Análisis de redes mediante Análisis Nodal Modificado
Por Mariano Llamedo Soria
Resumen
En este documento se presentan …
Funciones de análisis de cuadripolos: smna
De presentación algebraica: print_latex, a_equal_b_latex_s, print_subtitle
De manipulación de sistemas lineales: tf2sos_analog, parametrize_sos
Introducción
Documento en elaboración. Algunas referencias para este tema:
GIC bicuad
A partir de una red de referencia:
se dibuja y parametriza en LTspice:
y luego este archivo será lo que se toma como entrada a las funciones de análisis nodal modificado.
"""
Referencias:
------------
Cap. 5. Schaumann Rolf. Design of Analog Filters.
Cap. 17. Vlach, Jiri - Linear Circuit Theory Matrices in Computer Applications (2014)
"""
import sympy as sp
from pytc2.sistemas_lineales import parametrize_sos
from pytc2.general import s, print_latex, print_subtitle, a_equal_b_latex_s
from pytc2.cuadripolos import smna
from IPython.display import display, Markdown
fileName_asc = './schematics/GIC bicuad.asc'
# symbolic MNA
equ_smna, extra_results = smna(fileName_asc,
bAplicarValoresComponentes = True,
bAplicarParametros = False)
print_subtitle('Ecuación MNA')
display(equ_smna)
Utilizando netlist: ./schematics/GIC bicuad.net
Ecuación MNA
\[\begin{split}\displaystyle \left[\begin{matrix}v_{va} \left(a c_{0} s + c_{0} s \left(1 - a\right) + \frac{1}{r} + \frac{b + eps}{q r} + \frac{- b + eps + 1}{q r}\right) + v_{vi} \left(- a c_{0} s - \frac{b + eps}{q r}\right) - \frac{v_{vb}}{r}\\I_{V1} + v_{va} \left(- a c_{0} s - \frac{b + eps}{q r}\right) + v_{vi} \left(a c_{0} s + \frac{c + eps}{r} + \frac{b + eps}{q r}\right) - \frac{v_{vd} \left(c + eps\right)}{r}\\I_{OXu2} - c_{0} s v_{vc} + v_{vb} \left(c_{0} s + \frac{1}{r}\right) - \frac{v_{va}}{r}\\- c_{0} s v_{vb} + v_{vc} \left(c_{0} s + \frac{1}{r}\right) - \frac{v_{vo}}{r}\\I_{OXu1} - \frac{v_{vc}}{r} - \frac{v_{vd}}{r} + \frac{2 v_{vo}}{r}\\v_{vd} \left(\frac{c + eps}{r} + \frac{- c + eps + 1}{r} + \frac{1}{r}\right) - \frac{v_{vi} \left(c + eps\right)}{r} - \frac{v_{vo}}{r}\\v_{vi}\\v_{va} - v_{vc} - \frac{v_{vo}}{aop}\\- v_{vc} + v_{vd} - \frac{v_{vb}}{aop}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\\0\\0\\0\\V_{1}\\0\\0\end{matrix}\right]\end{split}\]
# tuning a mano de las ecuaciones
A0 = extra_results['A']
if extra_results['eps'] != 0:
A0 = A0.subs(extra_results['eps'], 0)
if extra_results['aop'] != 0:
A0 = A0.limit(extra_results['aop'], sp.oo)
equ_smna = sp.Eq(A0*extra_results['X'], extra_results['Z'])
print_subtitle('Ecuación MNA')
display(equ_smna)
Ecuación MNA
\[\begin{split}\displaystyle \left[\begin{matrix}v_{va} \left(a c_{0} s + \frac{b}{q r} + c_{0} s \left(1 - a\right) + \frac{1}{r} + \frac{1 - b}{q r}\right) + v_{vi} \left(- a c_{0} s - \frac{b}{q r}\right) - \frac{v_{vb}}{r}\\I_{V1} - \frac{c v_{vd}}{r} + v_{va} \left(- a c_{0} s - \frac{b}{q r}\right) + v_{vi} \left(a c_{0} s + \frac{b}{q r} + \frac{c}{r}\right)\\I_{OXu2} - c_{0} s v_{vc} + v_{vb} \left(c_{0} s + \frac{1}{r}\right) - \frac{v_{va}}{r}\\- c_{0} s v_{vb} + v_{vc} \left(c_{0} s + \frac{1}{r}\right) - \frac{v_{vo}}{r}\\I_{OXu1} - \frac{v_{vc}}{r} - \frac{v_{vd}}{r} + \frac{2 v_{vo}}{r}\\- \frac{c v_{vi}}{r} + v_{vd} \left(\frac{c}{r} + \frac{1 - c}{r} + \frac{1}{r}\right) - \frac{v_{vo}}{r}\\v_{vi}\\v_{va} - v_{vc}\\- v_{vc} + v_{vd}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\\0\\0\\0\\V_{1}\\0\\0\end{matrix}\right]\end{split}\]
