Conversión de modelos de cuadripolos

Por Mariano Llamedo Soria

Resumen

En este documento se presentan algunas de las capacidades que posee el módulo PyTC2 para operar con cuadripolos. Se muestran ejemplos de cómo definir redes y sus modelos Z, Y y \(T_{ABCD}\) asociados, como también algunas redes que implementan dichos modelos.

Funciones de conversión y definición de cuadripolos: Model_conversion, I2Tabcd_s, S2Tabcd_s, Ts2Tabcd_s
Funciones para presentación de markdown y latex: print_subtitle, print_latex, a_equal_b_latex_s

Introducción

A lo largo del curso se presentó una metodología sistemática para arribar a una función transferencia \(T(s)\) a partir de restricciones de la función de módulo \(\vert T(j\omega) \vert \) o retardo \( \tau(\omega) \). Si bien en primera instancia arribamos a una \(T_{LP}(s)\) pasabajos, es posible mediante núcleos de transformación el pasaje a otro tipo de transferencias (pasa-alto, pasabanda, etc).

Referencias

[1] Frickey, D.A. Conversions Between S, 2,Y , h, ABCD, and T Parameters which are Valid for Complex Source and Load Impedances. IEEE TRANSACTIONS ON MICROWAVE THEORY A N D TECHNIQUES. VOL 42, NO 2. FEBRUARY 1994.

from pytc2.cuadripolos import Model_conversion, I2Tabcd_s, S2Tabcd_s, Ts2Tabcd_s
from pytc2.general import print_latex, print_subtitle, a_equal_b_latex_s
import sympy as sp
import os

y11, y12, y21, y22 = sp.symbols('y11, y12, y21, y22', complex=True)
z11, z12, z21, z22 = sp.symbols('z11, z12, z21, z22', complex=True)
A, B, C, D = sp.symbols('A, B, C, D', complex=True)
Ai, Bi, Ci, Di = sp.symbols('Ai, Bi, Ci, Di', complex=True)
h11, h12, h21, h22 = sp.symbols('h11, h12, h21, h22', complex=True)
g11, g12, g21, g22 = sp.symbols('g11, g12, g21, g22', complex=True)
v1, v2, i1, i2 = sp.symbols('v1, v2, i1, i2', complex=True)

# Parámetros Z (impedancia - circ. abierto)
ZZ = sp.Matrix([[z11, z12], [z21, z22]])
# vars. dependientes
vv = sp.Matrix([[v1], [v2]])
# vars. INdependientes
ii = sp.Matrix([[i1], [i2]])

# Parámetros Y (admitancia - corto circ.)
YY = sp.Matrix([[y11, y12], [y21, y22]])
# vars. dependientes
# ii = sp.Matrix([[i1], [i2]])
# vars. INdependientes
# vv = sp.Matrix([[v1], [v2]])

# Parámetros H (híbridos h)
HH = sp.Matrix([[h11, h12], [h21, h22]])
# vars. dependientes
h_dep = sp.Matrix([[v1], [i2]])
# vars. INdependientes
h_ind = sp.Matrix([[i1], [v2]])

# Parámetros G (híbridos g)
GG = sp.Matrix([[g11, g12], [g21, g22]])
# vars. dependientes
g_dep = sp.Matrix([[i1], [v2]])
# vars. INdependientes
g_ind = sp.Matrix([[v1], [i2]])

# Parámetros Tabcd (Transmisión, ABCD)
TT = sp.Matrix([[A, -B], [C, -D]])
# vars. dependientes
t_dep = sp.Matrix([[v1], [i1]])
# vars. INdependientes.  (Signo negativo de corriente)
t_ind = sp.Matrix([[v2], [i2]])

