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Admittance parameters

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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 <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a>, such as <a href="Power_engineering" title="Power engineering">power</a>, <a href="Electronic_engineering" title="Electronic engineering">electronics</a>, and <a href="Telecommunications_engineering" title="Telecommunications engineering">telecommunications</a>. These parameters are used to describe the electrical behavior of <a href="Linear" class="mw-redirect" title="Linear">linear</a> <a href="Electrical_network" title="Electrical network">electrical networks</a>. They are also used to describe the <a href="Small-signal" class="mw-redirect" title="Small-signal">small-signal</a> (<a href="Linearization" title="Linearization">linearized</a>) 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: <a href="S-parameters" class="mw-redirect" title="S-parameters">S-parameters</a>,<a href="#cite_note-1">[1]</a> <a href="Z-parameters" class="mw-redirect" title="Z-parameters">Z-parameters</a>,<a href="#cite_note-2">[2]</a> <a href="Two_port" class="mw-redirect" title="Two port">H-parameters</a>, <a href="T-parameters" class="mw-redirect" title="T-parameters">T-parameters</a> or <a href="Two-port_network#ABCD-parameters" title="Two-port network">ABCD-parameters</a>.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="The_Y-parameter_matrix">The Y-parameter matrix</h2></div>

A Y-parameter matrix describes the behaviour of any linear electrical network that can be regarded as a <a href="Black_box" title="Black box">black box</a> with a number of <a href="Port_(circuit_theory)" title="Port (circuit theory)">ports</a>. A port in this context is a pair of <a href="Terminal_(electronics)" title="Terminal (electronics)">electrical terminals</a> carrying equal and opposite currents into and out of the network, and having a particular <a href="Voltage" title="Voltage">voltage</a> 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 n ranging from 1 to N, where N is the total number of ports. For port n, the associated Y-parameter definition is in terms of the port voltage and port current, Vn and In respectively.

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

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<annotation encoding="application/x-tex">{\displaystyle I=YV\,}</annotation>

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</math><img src="./53d5e61a4ffc0d259460dae08675be8a23a48224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.218ex; height:2.176ex;" alt="{\displaystyle I=YV\,}" loading="lazy"></dd></dl>

where Y is an N × N matrix the elements of which can be indexed using conventional <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> notation. In general the elements of the Y-parameter matrix are <a href="Complex_number" title="Complex number">complex numbers</a> and functions of frequency. For a one-port network, the Y-matrix reduces to a single element, being the ordinary <a href="Admittance" title="Admittance">admittance</a> measured between the two terminals.

<div class="mw-heading mw-heading2"><h2 id="Two-port_networks">Two-port networks</h2></div>

The Y-parameter matrix for the <a href="Two-port_network" title="Two-port network">two-port network</a> is probably the most common. In this case the relationship between the port voltages, port currents and the Y-parameter matrix is given by:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}I_{1}\\I_{2}\end{pmatrix}}={\begin{pmatrix}Y_{11}&Y_{12}\\Y_{21}&Y_{22}\end{pmatrix}}{\begin{pmatrix}V_{1}\\V_{2}\end{pmatrix}}}</annotation>

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</math><img src="./25172bc6b59177c99c355eea8b581925b9af351f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.879ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}I_{1}\\I_{2}\end{pmatrix}}={\begin{pmatrix}Y_{11}&Y_{12}\\Y_{21}&Y_{22}\end{pmatrix}}{\begin{pmatrix}V_{1}\\V_{2}\end{pmatrix}}}" loading="lazy">.</dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Y_{11}&={I_{1} \over V_{1}}{\bigg |}_{V_{2}=0}\qquad Y_{12}={I_{1} \over V_{2}}{\bigg |}_{V_{1}=0}\\[8pt]Y_{21}&={I_{2} \over V_{1}}{\bigg |}_{V_{2}=0}\qquad Y_{22}={I_{2} \over V_{2}}{\bigg |}_{V_{1}=0}\end{aligned}}}</annotation>

