How to solve two equations two unknowns for the equations with syms?

I am trying to calculate solutions for the two equation two unknown type scenario. But, the solver is not able to give any explicit solution. The unknows are syms type transfer functions H_mi_thetadcap and H_mi_vtest.

clc
clear

%% Parameters

Vm = 277 * sqrt(6) ;

ws = 2 * pi * 85000 ;

Lp = 28.3e-6 ;
RLp = 77.55e-3 ;
Zpss = ( ws * Lp * 1i ) + RLp ;

Cpp = 123.13e-9 ;
RCpp = 13.55e3 ;
Zppss = RCpp / ( 1 + RCpp * Cpp * ws * 1i ) ;

Cps = 160.32e-9 ;
RCps = 8e3 ;
Zpsss = RCps / ( 1 + RCps * Cps * ws * 1i ) ;

Ll = 37.45e-6 ;
RLl = 25.85e-3 ;
Zlss = ( ws * Ll * 1i ) + RLl ; 

Lm = 996.2e-6 ;
Zmss = ( ws * Lm * 1i ) ; 

Cdc = 161.466e-6 ;
RCdc = 20e3 ;
Zdcss = RCdc ;

Rload = 500e3 ; % 128.94 ;

Vbat = 730 ;
Rbat = 0.045 ;

Kp1ss = ( Zmss + Zpsss + Zlss ) / Zmss ;
Kp2ss = Zpsss + Zlss ;
Kp3ss = 1 + ( Zpss / Zppss ) + ( Zpss / ( Zpsss + Zlss ) ) ;
Kp4ss = Zpss / ( Zpsss + Zlss ) ;
Ko1ss = ( 1 / Zdcss ) + ( 1 / Rload ) ;
Ko2ss = 1 + ( Rbat / Rload ) ;

%% Steady State

Mi = 0.975 ;

Vtt = ( 2 * sqrt(3) / pi ) * Mi * Vm ;

thetad = 1.263077326 ;

Vdc = Vbat ;

Vd = ( 4 / pi ) * Vdc * ( cos( thetad ) - 1i * sin( thetad ) ) ;

Id = abs( ( Vtt - ( Vd * ( Kp1ss * Kp3ss - Kp4ss ) ) ) / ( Kp2ss * Kp3ss ) ) * ( cos( thetad ) - 1i * sin( thetad ) ) ;

for iterations = 1 : 1 : 10
    
    Iout = ( 2 / pi ) * abs( Id ) ;
    
    Idc = Iout ;
    
    Vdc = ( Vbat + Idc * Rbat ) / ( 1 + Rbat / Rload ) ; 
    
    Vd = ( 4 / pi ) * Vdc * ( cos( thetad ) - 1i * sin( thetad ) ) ;
    
    Id = abs( ( Vtt - ( Vd * ( Kp1ss * Kp3ss - Kp4ss ) ) ) / ( Kp2ss * Kp3ss ) ) * ( cos( thetad ) - 1i * sin( thetad ) ) ;
    
end

Pdc = Vdc * Idc ;

Rd = ( 8 / pi^2 ) * ( Vdc / Idc ) ;

Vtest = 339.254 ;

%% Small Signal

syms s H_mi_thetadcap H_mi_vtest real

mi = Mi ;

vtt = ( 2 * sqrt(3) / pi ) * mi * Vm ;

Zp = ( s * Lp ) + ( ws * Lp * 1i ) + RLp ;
Zpp = RCpp / ( 1 + ( s * Cpp * RCpp ) + ( ws * Cpp * 1i * RCpp ) ) ; 
Zps = RCps / ( 1 + ( s * Cps * RCps ) + ( ws * Cps * 1i * RCps ) ) ; 
Zl = ( s * Ll ) + ( ws * Ll * 1i ) + RLl ; 
Zm = ( s * Lm ) + ( ws * Lm * 1i ) ;  
Zdc = RCdc / ( 1 + ( s * RCdc * Cdc ) ) ;

Kp1 = ( Zm + Zps + Zl ) / Zm ;
Kp2 = Zps + Zl ;
Kp3 = 1 + ( Zp / Zpp ) + ( Zp / ( Zps + Zl ) ) ;
Kp4 = Zp / ( Zps + Zl ) ;
Ko1 = ( 1 / Zdc ) + ( 1 / Rload ) ;
Ko2 = 1 + ( Rbat / Rload ) ;

H_mi_idcap_V = ( ( 4 * sqrt(3) * Vtest ) + ( 4 * sqrt(3) * Mi * H_mi_vtest ) + ( 1i * Vdc * 4 * ( Kp1 * Kp3 - Kp4 ) * ( cos( thetad ) - 1i * sin( thetad ) ) * H_mi_thetadcap ) ) / ( pi * Kp2 * Kp3 ) ;

H_mi_idcap_real = real( H_mi_idcap_V ) ;
H_mi_idcap_imag = imag( H_mi_idcap_V ) ;
Id_real = real( Id ) ;
Id_imag = imag( Id ) ;
Id_abs = abs( Id ) ;

equation_one = H_mi_thetadcap + ( ( H_mi_idcap_imag * Id_real ) - ( H_mi_idcap_real * Id_imag ) ) / Id_abs^2 ;
equation_two = H_mi_idcap_real - ( ( -Id_imag / Id_real ) * H_mi_idcap_imag ) ;

solution = solve( [equation_one, equation_two], [H_mi_thetadcap, H_mi_vtest] ) 

Warning: Unable to find explicit solution. For options, see help.

solution = struct with fields:
    H_mi_thetadcap: [0×1 sym]
        H_mi_vtest: [0×1 sym]

H_mi_thetadcap = getfield( solution, 'H_mi_thetadcap' ) 

 
H_mi_thetadcap =
 
Empty sym: 0-by-1
 

H_mi_vtest = getfield( solution, 'H_mi_vtest' ) 

 
H_mi_vtest =
 
Empty sym: 0-by-1
 

Answer 1

[Dinesh]

Hi

The warning: unable to find explicit solution is because your equation is too complicated to solve. One way to solve this question would be to use numerical approach (vpasolve or fsolve).

Please refer to the following MATLAB documentations.

1. https://www.mathworks.com/help/symbolic/sym.vpasolve.html
2. https://www.mathworks.com/help/optim/ug/fsolve.html

Also refer to this MATLAB answer thread for more details on solving two equations with two unknowns.

https://www.mathworks.com/matlabcentral/answers/395460-how-can-i-solve-two-equations-with-two-unknown-variable-using-matlab

Hope this helps,

Thank you!