Impedance parameters
Encyclopedia
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 network
s. 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, Y-parameters, H-parameters, T-parameters or ABCD-parameters.
Z-parameters are also known as open-circuit impedance parameters as they are calculated under open circuit
conditions. ie., 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.
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 input currents and output voltages,
and 
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:
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 number
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 as for their determination we need to open circuit one port and analyze the currents and voltages through other ports.
is probably the most common. In this case the relationship between the input currents, output voltages and the Z-parameter matrix is given by:
.
where
For the general case of an N-port network,
where ZL is the impedance of the load connected to port two.
Similarly, the output impedance is given by:
where ZS is the impedance of the source connected to port one.
and
where
is the identity matrix
,
is a diagonal matrix
having the square root of the characteristic impedance
at each port as its non-zero elements,
and
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.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)
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.
where
is the determinant
of the Y-parameter matrix.
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, electronic engineering
Electronic engineering
Electronics engineering, also referred to as electronic engineering, is an engineering discipline where non-linear and active electrical components such as electron tubes, and semiconductor devices, especially transistors, diodes and integrated circuits, are utilized to design electronic...
, and communication systems
Communications system
In telecommunication, a communications system is a collection of individual communications networks, transmission systems, relay stations, tributary stations, and data terminal equipment usually capable of interconnection and interoperation to form an integrated whole...
engineering to describe the electrical behavior of linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
electrical network
Electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...
s. They are also used to describe the small-signal (linearized
Linearization
In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
) 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, H-parameters, T-parameters or ABCD-parameters.
Z-parameters are also known as open-circuit impedance parameters as they are calculated under open circuit
Open circuit
The term 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...
conditions. ie., Ix=0, where x=1,2 refer to input and output currents flowing through the ports (of a two-port network
Two-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
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 boxBlack box
A black box is a device, object, or system whose inner workings are unknown; only the input, transfer, and output are known characteristics.The term black box can also refer to:-In science and technology:*Black box theory, a philosophical theory...
with a number of ports. A port in this context is a pair of electrical terminals
Terminal (electronics)
A terminal is the point at which a conductor from an electrical component, device or network comes to an end and provides a point of connection to external circuits. A terminal may simply be the end of a wire or it may be fitted with a connector or fastener...
carrying equal and opposite currents into and out-of the network, and having a particular voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
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 input currents and output voltages,
and 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:
where Z is an N × N matrix the elements of which can be indexed using conventional matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
notation. In general the elements of the Z-parameter matrix are complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s and functions of frequency. For a one-port network, the Z-matrix reduces to a single element, being the ordinary impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
measured between the two terminals.The Z-parameters are also known as the open circuit parameters as for their determination we need to open circuit one port and analyze the currents and voltages through other ports.
Two-port networks
The Z-parameter matrix for the two-port networkTwo-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical 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:
.where
For the general case of an N-port network,
Impedance relations
The input impedance of a two-port network is given by:where ZL is the impedance of the load connected to port two.
Similarly, the output impedance is given by:
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 and
where
is the identity matrixIdentity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
,
is a diagonal matrixDiagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
having the square root of the characteristic impedance
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
at each port as its non-zero elements,
and
is the corresponding diagonal matrix of square roots of characteristic admittances. In these expressions the matrices represented by the bracketed factors commuteCommute
Commute, commutation or commutative may refer to:* Commuting, the process of travelling between a place of residence and a place of work* Commutative property, a property of a mathematical operation...
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-1BAB-1 = B-1ABB-1 = B-1A)
Two port
In the special case of a two-port network, with the same characteristic impedance at each port, the above expressions reduce toWhere
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 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
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the Y-parameter matrix.