Abstract
A filter is an electrical network that passes signals within a specified band of frequencies while attenuating those signals that fall outside of this band. Passive filters utilize passive components, namely, resistors, capacitors and inductors, while active filters contain passive as well as active components such as transistors and operational amplifiers. Passive filters have the advantage of not requiring an external power supply as do active filters. However, they often utilize inductors which tend to be bulky and costly, whereas active filters utilize mainly resistors and capacitors. Additionally, active filters can produce signal gain and have high input and low output impedances which allow simple cascading of systems with little or no interaction between stages. This chapter discusses the principles of active filter operation and treats with several types and configurations of active filters. At the end of the chapter, the student will be able to:
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Bibliography
R. Schaumann, M. Van Valkenburg, Design of Analog Filters, 1st edn. (Oxford University Press, 2001)
T. Kugelstadt, in Active filter design techniques in op amps for everyone, ed. by R. Mancini, (Texas Instruments, 2002)
A. Waters, Active filter design (Mc Graw Hill, 1991)
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Problems
Problems
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1.
Design a first-order low-pass non-inverting filter with unity gain and a break frequency of 5Â kHz.
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2.
Design a first-order non-inverting low-pass filter with gain of 8 and a cut-off frequency of 17Â kHz.
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3.
Design a first-order low-pass filter with gain of 12 and a cut-off frequency of 25Â kHz using the inverting configuration.
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4.
Show that in a first-order low-pass filter, the amplitude response is down 3Â dB at the cut-off frequency and determine the phase angle of the output relative to the input at that frequency.
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5.
Using the Sallen-Key configuration, design a second-order low-pass Butterworth filter with unity gain and a cut-off frequency of 50Â kHz. Ensure that offset voltage is minimized.
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6.
Show that in the Sallen-Key second-order low-pass Butterworth filter, for the case where the network resistors are equal and the network capacitors are equal, the gain must be 1.6. Hence repeat the design of Question 5 using this modified approach.
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7.
Show that in the Sallen-Key second-order unity-gain low-pass filter configuration shown in Fig. 11.61, if R1Â =Â R2 and C2Â =Â 2C1, then the transfer function reduces to a second-order Butterworth response
.
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8.
Design a VCVS second-order unity-gain low-pass Chebyshev filter with a cut-off frequency of 22 kHz and a ripple width RWdB = 1dB.
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9.
Design a VCVS second-order unity-gain low-pass Bessel filter with a cut-off frequency of 550Hz.
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10.
Design a MFB second-order low-pass Butterworth filter with a gain of 8 and a cut-off frequency of 3 kHz.
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11.
Design a VCVS third-order unity-gain low-pass Butterworth filter with a cut-off frequency of 33Â kHz. What is the ultimate roll-off rate of such a filter?
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12.
Show that in the Sallen-Key third-order unity-gain low-pass filter configuration shown in Fig. 11.62, if R1Â =Â R2Â =Â R3 and C2Â =Â 2C3 and C1Â =Â C3/2, then the transfer function reduces to a third-order Butterworth response. Hence repeat Question 11 using this simplified design procedure
.
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13.
Design a VCVS fourth-order unity-gain low-pass Chebyshev filter with a cut-off frequency of 12Â kHz and 3Â dB ripple width.
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14.
Design a VCVS fifth-order unity-gain low-pass Butterworth filter with a cut-off frequency of 1200Â Hz.
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15.
Design a first-order high-pass non-inverting filter with unity gain and a break frequency of 5Â kHz.
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16.
Design a first-order non-inverting high-pass filter with gain of 8 and a cut-off frequency of 17Â kHz.
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17.
Design a first-order high-pass filter with gain of 15 and a cut-off frequency of 10Â kHz using the inverting configuration.
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18.
Show that in a first-order high-pass filter, the amplitude response is down 3Â dB at the cut-off frequency and determine the phase angle of the output relative to the input at that frequency.
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19.
Using the Sallen-Key configuration, design a second-order high-pass Butterworth filter with unity gain and a cut-off frequency of 5Â kHz.
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20.
Show that in the Sallen-Key second-order high-pass Butterworth filter, for the case where the network resistors are equal and the network capacitors are equal, the gain must be 1.6. Hence repeat the design of Question 19 using this modified approach.
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21.
Show that in the Sallen-Key second-order unity-gain high-pass filter configuration shown in Fig. 11.63, if R1Â =Â 2R2 and C1Â =Â C2, then the transfer function reduces to a second-order high-pass Butterworth response
.
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22.
Design a VCVS second-order unity-gain high-pass Chebyshev filter with a cut-off frequency of 14 kHz and a ripple width RWdB = 1dB.
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23.
Design a VCVS second-order unity-gain high-pass Bessel filter with a cut-off frequency of 5 kHz.
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24.
Design a MFB second-order high-pass Butterworth filter with a gain of 6 and a cut-off frequency of 2 kHz.
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25.
Design a VCVS third-order unity-gain high-pass Butterworth filter with a cut-off frequency of 900Â Hz. What is the ultimate roll-off rate of such a filter?
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26.
Show that in the Sallen-Key third-order unity-gain low-pass filter configuration shown in Fig. 11.64, if C1Â =Â C2Â =Â C3 and R1Â =Â 2R3 and R2Â =Â R3/2, then the transfer function reduces to a third-order Butterworth response. Hence repeat Question 25 using this simplified design procedure
.
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27.
Design a VCVS fourth-order unity-gain high-pass Chebyshev filter with a cut-off frequency of 1200Â Hz and 2Â dB ripple width.
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28.
Design a VCVS fifth-order unity-gain high-pass Bessel filter with a cut-off frequency of 6Â kHz.
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29.
Using the VCVS band-pass filter shown in Fig. 11.65, design a Sallen-Key second-order band-pass filter having C1 = C2, R1 = R2 with a centre frequency of 13 kHz and a quality factor of 8
.
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30.
For the VCVS band-pass filter shown in Fig. 11.65, develop design equations for R1, R2 and R3 if C1 = C2 = C and 1 + Rb/Ra = 2. Hence, design a VCVS band-pass filter having fo = 20 kHz, Q = 6 and G = 10.
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31.
Design a second-order MFB band-pass filter with f0 = 2 kHz, Q = 9 and G = 5. Use the circuit shown in Fig. 11.66
.
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32.
For the second-order MFB band-pass filter shown in Fig. 11.66, develop design equations without imposing the condition C1 = C2. Hence, design a second-order MFB band-pass filter having fo = 1 kHz, Q = 5 and G = 10.
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33.
Design a second-order Wien band-pass filter with f0 = 60 Hz, Q = 10 and G = 2.
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34.
Design a twin-T notch filter having a notch frequency of 2.5Â kHz and QÂ =Â 25.
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35.
Using the circuit shown in Fig. 11.67, design a Wien notch filter to have a notch frequency of 200Â Hz, a Q of 25 and a gain of 2
.
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36.
By using a ganged potentiometer for R and two banks of switched capacitors for C, convert the Wien notch filter in Fig. 11.67 to a variable frequency circuit and specify the frequency range.
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37.
Develop the transfer function for a Tow-Thomas biquadratic filter.
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Gift, S.J.G., Maundy, B. (2021). Active Filters. In: Electronic Circuit Design and Application. Springer, Cham. https://doi.org/10.1007/978-3-030-46989-4_11
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DOI: https://doi.org/10.1007/978-3-030-46989-4_11
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