that the averaged simulation doesn't show the major effects of the MOSFET-limited drain current in its Vgs = +4v results, whereas the detailed simulation does.
In fact, the circuit values and components, such as the gate voltages I chose in this case, just happened to be "right on the edge" of giving either a more realistic or a less realistic result.  These "lucky" choices were extremely instructive regarding the potential dangers of accepting, without careful consideration, averaged model results.
We can't really fix this issue easily.  We can add more circuitry that will attempt to include more than just the averaged inductor current in the simulation.  For example, we could somehow incorporate both the maximum and minimum inductor current envelope values into the averaged model's switch currents.  These would require additional model components, however.  I hope to try this later when introducing techniques for nonlinear inductor modeling.

I've now created a somewhat-successful averaged model that works for nonlinear inductors, see Incorporate saturating inductor effects realistically.  I had hoped that this same technique might be used to solve the nonlinear switch issue, but it didn't.
Realistically incorporating detailed effects (including ripple) of nonlinear switches may be going a bit too far in the quest for averaged-model realism.  Hang on and we'll see how it turns out.
For now, though, it's more a case of "buyer beware".  The averaging process simply lacks some (sometimes important) information that a detailed transient simulation naturally incorporates.
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Copyright © 2012 Robert Steven Scott