# On Topic: ABCD (Transmission) Matrix Math

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• Posted on 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 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

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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Seven Transistor Labs, LLC
Electrical Engineering Consultation and Contract Design
We've slightly trimmed the long signature. Click to see the full one. Re: On Topic: ABCD (Transmission) Matrix Math
On Wednesday, July 5, 2017 at 5:53:41 PM UTC-4, Tim Williams wrote: 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.  Re: On Topic: ABCD (Transmission) Matrix Math
On 07/06/2017 08:07 AM, George Herold wrote: 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 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! 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'

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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Seven Transistor Labs, LLC
Electrical Engineering Consultation and Contract Design
We've slightly trimmed the long signature. Click to see the full one. Re: On Topic: ABCD (Transmission) Matrix Math
On 07/06/2017 08:57 PM, Tim Williams wrote: 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 Nice, thanks.  Does Hayward have definitions for the others, by any chance?

Tim

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Seven Transistor Labs, LLC
Electrical Engineering Consultation and Contract Design
We've slightly trimmed the long signature. Click to see the full one. Re: On Topic: ABCD (Transmission) Matrix Math 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

--
Seven Transistor Labs, LLC
Electrical Engineering Consultation and Contract Design
We've slightly trimmed the long signature. Click to see the full one. Re: On Topic: ABCD (Transmission) Matrix Math
I wrote up a page on ABCD matrices some time ago:

Neil Re: On Topic: ABCD (Transmission) Matrix Math 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

--
Seven Transistor Labs, LLC
Electrical Engineering Consultation and Contract Design
We've slightly trimmed the long signature. Click to see the full one. Re: On Topic: ABCD (Transmission) Matrix Math
On 05/07/17 23:53, Tim Williams wrote: 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: 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 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

--
Seven Transistor Labs, LLC
Electrical Engineering Consultation and Contract Design
We've slightly trimmed the long signature. Click to see the full one.

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