How do I need to configure Matlab's solver such that it returns the correct solutions when complex magnitudes are involved?

Hi community

I am trying to solve the equation

where g1, g2, Z, and C are positive, real numbers. w is the independent variable. It is real but can be negative. j is the imaginary number. The following Matlab code returns a imaginary result, although there should be a real solution:

clc; clear;

syms g1 g2 Z C positive
syms w real

expression = (g1+g2)*Z/(1+1i*w*Z*C)
solve(abs(expression)==1,w)

What am I missing? Here is my hand calculation:

Answer 1

[Torsten]

Better use

solve(expression*expression'==1,w)

instead of

solve(abs(expression)==1,w)

From your hand calculation it follows that you must assume that

(g1+g2)*Z >= 1 

to get a real solution w.

[Dominik Hiltbrunner]

Hi Torsten

Thank you for your answer. In my case, the term inside the square root is guaranteed to be larger than zero. So I modified the code as follows:

clc; clear;

syms g1 g2 Z C positive
syms w real
%assume((g1+g2)*Z >= 1)

expression = (g1+g2)*Z/(1+1i*w*Z*C)
solve(expression*expression'==1,w)

The assume statement givs some very strange results when used. If commented out, the solution is now correct; it is just not displayed nicely. The solution contains the imaginary number, but it could be simplified to a real solution (by taking the imaginary number into the square root). Using simplify() doesn't do the job. Any idea how to achieve this?

On a side note: What's wrong with abs()? Why is the solution not correct here? Somebody knows?

[Paul]

clc; clear;
syms g1 g2 Z C positive
syms w real
%assume((g1+g2)*Z >= 1)
expression = (g1+g2)*Z/(1+1i*w*Z*C)

expression = 

sol = solve(expression*expression'==1,w,'ReturnConditions',true);

There are two solutions, but the solutions obtain when two conditions are met, one of which was the commented assumption.

sol.w

ans = 

sol.conditions

ans = 

If we assume both conditions

assumeAlso(sol.conditions);

Then we get closer to the desired result

w = simplify(sol.w)

w = 

Don't know if there is way to combine each under a single radical and express as in the hand-written solution; I couldn't fine one easily.

[Sam Chak]

@Dominik Hiltbrunner, Is ω the frequency you are attempting to determine, where the amplitude ratio of the output sinusoid to the input sinusoid equals 1?

[Torsten]

On a side note: What's wrong with abs()? Why is the solution not correct here? Somebody knows?

I'd say it's a bug.

[Dominik Hiltbrunner]

@Sam Chak

Exactly.

[Paul]

I too am confused about a few things.

First, I think the desired solution obtains only when (g1_g2)*Z > 1 (strictly greater).

Second, I changed assume to assumeAlso, otherwise solve loses sight of the fact that g1, g2, and Z must be positive (and more importantly that they are real).

syms g1 g2 Z C positive
syms w real
assumeAlso((g1+g2)*Z > 1)
expression = (g1+g2)*Z/(1+1i*w*Z*C)

expression = 

sol = solve(expression*expression'==1,w,'ReturnConditions',true);

However, this doesn't change the solution posted above.

sol.w

ans = 

sol.conditions

ans = 

As I understand it, solve is supposed to respect assumptions, so I don't understand why it's considering Z*(g1+g2) = 1 as a possibility.

Furthermore, the conditions are always true (presumably under the given assumptions)

simplify(sol.conditions,100)

ans = 

so it's unclear why they are returned as conditions at all. The implication is that the conditions don't need to be assumed to simplify sol.w, and that is, in fact, the case

simplify(sol.w,100)

ans = 

[Torsten]

First, I think the desired solution obtains only when (g1_g2)*Z > 1 (strictly greater).

w = 0 is also real.