# Parámetros Tdcba (Transmisión inversos, DCBA)
TTi = sp.Matrix([[Ai, Bi], [-Ci, -Di]])
# vars. dependientes
ti_dep = sp.Matrix([[v2], [i2]])
# vars. INdependientes. (Signo negativo de corriente)
ti_ind = sp.Matrix([[v1], [i1]])


def do_conversion_table(model_dct):
    
    str_table = '$ \\begin{array}{ l ' + ' c'*len(model_dct) + ' }' + os.linesep 

    str_models = ''
    
    for src_model in model_dct:
        str_table +=  ' & ' + src_model['model_name']
        str_models +=  ', ' + src_model['model_name']

    print_subtitle('Tabla de conversión: ' + str_models[2:])
    
    str_table = str_table + ' \\\\ ' + os.linesep
    
    for dst_model in model_dct:
    
        str_table +=   dst_model['model_name']   + ' & '
        
        for src_model in model_dct:
            
            if src_model['model_name'] == dst_model['model_name']:
                
                str_table +=   dst_model['model_name']   + ' & '

            else:
                HH_z = Model_conversion( src_model, dst_model )
                str_table +=  sp.latex( HH_z['matrix'] )  + ' & '
            
        str_table = str_table[:-2] + ' \\\\ ' + os.linesep
        
        #print_latex( str_table )
        
    str_table = str_table[:-4] + os.linesep + '\\end{array} $'
    
    print_latex( str_table )

Conversión entre parámetros de cuadripolos

Con las funciones para crear la tabla de N x N modelos se procede a su uso, para ello solo basta definir un diccionario con las N definiciones de cada modelo.

# Definimos el diccionario  con la definición de cada modelo
model_dct = [ { 'model_name': 'Z', 'matrix': ZZ, 'dep_var': vv, 'indep_var':ii },
              { 'model_name': 'Y', 'matrix': YY, 'dep_var': ii, 'indep_var':vv },
              { 'model_name': 'H', 'matrix': HH, 'dep_var': h_dep, 'indep_var':h_ind },
              { 'model_name': 'G', 'matrix': GG, 'dep_var': g_dep, 'indep_var':g_ind },
              { 'model_name': 'T', 'matrix': TT, 'dep_var': t_dep, 'indep_var':t_ind, 'neg_i2_current': True },
              { 'model_name': 'Ti', 'matrix': TTi, 'dep_var': ti_dep, 'indep_var':ti_ind, 'neg_i2_current': True}
            ]

do_conversion_table(model_dct)

