A twocoil system has the selfinductances L1 and L2 with L1

A two-coil system has the self-inductances L_1 and L_2 with L_1 = 4 L_2 and the coupling coefficient k = 0.5. Suppose that coil #2 is open with an induced output voltage v_2(t) = 25 cos (120 pi t) (v). What is the input voltage v_1(t) on coil #1 ? If the two coils are connected in series and the total inductance is tested to be L = 7 H., determine L_1, L_2 and M; Determine both the input and output currents i_1(t) and i_2(t) of the two-coil system.

Solution

The amplitudes and phase shifts are only a distraction; the first function has the same period as cos(4t), which would be pi/2.

The second function might be rewritten by using the identity
cos(2A) = 2 cos^2(A) - 1, or
(1/2) cos(2A) + (1/2) = cos^2(A).
Hence, your function is
(1/2) cos(4t - 2 pi/3) + (1/2)
and again the amplitude and BOTH shifts are irrelevant; the period is [2 pi] divided by the coeffcient of t; it comes out to pi/2 again.