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**Canonical form of DC converter** on

Power Supply
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Dirk Kok

09-13-2013 04:17 AM

# Canonical form of DC converter

Hi,

I have been working through Erickson's book: fundamentals of power electronics (I know sacrilegious to mention here - ;-) )and I ran into the following, see attached document.

I was wondering if there is someone who could help me with understanding the difference between the canonical form I arrived at and the supposedly correct form from the book (page 5 in the document, last two equivalent circuits).

The difference between the circuits is minute: D*I*d(t) versus V/R*d(t) and probably have something to do with the transformer. My view is: If I would want to write the D*I*d(t) in terms of voltage and resistance I need to leave the factor D in place since this would disappear when I push the current source through the transformer to the other side. Yet, in the book it disappears, which I think means that if I push the current source to the other side during manipulations for transfer functions and the likes, I would introduce a factor 1/D when I do that.

Can someone explain, please?

09-13-2013 06:40 AM

Top #2

Ray Ridley

09-13-2013 06:40 AM

It's no problem to mention another book here, all education is good as long as it points in the same direction.

My concern with some of the modeling work after Vorperian is that it reverts back to state-space averaging again. Vorperian clearly showed the superiority of the switch model approach, both from a comprehension, teaching, and an accuracy point of view. Quite frankly, state-space averaging should have disappeared at that point (1986) but it lingers on.

The many lines of matrix equations in your reference that comprise state-space averaging are replaced with two lines of scalar equations with the switch modeling. I don't know why anyone would prefer to make this more complicated than it needs to be.

I know this is more of a comment than an answer to your question. Take a look at Vorperian's' final circuit model, perhaps that will give you the answer.

09-13-2013 09:05 AM

Top #3

Francisco Ramirez

09-13-2013 09:05 AM

Dirk,

In the state space averaged model of the buck converter, the dependent current source I*d is never "pushed" through the 1:D transformer. If you go back to Middlebrook's original paper you can see that the "outputs" of the state space averaged model of the buck converter are the output voltage Vo and the input current Ig.

From the average model of the buck converter:

ig=d*i

Perturbation of the previous equation gives:

Ig+ig^ = (D+d^)*(I+i^)

Therefore

Ig=D*I

And

ig^=I*d^

This last equation is the dependent source located in the primary of the 1:D transformer and it never gets "pushed" through the transformer.

Let me know if this helps

Francisco

09-13-2013 11:13 AM

Top #4

Francisco Ramirez

09-13-2013 11:13 AM

Correction to the previous e-mail:

Averaging the input current for the buck converter

ig=d*i

Perturbation of the previous equation

Ig+ig^ = (D+d^)*(I+i^)

Therefore

ig^ = I*d^ + D*i^

The first term in the last equation is the dependent source located in the primary side of the 1:D transformer. It never gets "pushed" through the transformer.

The second term in the last equation is the i^ current (small signal current in the secondary of the 1:D transformer) reflected to the primary

09-13-2013 01:29 PM

Top #5

Carlos Azevedo

09-13-2013 01:29 PM

Hi Dirk, Dr. Ray and Mr. Francisco.

I am new in this group and I have started to look into some of the discussions that exists and this one was one of the discussions that have caught my attention.

I am finishing my M.Sc. in ECE and I am thinking on doing my thesis in the field of Power Electronics. Yet, I don't know what to do but I hope to find something interesting and something where I can learn.

Meanwhile, I am reading some books and papers and I am starting my personal project on a Forward Converter and build the transformer.

Well this was a little introduction of myself.

About the discussion, I started to read Erickson's book, specially the part of the canonical form and I never thought that there was another approach to the teach the switching model. Dr. Ray, I never heard about Vorperian. So from what you said, it means that Ericksson approach is pretty much the same as Vorperian approach? In the sense that the Ericksson approach reverts back to state-space averaging again? Vorperian approach was the first one to appear?

Kind regards.

09-13-2013 04:05 PM

Top #6

Ray Ridley

09-13-2013 04:05 PM

The whole world was using state-space averaging up to 1986. Then, Vorperian came up with the PWM switch model. It was developed literally overnight, and he solved problems that had never before been modeled (eg the cuk converter in DCM). His was the first, is the most completely explored, and most accurate.