Tabla de conversión: Z, Y, H, G, T, Ti

\[\begin{split}\displaystyle \begin{array}{ l c c c c c c } & Z & Y & H & G & T & Ti \\ Z & Z & \left[\begin{matrix}\frac{y_{22}}{\Delta} & - \frac{y_{12}}{\Delta}\\- \frac{y_{21}}{\Delta} & \frac{y_{11}}{\Delta}\end{matrix}\right] & \left[\begin{matrix}\frac{\Delta}{h_{22}} & \frac{h_{12}}{h_{22}}\\- \frac{h_{21}}{h_{22}} & \frac{1}{h_{22}}\end{matrix}\right] & \left[\begin{matrix}\frac{1}{g_{11}} & - \frac{g_{12}}{g_{11}}\\\frac{g_{21}}{g_{11}} & \frac{\Delta}{g_{11}}\end{matrix}\right] & \left[\begin{matrix}\frac{A}{C} & \frac{\Delta}{C}\\\frac{1}{C} & \frac{D}{C}\end{matrix}\right] & \left[\begin{matrix}- \frac{Di}{Ci} & - \frac{1}{Ci}\\- \frac{\Delta}{Ci} & - \frac{Ai}{Ci}\end{matrix}\right] \\ Y & \left[\begin{matrix}\frac{z_{22}}{\Delta} & - \frac{z_{12}}{\Delta}\\- \frac{z_{21}}{\Delta} & \frac{z_{11}}{\Delta}\end{matrix}\right] & Y & \left[\begin{matrix}\frac{1}{h_{11}} & - \frac{h_{12}}{h_{11}}\\\frac{h_{21}}{h_{11}} & \frac{\Delta}{h_{11}}\end{matrix}\right] & \left[\begin{matrix}\frac{\Delta}{g_{22}} & \frac{g_{12}}{g_{22}}\\- \frac{g_{21}}{g_{22}} & \frac{1}{g_{22}}\end{matrix}\right] & \left[\begin{matrix}\frac{D}{B} & - \frac{\Delta}{B}\\- \frac{1}{B} & \frac{A}{B}\end{matrix}\right] & \left[\begin{matrix}- \frac{Ai}{Bi} & \frac{1}{Bi}\\\frac{\Delta}{Bi} & - \frac{Di}{Bi}\end{matrix}\right] \\ H & \left[\begin{matrix}\frac{\Delta}{z_{22}} & \frac{z_{12}}{z_{22}}\\- \frac{z_{21}}{z_{22}} & \frac{1}{z_{22}}\end{matrix}\right] & \left[\begin{matrix}\frac{1}{y_{11}} & - \frac{y_{12}}{y_{11}}\\\frac{y_{21}}{y_{11}} & \frac{\Delta}{y_{11}}\end{matrix}\right] & H & \left[\begin{matrix}\frac{g_{22}}{\Delta} & - \frac{g_{12}}{\Delta}\\- \frac{g_{21}}{\Delta} & \frac{g_{11}}{\Delta}\end{matrix}\right] & \left[\begin{matrix}\frac{B}{D} & \frac{\Delta}{D}\\- \frac{1}{D} & \frac{C}{D}\end{matrix}\right] & \left[\begin{matrix}- \frac{Bi}{Ai} & \frac{1}{Ai}\\- \frac{\Delta}{Ai} & - \frac{Ci}{Ai}\end{matrix}\right] \\ G & \left[\begin{matrix}\frac{1}{z_{11}} & - \frac{z_{12}}{z_{11}}\\\frac{z_{21}}{z_{11}} & \frac{\Delta}{z_{11}}\end{matrix}\right] & \left[\begin{matrix}\frac{\Delta}{y_{22}} & \frac{y_{12}}{y_{22}}\\- \frac{y_{21}}{y_{22}} & \frac{1}{y_{22}}\end{matrix}\right] & \left[\begin{matrix}\frac{h_{22}}{\Delta} & - \frac{h_{12}}{\Delta}\\- \frac{h_{21}}{\Delta} & \frac{h_{11}}{\Delta}\end{matrix}\right] & G & \left[\begin{matrix}\frac{C}{A} & - \frac{\Delta}{A}\\\frac{1}{A} & \frac{B}{A}\end{matrix}\right] & \left[\begin{matrix}- \frac{Ci}{Di} & - \frac{1}{Di}\\\frac{\Delta}{Di} & - \frac{Bi}{Di}\end{matrix}\right] \\ T & \left[\begin{matrix}\frac{z_{11}}{z_{21}} & \frac{\Delta}{z_{21}}\\\frac{1}{z_{21}} & \frac{z_{22}}{z_{21}}\end{matrix}\right] & \left[\begin{matrix}- \frac{y_{22}}{y_{21}} & - \frac{1}{y_{21}}\\- \frac{\Delta}{y_{21}} & - \frac{y_{11}}{y_{21}}\end{matrix}\right] & \left[\begin{matrix}- \frac{\Delta}{h_{21}} & - \frac{h_{11}}{h_{21}}\\- \frac{h_{22}}{h_{21}} & - \frac{1}{h_{21}}\end{matrix}\right] & \left[\begin{matrix}\frac{1}{g_{21}} & \frac{g_{22}}{g_{21}}\\\frac{g_{11}}{g_{21}} & \frac{\Delta}{g_{21}}\end{matrix}\right] & T & \left[\begin{matrix}\frac{Di}{\Delta} & - \frac{Bi}{\Delta}\\- \frac{Ci}{\Delta} & \frac{Ai}{\Delta}\end{matrix}\right] \\ Ti & \left[\begin{matrix}\frac{z_{22}}{z_{12}} & - \frac{\Delta}{z_{12}}\\- \frac{1}{z_{12}} & \frac{z_{11}}{z_{12}}\end{matrix}\right] & \left[\begin{matrix}- \frac{y_{11}}{y_{12}} & \frac{1}{y_{12}}\\\frac{\Delta}{y_{12}} & - \frac{y_{22}}{y_{12}}\end{matrix}\right] & \left[\begin{matrix}\frac{1}{h_{12}} & - \frac{h_{11}}{h_{12}}\\- \frac{h_{22}}{h_{12}} & \frac{\Delta}{h_{12}}\end{matrix}\right] & \left[\begin{matrix}- \frac{\Delta}{g_{12}} & \frac{g_{22}}{g_{12}}\\\frac{g_{11}}{g_{12}} & - \frac{1}{g_{12}}\end{matrix}\right] & \left[\begin{matrix}\frac{D}{\Delta} & - \frac{B}{\Delta}\\- \frac{C}{\Delta} & \frac{A}{\Delta}\end{matrix}\right] & Ti \end{array} \end{split}\]

