- Impedance parameters
Impedance parameters or Z-parameters are properties used in
electrical engineering ,electronics engineering , and communication systems engineering describe the electrical behavior oflinear electrical network s 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 , [David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc., 2005; pp. 170-174, ISBN 0-471-44878-8.]Y-parameters , [David M. Pozar, 2005 (op. cit); pp 170-174.]H-parameters ,T-parameters orABCD-parameters . [David M. Pozar, 2005 (op. cit); pp 183-186.] [A.H. Morton, " Advanced Electrical Engineering", Pitman Publishing Ltd., 1985; pp 33-72, ISBN 0-273-40172-6.]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 notation. In general the elements of the Z-parameter matrix are
complex number s.The phase part of a Z-parameter is the "spatial" phase at the test frequency, not the temporal (time-related) phase.
Two-Port Networks
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} = egin{pmatrix} Z_{11} & Z_{12} \ Z_{21} & Z_{22} end{pmatrix}{I_1 choose I_2} .
where
:Z_{11} = {V_1 over I_1 } igg|_{I_2 = 0} qquad Z_{12} = {V_1 over I_2 } igg|_{I_1 = 0}
:Z_{21} = {V_2 over I_1 } igg|_{I_2 = 0} qquad Z_{22} = {V_2 over I_2 } igg|_{I_1 = 0}
For the general case of an n-port network, it can be stated that:Z_{nm} = {V_n over I_m } igg|_{I_n = 0}
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.
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. [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.]
: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 S_{ij} and Z_{ij}. Note that the value of Delta can become 0 for specific values of Z_{ij} so the division by Delta in the calculations of S_{ij} may lead to a division by 0.
S-parameter conversions into other matrices by simply multiplying with e.g. Z_0 = 50Omega are only valid if the characteristic impedance Z_0 is not frequency dependent.
Conversion from
Y-parameters to Z-parameters is much simpler, as the Z-parameter matrix is basically thematrix 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 Delta_Y is the
determinant of the Y-parameter matrix.References
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.ee also
*
Scattering parameters
*Admittance parameters
*Two-port network
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