1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.4 • TRlCONOMETRIC AND HYPERBOLIC FUNCTIONS 185

Ee

~

I

'·{

c

R L

E(t)

~

O'\....

Figure 5 .9 An LRC circuit.

The voltages EL, ER, and Ee and tbe impressed voltage E (t) illustrated in
Figure 5.9 satisfy the equation

Ei +ER+ Ee = E (t).

Suppose that the current I (t) in the circuit is given by

I (t) =Io sin wt.

Using this in the equations for ER and EL gives

ER = Rio sin wt and

EL = wLiocoswt.

We then set to = ~ in the equation for Ee to obtain

t t

Ee=.!.. j1(r)dr = .!.. j1 0 sinwtdr = --

1

• C C wC
  .!!. 2 'I •

We rewrite the equation I (t) = Io sin wt as a "complex current,"

I*= Ioeiwt

(5-40)

(5-41)

(5-42)

(5-43)

with the understanding that the actual physical current I is the imaginary part
of r. Similarly, we rewrite Equations (5-41)-(5-43) as

ER. = Rloeiwt = RI*,

EI, = iwLioeiwt = iwLI*, and

Ee • = iwC^1 l oe iwt = iwC^1 1 • ·

Substituting these terms leads to an extension of Equation (5-40),

(5- 44)