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1-Newbie

## Expressing impedence in series/parallel combinatio

Hi,
I want to figure out the connection of a two-terminal circuit knowing impedance formula across the two terminals. I have the impedence of the circuit in terms of circuit parameters in symbolic form. If I can express it in the form of "continued fraction expansion", then I can figure out the connection between the components. Please see the attached file for an example. I wonder if this can be done in Mathcad.

Thanks.
32 REPLIES 32
1-Newbie
(To:mark_neil2)
It does not ring loud a bell.
Can you "Save as" version 11. [*.MCD]

jmG
1-Newbie
(To:ptc-1368288)
Here you go.

Mark.
1-Newbie
(To:mark_neil2)
On 2/18/2009 2:17:52 PM, mark_neil wrote:
>Here you go.
>
>Mark.
_____________________

Your CF expansion is correct as it simplifies to the long division "I have the expression below as computed by some software". Decomposing this long division into continued fraction is not immediately available. I get the first step only. For the next step, Mathcad/Maple does a final expansion based on the last convergent expansion. Your unknown software executes some or all of the CF reduction/expansion. Check if that software includes all 4 algorithms:

1. Rational ==> J
2. J ==> rational
3. Rational ==> CF
4. CF ==> Rational.

jmG

1-Newbie
(To:ptc-1368288)
1-Newbie
(To:mark_neil2)
Not very clear what you are trying to do about those circuits. A resulting XFR function is a graph, finally ... a graph depending upon at least one variable argument. If you would have such a graph in form of a data set, what is possible in Mathcad is to approximate with a Thiele or directly into a rational fraction. From there is it easy to expand in CF (Continued fraction). Thiele got me busy since many years and now with Mathcad/Maple (11 and lower), many projects turned into piece of cake. What we can do is generate the Thiele CF c/w as a scalar including the variable argument, and of any length. There is an immense collection of Thiele in this collab some pretty recent (few months or so).

jmG
10-Marble
(To:mark_neil2)
I suspsect that a continued fraction expansion of the impedance in terms of the circuit elements is possible only for ladder networks. A ladder network will indeed have such an expansion, but I don't believe that a more general network topology (e.g, a bridge network) will have such an expansion possible, since a CF expansion has a ladder realization.

If you know a priori that the network had a ladder structure, then you know that such a continued fraction expansion is possible. Otherwise, I don't think that such an expansion exists, since that would indicate that a ladder structure of the same named impedances gives the same result as another network topology. This can be true for specific values (where the network:impedance is many:one), but not for the functional dependencies.

My comments are on the network problem and the existence of a continued fraction expansion. If one exists, I don't know how to make mathcad do the expansion.

Lou
1-Newbie
(To:LouP)
On 2/18/2009 8:57:22 PM, lpoulo wrote:
>I suspsect that a continued
>fraction expansion of the
>impedance in terms of the
>circuit elements is possible
>have such an expansion, but I
>don't believe that a more
>general network topology (e.g,
>a bridge network) will have
>such an expansion possible,
>since a CF expansion has a
>
>If you know a priori that the
>structure, then you know that
>such a continued fraction
>expansion is possible.
>Otherwise, I don't think that
>such an expansion exists,
>since that would indicate that
>a ladder structure of the same
>named impedances gives the
>same result as another network
>topology. This can be true for
>specific values (where the
>network:impedance is
>many:one), but not for the
>functional dependencies.
>
>My comments are on the network
>problem and the existence of a
>continued fraction expansion.
>If one exists, I don't know
>how to make mathcad do the
>expansion.
>
>Lou

There may be certain exceptions that this may not work. But majority of the circuits are realized by some series/parallel connections of circuit elements. For example, R3 in the expression of P2 could be impedance of some network comprised of several circuit elements. The intension here is to figure out the connection of circuit elements by expressing the impedence in certain form, which the CFE seems to be the best choice.

JMG: you assumed the impedence is a graph eventually. Yes, if I assign values to circuit components. However, I don't assign any values to component values. So, I work in symbolic form, not numeric. There exists no graph...

