1. ELECTRIC CIRCUITS EIGHTH EDITION
JAMES W. NILSSON
&
SUSAN A. RIEDEL
ELECTRIC CIRCUITS EIGHTH EDITION

2. ACTIVE FILTER CIRCUITS
CHAPTER 15
ACTIVE FILTER CIRCUITS
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3. CONTENTS 15.1 First-Order Low-Pass and High-Pass Filters 15.2 Scaling
15.3 Op Amp Bandpass and Bandreject Filters
15.4 High Order Op Amp Filters
15.5 Narrowband Bandpass and Bandreject Filters
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4. 15.1 First-Order Low-Pass and High-Pass Filters
Active filters consist of op amps, resistors, and capacitors.
They can be configured as low-pass, high-pass, bandpass, and bandreject filters.
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5. 15.1 First-Order Low-Pass and High-Pass Filters
They overcome many of the disadvantages associated with passive filters.
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6. 15.1 First-Order Low-Pass and High-Pass Filters
A first-order low-pass filter
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7. 15.1 First-Order Low-Pass and High-Pass Filters
A general op amp circuit
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8. 15.1 First-Order Low-Pass and High-Pass Filters
A first-order high-pass filter
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9. 15.1 First-Order Low-Pass and High-Pass Filters
A prototype low-pass filter has component values of R1 = R2 = 1Ω and C = 1F, and it produces a unity passband gain and a cutoff frequency of 1 rad/s.
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10. 15.1 First-Order Low-Pass and High-Pass Filters
The prototype high-pass filter has same component values and also produces a unity passband gain and a cut-off frequency of 1 rad/s.
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11. 15.2 Scaling
Magnitude scaling can be used to alter component values without changing the frequency response of a circuit.
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12. 15.2 Scaling
For a magnitude scale factor of km, the scaled (primed) values of resistance, capacitance, and inductance are
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13. 15.2 Scaling
Frequency scaling can be used to shift the frequency response of a circuit to another frequency region without changing the overall shape of the frequency response.
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14. 15.2 Scaling
For a frequency scale factor of kf , the scaled (primed) values of resistance, capacitance, and inductance are
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15. 15.2 Scaling
Components can be scaled in both magnitude and frequency, with the scaled (primed) component values given by
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16. 15.2 Scaling
The design of active low-pass and high-pass filters can begin with a prototype filter circuit.
Scaling can then be applied to shift the frequency response to the desired cutoff frequency, using component values that are commercially available.
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17. 15.3 Op Amp Bandpass and Bandreject Filters
Constructing the Bode magnitude plot of a bandpass filter
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18. 15.3 Op Amp Bandpass and Bandreject Filters
A cascaded op amp bandpass filter
The block diagram
The circuit
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19. 15.3 Op Amp Bandpass and Bandreject Filters
An active broadband bandpass filter can be constructed using a cascade of a low-pass filter with the bandpass filter’s upper cutoff frequency, a high-pass filter with the bandpass filter’s lower cutoff frequency, and (optionally) an inverting amplifier gain stage to achieve nonunity gain in the passband.
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20. 15.3 Op Amp Bandpass and Bandreject Filters
Bandpass filters implemented in this fashion must be broadband filters (ωc2 » ωc1), so that the elements of the cascade can be specified independently of one another.
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21. 15.3 Op Amp Bandpass and Bandreject Filters
Example: Designing a Broadband Bandpass Op Amp Filter.
Design a bandpass filter for a graphic equalizer to provide an amplification of 2 within the band of frequencies between 100 and 10,000 Hz. Use 0.2µF capacitors.
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22. 15.3 Op Amp Bandpass and Bandreject Filters
Constructing the Bode magnitude plot of a bandreject filter
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23. 15.3 Op Amp Bandpass and Bandreject Filters
An active broadband bandreject filter can be constructed using a parallel combination of a low-pass filter with the bandreject filter’s lower cutoff frequency and a high-pass filter with the bandreject filter’s upper cutoff frequency.
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24. 15.3 Op Amp Bandpass and Bandreject Filters
The outputs are then fed into a summing amplifier, which can produce nonunity gain in the passband.
Bandreject filters implemented in this way must be broadband filters (ωc2 » ωc1), so that the low-pass and high-pass filter circuits can be designed independently of one another.
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25. 15.3 Op Amp Bandpass and Bandreject Filters
A parallel op amp bandreject filter
The block diagram
The circuit
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26. 15.4 Higher Order Op Amp Filters
The bode magnitude plot of a cascade of identical prototype first-order filters
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27. 15.4 Higher Order Op Amp Filters
Higher order active filters have multiple poles in their transfer functions, resulting in a sharper transition from the passband to the stopband and thus a more nearly ideal frequency response.
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28. 15.4 Higher Order Op Amp Filters
A cascade of identical unity-gain low-pass filters.
The block diagram
The circuit
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29. 15.4 Higher Order Op Amp Filters
The transfer function of an nth–order Butterworth low-pass filter with a cutoff frequency of 1 rad/s can be determined from the equation:
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30. 15.4 Higher Order Op Amp FiltersBy
Finding the roots of the denominator polynomial.
Assigning the left-half plane roots to H(s).
Writing the denominator of H(s) as a product of first- and second- order factors.
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31. 15.4 Higher Order Op Amp Filters
Defining the transition region for a low-pass filter
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32. 15.5 Narrowband Bandpass and Bandreject Filters
An active high-Q bandpass filter
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33. 15.5 Narrowband Bandpass and Bandreject Filters
A high-Q active bandreject filter
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34. 15.5 Narrowband Bandpass and Bandreject Filters
If a high-Q, or narrowband, bandpass, or bandreject filter is needed, the cascade or parallel combination will not work. Instead, the circuits shown previously are used with the appropriate design equations.
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35. THE END
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