Home‎ > ‎

SPICE-based Averaged PWM Models

Introduction to SPICE-based Averaged PWM Models

Why revisit averaged switchmode models?  They've been around for a long time!  What could possibly be new? 

Bottom line: I believe they can be made simpler to use yet also incorporate improved accuracy, NO matrices, and many new features.

Warning: this page, and some of the topics beneath it, can be quite "slow reading".  If you're looking for a "quick read", I suggest seeking it elsewhere!  However, I hope these pages just might spark some useful thinking amongst tenacious folks.

In the last few years I found myself getting interested in the "averaging" (state-space averaging) methods used to model switchmode converters.  It seemed to me that there were significant improvements that could be made to these averaging techniques as developed over the last few decades.  I set about trying to address some of my main objections to classical averaging analysis and I think I've pretty much succeeded in developing (slightly) alternative but useful techniques.

For example, I noticed that some of the assumptions used in most existing averaged models seemed to be oversimplified, such as the assumption of strictly linear components.  In my career I have found that such simplistic analyses sometimes hide significant potential sources of error.  At the very least, I feel that the consequences of simplified analyses should be understood, investigated, quantified, and weighed in an intelligent manner.  In these pages you'll see a few demonstrations and/or questions regarding the potential benefits and pitfalls of averaged analysis.

Historically a lot of effort has been put into simplifying switchmode converter circuits for state-space averaging (such as the "averaged inductor", "averaged switch", or "canonical switch model" techniques) so that the resulting circuits to be simulated would incorporate fewer components and thus be analyzed more easily or simulated more quickly.  I think these strategies were undoubtedly brilliant but largely misguided.  These circuit simplifications tended to eliminate a lot of the essential nonlinear characteristics of the usual time-domain SPICE models and thus their results became less trustworthy.  I now believe they "threw the baby out with the bath water".  Modern circuit simulators can now run so fast that a minor reduction of circuit component numbers is often of small importance in overall simulation speed.

As an example of my recent results,
 below is a plot showing simulated inductor current for a boost converter undergoing a transient load change.  One waveform (green) was generated using the usual SPICE detailed timestep transient simulation method. (Also note: you can find out how to download a very similar example from my webpage at Download an averaged model demo)
The other waveform (red) employs two different methods, but both were generated using my new averaged modeling techniques.  The "smooth" sections of the red waveform are the raw large signal output for my new averaged model inductor current.  The middle section of the red waveform, which closely matches the detailed transient result, was only "turned on" from 4 msec to 4.6 msec, but was also calculated from the smoother averaged model waveform using only analytical functions of that averaged model's results (the averaged circuit voltages, the inductor value, and time).  So, presumably, it required less detailed timestep circuit matrix solving than the detailed transient simulation.

Note that this technique can produce not only an "average" inductor current waveform, but also the peak and minimum values during a switching cycle (which is evident from the center section of the red averaged-based waveform and also exhibited explicitly in subsequent plots).

If you click on the plot it should expand to be a little more viewable (the same plot is at On-demand waveforms).  Click the browser back arrow to return to this page.

The pages in this "SPICE-Based Averaged PWM Models" website section are listed in the menu to the left "mostly" in the chronological order of my averaged model development progress.  These pages are basically a description of my personal investigation into enhanced averaged modeling techniques.  My modeling goals are listed, in detail, at My Averaging Model Goals, Results, and Comments.

Following are some of the features of my recently-developed averaged models:

· Intuitive SPICE implementation – no matrix math or unintuitive circuit transformations involved!

· Nonlinear components are easily incorporated using standard SPICE models.  Linearization of component models is not needed (but see additional comments re nonlinear switches and/or inductors)!  See Incorporate saturating inductor effects realistically and Nonlinear Switch Models.

Models DC/DC converters using a more accurate
averaging concept than conventional linear-algebra-based averaging methods.  See  Non-equilibrium Dynamics

Close agreement between
my averaged large signal results and a detailed timestep circuit simulation easily and reliably validates my averaged modeling techniques!  See Large Signal Startup ResultsOn-demand waveforms , Validating This Modeling Technique Using Real Circuit Measurements.

· Both small signal
frequency-domain AND large signal time-domain results, such as detailed inductor current waveforms, can be calculated from the averaged model.  See  Small Signal Averaged Results , Add avg-based IL waveform.

