On Topic: ABCD (Transmission) Matrix Math

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.

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

I wrote up a page on ABCD matrices some time ago:

Neil

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 
Website: http://seventransistorlabs.com

The Wiki page is a good enough introduction:

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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Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

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' 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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Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

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":

Nice, thanks. Does Hayward have definitions for the others, by any chance?

Tim

--
Seven Transistor Labs, LLC 
Electrical Engineering Consultation and Contract Design 
Website: http://seventransistorlabs.com

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

Please check out David Pozar's classic text

-- Microwave Engineering - chapter on Network Theory. The 4th edition is freely downloadable.

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 
Website: http://seventransistorlabs.com