Chapter 31, Problem 31.54P

The L-R-C Parallel Circuit. A resistor, an inductor, and a capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency ω. Let the source voltage be given by υ = Vcos ωt. (a) Show that each of the instantaneous voltages υ_R, υ_L, and υ_C at any instant is equal to u and that i = i_R + i_L + i_C, where i is the current through the source and i_R, i_L, and i_C are the currents through the resistor, inductor, and capacitor, respectively, (b) What are the phases of i_R, i_L, and i_C with respect to υ? Use current phasors to represent i, i_R, i_L, and i_C. In a phasor diagram, show the phases of these four currents with respect to υ. (c) Use the phasor diagram of part (b) to show that the current amplitude I for the current i through the source is $I=\sqrt{I_R^2+{(I_C-I_L)}^2}$. (d) Show that the result of part (c) can be written as I = V/Z, with 1/Z = $\sqrt{(1/R^2)+{[\omega C-(\text{1/}\omega L)]}^2}$.