A transformer (Prob. 7.57) takes an input AC voltage of amplitude V1, and delivers an ou...

A transformer (Prob. 7.57) takes an input AC voltage of amplitude V₁, and delivers an output voltage of amplitude V₂, which is determined by the turns ratio (V₂/V₁ = N₂/N₁). If N₂ > N₁, the output voltage is greater than the input voltage. Why doesn’t this violate conservation of energy? Answer: Power is the product of voltage and current; if the voltage goes up, the current must come down. The purpose of this problem is to see exactly how this works out, in a simplified model.

(a) In an ideal transformer, the same flux passes through all turns of the primary and of the secondary. Show that in this case M² = L₁L₂, where M is the mutual inductance of the coils, and L₁, L₂ are their individual self-inductances.

(b) Suppose the primary is driven with AC voltage V_in = V1 cos (ωt), and the secondary is connected to a resistor, R. Show that the two currents satisfy the relations

(c) Using the result in (a), solve these equations for I1(t) and I2(t). (Assume I1 has no DC component.)

(d) Show that the output voltage (V_out = I2R) divided by the input voltage (V_in) is equal to the turns ratio: V_out/V_in = N₂/N₁.

(e) Calculate the input power (P_in = V_in I1) and the output power (P_out = V_out I₂), and show that their averages over a full cycle are equal.