A vector loci method of treating coupled circuits

If in the first of two coupled circuits an electromotive force of constant amplitude and variable frequency is introduced, the currents in the primary and secondary respectively may be written i 1 = e /Z' and i 2 = e /Z" where Z' and Z" are complex impedance operators. The loci of these impedances ω as to is varied have definite geometrical forms. Z" is a parabola and Z' a cissoid family. If a parabola if y 2 = (— x ) p where p depends only on the inductances and resistances of the two circuits is drawn, and a pole O is taken a certain distance to the left of the vertex a , then OP represents the impedance Z" to a certain scale. The greater the coupling between the two circuits the longer is O a . As ω increases from a small value, P, starting on the lower arm of the parabola far to the left, moves counter clockwise round the parabola. If O is near a there will be a single minimum value of OP, and a single maximum value of the current i 2. But if the coupling and therefore O a is larger, there will be two minimum values separated by a maximum value, corresponding to the well-known double hump i 2/ω curve. The locus of Z' is the cissoid family of curves. The straight line of zero coupling bulges at the axis as the coupling is increased, and develops a loop as the coupling is still further increased. Here again a double minimum impedance appears, corresponding to the double hump resonance curves.