Many other switch models followed. None of them, in my opinion, added anything to the development, and none of them were necessary. I was quite frustrating to see these alternate models because Vorperian's work was so elegant in its simplicity.

I have no idea why state-space averaging persisted after this, except that people tend to be resistant to change, even when the new approach is so elegant. Forcing students to suffer through the contortions of state-space averaging obscures the meaning of the circuit.

Vorperian's model took 2 lines of scalar math, then showed the nonlinear circuit model. A further few lines and it was linearized, and there was a corresponding linear circuit model.

Why would anyone want to make it harder than this?

09-13-2013 06:07 PM

Top #7

Gavin Pinto

09-13-2013 06:07 PM

Dr. Ray,

This is something I want to understand properly.

In 1985 I bought a book on the Current-Injected-Equivalent-Circuit approach to modelling DC-DC converters by P.R.K. Chetty. That was my first introduction to modeling in Spice.

This was an early method of modeling. Why is it not used any longer?

09-13-2013 08:40 PM

Top #8

Ray Ridley

09-13-2013 08:40 PM

It didn't have the simplicity of Vorperian's switch model. It just never caught on in popularity.

09-13-2013 11:36 PM

Top #9

Rajaganesh Kasthuri

09-13-2013 11:36 PM

Dr Ray,

Can you tell some reference documents/book that will be useful for learning and applying PWM switch model?

09-14-2013 02:05 AM

Top #10

Dirk Kok

09-14-2013 02:05 AM

Thanks for the comments on my original question. I found the error and marked it in the document. I been staring myself blind at it.

@Ramirez thanks for the feedback. But I was looking at where I went wrong, and I found it. It still feels strange when you look at the final canonical model which shows, Id^ on the left hand side of the transformer and V/R on the right hand side. My feeling says that you can not just replace I with V/R because of the transformer in between and ignoring the ratio. But maybe I am just looking into it too much.

The reason for me to use average state space modelling is that I am also working on stability of converters using work done by: Giouaris and Banarjee and Lee as well. They all seem to be using it. They are using the Saltation Matrix and monodromy matrices to predict stability in converters. Very interesting stuff.

The Erickson book, I like because unlike most books and papers every step is almost in there including the steps that are often assumed you know (mathematical rewrites etc). It is useful to refresh my memory on those subjects at the same time instead of having to read other books as well.

The Vorperian method is mentioned in the book as well, I believe, but it is hidden in between various other methods of modelling converters but when you arrive at the equivalent circuit / final equations the outcome is the same any time. From the Erickson book it was not clear to me that it was a lot easier / simpler than State space. But that maybe down to the way of explaining. I might have another look at Vorperian stuff.

@Ray, which paper of vorperian do you recommend that explain his method, like you described it.

09-14-2013 04:18 AM

Top #11

Carlos Azevedo

09-14-2013 04:18 AM

Hi,

Like Dirk said, Ray could you recommend a paper, something like a chapter on a book or anything else where this method is explained with that simplicity? I'd like to take a look at it.

About the Erickson's book, I think that method is very popular because it allow us to make a parallel with the transformer theory and allow us to make stability analysis as Dirk said too. The Vorperian's method allows that too?

Let me take the advantage of this discussion to ask you guys the following: When we need to use those models that were mentioned here?

Let me try to explain what I mean:

For example, I am designing a forward converter (I am at the project of the transformer) and this converter is well studied. So, what I think about this is that I don't need to use any kind of model to design the converter, right? (because this converter is very well studied and documented)

But now, thinking in the Erickson's book and his method, there we have the method applied to some converters topologies but, right now I remember if he applies that to the forward converter. However, if I wanted to do an exercise I could use that method and follow the idea to have a little bit more insight of the converter and get the Transfer Functions to plot the Bode diagram in order to see the phase margin and gain margin, stability, etc.

So other way besides doing that little exercise, I don't need any model or method. So, as I said before, when we need to apply that model (in this case, we are talking about the Erickson model)?

