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R Ls V2 1 Cs R Ls V2 R V 1 H s 2 V1 2 L=R CR s LCs2
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FREQUENCY RESPONSE, FILTERS, AND RESONANCE
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p p (b) From LC 2=!2 and L=C R2 we get L 2R=!0 and C 2=R!0 . Substituting L and C into (4a) gives 0     1 1 1 1 p p H s or H j! 4b 2 1 2 s=!0 s=!0 2 2 1 j 2 !=!0 !=!0 2 jHj2   1 1 4 1 !=!0 4 and  tan 1 ! p 2!0 ! !2 !2 0
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Note that H j! is independent of R. The network passes low-frequency sinusoids and rejects, or attenuates, the high-frequency sinusoids. It is a low-pass lter with a half-power frequency of ! !0 and, in this case, the p p magnitude of the frequency response is jH j!0 j jH 0 j= 2 2=4 and its phase angle is H j!0 =2. (c) For !0 1, H s   1 1 p 2 1 2s s2 1 1 jHj 4 1 !4
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  1 1 p 2 1 j 2! !2
4c
 tan
p ! 2! 1 !2
The RC network of Fig 12-4 b was de ned as a rst-order low-pass lter with half-power frequency at !0 1=R1 C2 . The circuit of Fig. 12-13 is called a second-order Butterworth lter. It has a sharper cuto .
FREQUENCY RESPONSE FROM POLE-ZERO LOCATION
The frequency response of a network is the value of the network function H s at s j!. This observation can be used to evaluate H j! graphically. The graphical method can produce a quick sketch of H j! and bring to our attention its behavior near a pole or a zero without the need for a complete solution.
EXAMPLE 12.4 Find poles and zeros of H s 10s= s2 2s 26 . Place them in the s-domain and use the polezero plot to sketch H j! . H s has a zero at z1 0. Its poles p1 and p2 are found from s2 2s 26 0 so that p1 1 j5 and p2 1 j5. The pole-zero plot is shown in Fig. 12-14(a). The network function can then be written as H s 10 s z1 s p1 s p2
For each value of s, the term s z1 is a vector originating from the zero z1 and ending at point s in the s-domain. Similarly, s p1 and s p2 are vectors drawn from poles p1 and p2 , respectively, to the point s. Therefore, for any value of s, the network function may be expressed in terms of three vectors A, B, and C as follows: H s 10 A B C where A s z1 ; B s p1 , and C s p2
The magnitude and phase angle of H s at any point on the s-plane may be found from: jH s j 10 jAj jBj jCj H s A B C 5a 5b
By placing s on the j! axis [Fig. 12-14(a)], varying ! from 0 to 1, and measuring the magnitudes and phase angles of vectors A, B, and C, we can use (5a) and (5b) to nd the magnitude and phase angle plots. Figure 12-14(b) shows the magnitude plot.
IDEAL AND PRACTICAL FILTERS
In general, networks are frequency selective. Filters are a class of networks designed to possess speci c frequency selectivity characteristics. They pass certain frequencies una ected (the pass-band)
CHAP. 12]
FREQUENCY RESPONSE, FILTERS, AND RESONANCE
Fig. 12-14
and stop others (the stop-band). Ideally, in the pass-band, H j! 1 and in the stop-band, H j! 0. We therefore recognize the following classes of lters: low-pass [Fig. 12-15(a)], high-pass [Fig. 12-15(b)], bandpass [Fig. 12-15(c)], and bandstop [Fig. 12-15(d)]. Ideal lters are not physically realizable, but we can design and build practical lters as close to the ideal one as desired. The closer to the ideal characteristic, the more complex the circuit of a practical lter will be. The RC or RL circuits of Section 12.2 are rst-order lters. They are far from ideal lters. As illustrated in the following example, the frequency response can approach that of the ideal lters if we increase the order of the lter.
Fig. 12-15
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