Mark.
1-Newbie
(To:mark_neil2)
On 2/18/2009 9:53:49 PM, mark_neil wrote:
>On 2/18/2009 8:57:22 PM, lpoulo wrote:
...
>JMG: you assumed the impedance is a
>graph eventually. Yes, if I assign
>values to circuit components. However, I
>don't assign any values to component
>values. So, I work in symbolic form, not
>numeric. There exists no graph...
>
>Mark.
___________________

If the expansion is like Lou mentioned a "ladder expansion" and valid as much as it can except that some R you may want to substitute for a Z complex (suppose) ... if that expansion is like the true representation of your circuit analysis as replied by your special software, THEN the continued fraction ! continues !

Am I right ?

If so: the attached CF will generate your continued fraction, of any length accordingly to the number of components you have in the expansion from the software. Once you have that raw CF expansion, you can assign each element with the C vector of names indexed.

Mark,

At this stage, please make a nice picture of a typical circuit "resumed equivalent ladder" and expanded, in order to tune the collaboration and go to the next step. What I understand is that at some point in the ladder if you introduce a capacitor (per say), then in the CF you plug that capacitor and get a complex impedance value. And going further to your eventual next question: can we solve for the capacitor value given a desired complex impedance ? Answer is YES with the restriction to be proved. I have already Minerred the ISA (International Standard Atmosphere) CF, i.e: Minerred Thiele CF.

So, the olympic flame is back to you.

Jean
1-Newbie
(To:ptc-1368288)

jmG
1-Newbie
(To:mark_neil2)
How would you rewrite (R1*R3+R1*R4+R2*R3+R2*R4)/(R1+R2+R3+R4) in continued fraction form? What is the underlying topology?
__________________
� � � � Tom Gutman
1-Newbie
(To:TomGutman)
On 2/19/2009 1:07:32 AM, Tom_Gutman wrote:
>How would you rewrite
>(R1*R3+R1*R4+R2*R3+R2*R4)/(R1+
>R2+R3+R4) in continued
>fraction form? What is the
>underlying topology?
>__________________
>� � � � Tom Gutman
___________________

Good question for Watson !
Maybe "abstract topology".
About nothing to do with the example. More amazing is the parfrac R1 = R2 expansion ... R3 = R4 expansion. Looks like the "Ring of Fire" [Nemo].

jmG
1-Newbie
(To:ptc-1368288)
On 2/19/2009 2:17:15 AM, jmG wrote:
>On 2/19/2009 1:07:32 AM, Tom_Gutman
>wrote:
>>How would you rewrite
>>(R1*R3+R1*R4+R2*R3+R2*R4)/(R1+
>>R2+R3+R4) in continued
>>fraction form? What is the
>>underlying topology?
>>__________________
>>� � � � Tom Gutman
>___________________
>
>Good question for Watson !
>Maybe "abstract topology".
>About nothing to do with the example.
>More amazing is the parfrac R1 = R2
>expansion ... R3 = R4 expansion. Looks
>like the "Ring of Fire" [Nemo].
>
>jmG

ok, I felt that I need to make more explanations about my aim in this thread. Please see two attached files. Just read the Word file first and then the Mathcad file. These two files should be sufficient to explain my what I am trying to do.

Mark.
1-Newbie
(To:mark_neil2)
From your example, I doubt it.

You were unable to show how to decompose the simple four parameter system I proposed. If you cannot show any algorithm for that very simple case, how can you expect to handle more general cases.

You have moved rather far from a continued fraction form. You are now looking for general series/parallel decomposition. That is not in general a continued fraction, although it is a complex fraction nested fractions. Only in some special cases, as noted by Lou, will that be a continued fraction. Further, you have not even expressed the entire expression in this form. You are pulling out apparently arbitrary pieces of the equation to decompose.
__________________
� � � � Tom Gutman
1-Newbie
(To:mark_neil2)
On 2/19/2009 8:45:47 PM, mark_neil wrote:
>On 2/19/2009 2:17:15 AM, jmG wrote:
>>On 2/19/2009 1:07:32 AM, Tom_Gutman
...
>Mark.
___________________________

Assume nobody wrote anything !