# Parámetros Imagen
gamma, z1, z2 = sp.symbols('\gamma, z1, z2', complex=True)

Tim = I2Tabcd_s(gamma, z1, z2)
 #(Signo negativo de corriente)
Tim[:,1] = Tim[:,1] * -1
# vars. dependientes
tim_dep = t_dep
# vars. INdependientes.
tim_ind = t_ind

# Diccionario con la definición de cada modelo
model_dct = [ { 'model_name': 'Im', 'matrix': Tim, 'dep_var': tim_dep, 'indep_var':tim_ind, 'neg_i2_current': True }, # T_ABCD
              { 'model_name': 'T_{ABCD}', 'matrix': TT, 'dep_var': t_dep, 'indep_var':t_ind, 'neg_i2_current': True }, # T_ABCD
              { 'model_name': 'Z', 'matrix': ZZ, 'dep_var': vv, 'indep_var':ii },
              { 'model_name': 'Y', 'matrix': YY, 'dep_var': ii, 'indep_var':vv }
            ]

do_conversion_table(model_dct)

Tabla de conversión: Im, T_{ABCD}, Z, Y

\[\begin{split}\displaystyle \begin{array}{ l c c c c } & Im & T_{ABCD} & Z & Y \\ Im & Im & \left[\begin{matrix}A & B\\C & D\end{matrix}\right] & \left[\begin{matrix}\frac{z_{11}}{z_{21}} & \frac{\Delta}{z_{21}}\\\frac{1}{z_{21}} & \frac{z_{22}}{z_{21}}\end{matrix}\right] & \left[\begin{matrix}- \frac{y_{22}}{y_{21}} & - \frac{1}{y_{21}}\\- \frac{\Delta}{y_{21}} & - \frac{y_{11}}{y_{21}}\end{matrix}\right] \\ T_{ABCD} & \left[\begin{matrix}\sqrt{\frac{z_{1}}{z_{2}}} \cosh{\left(\gamma \right)} & \sqrt{z_{1} z_{2}} \sinh{\left(\gamma \right)}\\\frac{\sinh{\left(\gamma \right)}}{\sqrt{z_{1} z_{2}}} & \sqrt{\frac{z_{2}}{z_{1}}} \cosh{\left(\gamma \right)}\end{matrix}\right] & T_{ABCD} & \left[\begin{matrix}\frac{z_{11}}{z_{21}} & \frac{\Delta}{z_{21}}\\\frac{1}{z_{21}} & \frac{z_{22}}{z_{21}}\end{matrix}\right] & \left[\begin{matrix}- \frac{y_{22}}{y_{21}} & - \frac{1}{y_{21}}\\- \frac{\Delta}{y_{21}} & - \frac{y_{11}}{y_{21}}\end{matrix}\right] \\ Z & \left[\begin{matrix}\frac{\sqrt{\frac{z_{1}}{z_{2}}} \sqrt{z_{1} z_{2}} \cosh{\left(\gamma \right)}}{\sinh{\left(\gamma \right)}} & \frac{\sqrt{\frac{z_{2}}{z_{1}}} \sqrt{\frac{z_{1}}{z_{2}}} \sqrt{z_{1} z_{2}} \cosh^{2}{\left(\gamma \right)} - \sqrt{z_{1} z_{2}} \sinh^{2}{\left(\gamma \right)}}{\sinh{\left(\gamma \right)}}\\\frac{\sqrt{z_{1} z_{2}}}{\sinh{\left(\gamma \right)}} & \frac{\sqrt{\frac{z_{2}}{z_{1}}} \sqrt{z_{1} z_{2}} \cosh{\left(\gamma \right)}}{\sinh{\left(\gamma \right)}}\end{matrix}\right] & \left[\begin{matrix}\frac{A}{C} & \frac{\Delta}{C}\\\frac{1}{C} & \frac{D}{C}\end{matrix}\right] & Z & \left[\begin{matrix}\frac{y_{22}}{\Delta} & - \frac{y_{12}}{\Delta}\\- \frac{y_{21}}{\Delta} & \frac{y_{11}}{\Delta}\end{matrix}\right] \\ Y & \left[\begin{matrix}\frac{\sqrt{\frac{z_{2}}{z_{1}}} \cosh{\left(\gamma \right)}}{\sqrt{z_{1} z_{2}} \sinh{\left(\gamma \right)}} & - \frac{\Delta}{\sqrt{z_{1} z_{2}} \sinh{\left(\gamma \right)}}\\- \frac{1}{\sqrt{z_{1} z_{2}} \sinh{\left(\gamma \right)}} & \frac{\sqrt{\frac{z_{1}}{z_{2}}} \cosh{\left(\gamma \right)}}{\sqrt{z_{1} z_{2}} \sinh{\left(\gamma \right)}}\end{matrix}\right] & \left[\begin{matrix}\frac{D}{B} & - \frac{\Delta}{B}\\- \frac{1}{B} & \frac{A}{B}\end{matrix}\right] & \left[\begin{matrix}\frac{z_{22}}{\Delta} & - \frac{z_{12}}{\Delta}\\- \frac{z_{21}}{\Delta} & \frac{z_{11}}{\Delta}\end{matrix}\right] & Y \end{array} \end{split}\]

