Two identical two-port networks having transmission matrix 

\(\left[ {\begin{array}{*{20}{c}} A&B\\ C&D \end{array}} \right]\)

Concept:

When two identical two-port networks are cascaded, then their Transmission parameter matrices are to be multiplied.

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaWadaWdaeaafaqabeGacaaabaWdbiaadgeaa8aabaWdbiaadkea % a8aabaWdbiaadoeaa8aabaWdbiaadseaaaaacaGLBbGaayzxaaGaey % ypa0ZaamWaa8aabaqbaeqabiGaaaqaa8qacaWGbbGaai4jaaWdaeaa % peGaamOqaiaacEcaa8aabaWdbiaadoeacaGGNaaapaqaa8qacaWGeb % Gaai4jaaaaaiaawUfacaGLDbaadaWadaWdaeaafaqabeGacaaabaWd % biaadgeacaGGNaGaai4jaaWdaeaapeGaamOqaiaacEcacaGGNaaapa % qaa8qacaWGdbGaai4jaiaacEcaa8aabaWdbiaadseacaGGNaGaai4j % aaaaaiaawUfacaGLDbaaaaa!4FF2! \left[ {\begin{array}{*{20}{c}} A&B\\ C&D \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {A'}&{B'}\\ {C'}&{D'} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {A''}&{B''}\\ {C''}&{D''} \end{array}} \right]\)

Analysis:

Here two identical two-port networks are cascaded:

The resultant parameters will be the multiplication of the two parameters, i.e.

\(\left[ {\begin{array}{*{20}{c}} A&B\\ C&D \end{array}} \right]\)\(\left[ {\begin{array}{*{20}{c}} A&B\\ C&D \end{array}} \right]\)=\(\left[ {\begin{array}{*{20}{c}} {{A^2} + BC}&{AB + BD}\\ {AC + CD}&{BC + {D^2}} \end{array}} \right]\)

When two identical two-port networks are in series, then their Z- parameters are to be added.

\(\left[ {\begin{array}{*{20}{c}} z_{11}&z_{12}\\ z_{21}&z_{22} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} z_{11}'+z_{11}''&z_{12}'+z_{12}''\\ z_{21}'+z_{21}''&z_{22}'+z_{22}'' \end{array}} \right]\)

When two identical two-port networks are in parallel, then their Y- parameters are to be added.

\(\left[ {\begin{array}{*{20}{c}} y_{11}&y_{12}\\ y_{21}&y_{22} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} y_{11}'+y_{11}''&y_{12}'+y_{12}''\\ y_{21}'+y_{21}''&y_{22}'+y_{22}'' \end{array}} \right]\)

For series-parallel network combination, H-parameters are added:

\(\left[ {\begin{array}{*{20}{c}} h_{11}&h_{12}\\ h_{21}&h_{22} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} h_{11}'+h_{11}''&h_{12}'+h_{12}''\\ h_{21}'+h_{21}''&h_{22}'+h_{22}'' \end{array}} \right]\)

For parallel series network combination, g-parameters are added.

\(\left[ {\begin{array}{*{20}{c}} g_{11}&g_{12}\\ g_{21}&g_{22} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} g_{11}'+g_{11}''&g_{12}'+g_{12}''\\ g_{21}'+g_{21}''&g_{22}'+g_{22}'' \end{array}} \right]\)