u1 = sp.solve(equ_smna, extra_results['X'])
H = u1[extra_results['v_out']] / u1[extra_results['v_in']]
display(H)
\[\displaystyle \frac{2 V_{1} a c_{0}^{2} q r^{2} s^{2} + 2 V_{1} b c_{0} r s - V_{1} c c_{0}^{2} q r^{2} s^{2} - V_{1} c c_{0} r s + V_{1} c q}{V_{1} \left(c_{0}^{2} q r^{2} s^{2} + c_{0} r s + q\right)}\]
H0 = sp.collect(sp.simplify(sp.expand(H)),s)
H0 = parametrize_sos(H)[0]
display(H0)
\[\displaystyle \frac{\frac{c}{2 a c_{0}^{2} r^{2} - c c_{0}^{2} r^{2}} + s^{2} + s \left(\frac{2 b}{2 a c_{0} q r - c c_{0} q r} - \frac{c}{2 a c_{0} q r - c c_{0} q r}\right)}{s^{2} + \frac{s}{c_{0} q r} + \frac{1}{c_{0}^{2} r^{2}}} \left(2 a - c\right)\]
Ackerberg Mossberg bicuad
Repetimos el procedimiento, a partir de una red de referencia:
se dibuja y parametriza en LTspice:
y luego procedemos al análisis
fileName_asc = './schematics/ACKMOSS bicuad.asc'
# symbolic MNA
equ_smna, extra_results = smna(fileName_asc,
bAplicarValoresComponentes = True,
bAplicarParametros = False)
# tuning a mano de las ecuaciones
A0 = extra_results['A']
if extra_results['eps'] != 0:
A0 = A0.subs(extra_results['eps'], 0)
if extra_results['aop'] != 0:
A0 = A0.limit(extra_results['aop'], sp.oo)
equ_smna = sp.Eq(A0*extra_results['X'], extra_results['Z'])
print_subtitle('Ecuación MNA')
display(equ_smna)
Utilizando netlist: ./schematics/ACKMOSS bicuad.net
Ecuación MNA
\[\begin{split}\displaystyle \left[\begin{matrix}v_{va} \left(a c_{0} s + c_{0} s + \frac{d}{r} + \frac{1}{r} + \frac{1}{q r}\right) + v_{vi} \left(- a c_{0} s - \frac{d}{r}\right) + v_{vo} \left(- c_{0} s - \frac{1}{q r}\right) - \frac{v_{vd}}{r}\\I_{V2} - \frac{b v_{vc}}{r} - \frac{c v_{vb}}{r} + v_{va} \left(- a c_{0} s - \frac{d}{r}\right) + v_{vi} \left(a c_{0} s + \frac{b}{r} + \frac{c}{r} + \frac{d}{r}\right)\\I_{OXu1} + v_{va} \left(- c_{0} s - \frac{1}{q r}\right) + v_{vo} \left(c_{0} s + \frac{1}{r} + \frac{1}{q r}\right) - \frac{v_{vb}}{r}\\I_{OXu2} - \frac{v_{va}}{r} - \frac{v_{vc}}{r} + \frac{2 v_{vd}}{r}\\- \frac{c v_{vi}}{r} - c_{0} s v_{n001} + v_{vb} \left(\frac{c}{r} + c_{0} s + \frac{1}{r}\right) - \frac{v_{vo}}{r}\\- \frac{b v_{vi}}{r} + v_{vc} \left(\frac{b}{r} + \frac{2}{r}\right) - \frac{v_{n001}}{r} - \frac{v_{vd}}{r}\\I_{OXu3} - c_{0} s v_{vb} + v_{n001} \left(c_{0} s + \frac{1}{r}\right) - \frac{v_{vc}}{r}\\v_{vi}\\- v_{va}\\v_{vb}\\- v_{vc}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\\0\\0\\0\\0\\V_{2}\\0\\0\\0\end{matrix}\right]\end{split}\]
u1 = sp.solve(equ_smna, extra_results['X'])
H = u1[extra_results['v_out']] / u1[extra_results['v_in']]
display(H)
\[\displaystyle \frac{- V_{2} a c_{0}^{2} q r^{2} s^{2} + V_{2} b c_{0} q r s - V_{2} c q - V_{2} c_{0} d q r s}{V_{2} \left(c_{0}^{2} q r^{2} s^{2} + c_{0} r s + q\right)}\]
