Introduction to Electrodynamics · Problem 7.58

Conservation Laws — David J. Griffiths

Physics · transformer-energy-conservation

Problem

A transformer (Prob. 7.57) takes an input AC voltage of amplitude V1V_1, and delivers an output voltage of amplitude V2V_2, which is determined by the turns ratio (V2/V1=N2/N1V_2/V_1 = N_2/N_1). If N2>N1N_2 > N_1, the output voltage is greater than the input voltage. Why doesn't this violate conservation of energy? (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=L1L2M^2 = L_1 L_2, where MM is the mutual inductance of the coils, and L1L_1, L2L_2 are their individual self-inductances. (b) Suppose the primary is driven with AC voltage Vin=V1cos⁡(ωt)V_{\text{in}} = V_1 \cos(\omega t), and the secondary is connected to a resistor, RR. Show that the two currents satisfy the relations L1dI1dt+MdI2dt=V1cos⁡(ωt);L2dI2dt+MdI1dt=−I2R.L_1 \frac{dI_1}{dt} + M\frac{dI_2}{dt} = V_1 \cos(\omega t); \quad L_2 \frac{dI_2}{dt} + M\frac{dI_1}{dt} = -I_2 R. (c) Using the result in (a), solve these equations for I1(t)I_1(t) and I2(t)I_2(t). (Assume I1I_1 has no DC component.) (d) Show that the output voltage (Vout=I2RV_{\text{out}} = I_2 R) divided by the input voltage (VinV_{\text{in}}) is equal to the turns ratio: Vout/Vin=N2/N1V_{\text{out}}/V_{\text{in}} = N_2/N_1. (e) Calculate the input power (Pin=VinI1P_{\text{in}} = V_{\text{in}} I_1) and the output power (Pout=VoutI2P_{\text{out}} = V_{\text{out}} I_2), and show that their averages over a full cycle are equal.

Related theory

Browse the problem library