Understanding Total Gain and Phase Angle in Closed Loop Control Systems

[StripesUK]

Homework Statement

A closed loop control system has three components, Controller, Process, and Measuring System. Each has it's own gain and phase angle. I understand how to find the total of the gain but I'm unsure as to how to find the total phase angle?

Homework Equations

Total Gain=
\frac{A*B}{1+(A*B)*C}

The Attempt at a Solution

Using this formula on the phase angle doesn't give a sensible answer. My instincts tell me that it is just straight addition? I know I've not got very far on this myself, just a nudge in the right direction would be hugely useful.

Many thanks.

 

[FactChecker]

Every frequency corresponds to a complex number. That leads to a complex number for $$ \frac{A*B}{1+(A*B)*C} $$.
The modulus of the complex number is the gain and the argument is the phase shift.

 

[rude man]

You have
A = |A|exp(jθ_A)
B = |B|exp(jθ_B)
C = |C|exp(jθ_C)
then combine these in your formula to get the net θ.
You do have to know how to manipulate complex numbers both in polar and rectangular form.[/SUB]

 

[LvW]

For real values of A,B and C the phase shift is (of course) zero - trivial, but unrealistic solution.
Hence, all of these expressions must be considered to be frequency-dependent (complex expressions).
For finding the total phase shift you have nothing to do than (a) to rewrite the transfer function and split it into a real and an imag. part, and (b) apply the definition for phase shift.