Finding the Frequency and Critical Value of a Feedback Amplifier

[asdf12312]

Homework Statement

Consider a feedback amplifier for which the open loop gain A(s) is given by:
A(s) = X/ [(1+s/Y)*(1+s/Z)²] |s=jω
Where X = 7500, Y = 90000, and Z = 800000

a) If the feedback factor

is independent of frequency, find the frequency at which the phase shift is 180°.
b) Find the critical value of

at which oscillation will commence.


2. Homework Equations
A_f(s) = A(s)/ 1+A(s)B(s) (gain with feedback)

The Attempt at a Solution

I'm not sure how to start this problem. If

is constant value then I get -tan^-1(ω/90,000) - 2*tan^-1(ω/800,000) = tan(180) or (ω/90,000) + (ω/800,000) = tan(60) and when I try to solve I get ω=140,000 rad/s, but maybe I'm doing it wrong.

 

[milesyoung]

If

is constant value then I get -tan^-1(ω/90,000) - 2*tan^-1(ω/800,000) = tan(180) ...

If you want to convert -180° to radians, then that's not the way to do it. The rest is fine.

 

[rude man]

If

is constant value then I get -tan^-1(ω/90,000) - 2*tan^-1(ω/800,000) = tan(180)

This equation equates arc tangents to a tangent, which is incorrect. Change the rhs of your 1st equation to + or - π and it's OK then.

or (ω/90,000) + (ω/800,000) = tan(60) and when I try to solve I get ω=140,000 rad/s, but maybe I'm doing it wrong.

EDIT:
I don't see how you got this 2nd equation. Would like to see your derivation thereof.
Solving your 1st equation, which is transcendental, I got ω = 885,438 rad/s or f = 140,922 Hz.

This may be irrelevant, but
if arc tan a + arc tan b = θ
then it does not follow that
a + b = tan θ.
Try it with a = 0.25 and b = 0.35: tan θ = 0.25 + 0.35 = 0.60, so θ = arc tan 0.60 = 30.96°
whereas 14.04° + 19.29° = 33.33°
It's close but not right. I'm not saying you did this but in case you did ...