Closely Coupled Circuits

Assume that a transformer, or repeating coil, as it is often called in telephony, has a closed core of Permalloy and that all the magnetic flux φ produced by the primary winding N_p links with the secondary turns N_s; also, assume that the losses in the transformer are negligible. The magnitude of the back voltage induced in the primary is given by the fundamental expression e_p = N_pdφ/(10⁸ dt), and this will approximately equal the impressed voltage. The magnitude of the voltage induced in the secondary will be e_s = N_sdφ/(10⁸ dt). Dividing the second equation by the first, and in terms of effective instead of instantaneous values, gives

To determine the approximate impedance transforming equations, suppose that the transformer secondary is delivering power to a load. The magnitude of the load impedance will be Z_L = E_s/I_s and the impedance measured across the primary terminals will be Z_p = E_p/I_p. From equation 20, E_p = E_sN_p/N_s, and I_p = I_sN_s/N_p, and, when these are substituted in the equation for the primary impedance,

This equation shows that the transformer acts like an impedance changer because the impedance Z_p measured at the primary will be (N_p/N_s)² times the impedance connected as a load to the secondary.