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Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, such as power,
electronics Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...
, and
telecommunications Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of ...
. These parameters are used to describe the electrical behavior of
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
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 sou ...
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, such as diodes, transistors, vacuum tubes, and integrated circuits, with linea ...
( linearized) response of non-linear networks. Y parameters are also known as short circuited admittance parameters. They are members of a family of similar parameters used in electronic engineering, other examples being:
S-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 ...
, Z-parameters, H-parameters, T-parameters or ABCD-parameters.


The Y-parameter matrix

A Y-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 Hamburg, Manch ...
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 (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
between them. The Y-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 ranging from 1 to , where is the total number of ports. For port , the associated Y-parameter definition is in terms of the port voltage and port current, and respectively. For all ports the currents may be defined in terms of the Y-parameter matrix and the voltages by the following matrix equation: :I = Y V\, where Y is an matrix the elements of which can be indexed using conventional
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
notation. In general the elements of the Y-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 for ...
s and functions of frequency. For a one-port network, the Y-matrix reduces to a single element, being the ordinary
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the multiplicative inverse, reciprocal of Electrical impedance, impedance, analogous to how Electrical resistanc ...
measured between the two terminals.


Two-port networks

The Y-parameter matrix for the
two-port network In electronics, a two-port network (a kind of four-terminal network or quadripole) is an electrical network (i.e. a circuit) or device with two ''pairs'' of Terminal (electronics), terminals to connect to external circuits. Two terminals consti ...
is probably the most common. In this case the relationship between the port voltages, port currents and the Y-parameter matrix is given by: :\beginI_1 \\ I_2\end = \begin Y_ & Y_ \\ Y_ & Y_ \end\beginV_1 \\ V_2\end. where :\begin Y_ &= \bigg, _ \qquad Y_ = \bigg, _ \\ ptY_ &= \bigg, _ \qquad Y_ = \bigg, _ \end For the general case of an -port network, :Y_ = \bigg, _


Admittance relations

The input admittance of a two-port network is given by: :Y_ = Y_ - \frac where is the admittance of the load connected to port two. Similarly, the output admittance is given by: :Y_ = Y_ - \frac where is the admittance of the source connected to port one.


Relation to S-parameters

The Y-parameters of a network are related to its S-parameters by : \begin Y &= \sqrt (I_N - S) (I_N + S)^ \sqrt \\ &= \sqrt (I_N + S)^ (I_N - S) \sqrt \\ \end and : \begin S &= (I_N - \sqrtY\sqrt) (I_N + \sqrtY\sqrt)^ \\ &= (I_N + \sqrtY\sqrt)^ (I_N - \sqrtY\sqrt) \\ \end 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. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
, \sqrt 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 diagon ...
having the square root of the
characteristic admittance Characteristic admittance is the mathematical inverse of the characteristic impedance. The general expression for the characteristic admittance of a transmission line is as follows: :Y_0=\sqrt where :R is the resistance per unit length, :L is the ...
(the reciprocal 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 wave travelling in one direction along the line in the absence of reflections in th ...
) at each port as its non-zero elements, \sqrt = \begin \sqrt & \\ & \sqrt \\ & & \ddots \\ & & & \sqrt \end and \sqrt = (\sqrt)^ is the corresponding diagonal matrix of square roots of
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 wave travelling in one direction along the line in the absence of reflections in th ...
s. 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''−1''BAB''−1 = ''B''−1''ABB''−1 = ''B''−1''A'')


Two port

In the special case of a two-port network, with the same and real characteristic admittance y_ = y_ = Y_0 at each port, the above expressions reduce to :\begin Y_ &= Y_0 \\ Y_ &= Y_0 \\ ptY_ &= Y_0 \\ ptY_ &= Y_0 \end where :\Delta_S = (1 + S_) (1 + S_) - S_ S_ . The above expressions will generally use complex numbers for S_ and Y_. Note that the value of \Delta can become 0 for specific values of S_ so the division by \Delta in the calculations of Y_ may lead to a division by 0. The two-port S-parameters may also be obtained from the equivalent two-port Y-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, . :\begin S_ &= \\ S_ &= \\ ptS_ &= \\ ptS_ &= \end where :\Delta = (1 + Z_0 Y_) (1 + Z_0 Y_) - Z^2_0 Y_ Y_ \, and Z_0 is 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 wave travelling in one direction along the line in the absence of reflections in th ...
at each port (assumed the same for the two ports).


Relation to Z-parameters

Conversion from Z-parameters to Y-parameters is much simpler, as the Y-parameter matrix is just the inverse of the Z-parameter matrix. The following expressions show the applicable relations: :\begin Y_ &= \\ ptY_ &= \\ ptY_ &= \\ ptY_ &= \end where :, Z, = Z_ Z_ - Z_ Z_ \, In this case , Z, is the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of the Z-parameter matrix. Vice versa the Y-parameters can be used to determine the Z-parameters, essentially using the same expressions since :Y = Z^ \, and :Z = Y^{-1} .


See also

* Nodal admittance matrix *
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 ...
* Impedance parameters *
Two-port network In electronics, a two-port network (a kind of four-terminal network or quadripole) is an electrical network (i.e. a circuit) or device with two ''pairs'' of Terminal (electronics), terminals to connect to external circuits. Two terminals consti ...
* Hybrid-pi model * Power gain


Notes


References

Two-port networks Transfer functions de:Zweitor#Zweitorgleichungen und Parameter