Mathematics of Physics and Engineering

AC Circuits 189

the resonance frequency wo = l/VLC.

Now let us consider the parallel RCL circuit on the right-hand side of

Figure 4.1.2. This time, the voltage E(t) is the same across all compo-

nents; we take E(t) = EoeluJt. For the currents, we have /R(£) = E(t)/R

(in phase with E), Ic{t) = E(t)uiCein^^2 (ahead of E by TT/2), IL(t) =

(E(t)/(wL))e~™/^2 (behind E by 7r/2). The vector diagram corresponds to

t = 0; for t > 0, the diagram rotates counterclockwise with angular speed

w. The total current is I = IR + Ic + IL = (1/-R + iwC + l/{iuL))E{t).

Taking J(t) = / 0 ei(a"+*o), we conclude that

1 ( „ 1 N2

R2 + \wC- — ) E 0 , tan^o = fl(wC - l/(wL)). (4.1.9)

EXERCISE 4.1.11.^5 (a,) Ven/i/ (4.1.9). (b) Verify that, for every input

voltage E{t), the current I(t) in the parallel circuit satisfies

CE"(t) + E'(t)/R + E(t)/L = /'(*). (4.1.10)

Substitute E(t) = £ 0 eiM) and I{t) = I^^+M to recover (4.1.9).
Unlike the series circuit, the resonance frequency a>o = l/y/LC now cor-
responds to the smallest absolute value IQ = EQ/R of the current for given
EQ\ the corresponding phase shift o at the resonance frequency is zero. Ac-
cording to (4.1.10), a parallel circuit is a damped harmonic oscillator
with damping proportional to 1/R. The "ideal" parallel circuit with R = oo
is a pure harmonic oscillator and has zero total current at the resonance
frequency OJQ = 1/y/LC.
The above analysis also demonstrates that the effective resistance,
called reactance, of the capacitor and the inductor is, respectively,
Xc = l/(iuiC) and XL = iu>L. The usual laws for series or par-
allel connection apply: the total effective resistance Z, known as the
complex impedance, satisfies Z = R + Xc + X^ in the series circuit and
1/Z = (1/R) + (l/Xc) + (l/XL) in the parallel circuit.
EXERCISE 4.1.12.^3 Sketch the graph of \Z\ as a function of w for
(i) the series circuit; (ii) the parallel circuit.