Solve systems of 6 non-linear equations

I am trying to solve a sytem of 6 non-linear equations. I used vpasolve. One solution it gave me is I1=I2, V1=V2, and hence, my deltaT2 is roughly 0. So, I set the starting values of all to 1e-10 so that my deltaT2 will be non-zero. But after around one minute, I got an empty solution. I believe there should be a solution set of values for this system. So I wonder what could go wrong with my code/solution.

To make those equations I am solving more sense, I attached the screenshot I got from this paper below. Equation 2-3 are derived by using two points at t0 and t1 lying on one circle. Equation 4-5 are derived by using two points at t1 and t2 lying on an eclipse.

The only solution I got from my code so far is that V1=V2 and I1=I2. But as you can see, I want V1 and V2 to have different values. I1 and I2 should have differenet values as well.

clear;clc;close all;
%% Parameters
Vin = 400; Lr = 55e-6; Cr = 24e-9; Lm = 280e-6;
Zo = sqrt(Lr/Cr); Vo = 12; n = Vin/2/Vo; 
wo = 1/sqrt(Lr*Cr); fo = wo/(2*pi); 
Iheavy = 15; Rheavy = Vo/Iheavy;
Ilight = 5; Rlight = Vo/Ilight;
fs = 120e3; Ts = 1/fs;

Vin = 350;
Zr1 = sqrt(Lr/Cr); wr1 = sqrt(1/(Lr*Cr));
Zr2 = sqrt((Lr+Lm)/Cr); wr2 = sqrt(1/((Lr+Lm)*Cr));
R = Rheavy;
syms I1 V1 V2 I2 deltaT1 deltaT2 real

eq1 = I1 + I2 - (n*Vo/Lm)*deltaT1;
eq2 = (V1 - Vin + n*Vo)^2 + (I1*Zr1)^2 - (V2 - Vin + n*Vo)^2 - (I2*Zr1)^2;
eq3 = deltaT1 - (pi-atan(I2*Zr1/(V2-Vin+n*Vo)) + atan(I1*Zr1/(V1+Vin-n*Vo)))/wr1;
eq4 = (V1 - Vin)^2 + (I1*Zr2)^2 - (V2 - Vin)^2 - (I2*Zr2)^2;
eq5 = deltaT2 - (atan(I1*Zr2/(Vin-V1)) - atan(I2*Zr2/(Vin-V2)))/wr2;
eq6 = Vo*(deltaT1 + deltaT2)/(n*R) - (V1+V2)*Cr - ((I1-I2)/2)*deltaT1;
eqs = [eq1,eq2,eq3,eq4,eq5,eq6];

starting_value = [1e-10 Inf; 1e-10 Inf; 1e-10 Inf; ...
                 1e-10 Inf; 1e-10 Inf; 1e-10 Inf];
[I1,V1,V2,I2,deltaT1,deltaT2] = vpasolve(eqs,[I1,V1,V2,I2,deltaT1,deltaT2],... 
starting_value);
result = double([I1,V1,V2,I2,deltaT1,deltaT2])

Answer 1

[Tra Yen Nhu Phan]

I figured the problem out. The paper has a small syntax in their system of equations. Therefore, MATLAB cannot give the result that I want. The mistake was at equation 2 that Vin should be -Vin. Also, instead of using my own parameters, I used the paper's parameters, and everything just works perfectly. Thanks all for helping me this out.

