Editing Transmission line (section)

== Coupled transmission lines ==
Transmission lines may be placed in proximity to each other such that they electrically interact, such as two [[microstrip]] lines in close proximity.  Such transmission lines are said to be coupled transmission lines.  Coupled transmission lines are characterized by an even and odd mode analysis.  The even mode is characterized by excitation of the two conductors with a signal of equal amplitude and phase.  The odd mode is characterized by excitation with signals of equal and opposite magnitude.  The even and odd modes each have their own characteristic impedances (Zoe, Zoo) and phase constants (<math>\beta_e\text{, }\beta_o</math>).  Lossy coupled transmission lines have their own even and odd mode attenuation constants (<math>\alpha_e\text{, }\alpha_o</math>), which in turn leads to even and odd mode propagation constants (<math>\gamma_e\text{, }\gamma_o</math>).<ref>{{Cite book |last=Pozar |first=David M. |url=https://archive.org/details/microwaveenginee0000poza/mode/2up |title=Microwave Engineering |publisher=John Wiley and Sons, Inc. |year=1998 |isbn=0-471-17096-8 |edition=2nd |pages=383–388 |language=EN}}</ref><ref>{{Cite book |last1=Maththaei |first1=George L. |url=https://archive.org/details/microwavefilters0000matt/mode/ |title=Microwave filters, impedance-matching networks, and coupling structures |last2=Young |first2=Leo |last3=E. M. T. |first3=Jones |date=1964 |publisher=Artech House Books |isbn=0-89006-099-1 |location=Dedham, MA, US |publication-date=1964 |pages=174–196 |language=EN}}</ref><ref>{{Cite book |last=Rhea |first=Randall W. |url=https://archive.org/details/hffilterdesignco0000rhea/mode/2up |title=HF Filter Design and Computer Simulation |date=1995 |publisher=McGraw-Hill |isbn=0-07-052055-0 |location=New York, NY, US |publication-date=1995 |pages=85}}</ref><ref>{{Cite web |date=October 21, 2022 |title=5.6: Formulas for Impedance of Coupled Microstrip Lines |url=https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_II_-_Transmission_Lines_(Steer)/05%3A_Coupled_Lines_and_Applications/5.06%3A_Formulas_for_Impedance_of_Coupled_Microstrip_Lines |website=Engineering LibreTexts}}</ref><ref>{{Cite web |last1=Drakos |first1=Nikos |last2=Hennecke |first2=Marcus |last3=Moore |first3=Ross |last4=Herb |first4=Swan |date=November 22, 2013 |title=Parallel coupled microstrip lines |url=https://qucs.sourceforge.net/tech/node77.html |website=Quite universal circuit simulator}}</ref><ref>{{Cite book |last1=Garg |first1=Ramesh |url=https://ieeexplore.ieee.org/document/9101138 |title=Microstrip Lines and Slotlines |last2=Bahl |first2=Inder |last3=Bozzi |first3=Maurizio |date=2013 |publisher=Artech House |isbn=978-1-60807-535-5 |edition=3rd |location=Boston, London |publication-date=2013 |pages=462–473 |language=EN}}</ref>

=== Coupled matrix parameters ===
Coupled transmission lines may be modeled using even and odd mode transmission line parameters defined in the prior paragraph as shown with ports 1 and 2 on the input and ports 3 and 4 on the output,<ref>{{Cite web |date=October 21, 2020 |title=5.9: Models of Parallel Coupled Lines - Engineering LibreTexts |url=https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_II_-_Transmission_Lines_(Steer)/05%3A_Coupled_Lines_and_Applications/5.09%3A_Models_of_Parallel_Coupled_Lines |website=Libre Texts}}</ref>

<math>\begin{align}
Y  &= 
\begin{bmatrix} 
y11 & y12 & y13 & y14 \\ 
y21 & y22 & y23 & y24 \\ 
y31 & y32 & y33 & y34 \\ 
y41 & y42 & y43 & y44 \\ 
\end{bmatrix}\\
Z &= [Y]^{-1} \\
&\\
\text{Where:}& \\
\text{For lossless coupled lines:}& \\
y11 &= y22 = y33 = y44 = \frac{-j}{2} \bigg(\frac{cot(\beta_e l)}{Z_{oe}}+\frac{cot(\beta_o l)}{Z_{oo}}\bigg) \\
y12 &= y22 = y34 = y43 = \frac{-j}{2} \bigg(\frac{cot(\beta_e l)}{Z_{oe}}-\frac{cot(\beta_o l)}{Z_{oo}}\bigg) \\
y13 &= y31 = y24 = y42 = \frac{j}{2} \bigg(\frac{csc(\beta_e l)}{Z_{oe}}+\frac{csc(\beta_o l)}{Z_{oo}}\bigg) \\
y14 &= y41 = y23 = y32 = \frac{j}{2} \bigg(\frac{csc(\beta_e l)}{Z_{oe}}-\frac{csc(\beta_o l)}{Z_{oo}}\bigg) \\
\text{For lossy coupled lines:}& \\
y11 &= y22 = y33 = y44 = \frac{1}{2} \bigg(\frac{coth(\gamma_e l)}{Z_{oe}}+\frac{coth(\gamma_o l)}{Z_{oo}}\bigg) \\
y12 &= y22 = y34 = y43 = \frac{1}{2} \bigg(\frac{coth(\gamma_e l)}{Z_{oe}}-\frac{coth(\gamma_o l)}{Z_{oo}}\bigg) \\
y13 &= y31 = y24 = y42 = \frac{-1}{2} \bigg(\frac{csch(\gamma_e l)}{Z_{oe}}+\frac{csch(\gamma_o l)}{Z_{oo}}\bigg) \\
y14 &= y41 = y23 = y32 = \frac{-1}{2} \bigg(\frac{csch(\gamma_e l)}{Z_{oe}}-\frac{csch(\gamma_o l)}{Z_{oo}}\bigg) \\
\end{align}</math>..