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

754 Chapter 23 Fourier series
thevaluesofRandC.ThequantityRC isusuallyknownasthetimeconstantforthe
system.Consider the case whenRC = 0.3. Equations (23.11) and (23.12) reduce to
|G(jω)| =
1
√
1 +0.09ω
2
(23.13)
̸ G(jω) = −tan −1 0.3ω (23.14)
Letusexaminetheresponseofthissystemtoasquarewaveinputwithfundamental
angular frequency 1 and amplitude 1. This waveform is shown in Figure 23.29(a).
We note that T = 2π. The waveform function is odd and so will not contain any
cosine Fourier components. It has an average value of 0 and so will not have a zero
frequencycomponent;thatis,therewillbenod.c.component.Thereforecalculating
the Fourier components reduces toevaluating
f(t)=
∞∑
n=1
b n
sin 2πnt
T
b n
= 2 T
∫ T/2
−T/2
f(t)sin 2πnt
T
dt
forpositive integersn
SinceT = 2π, wefind
b n
= 1 (∫ 0 ∫ π
)
−1sinntdt + sinntdt
π −π 0
= 1 ([ [ ] cosnt −cosnt π )
+
π n n
0
] 0
−π
= 1 (cos0 −cosπn −cosπn +cos0)
nπ
= 1 (2 −2cosπn)
nπ
= 2 (1 −cosπn)
nπ
y i
y o
t
t
(a)
Figure23.29
(a) Input to low-pass filter;(b)outputfrom low-pass filter.
(b)