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

23.2 Periodic waveforms 723
23.2 PERIODICWAVEFORMS
Inthischapterweshallbeconcernedwithperiodicfunctions,especiallysineandcosine
functions. Let us recall some important definitions and properties ( already discussed in
Section 3.7. The function f (t) = Asin(ωt + φ) = Asin ω t + φ )
is a sine wave of
ω
amplitudeA, angular frequency ω, frequency ω 2π , periodT = 2π ω
and phase angle φ.
The time displacement isdefined tobe φ . These quantities areshown inFigure 23.1.
ω
SimilarremarkscanbemadeaboutthefunctionAcos(ωt + φ)andtogetherthesine
andcosinefunctionsformaclassoffunctionsknownassinusoidsorharmonics.Itwill
beparticularlyimportantforwhatfollowsthatyouhavemasteredtheskillsofintegrating
these functions. The following results can be found inTable 13.1 (see page 431):
∫
sinnωtdt = − cosnωt
nω +c ∫
cosnωtdt = sinnωt
nω +c
forn=±1,±2,...
Sometimesafunctionoccursasthesumofanumberofdifferentsineorcosinecomponents
such as
f (t) = 2sin ω 1
t +0.8sin2ω 1
t +0.7sin4ω 1
t (23.1)
The r.h.s.ofEquation (23.1) isalinear combinationof sinusoids.
Note in particular that the angular frequencies of all components in Equation (23.1)
are integer multiples of the angular frequency ω 1
. Functions like these can easily be
plottedusingagraphicscalculatororcomputergraph-plottingpackage.Thecomponent
with the lowest frequency, or largest period, is 2sinω 1
t. The quantity ω 1
is called the
fundamentalangularfrequencyandthiscomponentiscalledthefundamentalorfirst
harmonic. The component with angular frequency 2ω 1
is called the second harmonic
andsoon.Inwhatfollowsallangularfrequenciesareintegermultiplesofthefundamental
angular frequency as in Equation (23.1). A consequence of this is that the resulting
function, f (t), is periodic and has the same frequency as the fundamental. Some harmonicsmaybemissing.Forexample,inEquation(23.1)thethirdharmonicismissing.
Insomecases the first harmonic may be missing.For example, if
f(t) =cos2ω 1
t +0.5cos3ω 1
t +0.4cos4ω 1
t + ···
f(t)
A
2p
T = —
v , period
Amplitude
– f v
t
–A
Figure23.1
The function: f (t) =Asin(ωt + φ).