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**posted on**

- Tim Williams

July 5, 2017, 9:53 pm

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

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

--

Seven Transistor Labs, LLC

Electrical Engineering Consultation and Contract Design

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

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

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'

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

--

Seven Transistor Labs, LLC

Electrical Engineering Consultation and Contract Design

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":

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

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

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

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

I1= C

and dividing both you get

Zin=(A

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

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

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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