You told me before it was pure numbers, but now you show the dependence. It was clear to me it could not be pure numbers, anyway. Those transformations are not strange to me at all. In servomechanism we use these kind of circuit analysis for studying the response of the system at various perturbations and under the system own response ... call these circuits "Notch". But those things are about 45 years behind my back !

The software you are using builds that rational result from the Wallis algorithm, but the Wallis algo take the CF [Continued Fraction] as you have entered it from the components... and irony: Mathcad/Maple has that Wallis algo built-in !

Therefore: using Mathcad 11 you don't need the other software.

The topology of the CF therefore the topology of the components obeys some established rules that your book should be full of examples. Those things are not new, thus the Mathcad/Maple have implemented the Wallis, CF, rational expansions and partial fraction expansion. It might not be possible to cut the mustard as fine as you intend and why for, otherwise than for stability plot and/or filtering purpose. In the 11 resource, you will find a Qs PID, it shows about the undocumented symbolic numer, denom. You seem to dismantle/re-assemble the wrong way around, you must plot also. The other point in advance is about if you intend to filter. There are the numerical Chebyshev ... and the numerical Hilbert filters.

I have about no more to offer. You should consider what you can do rather than what you would like but going in the right direction, i.e: starting by the topology of the "ladder" circuit you are analyzing. No need for a Word document and the latent virus it carries.

jmG
1-Newbie
(To:ptc-1368288)

jmG
1-Newbie
(To:ptc-1368288)
... step by step:

1. If you can build the "ladder CF" from source, you don't need the other software.
2. Your "ladder CF": you seem to have checked equivalent to the first expansion.
3. The complete XFR fnct is the "ladder CF"
4. You just need to collect X for a new and more maneuverable expression.
6. Your solve 's' is incorrect.
7. Assign values to all components, plot the green and the olive.

If the two plots agree: project done !
Follow the book models, go as proposed, assign numerical values and study the root/locus as per the Qs PID [Mathcad 11.2a]

Obviously the long olive equation is for amusing the gallery vs the green XFR fnct. That's not all if you are in the frequency domain. Your 's' is 'f' and there may be some 'L' too, somewhere with C ...

Your project is all flexible and correct. The detail you didn't mentioned is about the other software: does it recognize a drawn circuit or do you feed it with the model "ladder" ?

jmG
1-Newbie
(To:ptc-1368288)
... so simple and elegant !
as long as your Ao, num/denom are correct.
It should be so going by the source book.
Now scrap the other software.

jmG
1-Newbie
(To:ptc-1368288)
... a beautiful piece of project, a model very similar to the Hydro transport lines. That gorgeous functionality is another proof Mathcad is the "baby" of Electrical Engineers.

Saved for the next collab.
Thanks again for the opportunity to collaborate.

jmG
5-Regular Member
(To:ptc-1368288)
The combination of series and parallel impedances will give a type of continued fraction, but not one in a canonical form.

I would expect that the continued fraction would have mixed terms at various levels, rather than single terms that we would expect from a pure maths viewpoint.

'n' impedances in series
Rs = R1 + R2 + R3 + ... Rn

in parallel
1/Rp = 1/r1 + 1/R2 + .. 1/Rn

The other theorem is Rosen's which simplifies a multiply connected network that would not fit simple series/parallel simplification. (see for example http://www.elect.mrt.ac.lk/EE201_network_theorems.pdf)

The end result is a repeated fraction. I think this is what is expected / produced by the other software.

Philip Oakley
>The end result is a repeated fraction. I think this is what is expected / produced by the other software < .
__________________________

What is a "repeated fraction" ?

From the work done on the project, from the immense work posted in this collab on Thiele applications, from project previous to "my Mathcad":

The other software expands what Mathcad expands. It expands CF, whatever their form and the particular project consists in dressing the CF appropriately.

jmG
10-Marble
(To:mark_neil2)
I still don't think that you will find a general algebraic method to do what you want. As Tom's simple example, of four resistors, there are many combinations of series-parallel configs (prob grow factorially with the number of components, but it grows quickly in any case). His example is (R1+R2)//(R3+R4), but I don't know how to write general rules to look for this type of behavior.

