On Topic: ABCD (Transmission) Matrix Math

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Suppose I have a source impedance, an ABCD matrix, and a load impedance.

What formulas yield these outputs?:

Input V(Source)
Input I(Source)
Input Z
Input Return Loss
Output V(Load)
Output I(Load)
Output Z
Output Return Loss
Power Gain
Voltage Gain
Current Gain

For example:

Zload applied to ABCD (on the right hand side) gives the input impedance.  
(I say "applied", because this operation doesn't have a clear meaning within  
linear algebra.)

Vin is given by the impedance divider between Zsrc and Zin.

For output-referred quantities, invert the matrix and apply the source  
impedance as "load" instead.

Etc.

I've drawn up likely definitions for these, but I'm suspicious, and would  
like someone to "check my homework".  (If you'd like me to "show your work  
first", it's all nicely implemented in JS here:  
https://www.seventransistorlabs.com/Calc/Filter1~.html :) )

Tim

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Re: On Topic: ABCD (Transmission) Matrix Math
On Wednesday, July 5, 2017 at 5:53:41 PM UTC-4, Tim Williams wrote:
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Hmm, Tim I'm not sure I can help.  And first you'd have to tell  
me what an ABCD matrix is.  (Way back as an undergrad I recall doing  
circuit analysis with matrices.. basically just shorthand way of writing  
down linear equations, but I've forgotten most of it.)  

George H.  
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Re: On Topic: ABCD (Transmission) Matrix Math
On 07/06/2017 08:07 AM, George Herold wrote:
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They're the "transmission matrix" parameters; AFAIK they're the only set  
of two-port matrix representations that allow you to model a more  
complex network by defining a matrix for a series connected component of  
a specific type, and a shunt connected component of a specific type, and  
when the two are multiplied (left to right, as in general A.B != B.A) it  
gives you the ABCD matrix for the series-shunt pair. And so on and so forth.

They make it a mechanical to find the transfer function of ladder  
networks that would be a bear to analyze just trying to grunge thru KCL  
or KVL, as all the properties hold when dealing with LTI systems in the  
"s" domain, too.

idk why Mr. Williams is trying to use them to find circuit properties  
other than the transfer function directly as there are transformations  
between ABCD parameters and all the other types (Z parameters, Y  
parameters, S parameters); for calculating quantities like input and  
output impedances it seems like it would be significantly easier to  
transform the complete ABCD matrix to parameters of a more appropriate type

Re: On Topic: ABCD (Transmission) Matrix Math
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Yeah, short of implementing a fully general matrix method -- which would be  
fine as well, but then I have to parse a netlist, AND one would hope,  
generate that netlist from a schematic capture.

Both of which exist in JS, or at least in free software packages I can  
borrow or port -- but that's at least double the work I've already done on  
this calculator tool so far, and I'm happy enough implementing a ladder  
network simulator just for starters!

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I'm aware of all the transformations; but I'm not aware of any references  
showing how any of these forms can be reduced to the types of 'measurements'  
I'm asking about.

In other words, given the ABCD matrix (or the H or G or Z or any other  
two-port matrix), and a source and a load impedance: how can I get input  
return loss, or power gain, or any of the others?

The impedance ones are straightforward, so I'm not worried about those, but  
I am concerned about the definitions I used on the gain and loss ones.

Just poking at a matrix, say to see all the poles and zeroes -- that's fine  
for analytical work, but my calculator is geared towards in-circuit  
parameters you'd measure in the real world.  You never measure A, B, C and D  
directly; you might measure port voltages, or currents, or impedances, or  
scattering, which have to be converted into any of the available forms.  I  
want to do the reverse: starting with a matrix, what are the useful  
parameters you'll measure in a real circuit?

Tim

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Re: On Topic: ABCD (Transmission) Matrix Math
On 07/06/2017 08:57 PM, Tim Williams wrote:

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The equations for two-port transducer power gain here aren't referenced  
to anything authoritative but appear to agree with what's in my copy of  
Wes Hayward's "Introduction to Radio Frequency Design":

<https://en.wikipedia.org/wiki/Power_gain#Transducer_power_gain


Re: On Topic: ABCD (Transmission) Matrix Math
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Nice, thanks.  Does Hayward have definitions for the others, by any chance?

Tim

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Re: On Topic: ABCD (Transmission) Matrix Math
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The Wiki page is a good enough introduction:
https://en.wikipedia.org/wiki/Two-port_network#ABCD-parameters
Note the diagram at the top of the page.  The box can be any circuit with  
the connections shown; the matrix is only concerned with the relations  
between the ports.

As in the examples I gave: by setting V2 = I2 * Z2, you apply a load  
resistance to the second port; you can then reduce the pair of equations to  
a single equation in V1/I1 = Z1, the input impedance.  And then you can play  
with this impedance as any old two-terminal (one-port) impedance, for  
instance, putting it in an impedance divider, against a source impedance, to  
get an overall gain.

But I'm not sure that I'm using the correct definitions, say for voltage or  
power gain, and would like to check them.

Tim

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Re: On Topic: ABCD (Transmission) Matrix Math
I wrote up a page on ABCD matrices some time ago:

http://www.milton.arachsys.com/nj71/index.php?menu=2&submenu=2&subsubmenu14%

Neil



Re: On Topic: ABCD (Transmission) Matrix Math
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Great!  I think I have the basic matrix math working well.

Could you provide formulas for the kinds of 'measurements' listed in the OP?

Tim

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Re: On Topic: ABCD (Transmission) Matrix Math
On 05/07/17 23:53, Tim Williams wrote:
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The ABCD matrix gives you a relation between V1, I1, V2 and I2, with V1  
(V2) the input (output) port voltages and I1 (I2) being the current  
flowing into the + terminal of port 1 (2)

V1   A B    V2
    =     *
I1   C D   -I2

There are several ways to solve this. For instance, a load impedance ZL  
introduces a new relation between V2 and -I2: V2=ZL*(-I2). In this case  
you get:

V1= A*ZL*(-I2) + B*(-I2)
I1= C*ZL*(-I2) + D*(-I2)

and dividing both you get

Zin=(A*ZL+B)/(C*ZL+D)

Knowing Vin, this gives you Iin and vice-versa. It gives you also the  
input return loss.

Once you have V1 and I1, matrix inversion gives you V2 and -I2 and from  
these you get everything you need!

Pere

Re: On Topic: ABCD (Transmission) Matrix Math
On Thursday, July 6, 2017 at 3:23:41 AM UTC+5:30, Tim Williams wrote:
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Please check out David Pozar's classic text
-- Microwave Engineering - chapter on Network
Theory. The 4th edition is freely downloadable.

Re: On Topic: ABCD (Transmission) Matrix Math
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Oh nice, a whole chapter too!

Having skimmed it, I'm still not seeing anything about my questions  
specifically though. :-\  I guess I'll have to convert it to scattering  
parameters then use the formulas for that.

Tim

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Electrical Engineering Consultation and Contract Design
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