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</math><img src="./0b522875ab5fda054459cb12be3247a416aee01c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:34.077ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}Y_{11}&={I_{1} \over V_{1}}{\bigg |}_{V_{2}=0}\qquad Y_{12}={I_{1} \over V_{2}}{\bigg |}_{V_{1}=0}\\[8pt]Y_{21}&={I_{2} \over V_{1}}{\bigg |}_{V_{2}=0}\qquad Y_{22}={I_{2} \over V_{2}}{\bigg |}_{V_{1}=0}\end{aligned}}}" loading="lazy"></dd></dl>

For the general case of an n-port network,

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<annotation encoding="application/x-tex">{\displaystyle Y_{nm}={I_{n} \over V_{m}}{\bigg |}_{V_{k}=0{\text{ for }}k\neq m}}</annotation>

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</math><img src="./e9b690b9efa818e98ac053b488a50fec57c80cef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.483ex; height:5.843ex;" alt="{\displaystyle Y_{nm}={I_{n} \over V_{m}}{\bigg |}_{V_{k}=0{\text{ for }}k\neq m}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Admittance_relations">Admittance relations</h3></div>

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

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<annotation encoding="application/x-tex">{\displaystyle Y_{in}=Y_{11}-{\frac {Y_{12}Y_{21}}{Y_{22}+Y_{L}}}}</annotation>

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</math><img src="./f9d9d7be2282317fce779ebe1c2cc29dd43771ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.908ex; height:5.509ex;" alt="{\displaystyle Y_{in}=Y_{11}-{\frac {Y_{12}Y_{21}}{Y_{22}+Y_{L}}}}" loading="lazy"></dd></dl>

where YL is the admittance of the load connected to port two.

Similarly, the output admittance is given by:

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<annotation encoding="application/x-tex">{\displaystyle Y_{out}=Y_{22}-{\frac {Y_{12}Y_{21}}{Y_{11}+Y_{S}}}}</annotation>

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</math><img src="./4fe071c98eeae9dbed35e00a284ce8fe83b90394.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.626ex; height:5.676ex;" alt="{\displaystyle Y_{out}=Y_{22}-{\frac {Y_{12}Y_{21}}{Y_{11}+Y_{S}}}}" loading="lazy"></dd></dl>

where YS is the admittance of the source connected to port one.

<div class="mw-heading mw-heading2"><h2 id="Relation_to_S-parameters">Relation to S-parameters</h2></div>

The Y-parameters of a network are related to its S-parameters by<a href="#cite_note-Russer-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Y&={\sqrt {y}}(I_{N}-S)(I_{N}+S)^{-1}{\sqrt {y}}\\&={\sqrt {y}}(I_{N}+S)^{-1}(I_{N}-S){\sqrt {y}}\\\end{aligned}}}</annotation>

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</math><img src="./5b298cbf26b5e63c1e2e9f2edd32254bea2fc471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.866ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}Y&={\sqrt {y}}(I_{N}-S)(I_{N}+S)^{-1}{\sqrt {y}}\\&={\sqrt {y}}(I_{N}+S)^{-1}(I_{N}-S){\sqrt {y}}\\\end{aligned}}}" loading="lazy"></dd></dl>

and<a href="#cite_note-Russer-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S&=(I_{N}-{\sqrt {z}}Y{\sqrt {z}})(I_{N}+{\sqrt {z}}Y{\sqrt {z}})^{-1}\\&=(I_{N}+{\sqrt {z}}Y{\sqrt {z}})^{-1}(I_{N}-{\sqrt {z}}Y{\sqrt {z}})\\\end{aligned}}}</annotation>

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</math><img src="./f7daf4a12cba87c1f010b15eedaa2727ad289a1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.053ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}S&=(I_{N}-{\sqrt {z}}Y{\sqrt {z}})(I_{N}+{\sqrt {z}}Y{\sqrt {z}})^{-1}\\&=(I_{N}+{\sqrt {z}}Y{\sqrt {z}})^{-1}(I_{N}-{\sqrt {z}}Y{\sqrt {z}})\\\end{aligned}}}" loading="lazy"></dd></dl>

where IN is the <a href="Identity_matrix" title="Identity matrix">identity matrix</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {y}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {y}}}</annotation>

