A transformer (Prob. 7.57) takes an input AC voltage of amplitude , and delivers an output voltage of amplitude , which is determined by the turns ratio . If , 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 , where is the mutual inductance of the coils, and , are their individual self-inductances.
(b) Suppose the primary is driven with AC voltage , and the secondary is connected to a resistor, . Show that the two currents satisfy the relations
(c) Using the result in (a), solve these equations for localid="1658292112247" and . (Assume has no DC component.)
(d) Show that the output voltage divided by the input voltage is equal to the turns ratio: .
(e) Calculate the input power localid="1658292395855" and the output power , and show that their averages over a full cycle are equal.