Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
,
electronic engineering
Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current ...
, and
communication systems
A communications system or communication system is a collection of individual telecommunications networks, transmission systems, relay stations, tributary stations, and terminal equipment usually capable of interconnection and interoperatio ...
engineering to describe the electrical behavior of
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
electrical network
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...
s. They are also used to describe the
small-signal
Small-signal modeling is a common analysis technique in electronics engineering used to approximate the behavior of electronic circuits containing nonlinear devices with linear equations. It is applicable to electronic circuits in which the AC ...
(
linearized) response of non-linear networks. They are members of a family of similar parameters used in electronic engineering, other examples being:
S-parameters,
Y-parameters
Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used to describe the el ...
,
H-parameters
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
,
T-parameters or
ABCD-parameters
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
.
Z-parameters are also known as ''open-circuit impedance parameters'' as they are calculated under
open circuit Open circuit may refer to:
*Open-circuit scuba, a type of SCUBA-diving equipment where the user breathes from the set and then exhales to the surroundings without recycling the exhaled air
* Open-circuit test, a method used in electrical engineerin ...
conditions. i.e., I
x=0, where x=1,2 refer to input and output currents flowing through the ports (of a
two-port network
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
in this case) respectively.
The Z-parameter matrix
A Z-parameter matrix describes the behaviour of any linear electrical network that can be regarded as a
black box
In science, computing, and engineering, a black box is a system which can be viewed in terms of its inputs and outputs (or transfer characteristics), without any knowledge of its internal workings. Its implementation is "opaque" (black). The te ...
with a number of
port
A port is a maritime facility comprising one or more wharves or loading areas, where ships load and discharge cargo and passengers. Although usually situated on a sea coast or estuary, ports can also be found far inland, such as Ham ...
s. 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
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
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,
and
respectively.
For all ports the voltages may be defined in terms of the Z-parameter matrix and the currents by the following matrix equation:
:
where Z is an ''N'' × ''N'' matrix the elements of which can be indexed using conventional
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
notation. In general the elements of the Z-parameter matrix are
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s 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
![Two-Port Z-parameters reciprocal equivalent](https://upload.wikimedia.org/wikipedia/commons/0/0f/Two-Port_Z-parameters_reciprocal_equivalent.svg)
The Z-parameter matrix for the
two-port network
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
is probably the most common. In this case the relationship between the port currents, port voltages and the Z-parameter matrix is given by:
:
.
where
:
:
For the general case of an ''N''-port network,
:
Impedance relations
The input impedance of a two-port network is given by:
:
where Z
L is the impedance of the load connected to port two.
Similarly, the output impedance is given by:
:
where Z
S 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
:
and
[
:
where is the ]identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, is a diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
having the square root of the characteristic impedance
The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in ...
at each port as its non-zero elements,
and is the corresponding diagonal matrix of square roots of characteristic admittance
Characteristic admittance is the mathematical inverse of the characteristic impedance.
The general expression for the characteristic admittance of a transmission line is:
:Y_0=\sqrt
where
:R is the resistance per unit length,
:L is the inductance ...
s. In these expressions the matrices represented by the bracketed factors commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
and so, as shown above, may be written in either order.[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''−1''BAB''−1 = ''B''−1''ABB''−1 = ''B''−1''A'')]
Two port
In the special case of a two-port network, with the same characteristic impedance at each port, the above expressions reduce to
:
:
:
:
Where
:
The two-port S-parameters may be obtained from the equivalent two-port Z-parameters by means of the following expressions
:
:
:
:
where
:
The above expressions will generally use complex numbers for and . Note that the value of can become 0 for specific values of so the division by in the calculations of may lead to a division by 0.
Relation to Y-parameters
Conversion from Y-parameters
Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used to describe the el ...
to Z-parameters is much simpler, as the Z-parameter matrix is just the inverse of the Y-parameter matrix. For a two-port:
:
:
:
:
where
:
is the determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the Y-parameter matrix.
Notes
References
Bibliography
*
* {{cite book, author = Simon Ramo, author2=John R. Whinnery , author3=Theodore Van Duzer , title = Fields and Waves in Communication Electronics, date = 1994-02-09, publisher = Wiley, isbn = 978-0-471-58551-0
See also
* Scattering parameters
Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals.
The parameters are useful f ...
* Admittance parameters
Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used to describe the ele ...
* Two-port network
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
Electrical parameters
Two-port networks
Transfer functions
de:Zweitor#Zweitorgleichungen und Parameter