As per your amplifier example, the denominator decomposition works because the left side of the network is a ladder section that has a simple two-terminal Thevenin equivalent looking to the left from the base. If there was a component - say the capacitor - that connected to the Rs-Rx-Rb node rather than the base, then this decomposition probably wouldn't work.

At a different level, the "simplifications" you are looking for are in fact tied to the network graph. If you start with the network, then by inspection one can see that Rs-Rb-Rx-Rp will show up in a fixed, Requiv form. This is not so obvious once you have a rational network function, and want to reverse the process without any knowledge of the network graph. If the process is to be automated, then I suspect that the network topology needs to guide the decomposition, and not just the algebra.

Are you looking to derive the netwrk based only on the given network function (the synthesis problem), or do you have the circuit and want to put the network function in some sort of canonical form to the extent possible?

Lou
1-Newbie
(To:LouP)
I disagree.
This kind of circuits don't change much except for the "notch" filtering that I have described previously. The circuit in demonstration is typical of a saturated amplifier for stability and independence about aging of the components and their tolerances, thus providing a predictable almost invariant gain. This circuit is typical of the analog computers circuitry. And the knowledge is to dress the CF for each case of application. That knowledge is what the collab is exercising from source and help from this collaboratory.
Your example of multiple combinations that don't have CF representation are just view of the mind, not applicable in designing circuits. There is nothing new because it's a transistors. Tube computers had the same basic concept as well as good HI_FI and typically the "Process control & Instrumentation" equipment ... Oh ! my goodness, how much they were refined and doing things that numerical maths hardly cope.

The XFR fnct Mark was looking for comes out immediately, Mathcad /Maple were initially very advanced in that kind of stuff: servomechanisms, XFR funct, InvereLaplace... If you have read some of my worksheets, you can see how quick and nicely it administrates the Laplace transform when directly expressed in their modular form. Engineers don't design from wild, rather from shared knowledge and skill acquired via their respective discipline and source.

It's like using Pi in functions:
you don't calculate it first, just get it.

jmG

1-Newbie
(To:ptc-1368288)
! Mark !... if you capture this thread:

There is an error:
1. EITHER in the other software
2. OR in your script in the olive formula
3. OR in the dressing of your CF

Good news: Mathcad CF expansion is correct.

jmG
1-Newbie
(To:LouP)
On 2/20/2009 9:03:45 AM, lpoulo wrote:
>I still don't think that you
>will find a general algebraic
>method to do what you want. As
>Tom's simple example, of four
>resistors, there are many
>combinations of
>series-parallel configs (prob
>grow factorially with the
>number of components, but it
>grows quickly in any case).
>His example is
>(R1+R2)//(R3+R4), but I don't
>know how to write general
>rules to look for this type of
>behavior.
>
>the denominator decomposition
>works because the left side of
>section that has a simple
>two-terminal Thevenin
>equivalent looking to the left
>from the base. If there was a
>component - say the capacitor
>- that connected to the
>Rs-Rx-Rb node rather than the
>base, then this decomposition
>probably wouldn't work.
>
>At a different level, the
>"simplifications" you are
>looking for are in fact tied
>to the network graph. If you
>by inspection one can see that
>Rs-Rb-Rx-Rp will show up in a
>fixed, Requiv form. This is
>not so obvious once you have a
>rational network function, and
>want to reverse the process
>without any knowledge of the
>network graph. If the process
>is to be automated, then I
>suspect that the network
>topology needs to guide the
>decomposition, and not just
>the algebra.
>
>Are you looking to derive the
>netwrk based only on the given
>network function (the
>synthesis problem), or do you
>have the circuit and want to
>put the network function in
>some sort of canonical form to
>the extent possible?
>
>Lou

Thanks to all who have participated in this discussion.
There were comments specific to the amplifier example. I used this example just because I had its analysis in my book that I can use to make my point, not that I am interested in the analysis of this amplifier.

