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  The Analysis and Design of Linear Circuits Seventh Edition 14 Active Filter Design 141 Exercise Solutions Exercise 14–1  Develop a second-order low-pass transfer function with a corner frequency of 50 rad/s, adc gain of 2, and a gain of 4 at the corner frequency Validate your result by using MATLAB to plot thetransfer function’s absolute gain versus frequencyThe transfer function for a second-order low-pass filter has the following form: T  ( s ) =  K  ( s/ω 0 ) 2 + 2 ζ  ( s/ω 0 ) + 1 =  Kω 20 s 2 + 2 ζω 0 s + ω 20 ω 0  = 50 rad/s K   = 2 | T  (  jω 0 ) | = 4 =  | K  | 2 ζ   = 22 ζ   = 1 ζ ζ   = 14 T  ( s ) = 5000 s 2 + 25 s + 2500The plot of the transfer function is shown below and it meets the specifications 10 0 10 1 10 2 10 3 005115225335445Frequency, (rad/sec)    A  m  p   l   i   t  u   d  e Exercise 14–2  Design circuits using both the equal element design and unity gain design techniques torealize the transfer function in Exercise 14–1 Use OrCAD to simulate your designs and compare them tothe MATLAB results shown in Figure 14–6For the equal element approach, pick  R  = 10 kΩ and solve for the other values as follows: ω 0  = 1 RC C   = 1 ω 0 R  = 1(50)(10000) = 2  µ F µ  = 3 − 2 ζ   = 3 − 2(0 25) = 2 5Design the circuit using the resistor and capacitor specified and then use a voltage divider to reduce the gainfrom 25 to 2 to meet the srcinal specifications in Exercise 14–1 The OrCAD circuit simulation is shownbelow and the output follows The output agrees with the MATLAB plotSolution Manual Chapter 14 Page 14-1  The Analysis and Design of Linear Circuits Seventh EditionFor the unity gain approach, pick  C  1  = 8  µ F and solve for the other values as follows: C  2  =  ζ  2 C  1  = 8 µ F16 = 0 5  µ F R  = 1 ω 0 √    C  1 C  2 = 10 kΩ µ  = 1Design the circuit using the resistor and capacitors specified and then use a noninverting amplifier to increasethe gain from 1 to 2 to meet the srcinal specifications in Exercise 14–1 The OrCAD circuit simulation isshown below and the output follows The output agrees with the MATLAB plotSolution Manual Chapter 14 Page 14-2  The Analysis and Design of Linear Circuits Seventh Edition Exercise 14–3  Construct a second-order high-pass transfer function with a corner frequency of 20 rad/s,an infinite-frequency gain of 4, and a gain of 2 at the corner frequencyThe transfer function for a second-order high-pass filter has the following form: T  ( s ) =  K  ( s/ω 0 ) 2 ( s/ω 0 ) 2 + 2 ζ  ( s/ω 0 ) + 1 =  Ks 2 s 2 + 2 ζω 0 s + ω 20 ω 0  = 20 rad/s K   = 4 | T  (  jω 0 ) | = 2 =  | K  | 2 ζ   = 42 ζ   = 2 ζ ζ   = 1 T  ( s ) = 4 s 2 s 2 + 40 s + 400The plot of the transfer function is shown below and it meets the specifications 10 0 10 1 10 2 10 3 0051152253354Frequency, (rad/sec)    A  m  p   l   i   t  u   d  e Exercise 14–4  Rework the design in Example 14–3, starting with  C   = 2000 pFSolution Manual Chapter 14 Page 14-3