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

726 Chapter 23 Fourier series
2 Expresseach ofthe following functionsasasingle
sinusoidandhence findtheir amplitudes and phases.
(a) f(t)=2cost−3sint
(b) f(t) =0.5cost +3.2sint
(c) f(t) =3cos3t
(d) f(t)=2cos2t+3sin2t
3 Sketch the graphs ofthe following functions:
(a) f(t) =t 2 ,−1t 1,period2
{
0 0 t < π/2 period π
(b) f(t) =
sint π/2tπ
{
−t −2t<0 period3
(c) f(t) =
t 0t<1
f(t)
f(t)
1
1
–2 –1 1 2 t –2 –1 0 1 2 t
(a)
(b)
f(t)
1
–3 –2 1 3 4 t
(c)
Figure23.4
4 Write down mathematical expressions to describe the
functionswhose graphs are shown in Figure23.4.
Solutions
1 (a) Fundamental frequency is50 Hz, amplitude3.
Second harmonic has frequency of100 Hz,
amplitude4. Thirdharmonic has frequency of
150 Hz,amplitude0.7.
(b) Fundamental frequency is 20 Hz, amplitude 1.
π
Second harmonic is missing.Third harmonic has
frequency of 60 Hz, amplitude 0.5. Fourth and
π
fifth harmonics are missing.Sixth harmonic has
frequency of 120 Hz, amplitude 0.3.
π
√ √
2 (a) 13cos(t +0.983);amplitude = 13,
phase = 0.983
(b) 3.24cos(t +4.867);amplitude = 3.24,
phase = 4.867
(c) 3cos3t;amplitude = 3,phase = 0
√ √
(d) 13cos(2t −0.983);amplitude = 13,
phase = −0.983
{
1 0 t1
4 (a) f(t)=
0 1<t<2 period =2
{
2t 0t1/2
(b) f(t) =
2−2t 1/2<t<1
period =1
{
0 0 t1
(c) f(t) =
t/2−1/2 1<t<3
period =3
23.3 ODDANDEVENFUNCTIONS
Thefunctionssint andcost eachpossesscertainpropertieswhichcanbegeneralizedto
otherfunctions.Figure23.5showsthegraphof f (t) = cost.Itisobviousfromthegraph
that the function value at a negativet value, say − π , will be the same as the function
4
value at the corresponding positivet value, in this case + π . This is true because the
4
graph is symmetrical about the vertical axis. We can therefore state that for any value
oft,cos(−t) =cost.