Lou: The only analysis method that I know of that will generate the transfer function in parallel/series combination of the curcuit elements is the EET or NEET method (http://www.rdmiddlebrook.com). I am trying to transform the transfer function into the form that the EET generates by starting from fully expanded transfer function that is generated by a symbolic circuit analysis software.
If the cap is connected to the node Rs, Rx, Rb, then the transfer function may not have parallel combination of these resistors even if I solve it with EET method. Then I don't expect to have parallel combinations of these resistors to show up in the transfer function. So, the algorith I am looking for may or may not find a parallel combination. It should find it only if it exists; it doesn't find it if such combination doesn't exist in the transfer fucntion.
If a fully automated algorithm can be written, it may take very long time for the algorithm to put the transfer function in series/parallel combinations depending on the number of components in the transfer function. The time it takes would increase exponentially with the number of components in the circuit. Therefore, it would be fine to give some help to the algorith as what components may have show up as parallel/series connections knowing the topology. Then the algorithm would look for such parallel combinations in the transfer function. So, this requires some symbolic keywords that look for specific combinations and extract those combinations and express the transfer function in terms of those combinations. For example, in the case of my amplifier example: assume I have Ao, numer(s) and denom(s) that can be easily obtained from the transfer function calculated by the symbolic simulator. And I know the circuit diagram. From the circuit diagram, I can guess that the transfer functions may be expressed as parallel combination Rb and Rs. Then I use this imaginary symbolic keyword (or some Mathcad program) to look for Rs*Rb/(Rs+Rb) or 1/(1/Rs+1/Rb) in the transfer function and it re-express the transfer function having this parallel combination in it. Then I can replace this parallel combination with a paralemer (say X). Then my new form of transfer function has less parameters in it and less complex (because it has reduced down to a simpler form). Then in the new form of the transfer function I look for another combination of series or parallel combinations. Therefore, the process is guided by the user, rather than being fully automatic. I guess all that is needed is some Mathcad program that looks for certain combinations of parameters in a given polynomial and it expresses it with the combination of these parameters. With such program, one can put the transfer function in series/parallel combination by a few number of trials/iterations.
I learn as post and read replies. So, I know I asked for a CFE in my first post and now I am asking about some other thing. Thanks for all your replies. I have now better picture of what I am looking for.

Mark.
5-Regular Member
(To:mark_neil2)
On 2/20/2009 2:32:57 PM, mark_neil wrote:
>Lou: The only analysis method that I
>know of that will generate the transfer
>function in parallel/series combination
>of the curcuit elements is the EET or
>NEET method
>(http://www.rdmiddlebrook.com). I am
>trying to transform the transfer
>function into the form that the EET
>generates by starting from fully
>expanded transfer function that is
>generated by a symbolic circuit analysis
>software.
>
>Mark.

Looks a useful site, specifically when you get to http://www.rdmiddlebrook.com/D_OA_Rules&Tools/index.asp which has all the slides as downloads.

I likes the motivation slides at the beginning which helped expalin the approach and issues for many cases.

I also have problems that are broader than that, where deciding the specification is part of the problem so you have to burn the candle at both ends to see if joins in the middle 😉

He still makes a lot of good points about approximations etc. The number of folk who won't, or can't, make first order calculations to see if they are in the right ball park is many (that'd be >>80% in my estimation, possibly >95%). E.g. when can you use a small angle approximation for sin, and when do you do you need the second order term for cos, or need to normalise a DCM so formed [That was on Thursday]

Philip Oakley
Thanks for the link, nice visit.
But it does not introduce to CF. It shows some partial fraction and that goes with the Mathcad CF package. I expected to see Mark's circuit, nowhere to see that model design.

"The key is Design-Oriented Analysis in terms of Low-Entropy Expressions, which enables you to solve "real life" design problems by keeping the algebra under control".