· Current Mode Control effects such as subharmonic instability (subharmonic oscillation) are demonstrated even in transient, large signal, averaged simulations (I believe, but have not yet fully proved!).  See Current Mode Control Intro .

· And I've explicitly explained, using only time domain arguments, the necessity for incorporation of a time delay function within the averaged model in order to model Current Mode Control circuits.  See  Current Mode Control Intro , Averaged Current Mode Control Model - A Solution , Evidence of need for historical info for CMC model.

· Discontinuous inductor current effects are automatically incorporated in my model. 

· Time-varying switching frequency, duty cycle, input voltage, load, and current mode control feedback effects are easily implemented.  See  Variable Switching FrequencyOn-demand waveforms , Averaged Current Mode Control Model - A Solution.

· No unintuitive model blocks, such as the "averaged switch" model, or a feed-forward function, are needed to create a very accurate current mode control model.

· My averaged model is based entirely on time domain arguments.  No frequency-domain or sampled data z-domain theory/approximations are needed (but small signal frequency-domain characteristics can still be easily obtained).  And, in fact, I don't even explicitly consider or use the concept of "sampling" (for the inductor current).  Given my model's apparent success this is interesting at the least!

· There's still some significant inaccuracies in the output voltage results of the averaged model that are not fully addressed.  But I've got a pretty good work-around.  And I suspect these inaccuracies are not always (at least in some published models) accounted for in conventional averaged models either.  See Vout Discrepancy Intro and Rethinking Averaged Vout calculation.
You may want to take a look at my web page here... to get an EXTREMELY brief explanation.  But you'll be better off researching this elsewhere...

Detailed Time-Step Simulation Strengths

During much of my career I believed that everything you needed to know about a power converter circuit, especially efficiency, could be answered by a detailed and accurate transient SPICE simulation of the circuit. 

The "gold standard" for circuit analysis is, in my opinion, such a transient timestep analysis using a Spice simulator (ASSUMING your component models are truly trustworthy!).  It can include component nonlinearities, arbitrary circuit designs, and time-dependent circuit conditions.  Of course, your SPICE models need to be very realistic.  Otherwise, you're probably wasting your time using SPICE simulation.  For that matter though, in order to be trustworthy, ANY circuit analysis method should incorporate very realistic component models.

Analytical (formula or equation-based) modeling techniques can not usually achieve as high a level of realism as the method demonstrated here.  When you ignore component or circuit "details" such as nonlinearities, in order to achieve simpler analytical formulations, you obviously lose some model realism and thus simulation result believability.  Furthermore, formula-based analyses (typically solved using Matlab, Excel, MathCad, etc.) always run the risk that the formulas developed (usually, individually for each circuit architecture) include some (human based) error or over-simplification.

Note, however, that Dr. Ray Ridley's Excel-based switchmode simulator Power 4-5-6 may be a VERY clever exception to this comment (his analysis techniques are not published but I suspect they are similar to a detailed SPICE time-domain simulation specialized to PWM circuits).

In addition my averaged SPICE time-step simulations demonstrate, in some detail, the important phenomenon of Current Mode Control instability in large signal time domain analysis.  Whereas conventional averaged PWM models don't!  Since many modern power converters use Current Mode Control techniques, this is a very large advantage for my new methods.

Conventional Averaged Model Strengths

What are the positive features of averaged models?

Gradually, I've begun to appreciate the usefulness of small signal (frequency domain) simulations for control and stability analysis, and the (potential) relative simulation speed of averaged models, even for large signal analysis.

Furthermore, the development of a traditional averaged model often produces an analytical (formula-based) description.  Such analytical models can result in frequency-domain transfer functions (such as a duty cycle to output voltage or inductor current) that explicitly include circuit parameters and component values in the formulas (identifying poles and zeros, for example).  Thus, for those who are comfortable with Laplace analysis, this yields a very direct connection between the component values and the circuit characteristics.  This can yield great insight into the effects of component values on circuit behavior.

My new averaged modeling techniques DO NOT, however, yield analytical frequency domain transfer functions that depend explicitly on component values.  Instead, they require the use of a SPICE simulator to yield results.  But they also do not require unintuitive transformations of the basic switching circuit architecture.
My recent modeling efforts have been aimed at improving both small signal and large signal "averaged" switchmode converter analysis.  The process has taught me a lot.  I think I've developed some techniques that make averaged switchmode model creation more intuitive, capable of both large and small signal analysis, are more accurate, and use existing and inexpensive software (LTSpice is free, actually). 

