Editing Transmission line (section)

== Matrix parameters ==
The [[Electronic circuit simulation|simulation]] of transmission lines embedded into larger systems generally utilize [[admittance parameters]] (Y matrix), [[impedance parameters]] (Z matrix), and/or [[scattering parameters]] (S matrix) that embodies the full transmission line model needed to support the simulation.

=== Admittance parameters ===
Admittance (Y) parameters may be defined by applying a fixed voltage to one port (V1) of a transmission line with the other end shorted to ground and measuring the resulting current running into each port (I1, I2)<ref name=":4">{{Cite book |last1=Leon |first1=Benjamin J. |url=https://archive.org/details/basiclinearnetwo0000leon/mode/2up |title=Basic Linear Networks for Electrical and Electronics Engineer |last2=Wintz |first2=Paul A. |date=1970 |publisher=Holt, Rinehart, and Winston |isbn=0030783259 |location=US |pages=127 to 129 |language=English}}</ref><ref name=":5">{{Cite book |last=Pozar |first=David M. |title=Microwave Engineering |date=2013 |publisher=John Wiley & Sones, Inc. |isbn=978-81-265-4190-4 |edition=4th |location=Hoboken, NJ, US |publication-date=2013 |pages=174, 175 |language=English}}</ref> and computing the admittance on each port as a ratio of I/V  The admittance parameter Y11 is I1/V1, and the admittance parameter Y12 is I2/V1.  Since transmission lines are electrically passive and symmetric devices, Y12 = Y21, and Y11 = Y22.

For lossless and lossy transmission lines respectively, the Y parameter matrix is as follows:<ref name=":2">{{Cite book |last1=Matthaei |first1=George L. |url=https://archive.org/details/microwavefilters0000matt |title=Microwave Filters, Impudence-Matching Networks, and Coupling Structures |last2=Young |first2=Leo |last3=Jones |first3=E. M. T. |date=1984 |publisher=Artech House, Inc. |isbn=0-89006-099-1 |location=610 Washington Street, Dedham, Massachusetts, US |publication-date=1985 |pages=30 |language=English}}</ref><ref name=":0">{{Cite web |last1=Drakos |first1=Nikos |last2=Hennecke |first2=Marcus |last3=Moore |first3=Ross |last4=Swan |first4=Herb |date=November 22, 2013 |title=Transmission Line |url=https://qucs.sourceforge.net/tech/node61.html |website=Quite Universal Circuit Simulator (Qucs)}}</ref>

<math>Y_{\text{Lossless}}=\begin{bmatrix} 
\frac{-jcot(\beta l)}{Z_o} & \frac{jcsc(\beta l)}{Z_o} \\ 
\frac{jcsc(\beta l)}{Z_o}  & \frac{-jcot(\beta l)}{Z_o}
\end{bmatrix}
\text{    }
Y_{\text{Lossy}}=\begin{bmatrix} 
\frac{coth(\gamma l)}{Z_o} & \frac{-csch(\gamma l)}{Z_o} \\ 
\frac{-csch(\gamma l)}{Z_o} & \frac{coth(\gamma l)}{Z_o} 
\end{bmatrix}</math>

=== Impedance parameters ===
Impedance (Z) parameter may defines by applying a fixed current into one port (I1) of a transmission line with the other port open and measuring the resulting voltage on each port (V1, V2)<ref name=":4" /><ref name=":5" /> and computing the impedance parameter Z11 is V1/I1, and the impedance parameter Z12 is V2/I1.  Since transmission lines are electrically passive and symmetric devices, V12 = V21, and V11 = V22.

In the Y and Z matrix definitions, <math>Y = Z^{-1}</math> and <math>Z = Y^{-1}</math>.<ref>{{Cite book |last=Pozar |first=David M. |url=https://archive.org/details/microwaveenginee0000poza/mode/2up |title=Microwave Engineering |date=1998 |publisher=John Wiley & Sons, Inc. |isbn=0-471-17096-8 |edition=2nd |location=Canada |publication-date=1998 |pages=192 |language=English}}</ref>  Unlike ideal [[Lumped-element model|lumped 2 port elements]] ([[Resistor|resistors]], [[Capacitor|capacitors]], [[Inductor|inductors]], etc.) which do not have defined Z parameters, transmission lines have an internal path to ground, which permits the definition of Z parameters.

