Control Systems: Frequency response Bode plots and analytical expressions

[VinnyCee]

Homework Statement

Find analytical expressions for the magnitude and phase of the frequency response for each G(s) below:

(a) $$G(s)\,=\,\frac{20}{s\,\left(s\,+\,5\right)\left(s\,+\,5\right)}$$(b) $$G(s)\,=\,\frac{2\,\left(s\,+\,5\right)}{\left(s\,+\,1\right)\,\left(s\,+\,10\right)}$$(c) $$G(s)\,=\,\frac{100}{s\left(s^2\,+\,10\,s\,+\,100\right)}$$

Homework Equations

$$M\left(\omega\right)\,=\,\vert\,G\left(j\omega\right)\vert$$

Also need to find the angle expression.

Complex number operations are quite intensive, this is probably why I can't figure it out!

The Attempt at a Solution

Prof. told us to first replace all s with jw.

(a) $$G(j\omega)\,=\,\frac{20}{\left(j\omega\right)\,\left[\left(j\omega\right)\,+\,1\right]\,\left[\left(j\omega\right)\,+\,5\right]}$$

Then I do some complex number manipulation to slightly simplify the expression.

$$G(j\omega)\,=\,\frac{20}{\left(-j\omega^3\,-\,6\omega^2\,+\,5j\omega\right)}$$

Does that look right? I've tried to get the magnitude expression multiple times, but it just doesn't seem right! I did the Bode plot for this transfer function in MATlab and it reports that at w=1 the magnitude should be 8.83dB. NONE of my magnitude expressions produce that data point. What am I doing wrong?EDIT:

I figured out the magnitude part. But I still don't understand how to get the phase part.

$$M(\omega)\,=\,\frac{20}{\sqrt{\left(5\omega\,-\,\omega^3\right)^2\,+\,\left(6\omega^2\right)^2}}$$

And then to get the dB magnitude...

$$20\,log\left(M(\omega)\right)$$

But now how do I get the phase?

 

[DaveE]

Your alegra looks good to me.
G(ω) is just a complex number for any given value of ω. So you can find the phase angle with the atan function. I suggest you review the polar form of complex numbers if this is still confusing.