Center of Ossanna Circle

Question

The Ossanna circle is given by all points that $I = \dfrac{U_1}{R_1 + jX_1 + \dfrac{X_h^2}{jX_2 + \dfrac{R_2}{s}}}$ will reach (in the complex plane) while the parameter s goes from negative infinity to positve infinity. Only $s$ is variable, all other values are constants. By drawing the location curve with tools like Matlab, one can see, that it forms a circle (Ossanna Circle know in electrical engineering). The center of this circle is somewhere but not the origin of the coordinate system. How can the coordinates of the center of this circle be calculated?

Answer 1

I'm converting variables to lower-case, to save space (and reduce eye strain); I'm also using "$i$" for the imaginary unit.

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Begin by calculating the modulus of $I$: $$|I|^2 = I \cdot \overline{I} = \frac{u_1^2 \left(\;s^2 x_2^2 + r_2^2 \;\right)}{ s^2( r_1^2 x_2^2 + ( x_h^2 - x_1 x_2)^2) + 2 s r_1 r_2 x_h^2 + r_2^2( r_1^2 + x_1^2) }$$ This (real) expression attains its maximum and minimum values at the endpoints of a diameter of the Ossanna circle that (extended as necessary) passes through the origin. We find the corresponding values of $s$ by checking where the numerator of the derivative (with respect to $s$) vanishes: $$s^2 r_1 x_2^2 - s r_2 ( x_h^2 - 2 x_1 x_2 ) - r_1 r_2^2 = 0$$ $$\implies\qquad s = \frac{r_2}{2r_1 x_2^2}\left( x_h^2 - 2 x_1 x_2 \pm \sqrt{ 4 r_1^2 x_2^2 + ( x_h^2 - 2 x_1 x_2 )^2}\right)$$

The center of the Ossanna circle, as a complex number, is the average of $I$ evaluated at the two roots $s$. The radius is the modulus of the half-difference of those two values. Intermediate calculations are messy, so I'll jump right to the results:

$$\text{center} = \frac{u_1 \left(\; 2 r_1 x_2 + i ( x_h^2 - 2 x_1 x_2)\;\right)}{2 \left(\; r_1^2 x_2 + x_1^2 x_2 - x_1 x_h^2 \;\right)} \qquad\text{radius} = \frac{u_1 x_h^2}{2 \left|\;r_1^2 x_2 + x_1^2 x_2 - x_1 x_h^2\;\right|}$$