Effective (r.m.s.) value

• The average value of an alternating current over a cycle is zero because the current goes as far in the negative direction as in the positive one.
• But the current still heats a resistor, and the rate of heating depends on I², which is never negative.
• So we define the effective value of an alternating current as the value of the steady DC which produces the same heat in the same resistance in the same time (the same power).

I_eff = 0.707 I_max (= I_max/√2) and V_eff = 0.707 V_max

Example 6. A mains supply is 220 V (effective). Its peak value is 220/0.707 = 311 V. If I_eff = 5 A, I_max = 7.07 A. Meters (hot-wire ammeters, AC voltmeters) read effective values.

Example 6a. A circuit carries an effective current of 8 A at an effective voltage of 110 V. I_max = 8 / 0.707 = 11.3 A and V_max = 110 / 0.707 = 155.6 V.

Example 6b. A 100 Ω heater is connected to an AC supply of peak value 141 V. V_eff = 0.707 × 141 = 100 V, so the power is V_eff²/R = 10 000 / 100 = 100 W.