H0 = sp.collect(sp.simplify(sp.expand(H)),s)
H0 = parametrize_sos(H)[0]
display(H0)
\[\displaystyle - a \frac{s^{2} + \frac{c}{a c_{0}^{2} r^{2}} + \frac{s \left(- b + d\right)}{a c_{0} r}}{s^{2} + \frac{s}{c_{0} q r} + \frac{1}{c_{0}^{2} r^{2}}}\]
Lattice como ecualizador de fase de primer orden
Analizamos ahora una red pasiva balanceada de primer orden:
fileName_asc = './schematics/lattice_1ord_delay_eq.asc'
# symbolic MNA
equ_smna, extra_results = smna(fileName_asc,
bAplicarValoresComponentes = True,
bAplicarParametros = True)
print_subtitle('Ecuación MNA')
display(equ_smna)
Utilizando netlist: ./schematics/lattice_1ord_delay_eq.net
Ecuación MNA
\[\begin{split}\displaystyle \left[\begin{matrix}I_{V1} + v_{n001} - v_{v1}\\I_{L1} - s v_{4} - v_{n001} + v_{v1} \left(s + 1\right)\\I_{L2} - s v_{v1} - v_{3} + v_{4} \left(s + 1\right)\\- I_{L1} + v_{3} \left(s + 1\right) - v_{4}\\v_{n001}\\- I_{L1} s - v_{3} + v_{v1}\\- I_{L2} s + v_{4}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\\0\\V_{1}\\0\\0\end{matrix}\right]\end{split}\]
Notar que al tratarse de una salida desbalanceada debemos calcular: $\( H = \frac{v_3-v_4}{v1-v0} \)$
y qye \(v_0 = 0\)
u1 = sp.solve(equ_smna, extra_results['X'])
H = (u1[extra_results['X'][2]] - u1[extra_results['X'][3]]) / u1[extra_results['X'][1]]
H = sp.simplify(sp.expand(H))
display(H)
\[\displaystyle \frac{s - 1}{s + 1}\]
Tee con trafo como ecualizador de fase de primer orden
Analizamos ahora la versión desbalanceada del lattice anteriormente vista:
fileName_asc = './schematics/tee_1ord_delay_eq.asc'
# symbolic MNA
equ_smna, extra_results = smna(fileName_asc,
bAplicarValoresComponentes = True,
bAplicarParametros = True)
print_subtitle('Ecuación MNA')
display(equ_smna)
Utilizando netlist: ./schematics/tee_1ord_delay_eq.net
Ecuación MNA
\[\begin{split}\displaystyle \left[\begin{matrix}I_{V1} + v_{n001} - v_{vi}\\- I_{L1} - v_{n001} + v_{vi}\\I_{L2} + v_{vo}\\I_{L1} - I_{L2} + 2 s v_{n002}\\v_{n001}\\- \frac{I_{L1} s}{2} - \frac{I_{L2} s}{2} + v_{n002} - v_{vi}\\- \frac{I_{L1} s}{2} - \frac{I_{L2} s}{2} - v_{n002} + v_{vo}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\\0\\V_{1}\\0\\0\end{matrix}\right]\end{split}\]
u1 = sp.solve(equ_smna, extra_results['X'])
H = u1[extra_results['v_out']] / u1[extra_results['v_in']]
H = sp.simplify(sp.expand(H))
display(H)
\[\displaystyle \frac{1 - s}{s + 1}\]
Lattice como ecualizador de fase de segundo orden
Analizamos ahora una red pasiva balanceada de segundo orden:
fileName_asc = './schematics/lattice_2ord_delay_eq.asc'
# symbolic MNA
equ_smna, extra_results = smna(fileName_asc,
bAplicarValoresComponentes = True,
bAplicarParametros = True)
print_subtitle('Ecuación MNA')
display(equ_smna)
Utilizando netlist: ./schematics/lattice_2ord_delay_eq.net
Ecuación MNA