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</math><img src="./da57099cd9dddb97c99bcaca0f5681dcbd4c7011.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.091ex; height:3.009ex;" alt="{\displaystyle {\sqrt {y}}}" loading="lazy"> is a <a href="Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> having the square root of the <a href="Characteristic_admittance" title="Characteristic admittance">characteristic admittance</a> (the reciprocal of the <a href="Characteristic_impedance" title="Characteristic impedance">characteristic impedance</a>) at each port as its non-zero elements,

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {y}}={\begin{pmatrix}{\sqrt {y_{01}}}&\\&{\sqrt {y_{02}}}\\&&\ddots \\&&&{\sqrt {y_{0N}}}\end{pmatrix}}}</annotation>

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</math><img src="./4ff8dbb8823f5f633152c52b4ec5e845e8c27aa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.908ex; margin-bottom: -0.263ex; width:36.446ex; height:15.509ex;" alt="{\displaystyle {\sqrt {y}}={\begin{pmatrix}{\sqrt {y_{01}}}&\\&{\sqrt {y_{02}}}\\&&\ddots \\&&&{\sqrt {y_{0N}}}\end{pmatrix}}}" loading="lazy">

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {z}}=({\sqrt {y}})^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {z}}=({\sqrt {y}})^{-1}}</annotation>

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</math><img src="./259eee22574f50b86df55059aead180021096c76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.356ex; height:3.509ex;" alt="{\displaystyle {\sqrt {z}}=({\sqrt {y}})^{-1}}" loading="lazy"> is the corresponding diagonal matrix of square roots of <a href="Characteristic_impedance" title="Characteristic impedance">characteristic impedances</a>. In these expressions the matrices represented by the bracketed factors <a href="Commuting_matrices" title="Commuting matrices">commute</a> and so, as shown above, may be written in either order.<a href="#cite_note-Russer-5">[5]</a><a href="#cite_note-6">[note 1]</a>

In the special case of a two-port network, with the same and <a href="Real_number" title="Real number">real</a> characteristic admittance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{01}=y_{02}=Y_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{01}=y_{02}=Y_{0}}</annotation>

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</math><img src="./02e7e4a69afdc1096690605dd91392b4a3866184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.633ex; height:2.509ex;" alt="{\displaystyle y_{01}=y_{02}=Y_{0}}" loading="lazy"> at each port, the above expressions reduce to <a href="#cite_note-7">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Y_{11}&={(1-S_{11})(1+S_{22})+S_{12}S_{21} \over \Delta _{S}}Y_{0}\\Y_{12}&={-2S_{12} \over \Delta _{S}}Y_{0}\\[4pt]Y_{21}&={-2S_{21} \over \Delta _{S}}Y_{0}\\[4pt]Y_{22}&={(1+S_{11})(1-S_{22})+S_{12}S_{21} \over \Delta _{S}}Y_{0}\end{aligned}}}</annotation>

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</math><img src="./b3a616cadca1cf9f8371d9bfe07b8c7d5390ccc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:37.987ex; height:26.509ex;" alt="{\displaystyle {\begin{aligned}Y_{11}&={(1-S_{11})(1+S_{22})+S_{12}S_{21} \over \Delta _{S}}Y_{0}\\Y_{12}&={-2S_{12} \over \Delta _{S}}Y_{0}\\[4pt]Y_{21}&={-2S_{21} \over \Delta _{S}}Y_{0}\\[4pt]Y_{22}&={(1+S_{11})(1-S_{22})+S_{12}S_{21} \over \Delta _{S}}Y_{0}\end{aligned}}}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{S}=(1+S_{11})(1+S_{22})-S_{12}S_{21}.}</annotation>

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</math><img src="./f763219cd97cc723f6f67bc0979b2c680aeb3293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.643ex; height:2.843ex;" alt="{\displaystyle \Delta _{S}=(1+S_{11})(1+S_{22})-S_{12}S_{21}.}" loading="lazy"></dd></dl>

