RLC Locus Diagrams

[ghoti]

1. The problem statement, all variables and given/known data


In a series circuit V = V<0 @50hz with an R and C, show that the graph of the loci or the port impedance, Z, and port admittance, Y, as the resistance R is varied from 0 to inf ohms are as shown.
http://ivila.net/prob.png


http://ivila.net/loci.png

2. Relevant equations



3. The attempt at a solution

Z = R + j*X;
Y = Z-1;

Map to Y = G + jB (Separate Real and Complex)
G = R/(R2 + X2);
B = -X/(R2 + X2);

G2 + B2 = 1/(R2+X2) = B/-X

Then I get lost,

I know I need to arrange into the format below with the condition that the R term (varying from 0 to inf) remains in the LHS, ie, out of the radius. however I am stuck .
(G-x)2 + (B-y)2 = radius2

Any help would be much appreciated.

[ghoti]

I found the solution!

Firstly.

Z = R - jX (incorrectly stated in my first post)
Y = G + jB
G = R/(R^2 + X^2)
B = X/(R^2 + X^2)
G^2 + B^2 = 1/(R^2 + X^2) this is very close to B, infact B/X = G^2 + B^2

using this we can say.

G^2 + B^2 = 1/X * B
G^2 + (B^2 - B/X) = 0

Now Complete the Square

G^2 + (B^2 - B/X + a^2) = a^2
2a = -1/x, a = -1/2x

Therefore

(G^2) + (B + -1/2x)^2 = (-1/2x)^2

or, a circle, origin at 0,-1/2x radius -1/2x

Thanks!
Alex