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Unformatted text preview: suppresses t he time-varying component and yields a dc component with some
residual ripple. Find t he filter o utput y(t). F ind also t he dc o utput a nd t he rms value of
the ripple voltage.
First, we shall find t he Fourier series for the rectified signal J (t), whose period is
To = 1l'. Consequently, Wo = 2, a nd where Dn ='!'
1l' 0 L I n Sec. 2.4-3, w e s howed t hat t he r esponse o f a n L TIC s ystem w ith t ransfer f unction H (s) t o a n e verlasting e xponential i nput e st is also a n e verlasting e xponential
H (s ) e st . T herefore, t he s ystem r esponse t o e verlasting e xponential e jwt is a n e verlasting e xponential H ( jw )e jwt . T his i nput-output p air c an b e d isplayed as+ To compute trigonometric Fourier series coefficients, we recall t he p rogram c31.m
along with c ommands t o convert D n into C n a nd en.
C O=Dnmag(l); C n=2*Dnmag(2:M);
A mplitudes= [CO;Cn]
d isp(' A mplitudes A ngles')
[ Amplitudes A ngles]
% T o P lot t he F ourier c oefficients
k =O:length(Amplitudes)-l; k =k';
s Ubplot(211 ) ,stem(k,Amplitudes)
s ubplot(212), s tem(k,Angles) 219 3.7 L TIC S ystem R esponse t o P eriodic I nputs "
0 s inte- j2n 'dt= (22) 1l' 1 - 4n (3.88) +This result applies only to the asymptotically stable systems. This is because when s = j w, the
integral on the right-hand side of Eq. (2.48) does not converge for unstable systems. Moreover, for
marginally stable systems also, t hat integral does not converge in t he ordinary sense, and H (jw)
cannot be obtamed by replacing s in H (s) with jw. 3 Signal Representation by O rthogonal Sets 220 1 50
+ + Full-Wave
Rectifier s in f y (t) f (t) (a) 3.7 221 L TIC S ystem Response t o P eriodic I nputs The output Fourier series coefficient corresponding to n = 0 is the dc component
of the output, given by 2 /". T he remaining terms in the Fourier series constitute the
unwanted component called the ripple. We can determine the rms value of the ripple
voltage by finding the power of the ripple component using Eq. (3.83). The power of
the ripple is the power of all the components except the dc (n = 0). Note t hat Dn, the
exponential Fourier coefficient for the output y (t) is Dn= 2
) - "7 :( Therefore, from Eq. (3.83b), we have
( a) y (t) 00 1
- 2 00
Proppl. - ~ IDnl - 2 ~ , ,(I _ 4 n2)(j6n ,--- o - 31t Prippl. = 0.0025,
31t f (t) 2 = L..,.. ,,(1 _ 4n2) e j2nt - 00 Next we find t he transfer function of the R C filter in Fig. 3.20a. This filter is identical
circuit i n Example 1.11 (Fig. 1.32) for which the differential equation relating
the output (capacitor voltage) to the input f (t) was found to be [Eq. (1.60)] RC (3D + l )y(t) = f (t) The transfer function H (s) for this system is found from Eq. (2.50) as
1 H (s)=3s+1 and 1
H (jw)=-3jw + 1 (3.89) From Eq. (3.87), t he filter output y (t) can be expressed as (with wo = 2) L
00 y (t) = DnH(jnwo)ejnwot = L D nH(j2n)ej2nt Substituting D n a nd H (j2n) f...
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