The above expressions will generally use complex numbers for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{ij}}">

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</math><img src="./7207b997c68f09a399b687baba2430393420dbba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.902ex; height:2.843ex;" alt="{\displaystyle S_{ij}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{ij}}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{ij}}</annotation>

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</math><img src="./6035187f7e2cee387277f07091bb7827e8e66818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.828ex; height:2.843ex;" alt="{\displaystyle Y_{ij}}" loading="lazy">. Note that the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

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</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> can become 0 for specific values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{ij}}">

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{ij}}</annotation>

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</math><img src="./7207b997c68f09a399b687baba2430393420dbba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.902ex; height:2.843ex;" alt="{\displaystyle S_{ij}}" loading="lazy"> so the division by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

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</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> in the calculations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{ij}}</annotation>

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The two-port S-parameters may also be obtained from the equivalent two-port Y-parameters by means of the following expressions.<a href="#cite_note-8">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{11}&={(1-Z_{0}Y_{11})(1+Z_{0}Y_{22})+Z_{0}^{2}Y_{12}Y_{21} \over \Delta }\\S_{12}&={-2Z_{0}Y_{12} \over \Delta }\\[4pt]S_{21}&={-2Z_{0}Y_{21} \over \Delta }\\[4pt]S_{22}&={(1+Z_{0}Y_{11})(1-Z_{0}Y_{22})+Z_{0}^{2}Y_{12}Y_{21} \over \Delta }\end{aligned}}}</annotation>

</semantics>

</math><img src="./0c7c42c659271a252e92fd9c46d2c57a5435b835.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:43.405ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}S_{11}&={(1-Z_{0}Y_{11})(1+Z_{0}Y_{22})+Z_{0}^{2}Y_{12}Y_{21} \over \Delta }\\S_{12}&={-2Z_{0}Y_{12} \over \Delta }\\[4pt]S_{21}&={-2Z_{0}Y_{21} \over \Delta }\\[4pt]S_{22}&={(1+Z_{0}Y_{11})(1-Z_{0}Y_{22})+Z_{0}^{2}Y_{12}Y_{21} \over \Delta }\end{aligned}}}" loading="lazy"></dd></dl>

where

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Delta =(1+Z_{0}Y_{11})(1+Z_{0}Y_{22})-Z_{0}^{2}Y_{12}Y_{21}\,}</annotation>

</semantics>

</math><img src="./b3dc16d834858082366e9dda525f6cfc18e7caac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.839ex; height:3.176ex;" alt="{\displaystyle \Delta =(1+Z_{0}Y_{11})(1+Z_{0}Y_{22})-Z_{0}^{2}Y_{12}Y_{21}\,}" loading="lazy"></dd></dl>

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Z_{0}}</annotation>

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</math><img src="./bfcd49ab63d30163ac54e60a8e24ff9ccd7bcd44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{0}}" loading="lazy"> is the <a href="Characteristic_impedance" title="Characteristic impedance">characteristic impedance</a> at each port (assumed the same for the two ports).

<div class="mw-heading mw-heading2"><h2 id="Relation_to_Z-parameters">Relation to Z-parameters</h2></div>

Conversion from <a href="Z-parameters" class="mw-redirect" title="Z-parameters">Z-parameters</a> to Y-parameters is much simpler, as the Y-parameter matrix is just the <a href="Matrix_inverse" class="mw-redirect" title="Matrix inverse">inverse</a> of the Z-parameter matrix. The following expressions show the applicable relations:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Y_{11}&={Z_{22} \over |Z|}\\[4pt]Y_{12}&={-Z_{12} \over |Z|}\\[4pt]Y_{21}&={-Z_{21} \over |Z|}\\[4pt]Y_{22}&={Z_{11} \over |Z|}\end{aligned}}}</annotation>

