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.
>
>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
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.