Find the Phase of the Plant which has imaginary poles

Question

I had Given a Continuous Time Plant in which I need to design a PID - Discrete Controller. Given a Bandwidth and a Phase Margin, My approach was to include a delay due to the zoh. However the Plant has imaginary poles which threw me off from my normal procedure. I am not suppose to used mathlab. $BW=1 Hz; Wc=4.18 rad/s zohlad = -T*Wc/s = 12 degrees; PM=60; Fs=10Hz$

$G(s)=(-0.2s+4)/(s^2+3s+3)$ I have calculated poles to be $(s-3/2-0.88j)(s-3/2+0.866j)$ and simplified my zero to $-0.2(s-20)$ I was shown to calculate the phase of a pole to do $-tan_inv(Wc/a)$

How can I find the Phase of the Poles with Imaginary number

Thanks in advacned

Answer 1

Given $G(s) = (a_1 s + a_0)/(s^2+b_1s+b_0)$ the Bode plot is made over

$20\log_{10}\vert G(j\omega)\vert$ for amplitude and $\angle{G (j\omega)}$ for phase

but

$$ G(j\omega) = \vert G(j\omega)\vert e^{\angle{G (j\omega)}} $$

and here $\vert G(j\omega)\vert = \frac{\sqrt{(a_1\omega)^2+a_0^2}}{\sqrt{(b_0-\omega^2)^2+(b_1\omega)^2}}$

and

$\angle{G (j\omega)} = \arctan\left(\frac{a_1\omega}{a_0}\right)-\arctan\left(\frac{b_1\omega}{b_0-\omega^2}\right)$