Impedance parameters

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Impedance parameters or Z-parameters are properties used in electrical engineering, electronics engineering, and communication systems engineering describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by small signals. They are members of a family of similar parameters used in electronics engineering, other examples being: S-parameters,[1] Y-parameters,[2] H-parameters, T-parameters or ABCD-parameters.[3][4]

Contents

[edit] The General Z-Parameter Matrix

For a generic multi-port network definition, it is assumed that each of the ports is allocated an integer 'n' ranging from 1 to N, where N is the total number of ports. For port n, the associated Z-parameter definition is in terms of input currents and output voltages, I_n\, and V_n\, respectively.

For all ports the output voltages may be defined in terms of the Z-parameter matrix and the input currents by the following matrix equation:

V = Z I\,

where Z is an N x N matrix the elements of which can be indexed using conventional matrix(mathematics) notation. In general the elements of the Z-parameter matrix are complex numbers.

The phase part of a Z-parameter is the spatial phase at the test frequency, not the temporal (time-related) phase.

[edit] Two-Port Networks

Schematic
Schematic

The Z-parameter matrix for the two-port network is probably the most common. In this case the relationship between the input currents, output voltages and the Z-parameter matrix is given by:

{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}

For the general case of an n-port network, it can be stated that

Z_{nm} = {V_n \over I_m } \bigg|_{I_n = 0}

[edit] Impedance relations

The input impedance of a two-port network is given by:

Z_{in} = z_{11} - \frac{z_{12}z_{21}}{z_{22}+Z_L}

where ZL is the impedance of the load connected to port two.

Similarly, the output impedance is given by:

Z_{out} = z_{22} - \frac{z_{12}z_{21}}{z_{11}+Z_S}

where ZS is the impedance of the source connected to port one.

[edit] Converting Two-Port Parameters

The two-port S-parameters may be obtained from the equivalent two-port Z-parameters by means of the following expressions.[5]

S_{11} = {(Z_{11} - Z_0) (Z_{22} + Z_0) - Z_{12} Z_{21} \over \Delta} \,
S_{12} = {2 Z_0 Z_{12} \over \Delta} \,
S_{21} = {2 Z_0 Z_{21} \over \Delta} \,
S_{22} = {(Z_{11} + Z_0) (Z_{22} - Z_0) - Z_{12} Z_{21} \over \Delta} \,

Where

\Delta = (Z_{11} + Z_0) (Z_{22} + Z_0) - Z_{12} Z_{21} \,

The above expressions will generally use complex numbers for Sij and Zij. Note that the value of Δ can become 0 for specific values of Zij so the division by Δ in the calculations of Sij may lead to a division by 0.

S-parameter conversions into other matrices by simply multiplying with e.g. Z0 = 50Ω are only valid if the characteristic impedance Z0 is not frequency dependent.

Conversion from Y-parameters to Z-parameters is much simpler, as the Z-parameter matrix is basically the matrix inverse of the Y-parameter matrix. The following expressions show the applicable relations:

Z_{11} = {Y_{22} \over \Delta_Y} \,
Z_{12} = {-Y_{12} \over \Delta_Y} \,
Z_{21} = {-Y_{21} \over \Delta_Y} \,
Z_{22} = {Y_{11} \over \Delta_Y} \,

Where

\Delta_Y = Y_{11} Y_{22} - Y_{12} Y_{21} \,

In this case ΔY is the determinant of the Y-parameter matrix.

[edit] References

  1. ^ David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc., 2005; pp. 170-174, ISBN 0-471-44878-8.
  2. ^ David M. Pozar, 2005 (op. cit); pp 170-174.
  3. ^ David M. Pozar, 2005 (op. cit); pp 183-186.
  4. ^ A.H. Morton, Advanced Electrical Engineering, Pitman Publishing Ltd., 1985; pp 33-72, ISBN 0-273-40172-6.
  5. ^ Simon Ramo, John R. Whinnery, Theodore Van Duzer, "Fields and Waves in Communication Electronics", Third Edition, John Wiley & Sons Inc.; 1993, pp. 537-541, ISBN 0-471-58551-3.

[edit] Bibliography

  • David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc.; ISBN 0-471-44878-8.
  • Simon Ramo, John R. Whinnery, Theodore Van Duzer, "Fields and Waves in Communication Electronics", Third Edition, John Wiley & Sons Inc.; ISBN 0-471-58551-3.

[edit] See also