NSolve gives correct solution after splitting real and imaginary equations

[bertramaerts (Bert Aerts)]

I have a second order continuous time system transfer function denominator with as roots:

Now I want to go to a discrete time system with the equation:

In fact I want to solve following complex equation, making z equal to the pole of a discrete time system with sampling time 0.01 sec:

But Mathematica 13.2 gives a huge answer depending on a constant c1.

When I help Mathematica as follows, by splitting real and imaginary equations, it does find the correct solution:

How can I adapt the first formula?

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[bertramaerts (Bert Aerts)]

I solved it by giving 2 complex equations for 2 unknowns zeta and omegaN, as there are 2 complex conjugate poles. And now Mathematica immediately finds the solution.

This method also works for 2 real poles:

Full Mathematica 13.2 notebook:

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