AC Circuits 1874.1.3 Applications to Analysis of AC Circuits
In this section, we continue to use i as the notation for the imaginary unit.
The basic components of an alternating current (AC) circuit are the resistor
R, the capacitor C, and the inductor L; see Figure 4.1.2. The main facts
about electric circuits are summarized in Section 8.4 in Appendix. If A is
a quantity, such as current or voltage, changing periodically in time with
period T = l/v, then we represent this quantity as A(t) = Aoe%^tJr^a\
where 4>o is the initial phase, and u> is the underlying angular frequency,
related to the usual frequency v by w = 2nv (a household outlet in the US
has v = 60 Hz or 60 cycles per second; in Europe, the standard is / = 50
Hz). Since taking the real part of A(t) brings us back to physical reality and
can be done at any moment, we will work only with complex currents and
voltages.
Denote by Iy and Vy the current through and the voltage across the
element Y, respectively, with Y being a resistor R, a capacitor C, or an
inductor L. We assume that all the elements are linear, so that
- by Ohm's Law, see page 173, IR — VR/R;
- by the definition of the capacitance, C = qc/Vc, where qc{t) =
JQ Ic(s)ds is the charge; thus, Ic = CdVc/dt; - by Faraday's Law, see page 165, VL = Ldli,{t)/dt.
EXERCISE 4.1.9. c Taking I(t) - I 0 ei"t+i4'^0 , verify that (a) The current
through the resistor is in phase with the voltage; (b) The current though
the capacitor is ahead of, or leads, the voltage by the phase n/2; (c) The
current through the inductor is behind, or lags, the voltage by the phase
7T/2. Hint: i = ei7r/2.
Consider the series circuit on the left-hand side of Figure 4.1.2, with
E(t) = Eoei<-UJt+'t'^0 K All the elements of the circuit have the same cur-
rent I(t) passing through them; we take I(t) = Joe""'*. Then we have
VR(£) = I{t)R (the voltage across R is in phase with the current),
Vc(t) = (l/wC)I(t)e~in/^2 (the voltage across C is behind the current by
7r/2), and Vi(t) = Lu>I(t)en/^2 (the voltage across L is ahead of the cur-
rent by 7r/2). The vector diagram corresponds to time t = 0; for t > 0
the diagram rotates counterclockwise with angular speed u> = 2-KV; lin-
ear frequency v = 60 Hz, corresponds to 60 full turns per second. Since