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Root-Mean-Square Voltages When a time-varying voltage V(t) is applied to a resistor, the power dissipated in it, V2/R is also time-dependent. The average power is the energy consumed in a cycle over the time of one period. If we were to use a steady d.c. voltage to give the same average power, then this “d.c. equivalent steady voltage” is called the root mean square voltage of the a.c. I. Complete sinusoidal waveform: A sinusoidal a.c. is applied to a resistor V = V0 sin t V 2 The instantaneous power dissipated in R is P = V 2 V0 sin2t R R 0 T The average power , P 2 0 1 T 1V Pdt T 0 T R T 0 sin 2ω t 2 0 1V 2 R ( V0 2 R )2 ………..(1) The power of a steady d.c. voltage Vdc is Vdc2 P R ………(2) If (2) is exactly equal to (1), then V 0 2 Vdc2 ( 2 ) R R This implies Vdc . V0 2 This “d.c. equivalent steady voltage” Vdc is called the root-mean-square voltage (Vrms) of the a.c. because it is calculated from the procedures “root”, “mean” and “square”, but in the reverse order. 1. Squaring the time-depending a.c. voltage. 2. Taking the mean of the squared V over a cycle. 3. Taking the square root of the mean. Vrms V 2 2 Vrms I 2rms R Vrms I rms In a pure R, P R II. Half sinusoidal waveform: V V0 Vrms V0 2 III. Square waveform: V Va -Vb T 2 t T Vrms = Va2 Vb2 2