Finding points on root-locus by hand

[Jacer2]

Homework Statement

[/B]

I am trying to answer two questions:

1. Solve part (b) in the image above by hand.

2. What are the differences between transfer function (a) and transfer function (b).

Homework Equations

cos(Im(s) / Re(s)) = ζ

The Attempt at a Solution

1. For part one given the damp I know that cos^-1(ζ) = 45. Thus the magnitude of the real and imaginary parts of the root has to be the same. However I can't seem to think of a way to do this by hand. I can plot the root locus in Matlab.

Once I find the root locus I know I can find K by using:
K = \frac{|denominator \quad of \quad tf|}{|numerator \quad of \quad tf|}2. The transfer functions are K = \frac{KGH}{1+KGH} and K = \frac{KG}{1+KGH} . Besides a different output response would the root locus and bode plots of the both transfer functions be the same?

 

[Jacer2]

The only solution I could think of is using the angle condition:

The phase should all add up to 180 degrees + a multiple of 360:
SInce the damping results in a 45 degree angle, the real and imaginary parts of s are the same. Thus:
s = -x + xj

I can then apply the angle criterion by getting the phase of each pole:
\sum\angle(zeroes) - \sum\angle(poles) = 180 \pm 360*n
-2*tan^{-1}(\frac{x}{-x+2})-tan^{-1}(\frac{x}{-x+4})-tan^{-1}(\frac{x}{-x+1}) = 180

I had to brute force and solve this in my calculator and got x = 2.203. I had to then try setting the equation equal to -180 as opposed to 180:
-2*tan^{-1}(\frac{x}{-x+2})-tan^{-1}(\frac{x}{-x+4})-tan^{-1}(\frac{x}{-x+1}) = -180
and got x = .907672 which is the right answer.

Is there a better way to do this without numerically solving it?