082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

23.8 Complex notation 749
Engineeringapplication23.2
Averagepowerofasignal
Find the average power developed across a 1 resistor by a voltage signal with
period 2πgiven by
v(t) =cost − 1 3 sin2t + 1 2 cos3t
Solution
We note that v(t) is periodic with period T = 2π; v(t) is already expressed as a
Fourierserieswitha 1
= 1,a 3
= 1 2 andb 2 =−1 3 .AllotherFouriercoefficientsare0.
The instantaneous power is (v(t)) 2 and hence the average power over one period is
given by
∫ 2π
P av
= 1 (v(t)) 2 dt
2π 0
Therefore, using Parseval’s theorem wefind
P av
= 1 ( (
1 2 + − 1 2 ( 1 2 )
+ = 0.68 W
2 3)
2)
23.8 COMPLEXNOTATION
AnalternativenotationforFourierseriesinvolvingcomplexnumbersisavailablewhich
leadsnaturallyintothemoregeneraltopicofFouriertransforms.RecallfromChapter9
the Euler relations
e ±jθ =cosθ±jsinθ
fromwhich wecan obtain expressions forcosθ and sinθ:
cosθ = ejθ +e −jθ
2
sinθ = ejθ −e −jθ
2j
which enable us torewrite the Fourier representation
f(t)= a ∞
0
2 + ∑
(
a n
cos 2nπt +b
T n
sin 2nπt )
T
as
n=1
f(t)= a ∞
0
2 + ∑ e
(a j2nπt/T +e −j2nπt/T
n
2
n=1
= a ∞
0
2 + ∑
(
an −jb n
2
n=1
e j2nπt/T + a n +jb n
2
+b n
e j2nπt/T −e −j2nπt/T
2j
e −j2nπt/T )
)