</semantics>

</math><img src="./fdc62a4d7e9e726981a4ac2dbe7b5d45319ea885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.171ex; width:13.185ex; height:27.509ex;" alt="{\displaystyle {\begin{aligned}Y_{11}&={Z_{22} \over |Z|}\\[4pt]Y_{12}&={-Z_{12} \over |Z|}\\[4pt]Y_{21}&={-Z_{21} \over |Z|}\\[4pt]Y_{22}&={Z_{11} \over |Z|}\end{aligned}}}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle |Z|=Z_{11}Z_{22}-Z_{12}Z_{21}\,}</annotation>

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</math><img src="./607d92b99ec2e552b0bb40cdacf0021d1c73be6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.155ex; height:2.843ex;" alt="{\displaystyle |Z|=Z_{11}Z_{22}-Z_{12}Z_{21}\,}" loading="lazy"></dd></dl>

In this case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |Z|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |Z|}</annotation>

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</math><img src="./76645902480f5afcbd4ebfe247c30268a3e16d69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.974ex; height:2.843ex;" alt="{\displaystyle |Z|}" loading="lazy"> is the <a href="Determinant" title="Determinant">determinant</a> of the Z-parameter matrix.

Vice versa the Y-parameters can be used to determine the Z-parameters, essentially using the

same expressions since

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle Y=Z^{-1}\,}</annotation>

</semantics>

</math><img src="./67b1e9a579df164f642804b47c1010a3f40c534e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.3ex; height:2.676ex;" alt="{\displaystyle Y=Z^{-1}\,}" loading="lazy"></dd></dl>

and

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle Z=Y^{-1}.}</annotation>

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</math><img src="./ec19a3b6a04e3d02b359459fa68c5a56a14d0d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.659ex; height:2.676ex;" alt="{\displaystyle Z=Y^{-1}.}" loading="lazy"></dd></dl>

<ul><li><a href="Nodal_admittance_matrix" title="Nodal admittance matrix">Nodal admittance matrix</a></li>

<li><a href="Scattering_parameters" title="Scattering parameters">Scattering parameters</a></li>

<li><a href="Impedance_parameters" title="Impedance parameters">Impedance parameters</a></li>

<li><a href="Two-port_network" title="Two-port network">Two-port network</a></li>

<li><a href="Hybrid-pi_model" title="Hybrid-pi model">Hybrid-pi model</a></li>

<li><a href="Power_gain" title="Power gain">Power gain</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> 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)

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="David_M._Pozar" title="David M. Pozar">Pozar, David M.</a> (2005); Microwave Engineering, Third Edition (Intl. Ed.); John Wiley & Sons; pp. 170-174. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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/* end https://en.wikipedia.org/ */

</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-44878-8</bdi>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Pozar, David M. (2005) (op. cit); pp. 170-174.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Pozar, David M. (2005) (op. cit); pp. 183-186.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Morton, A. H. (1985); Advanced Electrical Engineering;Pitman Publishing Ltd.; pp. 33-72. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-273-40172-6</bdi>

</li>

<li id="cite_note-Russer-5">^ <a href="#cite_ref-Russer_5-0">a</a> <a href="#cite_ref-Russer_5-1">b</a> <a href="#cite_ref-Russer_5-2">c</a> <cite id="CITEREFRusser2003" class="citation book cs1">Russer, Peter (2003). Electromagnetics, microwave circuit and antenna design for communications engineering. Artech House. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-58053-532-8</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFFrickey1994" class="citation journal cs1">Frickey, D. A. (February 1994). "Conversions between S, Z, Y, H, ABCD, and T parameters which are valid for complex source and load impedances". IEEE Transactions on Microwave Theory and Techniques. 42 (2): 205–211. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994ITMTT..42..205F">1994ITMTT..42..205F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F22.275248">10.1109/22.275248</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9480">0018-9480</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Simon Ramo, John R. Whinnery, Theodore Van Duzer, "Fields and Waves in Communication Electronics", Third Edition, John Wiley & Sons Inc.; 1993, pp. 537-541, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-58551-3</bdi>.

</li>

</ol></div></div><div><div class="zim-footer">

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