[SI-LIST] Input impedance expression for 3 port system

Discussion:

Sumant Srikant

2018-11-05 07:35:29 UTC

Hello All,
I am trying to derive a expression for calculating the input impedance of a
3 port network to use as direct calculating code and avoid SPICE/simulator
solving of the same.

I am able to solve the input impedance of a 2 port system with a
load,Zload, connected to the port2. The impedance looking into port 1 would
be( by solving basic 2 port theory) :

Zin= Z11 - (Z21*Z12)/(Z22+Zload)

If i have a 3 port network with 2 ports connected to different loads ,
Zload1 and Zload2, what would be the generated simplified expression for
Zin looking in from port1 and having only the terms of the Z-matrix,Zload1
and Zload2?
--
Regards
Sumant

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Dmitriev-Zdorov, Vladimir

2018-11-07 00:41:25 UTC

Permalink

Hi Sumant,

I'm not aware about ready-to-use expressions for 3+ port systems. Even if they are published somewhere, they could be complicated and difficult to use. The best way is learn how to find them for any number of ports, and you'll need to apply some algebra to do that. Let's think in terms of equations and unknowns. N-port Z-parameters provide N equations that relate N input port currents and N port voltages. (These equations are linear and without constant terms). Hence, you have N equations and 2N unknowns. In case N=1, it is a single relation between one port voltage and current, from where you can find the ratio V/I = z1.

If your model has more ports, voltages and currents referring to other ports could be excluded by using additional equations given by termination conditions on those ports. For N-port model, with N-1 terminations on the remaining N-1 ports you have additional N-1 equations: Vn = - Zload_n * In. Sign "-" is because you measure current flowing into the port, and positive voltage Vn would create negative port current with a passive load. In total, you have 2N variables, and N + (N-1) equations. This will allow you to exclude all variables related to other ports and leave just one equation that includes voltage and current of the port of interest. Then, you can find their ratio, which is an input impedance of admittance.

Vladimir

From: Sumant Srikant <***@gmail.com>
Date: Mon, 5 Nov 2018 16:35:29 +0900
Subject: [SI-LIST] Input impedance expression for 3 port system

Hello All,
I am trying to derive a expression for calculating the input impedance of a
3 port network to use as direct calculating code and avoid SPICE/simulator
solving of the same.

I am able to solve the input impedance of a 2 port system with a
load,Zload, connected to the port2. The impedance looking into port 1 would
be( by solving basic 2 port theory) :

Zin= Z11 - (Z21*Z12)/(Z22+Zload)

If i have a 3 port network with 2 ports connected to different loads ,
Zload1 and Zload2, what would be the generated simplified expression for
Zin looking in from port1 and having only the terms of the Z-matrix,Zload1
and Zload2?

--
Regards
Sumant

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Ihsan Erdin

2018-11-08 14:20:15 UTC

Permalink

All you need is to put the scalar expression into matrix format:
zin=z11-[Z12]*inv( [Z22]+[ZL] ) *[Z21], where z11 is a scalar term as the
first entry of your original 3x3 matrix. [Z12]=[z12 z13],
[Z21]=transpose(Z12), [Z22]=[z22 z23; z32 z33] and [ZL]=diagonal(zL1, zL2).
The final expression for zin can easily be obtained through elementary
matrix operations but unlike the 2-port case, it will be a mess of
z-parameter terms with little practical use.
If you want to dig further, the idea has already been published for an
N-load case in last year's EDAPS conference and titled as "Analysis of
Decoupling Capacitors Inside Via Arrays with Mutual Interaction."

Regards.

Ihsan Erdin

Post by Sumant Srikant
Hello All,
I am trying to derive a expression for calculating the input impedance of a
3 port network to use as direct calculating code and avoid SPICE/simulator
solving of the same.
I am able to solve the input impedance of a 2 port system with a
load,Zload, connected to the port2. The impedance looking into port 1 would
Zin= Z11 - (Z21*Z12)/(Z22+Zload)
If i have a 3 port network with 2 ports connected to different loads ,
Zload1 and Zload2, what would be the generated simplified expression for
Zin looking in from port1 and having only the terms of the Z-matrix,Zload1
and Zload2?
--
Regards
Sumant
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Sumant Srikant

2018-11-09 09:16:51 UTC

Permalink

Thank you very much for pointing me to the paper.
It was very useful and a N port input impedance calculator can be very
easily implemented using the paper data

Regards
Sumant

Post by Ihsan Erdin
zin=z11-[Z12]*inv( [Z22]+[ZL] ) *[Z21], where z11 is a scalar term as the
first entry of your original 3x3 matrix. [Z12]=[z12 z13],
[Z21]=transpose(Z12), [Z22]=[z22 z23; z32 z33] and [ZL]=diagonal(zL1, zL2).
The final expression for zin can easily be obtained through elementary
matrix operations but unlike the 2-port case, it will be a mess of
z-parameter terms with little practical use.
If you want to dig further, the idea has already been published for an
N-load case in last year's EDAPS conference and titled as "Analysis of
Decoupling Capacitors Inside Via Arrays with Mutual Interaction."
Regards.
Ihsan Erdin