For lossless and lossy transmission lines respectively, the Z parameter matrix is as follows:<ref name=":2" /><ref name=":0" />

<math>Z_{\text{Lossless}}=\begin{bmatrix} 
-jZ_ocot(\beta l) & -jZ_ocsc(\beta l) \\ 
-jZ_ocsc(\beta l) & -jZ_ocot(\beta l)
\end{bmatrix}
\text{     }
Z_{\text{Lossy}}=\begin{bmatrix} 
Z_ocoth(\gamma l) & Z_ocsch(\gamma l) \\ 
Z_ocsch(\gamma l)  & Z_ocoth(\gamma l)
\end{bmatrix}</math>

=== Scattering parameters ===
Scattering (S) matrix parameters model the electrical behavior of the transmission line with [[Impedance matching|matched loads]] at each [[Electrical termination|termination]].<ref name=":2" />

For lossless and lossy transmission lines respectively, the S parameter matrix is as follows,<ref>{{Cite web |last=University of Texas at Austin |date=December 14, 2015 |title=Microsoft Word - dissertation_def_rev.doc - ch_2.pdf |url=http://weewave.mer.utexas.edu/MED_files/Former_Students/thesis_dssrtns/Friar_R_diss/ch_2.pdf }}</ref><ref>{{Cite web |date=October 21, 2020 |title=2.3: Scattering Parameters - Engineering LibreTexts |url=https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_III_-_Networks_(Steer)/02%3A_Chapter_2/2.3%3A_Scattering_Parameters |website=Engineering LibreTexts}}</ref> using standard [[Hyperbolic functions#Complex trigonometric definitions|hyperbolic to circular complex translations]].

<math>S_{\text{Lossless}}=\begin{bmatrix} 
\frac{(Z_o^2-Z_p^2)sin(\beta l)}{(Z_o^2+Z_p^2)sin(\beta l)-j2Z_oZ_pcos(\beta l)} 
& 
\frac{2Z_oZ_p}{j(Z_o^2+Z_p^2)sin(\beta l)+2Z_oZ_pcos(\beta l)} 
\\ 
\frac{2Z_oZ_p}{j(Z_o^2+Z_p^2)sin(\beta l)+2Z_oZ_pcos(\beta l)}
&
\frac{(Z_o^2-Z_p^2)sin(\beta l)}{(Z_o^2+Z_p^2)sin(\beta l)-j2Z_oZ_pcos(\beta l)} 
\end{bmatrix}
\text{    }
S_{\text{Lossy}}=\begin{bmatrix} 
\frac{(Z_o^2-Z_p^2)sinh(\gamma l)}{(Z_o^2+Z_p^2)sinh(\gamma l)+2Z_oZ_pcosh(\gamma l)} 
& 
\frac{2Z_oZ_p}{(Z_o^2+Z_p^2)sinh(\gamma l)+2Z_oZ_pcosh(\gamma l)} 
\\ 
\frac{2Z_oZ_p}{(Z_o^2+Z_p^2)sinh(\gamma l)+2Z_oZ_pcosh(\gamma l)} 
& 
\frac{(Z_o^2-Z_p^2)sinh(\gamma l)}{(Z_o^2+Z_p^2)sinh(\gamma l)+2Z_oZ_pcosh(\gamma l)} 
\end{bmatrix}</math>

=== Variable definitions ===
In all matrix parameters above, the following variable definitions apply:

<math>Z_o</math> = [[characteristic impedance]]

Zp = port impedance, or [[Electrical termination|termination impedance]]

<math>\gamma = \alpha + j\beta</math> = the [[propagation constant]] per unit length

<math>\alpha</math> = [[attenuation constant]] in [[Neper|nepers]] per unit length

<math>\beta = \frac{2\pi}{\lambda} = \frac{\omega}{V}</math> = [[Wavenumber|wave number]] or [[phase constant]]  radians per unit length

<math>\omega</math> = [[frequency]] radians / second

<math>V=\frac{1}{\sqrt{LC}} = \frac{V_C}{\sqrt{E_{re}}}</math> = [[Signal velocity|Speed of propagation]]

<math>\lambda</math> = [[Wavelength|wave length]] in unit length

L = [[inductance]] per unit length

C = [[capacitance]] per unit length

<math>E_{re}</math> = [[Relative permittivity|effective dielectric constant]]

<math>V_C</math> = 299,792,458 meters / second = [[Speed of light]] in a vacuum