Two-port network

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Example two-port network
Example two-port network

A two-port network (or four-terminal network, or quadripole) is an electrical circuit or device with two pairs of terminals. Examples include transistors, filters and matching networks. The analysis of two-port networks was pioneered in the 1920s by Franz Breisig, a German mathematician.

A two-port network basically consists in isolating either a complete circuit or part of it and finding its characteristic parameters. Once this is done, the isolated part of the circuit becomes a "black box" with a set of distinctive properties, enabling us to abstract away its specific physical buildup, thus simplifying analysis. Any circuit can be transformed into a two-port network provided that it does not contain an independent source.

The parameters used in order to describe a two-port network are the following: Z, Y, h, g, T. They are usually expressed in matrix notation and they establish relations between the following parameters:

Input voltage V1
Output voltage V2
Input current I1
Output current I2

Contents

[edit] Z-parameters (impedance parameters)

{V_1 \choose V_2} = \begin{pmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{pmatrix}{I_1 \choose I_2}.

where

Z_{11} = {V_1 \over I_1 } \bigg|_{I_2 = 0} \qquad Z_{12} = {V_1 \over I_2 } \bigg|_{I_1 = 0}
Z_{21} = {V_2 \over I_1 } \bigg|_{I_2 = 0} \qquad Z_{22} = {V_2 \over I_2 } \bigg|_{I_1 = 0}

[edit] Y-parameters (admittance parameters)

{I_1 \choose I_2} = \begin{pmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{pmatrix}{V_1 \choose V_2}.

where

Y_{11} = {I_1 \over V_1 } \bigg|_{V_2 = 0} \qquad Y_{12} = {I_1 \over V_2 } \bigg|_{V_1 = 0}
Y_{21} = {I_2 \over V_1 } \bigg|_{V_2 = 0} \qquad Y_{22} = {I_2 \over V_2 } \bigg|_{V_1 = 0}

[edit] h-parameters (hybrid parameters)

{V_1 \choose I_2} = \begin{pmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{pmatrix}{I_1 \choose V_2}.

where

h_{11} = {V_1 \over I_1 } \bigg|_{V_2 = 0} \qquad h_{12} = {V_1 \over V_2 } \bigg|_{I_1 = 0}
h_{21} = {I_2 \over I_1 } \bigg|_{V_2 = 0} \qquad h_{22} = {I_2 \over V_2 } \bigg|_{I_1 = 0}

[edit] g-parameters (inverse hybrid parameters)

{I_1 \choose V_2} = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix}{V_1 \choose I_2}.

where

g_{11} = {I_1 \over V_1 } \bigg|_{I_2 = 0} \qquad g_{12} = {I_1 \over I_2 } \bigg|_{V_1 = 0}
g_{21} = {V_2 \over V_1 } \bigg|_{I_2 = 0} \qquad g_{22} = {V_2 \over I_2 } \bigg|_{V_1 = 0}

[edit] ABCD-parameters

The ABCD-parameters are known variously as chain, cascade, or transmission parameters.

{V_2 \choose I_2} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}{V_1 \choose I_1}.

where

A = {V_2 \over V_1 } \bigg|_{I_1 = 0} \qquad B = {V_2 \over I_1 } \bigg|_{V_1 = 0}
C = -{I_2 \over V_1 } \bigg|_{I_1 = 0} \qquad D = -{I_2 \over I_1 } \bigg|_{V_1 = 0}

Note that we have inserted negative signs in front of the fractions in the definitions of parameters C and D. The reason for adpoting this convention (as opposed to the convention adopted above for the other sets of parameters) is that it allows us to represent the transmission matrix of cascades of two or more two-port networks as simple matrix multiplications of the matrices of the individual networks. This convention is equivalent to reversing the direction of I2 so that it points in the same direction as the input current to the next stage in the cascaded network.

This technique is exactly analogous to the use of ABCD matrices for ray tracing in the science of optics. See also ray transfer matrix.

[edit] Table of transmission parameters

for simple network elements

Element Matrix Remarks
Series resistor \begin{pmatrix} 1 & -R \\ 0 & 1 \end{pmatrix} R = resistance
Shunt resistor \begin{pmatrix} 1 & 0 \\ -1/R & 1 \end{pmatrix} R = resistance
Series conductor \begin{pmatrix} 1 & -1/G \\ 0 & 1 \end{pmatrix} G = conductance
Shunt conductor \begin{pmatrix} 1 & 0 \\ -G & 1 \end{pmatrix} G = conductance
Series inductor \begin{pmatrix} 1 & -Ls \\ 0 & 1 \end{pmatrix} L = inductance
s = complex angular frequency
Shunt capacitor \begin{pmatrix} 1 & 0 \\ -Cs & 1 \end{pmatrix} C = capacitance
s = complex angular frequency

[edit] Example: Cascading two networks

Suppose we have a two-port network consisting of a series resistor R followed by a shunt capacitor C. We can model the entire network as a cascade of two simpler networks:

\mathbf{T}_1 = \begin{pmatrix} 1 & -R \\ 0 & 1 \end{pmatrix}
\mathbf{T}_2 = \begin{pmatrix} 1 & 0 \\ -Cs & 1 \end{pmatrix}

The transmission matrix for the entire network T is simply the matrix multiplication of the transmission matrices for the two network elements:

\mathbf{T} = \mathbf{T}_2 \cdot \mathbf{T}_1
= \begin{pmatrix} 1 & 0 \\ -Cs & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & -R \\ 0 & 1 \end{pmatrix}
= \begin{pmatrix} 1 & -R \\ -Cs & 1+RCs \end{pmatrix}

Thus:

\begin{pmatrix} V_2 \\ I_2 \end{pmatrix} = \begin{pmatrix} 1 & -R \\ -Cs & 1+RCs \end{pmatrix} \begin{pmatrix} V_1 \\ I_1 \end{pmatrix}

[edit] Note regarding definition of transmission parameters

It should be noted that all these examples are specific to the definition of transmission parameters given here. Other definitions exist in the literature, such as:

{V_1 \choose I_1} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}{V_2 \choose -I_2}

[edit] See also

[edit] References

    Pozar, David M. (2005). Microwave Engineering, 3rd Edition. John Wiley & Sons, 161-221. 

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