\[\begin{split}\displaystyle \left[\begin{matrix}I_{V1} - v_{1} + v_{n001}\\I_{L1} + I_{L2} + v_{1} \left(1 + \frac{s}{a}\right) - v_{n001} - \frac{s v_{3}}{a}\\- I_{L3} - \frac{a s v_{n002}}{b} - v_{3} + v_{4} \left(\frac{a s}{b} + 1 + \frac{s}{a}\right)\\- I_{L1} - \frac{a s v_{n003}}{b} + v_{3} \left(\frac{a s}{b} + 1 + \frac{s}{a}\right) - v_{4} - \frac{s v_{1}}{a}\\- I_{L2} - \frac{a s v_{4}}{b} + \frac{a s v_{n002}}{b}\\- I_{L4} - \frac{a s v_{3}}{b} + \frac{a s v_{n003}}{b}\\v_{n001}\\- \frac{I_{L1} a s}{b} + v_{1} - v_{3}\\- \frac{I_{L2} s}{a} + v_{1} - v_{n002}\\- \frac{I_{L3} a s}{b} - v_{4}\\- \frac{I_{L4} s}{a} - v_{n003}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\\0\\0\\0\\V_{1}\\0\\0\\0\\0\end{matrix}\right]\end{split}\]
u1 = sp.solve(equ_smna, extra_results['X'])
H = (u1[extra_results['X'][2]] - u1[extra_results['X'][3]]) / u1[extra_results['X'][1]]
H = sp.simplify(sp.expand(H))
display(H)
\[\displaystyle \frac{a s - b - s^{2}}{a s + b + s^{2}}\]
Tee puenteada como ecualizador de demora de segundo orden
Analizamos ahora una red desbalanceada equivalente sin transformador:
fileName_asc = './schematics/tee_puen_2ord_delay_eq.asc'
# symbolic MNA
equ_smna, extra_results = smna(fileName_asc,
bAplicarValoresComponentes = True,
bAplicarParametros = True)
# tuning a mano de las ecuaciones
A0 = extra_results['A']
if extra_results['eps'] != 0:
A0 = A0.subs(extra_results['eps'], 0)
if extra_results['aop'] != 0:
A0 = A0.limit(extra_results['aop'], sp.oo)
equ_smna = sp.Eq(A0*extra_results['X'], extra_results['Z'])
print_subtitle('Ecuación MNA')
display(equ_smna)
Utilizando netlist: ./schematics/tee_puen_2ord_delay_eq.net
Ecuación MNA
\[\begin{split}\displaystyle \left[\begin{matrix}I_{V1} + v_{n001} - v_{vi}\\I_{L1} - v_{n001} + v_{vi} \left(1 + \frac{s}{a}\right) - \frac{s v_{va}}{a}\\- I_{L1} + v_{vo} \left(1 + \frac{s}{a}\right) - \frac{s v_{va}}{a}\\- I_{L2} + \frac{2 s v_{n002}}{b \left(- \frac{a}{b} + \frac{1}{a}\right)}\\I_{L2} + \frac{2 s v_{va}}{a} - \frac{s v_{vi}}{a} - \frac{s v_{vo}}{a}\\v_{n001}\\- \frac{2 I_{L1} a s}{b} + v_{vi} - v_{vo}\\- \frac{I_{L2} s}{2 a} - v_{n002} + v_{va}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\\0\\0\\V_{1}\\0\\0\end{matrix}\right]\end{split}\]
u1 = sp.solve(equ_smna, extra_results['X'])
H = u1[extra_results['v_out']] / u1[extra_results['v_in']]
H0 = sp.collect(sp.simplify(sp.expand(H)),s)
H0 = parametrize_sos(H)[0]
display(H0)
\[\displaystyle 1 \frac{- a s + b + s^{2}}{a s + b + s^{2}}\]
Tee puenteada con trafo
Ahora la misma red pero con un transformador para elevar un poco la complejidad:
fileName_asc = './schematics/tee_puen_2ord_delay_eq2.asc'
# symbolic MNA
equ_smna, extra_results = smna(fileName_asc,
bAplicarValoresComponentes = True,
bAplicarParametros = True)
# tuning a mano de las ecuaciones
A0 = extra_results['A']
if extra_results['eps'] != 0:
A0 = A0.subs(extra_results['eps'], 0)
if extra_results['aop'] != 0:
A0 = A0.limit(extra_results['aop'], sp.oo)
equ_smna = sp.Eq(A0*extra_results['X'], extra_results['Z'])
print_subtitle('Ecuación MNA')
display(equ_smna)
Utilizando netlist: ./schematics/tee_puen_2ord_delay_eq2.net
Ecuación MNA