s11, s12, s21, s22 = sp.symbols('s11, s12, s21, s22', complex=True)
ts11, ts12, ts21, ts22 = sp.symbols('t11, t12, t21, t22', complex=True)

# ondas normalizadas de tensión
a1, a2, b1, b2 = sp.symbols('a1, a2, b1, b2', complex=True)

# impedancia característica
Z01, Z02 = sp.symbols('Z01, Z02', complex=True)
Zo = sp.symbols('Zo', real=True)

# Parámetros dispersión (scattering - S)
Spar = sp.Matrix([[s11, s12], [s21, s22]])
# vars. dependientes
bb = sp.Matrix([[b1], [b2]])
# vars. INdependientes
aa = sp.Matrix([[a1], [a2]])

# a y b en función de tensiones y corrientes, para vincular con cuadripolos convencionales
aa_iv = aa.subs({a1:(v1 + Z01*i1)/sp.sqrt(2*(Z01 + sp.conjugate(Z01))), a2:(v2 + Z02*i2)/sp.sqrt(2*(Z02 + sp.conjugate(Z02)))})
bb_iv = bb.subs({b1:(v1 - sp.conjugate(Z01)*i1)/sp.sqrt(2*(Z01 + sp.conjugate(Z01))), b2:(v2 - sp.conjugate(Z02)*i2)/sp.sqrt(2*(Z02 + sp.conjugate(Z02)))})

# Parámetros transmisión dispersión (scattering transfer param.)
TSpar = sp.Matrix([[ts11, ts12], [ts21, ts22]])
# vars. dependientes
ts1 = sp.Matrix([[a1], [b1]])
# vars. INdependientes
ts2 = sp.Matrix([[b2], [a2]])

# ts1,2 en función de tensiones y corrientes, para vincular con cuadripolos convencionales
ts1_iv = aa.subs({a1:(v1 + Z01*i1)/sp.sqrt(2*(Z01 + sp.conjugate(Z01))), b1:(v1 - sp.conjugate(Z01)*i1)/sp.sqrt(2*(Z01 + sp.conjugate(Z01)))})
ts2_iv = bb.subs({b2:(v2 - sp.conjugate(Z02)*i2)/sp.sqrt(2*(Z02 + sp.conjugate(Z02))), a2:(v2 + Z02*i2)/sp.sqrt(2*(Z02 + sp.conjugate(Z02)))})

#Tabcd_proxy_model = { 'model_name': 'T_{ABCD}', 'matrix': TT, 'dep_var': t_dep, 'indep_var':t_ind, 'neg_i2_current': True }

# Diccionario con la definición de cada modelo
model_dct = [ { 'model_name': 'S', 'matrix': Spar, 'dep_var': bb, 'indep_var':aa },
              { 'model_name': 'T_S', 'matrix': TSpar, 'dep_var': ts1, 'indep_var':ts2 },
            ]
do_conversion_table(model_dct)

# Diccionario con la definición de cada modelo:
# En este caso convertimos el modelo S a un ABCD primero, con sus respectivas variables
model_dct = [ { 'model_name': 'S', 'matrix': Spar, 'proxy_matrix': S2Tabcd_s( Spar, sp.Rational(1), sp.Rational(1)), 'dep_var': t_dep, 'indep_var':t_ind, 'neg_i2_current': True },              
              { 'model_name': 'Z', 'matrix': ZZ, 'dep_var': vv, 'indep_var':ii },
              { 'model_name': 'Y', 'matrix': YY, 'dep_var': ii, 'indep_var':vv }
            ]

do_conversion_table(model_dct)