It seems difficult in this collab to reduce entropy. Solving one problem generates 2 more and so on and often deviates from the original question.

jmG

1-Newbie
(To:mark_neil2)
>I guess all that is needed is some Mathcad program that looks for certain combinations of parameters in a given polynomial and it expresses it with the combination of these parameters. With such program, one can put the transfer function in series/parallel combination by a few number of trials/iterations <.
____________________

You have missed the point I made in one of the very early reply and exemplified in the individual sheet *.... Continued fraction. Whichever way you consider this kind of "XFR analysis" it resumes in terms of Laplace algebra, and consequently results in the CF that it needs be interpreted correctly as per some helpful reference like "Handbooks". Once that done and assigned numerical values you get the minimum numerical equivalent circuit, i.e: the "Unit XFR".
From there and purporting some specifics of the project and giving yourself an XFR that would have to meet a specific graph from a client, Mathcad will find the corresponding values of all the components

If you have other wish about rearranging the components, you just need to rearrange the CF. I'm sorry if it looks dense as it reads, the attached work sheet is there to help. The only skill needed is the CF representation of the circuit, but you have source for that.

I tried to visit the site but the *.PDF crashed my PC. It didn't capture my interest. I have more about Mathcad/Maple CF, but some are reverse engineered and other arguably reverse engineered therefore not for this collab or otherwise by e-mail.

For unknown reason, though Mathcad does the project nicely, it does NOT like playing too much with it. The trick is to go on a new sheet.

You can contribute if you wish:
If you have some kind of library of basic circuits and the CF, welcome. They will go in he main work sheet and will serve again the Mathcad user community. Read again the other thread where it points an error in some of the terms Ao*[num/denom]

jmG
1-Newbie
(To:ptc-1368288)
On 2/20/2009 6:17:25 PM, jmG wrote:

>
>You have missed the point I
>made in one of the very early
>individual sheet *....
>Continued fraction. Whichever
>way you consider this kind of
>"XFR analysis" it resumes in
>terms of Laplace algebra, and
>consequently results in the CF
>that it needs be interpreted
>reference like "Handbooks".
>Once that done and assigned
>numerical values you get the
>minimum numerical equivalent
>circuit, i.e: the "Unit XFR".
>From there and purporting some
>specifics of the project and
>giving yourself an XFR that
>would have to meet a specific
>will find the corresponding
>values of all the components
>jmG

jmG,
I have seen your Mathcad file but found it irrelevant to my question (maybe I didn't understand it). Your file uses the A0, numer(s) and denom(s) as given by the textbook. Just keep in mind that I don't have them as nicely arranged with series/parallel combinations of the circuit components but instead fully expanded out. I am trying to transform my tranfer function into the form in your Mathcad file (as arranged my series/parallel combinations). You have assumed some numeric values and did more stuff. Assigning numeric values is the last thing to do it is no challenge. Just to let you know, I use MC14. Some lines on your files do not execute on my MC14; I think due to the differences bewen MC11 and MC14.

Mark.
1-Newbie
(To:mark_neil2)
On 2/20/2009 6:45:53 PM, mark_neil wrote:
>jmG,
I have seen your Mathcad file but found it irrelevant to my question (maybe I didn't understand it). Your file uses the A0, numer(s) and denom(s) as given by the textbook. Just keep in mind that I don't have them as nicely arranged with series/parallel combinations of the circuit components but instead fully expanded out. I am trying to transform my transfer function into the form in your Mathcad file (as arranged my series/parallel combinations). You have assumed some numeric values and did more stuff. Assigning numeric values is the last thing to do it is no challenge. Just to let you know, I use MC14. Some lines on your files do not execute on my MC14; I think due to the differences between MC11 and MC14.

Mark.
>_______________________

What's red in there ? see the *.gif attached.

"maybe I didn't understand it"
==> That was obvious from the very beginning, thus I insisted and did my best to gear you in the track. The XFR fnct is as simple as Mathcad does and as in the site in reference, A + num/[x+denom]. Mathcad solves for the CF elements as per the given XFR, but you must read the latest work sheet "CF XFR Topology SOLVE" [*], and that's why you MUST assign numeric values. You arrange, i.e: make both circuits and the CF, Mathcd does the rest.

If you can't read the works [*], sorry, I'm out because of version incompatibility. But I doubt your version does not execute the given/Find.

jmG

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