Mathematics of Physics and Engineering

188 Algebra of Complex Numbers

"VL

E = VR + Vc + VL

Vc

•+J •—

VR I

Series

L

Ic

IR E

•+T *—

II

I = IR + IC + IL

Parallel

Fig. 4.1.2 RCL Circuits

VR + VC + VL = E, we have E(t) = (R+l/(iu}C) + iuL)I(t), and therefore

E 0

h

^/R (^2) + ^L2_i 7 y
-., tan^>o = ^-. (4.1.7)
EXERCISE 4.1.10.B (a) Verify (4-1.7). Hint: Write (R+l/(iwC) + iu>L) in
the complex exponential form; keep in mind that \/i = —i. (b) Verify that, for
every input E(t), the current I(t) in the series circuit satisfies
LI"(t) + RI'{t) + I(t)/C = E'(t). (4.1.8)
Substitute E(t) = E 0 ei{"t+'t'o) and I(t) = I 0 eiuJt to recover (4.1.7). Hint: if
q is the charge, then E = EL + ER + EC = Lq" + Rq' + q/C; I = q'.
For fixed Eo, the largest value of IQ = EQ/R is achieved at the reso-
nance frequency U>Q — \/\/LC; the corresponding phase shift <po at this res-
onance frequency is zero. According to (4.1.8), a series circuit is a damped
harmonic oscillator, with damping proportional to R. The "ideal" series
circuit with R = 0 is a pure harmonic oscillator and has infinite current at