# Diccionario con la definición de cada modelo:
# En este caso convertimos el modelo Ts a un ABCD primero, con sus respectivas variables
model_dct = [ { 'model_name': 'T_S', 'matrix': TSpar, 'proxy_matrix': Ts2Tabcd_s( TSpar, Z01, Z02), 'dep_var': t_dep, 'indep_var':t_ind, 'neg_i2_current': True },
              { 'model_name': 'Z', 'matrix': ZZ, 'dep_var': vv, 'indep_var':ii },
              { 'model_name': 'Y', 'matrix': YY, 'dep_var': ii, 'indep_var':vv }
            ]

do_conversion_table(model_dct)

Tabla de conversión: S, T_S

\[\begin{split}\displaystyle \begin{array}{ l c c } & S & T_S \\ S & S & \left[\begin{matrix}\frac{t_{21}}{t_{11}} & \frac{\Delta}{t_{11}}\\\frac{1}{t_{11}} & - \frac{t_{12}}{t_{11}}\end{matrix}\right] \\ T_S & \left[\begin{matrix}\frac{1}{s_{21}} & - \frac{s_{22}}{s_{21}}\\\frac{s_{11}}{s_{21}} & - \frac{\Delta}{s_{21}}\end{matrix}\right] & T_S \end{array} \end{split}\]

Tabla de conversión: S, Z, Y

\[\begin{split}\displaystyle \begin{array}{ l c c c } & S & Z & Y \\ S & S & \left[\begin{matrix}\frac{z_{11}}{z_{21}} & \frac{\Delta}{z_{21}}\\\frac{1}{z_{21}} & \frac{z_{22}}{z_{21}}\end{matrix}\right] & \left[\begin{matrix}- \frac{y_{22}}{y_{21}} & - \frac{1}{y_{21}}\\- \frac{\Delta}{y_{21}} & - \frac{y_{11}}{y_{21}}\end{matrix}\right] \\ Z & \left[\begin{matrix}\frac{- s_{11} s_{22} + s_{11} + s_{12} s_{21} - s_{22} + 1}{s_{11} s_{22} - s_{11} - s_{12} s_{21} - s_{22} + 1} & - \frac{2 s_{12}}{s_{11} s_{22} - s_{11} - s_{12} s_{21} - s_{22} + 1}\\\frac{2 s_{21}}{s_{11} s_{22} - s_{11} - s_{12} s_{21} - s_{22} + 1} & \frac{s_{11} s_{22} + s_{11} - s_{12} s_{21} - s_{22} - 1}{s_{11} s_{22} - s_{11} - s_{12} s_{21} - s_{22} + 1}\end{matrix}\right] & Z & \left[\begin{matrix}\frac{y_{22}}{\Delta} & - \frac{y_{12}}{\Delta}\\- \frac{y_{21}}{\Delta} & \frac{y_{11}}{\Delta}\end{matrix}\right] \\ Y & \left[\begin{matrix}\frac{- s_{11} s_{22} - s_{11} + s_{12} s_{21} + s_{22} + 1}{s_{11} s_{22} + s_{11} - s_{12} s_{21} + s_{22} + 1} & - \frac{2 s_{12}}{s_{11} s_{22} + s_{11} - s_{12} s_{21} + s_{22} + 1}\\\frac{2 s_{21}}{s_{11} s_{22} + s_{11} - s_{12} s_{21} + s_{22} + 1} & \frac{s_{11} s_{22} - s_{11} - s_{12} s_{21} + s_{22} - 1}{s_{11} s_{22} + s_{11} - s_{12} s_{21} + s_{22} + 1}\end{matrix}\right] & \left[\begin{matrix}\frac{z_{22}}{\Delta} & - \frac{z_{12}}{\Delta}\\- \frac{z_{21}}{\Delta} & \frac{z_{11}}{\Delta}\end{matrix}\right] & Y \end{array} \end{split}\]