\[\begin{split}\displaystyle \left[\begin{matrix}I_{V1} + v_{n001} - v_{vi}\\- I_{L1} - v_{n001} + v_{vi} \left(1 + \frac{s}{2 a}\right) - \frac{s v_{vo}}{2 a}\\- I_{L2} + v_{vo} \left(1 + \frac{s}{2 a}\right) - \frac{s v_{vi}}{2 a}\\I_{L1} + I_{L2} + \frac{2 a s v_{va}}{b}\\v_{n001}\\- I_{L1} s \left(\frac{a}{2 b} + \frac{1}{2 a}\right) - \frac{I_{L2} s \left(- \frac{a}{b} + \frac{1}{a}\right) \sqrt{\left(\frac{a}{2 b} + \frac{1}{2 a}\right)^{2}}}{\frac{a}{b} + \frac{1}{a}} + v_{va} - v_{vi}\\- \frac{I_{L1} s \left(- \frac{a}{b} + \frac{1}{a}\right) \sqrt{\left(\frac{a}{2 b} + \frac{1}{2 a}\right)^{2}}}{\frac{a}{b} + \frac{1}{a}} - I_{L2} s \left(\frac{a}{2 b} + \frac{1}{2 a}\right) + v_{va} - v_{vo}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\\0\\V_{1}\\0\\0\end{matrix}\right]\end{split}\]
u1 = sp.solve(equ_smna, extra_results['X'])
H = u1[extra_results['v_out']] / u1[extra_results['v_in']]
display(H)
\[\displaystyle \frac{- \frac{2 V_{1} a^{5} b s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{2 V_{1} a^{4} b^{2}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{V_{1} a^{4} b s^{2}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{2 V_{1} a^{4} s^{4}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{3 V_{1} a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 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2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{2 V_{1} a^{4} s^{4}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{V_{1} a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{4 V_{1} a^{3} b^{2} s}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{4 V_{1} a^{3} b s^{3}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} - \frac{2 V_{1} a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{2 V_{1} a^{2} b^{3}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{4 V_{1} a^{2} b^{2} s^{2}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{2 V_{1} a^{2} b s^{4}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} - \frac{V_{1} a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{2 V_{1} a b^{3} s}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}} + \frac{V_{1} b^{3} s^{2}}{4 a^{6} s^{2} + 6 a^{5} b s + 6 a^{5} s^{3} + 2 a^{4} b^{2} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{4} b^{2} + 2 a^{4} b s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 10 a^{4} b s^{2} + 4 a^{4} s^{4} + 2 a^{3} b^{2} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{3} b^{2} s + 8 a^{3} b s^{3} - 2 a^{2} b^{3} s \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 4 a^{2} b^{3} - 2 a^{2} b^{2} s^{3} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 8 a^{2} b^{2} s^{2} + 4 a^{2} b s^{4} - 2 a b^{3} s^{2} \sqrt{\frac{a^{2}}{b^{2}} + \frac{2}{b} + \frac{1}{a^{2}}} + 2 a b^{3} s + 2 a b^{2} s^{3} + 2 b^{3} s^{2}}}\]
H0 = H.subs(extra_results['dic_params'])
H0 = sp.collect(sp.simplify(sp.expand(H0)),s)
#H0 = parametrize_sos(H)[0]
display(H0)
\[\displaystyle \frac{\left(s^{4} + 1\right) \left(s^{4} + 2 \sqrt{2} s^{3} + 4 s^{2} + 2 \sqrt{2} s + 1\right)}{s^{8} + 4 \sqrt{2} s^{7} + 16 s^{6} + 20 \sqrt{2} s^{5} + 34 s^{4} + 20 \sqrt{2} s^{3} + 16 s^{2} + 4 \sqrt{2} s + 1}\]
No es el resultado que esperábamos, habrá que ver dónde quedó un error …