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chapter 16 - Chapter 16 Filter Circuits Exercises Ex 16.3-1...

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Chapter 16: Filter Circuits Exercises Ex. 16.3-1 1 ( ) 1 1 1250 ( ) 1250 1250 1 1250 n n T s s s T s T s s = + = = = + + Problems Se ction 16.3: Filters P16.3-1 Equation 16-3.2 and Table 16-3.2 provide a third-order Butterworth low-pass filter having a cutoff frequency equal to 1 rad/s. 2 1 ( ) ( 1)( 1) n H s s s s = + + + Frequency scaling so that c = 2 100=628 rad/s ω π : 3 2 2 2 2 1 628 247673152 ( ) ( 628)( 628 628 ) ( 628)( 628 394384) 1 1 628 628 628 L H s s s s s s s s s s = = = + + + + + +   + + +     P16.3-2 Equation 16-3.2 and Table 16-3.2 provide a third-order Butterworth low-pass filter having a cutoff frequency equal to 1 rad/s and a dc gain equal to 1. 2 1 ( ) ( 1)( 1 n H s s s s = ) + + + Multiplying by 5 to change the dc gain to 5 and frequency scaling to change the cutoff frequency to ω c = 100 rad/s: 3 2 2 2 2 5 5100 5000000 ( ) ( 100)( 100 100 ) ( 100)( 100 10000) 1 1 100 100 100 L H s s s s s s s s s s = = = + + + + + + + + + 16-1
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P16.3-3 Use Table 16-3.2 to obtain the transfer function of a third-order Butterworth high-pass filter having a cutoff frequency equal to 1 rad/s and a dc gain equal to 5. 3 2 5 ( ) ( 1)( 1) n s H s s s s = + + + Frequency scaling to change the cutoff frequency to c 100 rad/s ω = 3 3 3 2 2 2 2 5 5 5 100 ( ) ( 100)( 100 100 ) ( 100)( 100 10000) 1 1 100 100 100 H s s s H s s s s s s s s s s = = = + + + + + + + + + P16.3-4 Use Table 16-3.2 to obtain the transfer function of a fourth-order Butterworth high-pass filter having a cutoff frequency equal to 1 rad/s and a dc gain equal to 5. ( ) ( ) ( ) 4 2 2 5 0.765 1 1.848 1 s H s n s s s s = + + + + Frequency scaling can be used to adjust the cutoff frequency 500 hertz = 3142 rad/s: ( ) ( ) ( ) 4 2 2 4 2 2 2 2 5 3142 0.765 1 1.848 1 3142 3142 3142 3142 5 2403.6 3142 5806.4 3142 H s H s s s s s s s s s s =   + + +       = + + + + + 16-2
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P16.3-5 First, obtain the transfer function of a second-order Butterworth low-pass filter having a dc gain equal to 2 and a cutoff frequency equal to 2000 rad/s: ( ) 2 2 2 8000000 2828 4000000 1.414 1 2000 2000 L H s s s s s = = + + + + Next, obtain the transfer function of a second-order Butterworth high-pass filter having a passband gain equal to 2 and a cutoff frequency equal to 100 rad/s: ( ) 2 2 2 2 2 2 100 141.4 10000 1.414 1 100 100 H s s H s s s s s = = + + + + Finally, the transfer function of the bandpass filter is ( ) ( ) ( ) ( ) ( ) 2 2 2 16000000 141.4 10000 2828 4000000 B L s H s H s H s H s s s s = = + + + + P16.3-6 ( ) ( ) 2 2 2 2 2 2 250 250000 1 4 250 250 62500 250 1 B s s H s s s s s = = + + + + P16.3-7 First, obtain the transfer function of a second-order Butterworth high-pass filter having a dc gain equal to 2 and a cutoff frequency equal to 2000 rad/s: ( ) 2 2 2 2 2 2 2000 2828 4000000 1.414 1 2000 2000 L s s H s s s s s = = + + + + Next, obtain the transfer function of a second-order Butterworth low-pass filter having a pass-
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