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# Solve equation that has a complex subexpression

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Bill Tubbs on 30 Jul 2020
Commented: Star Strider on 2 Dec 2020
I want to solve the following equation for omega: where So I tried this:
syms s omega G(s)
G(s) = 10/(s*(1+s)*(1+0.2*s));
% Try to find omega that satisfies the equation:
solve(angle(subs(G(s),s,omega*j))-deg2rad(-135),omega,'Real',true)
Result:
Error using mupadengine/feval_internal (line 172)
No complex subexpressions allowed in real mode.
Error in solve (line 293)
sol = eng.feval_internal('solve', eqns, vars, solveOptions);
Although there is an imaginary number in the expression, the decision variable is real and the expression evaluates to a real number (due to angle) so I don't see why it should have a problem solving this.
Obviously, I can think of other ways to solve the problem, but it would be nice to just use angle on the whole transfer function.
% Get solution a different way:
omega_sol = solve(-pi/2-atan2(omega,1)-atan2(omega,5)-deg2rad(-135),omega)
% Confirm solution:
subs(angle(subs(G(s),s,omega*j))-deg2rad(-135),omega,omega_sol)
omega_sol =
0.7417
ans =
-1.8367e-40
In summary, is there any way to solve the original expression for omega directly:
angle(subs(G(s),s,omega*j)) == deg2rad(-135)
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### Accepted Answer

Star Strider on 30 Jul 2020
Solving for the tangent of the phase angle, rather than using the arctangent of the transfer function, appears to produce the correct result:
syms s omega G(s)
assume(omega > 0)
G(s) = 10/(s*(1+s)*(1+0.2*s));
G = subs(G, s, 1j*omega)
OMG = solve(imag(G)/real(G) == tan(deg2rad(-135)), omega)
vpaOMG = vpa(OMG)
producing:
vpaOMG =
0.74165738677394138558374873231655
.
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Star Strider on 30 Jul 2020
As always, my pleasure!
I thought about using ‘1j*omega’ as a function argument, however went with subs because that was in your original code, and there was some reason you specifically used it.

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### More Answers (1)

Bill Tubbs on 2 Dec 2020
Edited: Bill Tubbs on 2 Dec 2020
I just discovered that you can also solve this numerically with vpasolve:
syms s omega G(s)
assume(omega > 0)
G(s) = 10/(s*(1+s)*(1+0.2*s));
% Try to find omega that satisfies the equation:
vpasolve(angle(G(omega*j)) == deg2rad(-135),omega)
ans =
0.74165738677394138558374873231655
This is a more robust solution as it can handle more complex functions such as this:
G(s) = exp(-4*s)/(1+s);
vpasolve(angle(G(omega*j)) == deg2rad(-135),omega)
ans =
0.47764713626095932403299027979129
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Star Strider on 2 Dec 2020
I’m not certain what you’re plotting.
Experiment with something like this:
ad = -180:20:180;
ad360 = mod(ad+360,360);
ar = -pi:0.31:pi;
ar2pi = mod(ar+2*pi,2*pi);
.

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