And I believe I have been able to reproduce Current Mode Control instability effects in large signal time-domain analysis using my averaged modeling techniques, see Averaged Current Mode Control Model - A Solution.

During this model development I've learned considerably more about the limitations of both the conventional "averaging" and my own SPICE averaging techniques.

Motivation For Enhanced Averaged Model Development 
I found I wasn't really satisfied with the traditional averaged models that used linearized switch models and required the use of matrix math to solve (the original averaging concept), or used a circuit simulator but modified the circuit so much it was unrecognizable (such as the "averaged switch model").  I believe these methods are very brilliant and very useful, but they just don't yield a model that's intuitive to me.  Plus, they may need special software (e.g., Matlab), almost always employ simplified linear circuit component models (unless very arduous techniques are applied) and, I believe, actually contain some fixable minor inaccuracies (see Non-equilibrium Dynamics). 
I decided I'd try to develop enhanced switchmode modeling techniques that significantly reduce the limitations of conventional averaging methods, use MORE INTUITIVE circuit averaging methods, and are based on SPICE.
In addition, I thought it ought to be possible to generate detailed large signal switching waveforms, at least approximately, from the averaged simulation results.  See Add avg-based IL waveform .
Some simulation experts I've met believe that averaged models can only be used for small signal simulations.  Actually, this isn't true.  Perhaps this misconception arose from the early papers on averaged models from several decades ago, which originally described only small signal averaged simulations.  In the meantime, many authors have extended the basic concepts of averaging to large signal models.  A good example is the "averaged switch model".
Back in 1995 I developed a successful averaged large signal model for an automotive H-bridge DC motor drive project using the Saber simulator.  It didn't generate detailed waveforms (within each cycle), but it did include nonlinear switches, averaged large signal analysis, and automatic discontinuous mode calculations.  At the time, though, I was basically ignorant of the existing averaging modeling techniques and theory (except that I had a vague idea of the concept of state averaging).  So I didn't even know that I was actually using pretty advanced techniques back then.  My recent efforts go a bit further, such as producing detailed waveforms (see plot above or Add avg-based IL waveform), and are more carefully thought out and more realistic than my 1995 models. 
So far the results of my recent efforts have been both very interesting and very educational for me.  Perhaps they may help to illustrate some averaged model issues for you also.

The following link takes you to a page that describes the basic boost switchmode converter I am using as a test case for this model development effort.  You'll probably want to take look at it in order to understand the links in my "Goals and Results" list (at My Averaging Model Goals, Results, and Comments):

Some Terminology

On this website, if I refer to "detailed" simulation results, "detailed" means that the SPICE simulation was done in the usual small-timestep transient mode, that is, a "rectangular" Pulse Width Modulated waveform drives the switch and detailed periodic waveforms are generated by SPICE at every time point.  This is in contrast to waveforms generated by an "averaged" model, which has no changes in any inputs that occur at the switching frequency. 
But PLEASE NOTE: the "averaged-model-generated" detailed large signal switching waveforms I've recently developed are a special case and are created by adding some additional assumptions about the switching frequency and phase to the (strictly) averaged model results.  They CAN exhibit waveforms that vary at the switching frequency but they are generated using analytical calculations from the averaged simulation results that have no variation at the switching frequency.  And, presumably, they should require less iterative matrix solving by the simulator.

Please see this link for a comment about how simulation efforts are often, in my opinion, insufficiently valued:  Comment: Simulation vs. Reality 
There's nothing in my averaged models that needs more than algebra and simple calculus (derivatives, integrals) to understand.  So it's not "rocket science" or "string theory".  Pretty simple stuff, actually.  Yet sometimes it may require a little different thinking (compared to the traditional linearized averaging techniques).  But very little math!
Note the packing paper.  The One Who Must Be Served believes that string theory can be
experimentally confirmed by whacking a string against packing paper.  This produces a sound
that can be detected (and is extremely satisfying to The One Who Must Be Served).  Stay tuned for
further results of her experimental evidence.
Copyright © 2011-2014 Robert Steven Scott
Back to my home page: Home