Tabla de conversión: T_S, Z, Y

\[\begin{split}\displaystyle \begin{array}{ l c c c } & T_S & Z & Y \\ T_S & T_S & \left[\begin{matrix}\frac{z_{11}}{z_{21}} & \frac{\Delta}{z_{21}}\\\frac{1}{z_{21}} & \frac{z_{22}}{z_{21}}\end{matrix}\right] & \left[\begin{matrix}- \frac{y_{22}}{y_{21}} & - \frac{1}{y_{21}}\\- \frac{\Delta}{y_{21}} & - \frac{y_{11}}{y_{21}}\end{matrix}\right] \\ Z & \left[\begin{matrix}\frac{Z_{01} t_{21} + Z_{01} t_{22} + t_{11} \overline{Z_{01}} + t_{12} \overline{Z_{01}}}{t_{11} + t_{12} - t_{21} - t_{22}} & \frac{- Z_{01} Z_{02} t_{11} t_{22} + Z_{01} Z_{02} t_{12} t_{21} - Z_{01} t_{11} t_{22} \overline{Z_{02}} + Z_{01} t_{12} t_{21} \overline{Z_{02}} - Z_{02} t_{11} t_{22} \overline{Z_{01}} + Z_{02} t_{12} t_{21} \overline{Z_{01}} - t_{11} t_{22} \overline{Z_{01}} \overline{Z_{02}} + t_{12} t_{21} \overline{Z_{01}} \overline{Z_{02}}}{2 t_{11} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} + 2 t_{12} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} - 2 t_{21} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} - 2 t_{22} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}}}\\\frac{2 \operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}}{t_{11} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} + t_{12} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} - t_{21} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} - t_{22} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}}} & \frac{Z_{02} t_{12} - Z_{02} t_{22} - t_{11} \overline{Z_{02}} + t_{21} \overline{Z_{02}}}{t_{11} + t_{12} - t_{21} - t_{22}}\end{matrix}\right] & Z & \left[\begin{matrix}\frac{y_{22}}{\Delta} & - \frac{y_{12}}{\Delta}\\- \frac{y_{21}}{\Delta} & \frac{y_{11}}{\Delta}\end{matrix}\right] \\ Y & \left[\begin{matrix}\frac{- Z_{02} t_{12} + Z_{02} t_{22} + t_{11} \overline{Z_{02}} - t_{21} \overline{Z_{02}}}{- Z_{01} Z_{02} t_{22} + Z_{01} t_{21} \overline{Z_{02}} - Z_{02} t_{12} \overline{Z_{01}} + t_{11} \overline{Z_{01}} \overline{Z_{02}}} & \frac{- Z_{01} Z_{02} t_{11} t_{22} + Z_{01} Z_{02} t_{12} t_{21} - Z_{01} t_{11} t_{22} \overline{Z_{02}} + Z_{01} t_{12} t_{21} \overline{Z_{02}} - Z_{02} t_{11} t_{22} \overline{Z_{01}} + Z_{02} t_{12} t_{21} \overline{Z_{01}} - t_{11} t_{22} \overline{Z_{01}} \overline{Z_{02}} + t_{12} t_{21} \overline{Z_{01}} \overline{Z_{02}}}{- 2 Z_{01} Z_{02} t_{22} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} + 2 Z_{01} t_{21} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} \overline{Z_{02}} - 2 Z_{02} t_{12} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} \overline{Z_{01}} + 2 t_{11} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} \overline{Z_{01}} \overline{Z_{02}}}\\\frac{2 \operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}}{- Z_{01} Z_{02} t_{22} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} + Z_{01} t_{21} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} \overline{Z_{02}} - Z_{02} t_{12} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} \overline{Z_{01}} + t_{11} \sqrt{\operatorname{re}{\left(Z_{01}\right)} \operatorname{re}{\left(Z_{02}\right)}} \overline{Z_{01}} \overline{Z_{02}}} & \frac{- Z_{01} t_{21} - Z_{01} t_{22} - t_{11} \overline{Z_{01}} - t_{12} \overline{Z_{01}}}{- Z_{01} Z_{02} t_{22} + Z_{01} t_{21} \overline{Z_{02}} - Z_{02} t_{12} \overline{Z_{01}} + t_{11} \overline{Z_{01}} \overline{Z_{02}}}\end{matrix}\right] & \left[\begin{matrix}\frac{z_{22}}{\Delta} & - \frac{z_{12}}{\Delta}\\- \frac{z_{21}}{\Delta} & \frac{z_{11}}{\Delta}\end{matrix}\right] & Y \end{array} \end{split}\]