Impedance parameters

Short description: Set of properties used in electrical engineering

Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal (linearized) response of non-linear networks. They are members of a family of similar parameters used in electronic engineering, other examples being: S-parameters,[1] Y-parameters,[2] H-parameters, T-parameters or ABCD-parameters.[3][4]

Z-parameters are also known as open-circuit impedance parameters as they are calculated under open circuit conditions. i.e., Ix=0, where x=1,2 refer to input and output currents flowing through the ports (of a two-port network in this case) respectively.

## Contents

* 1 The Z-parameter matrix
* 2 Two-port networks
* 2.1 Impedance relations
* 3 Relation to S-parameters
* 3.1 Two port
* 4 Relation to Y-parameters
* 5 Notes
* 6 References
* 7 Bibliography
* 8 See also

## The Z-parameter matrix

A Z-parameter matrix describes the behaviour of any linear electrical network that can be regarded as a black box with a number of ports. A port in this context is a pair of electrical terminals carrying equal and opposite currents into and out-of the network, and having a particular voltage between them. The Z-matrix gives no information about the behaviour of the network when the currents at any port are not balanced in this way (should this be possible), nor does it give any information about the voltage between terminals not belonging to the same port. Typically, it is intended that each external connection to the network is between the terminals of just one port, so that these limitations are appropriate.

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 the port current and port voltage, In and Vn respectively.

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

V=ZI

where Z is an N × N matrix the elements of which can be indexed using conventional matrix notation. In general the elements of the Z-parameter matrix are complex numbers and functions of frequency. For a one-port network, the Z-matrix reduces to a single element, being the ordinary impedance measured between the two terminals. The Z-parameters are also known as the open circuit parameters because they are measured or calculated by applying current to one port and determining the resulting voltages at all the ports while the undriven ports are terminated into open circuits.

## Two-port networks

The equivalent circuit for Z-parameters of a two-port network. The equivalent circuit for Z-parameters of a reciprocal two-port network.

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

(V1V2)=(Z11Z12Z21Z22)(I1I2).

where

Z11=V1I1|I2=0Z12=V1I2|I1=0

Z21=V2I1|I2=0Z22=V2I2|I1=0

For the general case of an N-port network,

Znm=VnIm|Ik=0 for k≠m

### Impedance relations

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

Zin=Z11−Z12Z21Z22+ZL

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

Similarly, the output impedance is given by:

Zout=Z22−Z12Z21Z11+ZS

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

## Relation to S-parameters

The Z-parameters of a network are related to its S-parameters by[5]

Z=z(1N+S)(1N−S)−1z=z(1N−S)−1(1N+S)z

and[5]

S=(yZy−1N)(yZy+1N)−1=(yZy+1N)−1(yZy−1N)

where 1N is the identity matrix, z is a diagonal matrix having the square root of the characteristic impedance at each port as its non-zero elements,

z=(z01z02⋱z0N)

and y=(z)−1 is the corresponding diagonal matrix of square roots of characteristic admittances. In these expressions the matrices represented by the bracketed factors commute and so, as shown above, may be written in either order.[5][note 1]

### Two port

In the special case of a two-port network, with the same characteristic impedance z01=z02=Z0 at each port, the above expressions reduce to

Z11=((1+S11)(1−S22)+S12S21)ΔSZ0

Z12=2S12ΔSZ0

Z21=2S21ΔSZ0

Z22=((1−S11)(1+S22)+S12S21)ΔSZ0

Where

ΔS=(1−S11)(1−S22)−S12S21

The two-port S-parameters may be obtained from the equivalent two-port Z-parameters by means of the following expressions[6]

S11=(Z11−Z0)(Z22+Z0)−Z12Z21Δ

S12=2Z0Z12Δ

S21=2Z0Z21Δ

S22=(Z11+Z0)(Z22−Z0)−Z12Z21Δ

where

Δ=(Z11+Z0)(Z22+Z0)−Z12Z21

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.

## Relation to Y-parameters

Conversion from Y-parameters to Z-parameters is much simpler, as the Z-parameter matrix is just the inverse of the Y-parameter matrix. For a two-port:

Z11=Y22ΔY

Z12=−Y12ΔY

Z21=−Y21ΔY

Z22=Y11ΔY

where

ΔY=Y11Y22−Y12Y21

is the determinant of the Y-parameter matrix.

## Notes

1. ↑ Any square matrix commutes with itself and with the identity matrix, and if two matrices A and B commute, then so do A and B−1 (since AB−1 = B−1BAB−1 = B−1ABB−1 = B−1A)

## References

1. ↑ David M. Pozar (2004-02-05). Microwave Engineering. Wiley. pp. 170–174. ISBN 978-0-471-44878-5.
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. ↑ 5.0 5.1 5.2 Russer, Peter (2003). Electromagnetics, microwave circuit and antenna design for communications engineering. Artech House. pp. 420. ISBN 1-58053-532-1.
6. ↑ Simon Ramo; John R. Whinnery; Theodore Van Duzer (1994-02-09). Fields and Waves in Communication Electronics. Wiley. pp. 537–541. ISBN 978-0-471-58551-0.

## Bibliography

* David M. Pozar (2004-02-05). Microwave Engineering. Wiley. ISBN 978-0-471-44878-5.
* Simon Ramo; John R. Whinnery; Theodore Van Duzer (1994-02-09). Fields and Waves in Communication Electronics. Wiley. ISBN 978-0-471-58551-0.

## See also

* Scattering parameters
* Admittance parameters
* Two-port network

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