Below is the code that works for me. The modification includes:

1) Correct the error in the equation system

2) Use parameters in the paper instead

3) Add the range for all parameters as positive numbers

clear;clc; close all;
Vo = 48; Po = 7200; figure; 
N = 18; Lm = 111.4e-6; Lr = 19.18e-6;
Cr = 20.25e-9; Co = 1e-3;
Zr1 = sqrt(Lr/Cr); wr1 = 1/sqrt(Lr*Cr);     
fr1 = wr1/(2*pi); Tr1 = 1/fr1;
Zr2 = sqrt((Lr + Lm)/Cr); wr2 = sqrt(1/((Lr + Lm)*Cr));
fr2 = wr2/(2*pi); Tr2 = 1/fr2;
select = 1;
Q1 = 0.1831; Q2 = 0.3662;
if (select == 1)
    Q = Q1;
else 
    Q = Q2;
end
Rac = Zr1/Q; Ro = Rac*(pi^2)/8/(N^2);

%% Below Resonance
syms I1 V1 I2 V2 deltaT1 deltaT2 fs
Vin = 800;
eq1 = (I1 + I2) -(N*Vo/Lm)*deltaT1;
eq2 = (-V1 - Vin + N*Vo)^2 +(I1*Zr1)^2 - (V2 - Vin + N*Vo)^2 - (I2*Zr1)^2 ;
eq3 = deltaT1 - ((pi - atan(I2*Zr1/(V2-Vin+N*Vo)) + atan(I1*Zr1/(V1+Vin-N*Vo)))/wr1);
eq4 = (V1 - Vin)^2 +(I1*Zr2)^2 - (V2 - Vin)^2 - (I2*Zr2)^2 ;
eq5 = deltaT2 - ((atan(I1*Zr2/(Vin - V1)) - atan(I2*Zr2/(Vin - V2)))/wr2);
eq6 = (Vo*(deltaT1 + deltaT2)/(N*Ro)) - (V1+V2)*Cr - (deltaT1*(I1-I2))/2;
eqs = [eq1,eq2,eq3,eq4,eq5,eq6];
range = [0, Inf;                        % I1
         0, Inf;                        % V1
         0, Inf;                        % I2
         0, Inf;                        % V2
         0, Inf;                        % deltaT1
         0, Inf];                       % deltaT2
[I1,V1,V2,I2,deltaT1,deltaT2] = vpasolve(eqs,[I1,V1,V2,I2,deltaT1,deltaT2],range)

I1 = 

8.483487211088991203528612806318

V1 = 

253.64132442119089964849356851349

V2 = 

163.29575706350444322315252328323

I2 = 

7.4427274570926256928348488481962

deltaT1 = 

0.0000020534494375410093334689879526685

deltaT2 = 

0.00000022936567635361382499350827474835

fs = 1/(deltaT1*2 + deltaT2*2)     

fs = 

219027.81217659331611383356783722

[Sam Chak]

Hi @Tra Yen Nhu Phan

I'm glad to hear that you were able to resolve the issue on your own. However, your "self-Accepted Answer" doesn't technically qualify as a complete "Answer." It appears more as an update to inform the community that you've found a solution, without providing the updated parameters and code.

Given that your question involves a rather specialized domain in LLC resonant converter, comprehending the paper and validating the equation error would require significant expertise. Consequently, it's likely that your "Answer" won't be of much benefit to those outside the domain.

Nonetheless, you have the option to edit your question or "Answer" or even a "Comment" to include the updated parameters and code, which would greatly enhance its value. Your contribution would be more informative and helpful to others in the community.

[Tra Yen Nhu Phan]

Hi @Sam Chak, I understand and will add the updated code. Thanks for your suggestion ! I appreciated your help and time.

[Sam Chak]

Hi @Tra Yen Nhu Phan

Thank you for updating your "self-accepted answer" with the MATLAB code; it's a valuable contribution to the community.

In addition, I'd like to encourage you to consider upvoting @Star Strider's answer if he was helpful in resolving your issue. It's a way to show gratitude for his and others voluntary assistance and can further foster a sense of community support here. Your acknowledgment can be a simple yet meaningful gesture.

Thank you for your active participation, and please don't hesitate to reach out if you have any more questions or need further assistance.

[Tra Yen Nhu Phan]

I absolutely can do that. Thanks for letting me know what I should do to express my appreciation for all's help.

[Sam Chak]

@Tra Yen Nhu Phan, 👍.