Appendix B Time Variation

Exponential Notation of Time

The widespread use of complex notation in electromagnetics (and in other branches of physics) and Euler’s formula leads to an effective shorthand for expressing sinusoidal variations of time. Using Euler’s formula we can write

e+iωt = cos ωt + i sin ωt

(B.1)

which leads to

sin ωt = Im e+iωt

(B.2)

and

cos ωt = Re e+iωt

(B.3)

where Im and Re refer to the imaginary and real parts, respectively.

These “sinusoidal” time variations are often simply expressed by saying that the time dependency is e^+iωt (or e^-iωt ). It does not mean that the time dependency is complex. The resulting expression can be returned to a real one by extracting the real or imaginary part in the end. This is acceptable as long as various manipulations involve only addition, subtraction, differentiation, or integration where the real and imaginary parts do not mix. However, usually the final expression is simply left in exponential form.

Using the exponential form is preferred because the algebra of complex exponentials is easier to work with compared to sines and cosines. For example, the differentiation operation d/dt is just a multiplication by +iω. That is,

d/dt (e+iωt ) = iω e+iωt .

(B.4)

To illustrate the use of complex exponential notation, again consider the earlier equation 1.2.6, i.e.,

V = IR + L dI/dt + q/C

(B.5)

with

I = I0 e+iωt.

(B.6)

Equation B.5 becomes

V = IR +I iωL – iI/ωC

(B.7)

after the integration

	q = ∫I0eiωtdt
	= -iI/ω	(B.8)

Now, upon taking the imaginary part of the expression for V after inserting

I = I0 e+iωt = I0 (cos ωt + i sin ωt)

(B.9)

equation B.7 becomes

V = I0 R sin ωt + I0 ωL cos ωt - I0/ωC cos ωt

(B.10)

which is the identical to the expression we derived earlier (equation 1.2.10) using I = I₀ sin ωt.