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

23.2 Periodic waveforms 725
f(t)
1
f(t)
1
–1 0 1 2 3 t
Figure23.2
Graph forExample 23.4.
–1 0 1
Figure23.3
Graph forExample 23.5.
2 3 4 5 6 t
Many other periodic functions arise in engineering applications as well as the more familiar
harmonic waves. Remember, to be periodic the function values must repeat at
regularintervalsknownastheperiod,T.Theangularfrequency ω isgivenby ω = 2π
T .
To describe a periodic function mathematically it is sufficient to give its equation over
one full period and state that period. From this information the complete graph can be
drawn as Examples 23.4 and 23.5 show.
Example23.4 Sketch the graph ofthe periodicfunctiondefined by
f(t)=t 0t<1 period1
Solution To proceed we first sketch the graph on the given interval 0 t < 1 (Figure 23.2), and
thenusethefactthatthefunctionrepeatsregularlywithperiod1tocompletethepicture.
Example23.5 Write down a mathematical expression for the function whose graph is shown in
Figure 23.3.
Solution We first note that the interval over which the function repeats itself is 2; that is, period
= 2. It is then sufficient to describe the function over any interval of length 2. The
simplest interval to take is 0 t < 2. We note in this example that a single formula is
insufficienttodescribethefunctionfor0 t < 2sincedifferentbehaviourisexhibited
inthetwointervals0 t < 1and1 t < 2.For0 t < 1thefunctionisarampwith
slope1andpassesthroughtheorigin,thatisithasequation f (t) =t.For1 t < 2the
function value remains constant at 1. Therefore this periodic function can be described
by the expression
{
t 0t<1 period2
f(t)=
1 1t<2
EXERCISES23.2
1 Describethe frequency andamplitudecharacteristics
ofthedifferentharmoniccomponentsofthefollowing
waveforms:
(a)f(t) =3sin100πt −4sin200πt
+0.7sin300πt
(b) f (t) = sin40t −0.5cos120t
+0.3cos240t
Use agraph-plottingcomputer packageorgraphics
calculator to graphthese waveforms.