N Circles

Constant phase-angle loci (N-circles):

Fig 1: A family of constant N circles

We shall obtain the phase angle α in terms of X and Y. Since


This is an equation of a circle with center at X = -1/2 , y = 1/(2N) and with radius . For example, if α = 30°, then N = tan α = 0.577, and the center and the radius of the circle corresponding to α = 30° are found to be (-0.5,0.866) and unity, respectively. Since the above equation is satisfied when X = Y = 0 and X = -1, Y = 0 regardless of the value of N, each circle passes through the origin and the -1 j0 point. The constant a loci can be drawn easily once the value of N is given. A family of constant N circles is shown in Figure 1 with α as a parameter.

It should be noted that the constant N locus for a given value of a is actually not the entire circle but only an arc. In other words, α = 30° and α = -150° arcs are parts of the same circle. This is so because the tangent of an angle remains the same if ± 180° (or multiplies thereof) is added to the angle.