Just one more thing (I started to read Erickson's book recently so I am trying to organize the ideas): The Erickson approach doesn't have nothing to do with state-space averaging method? I mean, state-space averaging is a modeling method and Erickson approach is another method?

Thanks in advance.

Kind regards.

09-14-2013 07:05 AM

Top #12

Ray Ridley

09-14-2013 07:05 AM

Here is Vorperian's original IEEE paper:

"Simplified analysis of PWM converters using model of PWM switch. Continuous conduction mode" Aerospace and Electronic Systems, IEEE Transactions on

Date of Publication: May 1990.

On my first day of my first job, I was given Cuk's dissertation by my roommate, and told to show him I could derive state space averaging. 2 gruelling days later, I could do it.

It was such a relief when Vorperian's work came along to realize none of the state-space averaging was necessary, and the model for all converters was so trivial to derive. It was brilliant work.

09-14-2013 09:22 AM

Top #13

Brian Ernzen

09-14-2013 09:22 AM

I have often been confused about the timeline. Dr. Vorperian invented the switch model in 1986 and his paper is from 1990? It is also confusing with all the different forms of the switch model. Many (actually SOME) books talk about switch modeling but they don't usually mention which model they are using. Does the new Basso book which I plan to buy soon cover the Vorperian models or does he make some modifications? Are there any good books on the subject that cover Vorperian's models (other than Fast Analytical Techniques = it is a little difficult right now)?

09-14-2013 12:02 PM

Top #14

Ray Ridley

09-14-2013 12:02 PM

Vorperian was definitely the first. It takes quite a few years to get into the IEEE transactions. Sometimes it is hard for people to accept a new idea, especially one as profound as this that eliminated the need for state-space averaging. There was quite a lot of political will not to accept that, and that endures today. 27 years later, they are still doing it the hard way!

The thing about the switch model is that it is so simple to derive, it can be done in one page. It doesn't need a book to cover it, unlike the contortions of state-space averaging. Some people seem to think that SSA is better because its more complicated, which is nonsense.

To get the earliest references to the work, see the paper

http://www.dianyuan.com/bbs/u/49/1169958395.pdf, and check the references at the end.

09-14-2013 02:59 PM

Top #15

Brian Ernzen

09-14-2013 02:59 PM

Oh, I see! I don't think I've ever noticed those two papers, [6] and [7] from Vorperian before. I had only know of Vorperian's switch in his 1990 paper(s). I'll have to compare Vorperian's 1986 papers, the 1989 Tymerski/Vorperian paper, and the 1990 Vorperian papers. Thanks for the insight!

09-14-2013 05:25 PM

Top #16

Julio Cesar Rosas Caro

09-14-2013 05:25 PM

Hi guys

It is very nice reading that many people with expertise on this topic, Dirk what you are doing, going from the state space averaging to the canonical model, is a very good exercise and I guess it should be important for what you are doing in your research topic, but agree that most of the time that is not necessary, the canonical model may be obtained also from the circuit averaging (switch model).

I also like the book (Fundamentals of Power electronics, Erickson and Maksimovic) and it worth mentioning that they present three approaches to model converters in chapter 7, they are not married with state space averaging, actually that is the method they put less attention, the papers we have discuss here are in their references and they provide a very good explanation for the switch model (circuit averaging).

I am almost a starter (5 years on the field) and I actually recommend to people new in the field to start with Erickson’s book, because papers are for initiated.

Best regards

Julio Cesar

(English is my second language I apologies for possible mistakes.)

02-06-2015 04:26 PM

Top #17

Peter Franks

02-06-2015 04:26 PM

Ray Ridley wrote above:

"Vorperian clearly showed the superiority of the switch model approach, both from a comprehension, teaching, and an accuracy point of view."

I'm not sure I agree with the statement that the switch model is more accurate. Where exactly has Vorperian proved this?

06-22-2015 11:19 AM

Top #18

Dirk
Kok, you’re reading into it way more than you really should. In my opinion it
is just a convention that we use V/R on the other side and I completely agree
that 1:D is not something that gets pushed through. It is also important to
note here that in books, conventions are usually not explained a lot. For
example, the “V” might be a key factor is general application on the other side
of the transformer.

Regards,

Levent Caliskan