CHAPTER 15: ACTIVE FILTER CIRCUITS
1 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
Electronic Circuits, Tenth Edition J ames W. Nilsson | Susan A. Riedel 2 15.1 1st-Order Low-Pass and High-Pass Filters
• Active filters consist of op amps, resistors, and capacitors. • They overcome many of the disadvantages associated with passive filters.
A first-order low-pass filter.
A first-order low-pass filter. A general op amp circuit.
• At very low frequencies, the capacitor acts like an open circuit, and the op amp circuit acts like an amplifier with a gain. • At very high frequencies, the capacitor acts like a short circuit, thereby connecting the output of the op amp circuit to ground.
Electronic Circuits, Tenth Edition J ames W. Nilsson | Susan A. Riedel 3 15.1 1st-Order Low-Pass and High-Pass Filters
Transfer function for the circuit
1 ‖
Where and =
The gain in the passband, K, is set by the ratio R2/R1. The op amp low-pass filter thus permits the passband gain and the cutoff frequency to be specified independently.
Electronic Circuits, Tenth Edition J ames W. Nilsson | Susan A. Riedel 4 15.1 1st-Order Low-Pass and High-Pass Filters
• Bode plot
(1) uses a logarithmic axis, instead of using a linear axis for the frequency values (2) plotted in decibels (dB), instead of plo tting the absolute magnitude of the tra nsfer function vs. frequency
20 10 1 20 3 2 15.1 1st-Order Low-Pass and High-Pass Filters
• A prototype low-pass filter : component values of 1 2 1Ω and 1 , : a unity passband gain and a cutoff frequency of 1 / .
: Providing a useful starting point for the design of filters by using more realistic component values to achieve a desired frequency response
Electronic Circuits, Tenth Edition J 6 ames W. Nilsson | Susan A. Riedel 15.1 1st-Order Low-Pass and High-Pass Filters
A first-order high-pass filter.
Transfer function Equation for passive high-pass filters.
1 ‐K Where , and = 15.2 Scaling
Scaling : transforming the convenient values into realistic values
1. Magnitude scaling altering component values without changing the frequency response of a circuit.
•For a magnitude scale factor of , Let primed variables represent the scaled values of the variables
, , / 2. Frequency scaling shifting the frequency response of a circuit to another frequency region without changing the overall shape of the frequency response.
• For a frequency scale factor of kf ,
, / , /
Electronic Circuits, Tenth Edition J 8 ames W. Nilsson | Susan A. Riedel 15.2 Scaling
• Components can be scaled in both magnitude and frequency, with the scaled (primed) component values given by
, ,
• The design of active low-pass and high-pass filters
1. an a prototype filter circuit. : component values of 1 2 1Ω and 1 , : a unity passband gain and a cutoff frequency of 1 / . 2. Scaling can then be applied to shift the frequency response to the desired cutoff frequency, using component values that are commercially available.
Electronic Circuits, Tenth Edition J 9 ames W. Nilsson | Susan A. Riedel 15.2 Scaling
The Use of Scaling in the Design of Op Amp Filters
1. Selecting the cutoff frequency , to be 1 rad/s (if you are designing low- or highpass filters), or selecting the center frequency 0, to be 1 rad/s (if you are designing bandpass or bandreject filters).
2. Selecting a 1 capacitor and calculating the values of the resistors needed to give the desired passband gain and the 1 rad/s cutoff or center frequency.
3. Using scaling to compute more realistic component values that give the desired cutoff or center frequency. 15.3 Op Amp Bandpass and Bandreject Filters
Bandpass filters Three separate components consisting of the bandpass filter 1. A unity-gain low-pass filter whose cutoff frequ
ency is 2, the larger of the two cutoff freque ncies 2. A unity-gain high-pass filter whose cutoff fr
equency is ωc1. the smaller of the two cuto ff frequencies 3. A gain component to provide the desired l evel of gain in the passband
These three components are cascaded in series (combine multiplicatively in the s domain)
The formal definition of a broadband filter Constructing the Bode magni tude plot of a bandpass filter. 2
Electronic Circuits, Tenth Edition J 11 ames W. Nilsson | Susan A. Riedel 15.3 Op Amp Bandpass and Bandreject Filters
A cascaded op amp bandpass filter. (a) The block diagram. (b) The circuit.
*Establishing the relationship between and that will permit each 1 2 subcircuit to be designed independently, without concern for the other subcircuits in the cascade.
Electronic Circuits, Tenth Edition J 12 ames W. Nilsson | Susan A. Riedel 15.3 Op Amp Bandpass and Bandreject Filters
• The transfer function of the cascaded bandpass filter is the product of the transfer functions of the three cascaded components:
• Standard form for the transfer function of a bandpass filter
2 15.3 Op Amp Bandpass and Bandreject Filters
Converting into the form of the standard transfer function for a bandpass filter, Assumption ≫
Transfer function for the cascaded bandpass filter
Upper cutoff frequency, ωc2 1
lower cutoff frequency, ωc1 1 15.3 Op Amp Bandpass and Bandreject Filters
Computing the values of Ri and Rf in the inverting amplifier to provide the desired passband gain.
the magnitude of the bandpass filter’s transfer function,
evaluated at the center frequency, ω0
K.
Gain of the inverting amplifier is Rf / Ri
15.3 Op Amp Bandpass and Bandreject Filters
Bandreject filters Bandreject filter configuration
1. A unity-gain low-pass filter whose cutoff
frequency is 1, the smaller of the two cutoff frequencies 2. A unity-gain high-pass filter whose
cutoff frequency is 2. the larger of the two cutoff frequencies 3. A gain component to provide the desired level of gain in the passband
: these three components cannot be cascaded in series (we use a parallel connection and a summing amplifier) Constructing the Bode magnitude plot of a bandreject filter.
Electronic Circuits, Tenth Edition J 16 ames W. Nilsson | Susan A. Riedel 15.3 Op Amp Bandpass and Bandreject Filters
The transfer function of the resulting circuit : the sum of the low-pass and high-pass filter transfer functions.
= ( )
= ( )
A parallel op amp bandreject filter. (a) The block diagram. (b) The circuit.
Electronic Circuits, Tenth Edition J 17 ames W. Nilsson | Susan A. Riedel 15.3 Op Amp Bandpass and Bandreject Filters
1 Cutoff frequencies 1 In the two passbands (as s 0 ands ), the gain of the transfer function is Rf / Ri
Note the magnitude of the transfer function , at the center frequency 2
If ≫ , then ≪2 / (as / ≪1) the magnitude at the center frequency is much smaller than the passband magnitude. 15.4 Higher Order Op Amp Filters
•An ideal filter has a discontinuity at the point of cutoff, which sharply divides the passband and the stopband. • 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.
The Bode magnitude plot of a cascade of identical prototype first-order filters.
•With one filter, the transition generally occurs with an asymptotic slope of 20 (dB/ dec).
•With two filters, a transition with an asymptotic slope of 20 + 20 = 40 dB/ dec.
Electronic Circuits, Tenth Edition J 19 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
• The transfer function for a cascade of n prototype low-pass filters—we just multiply the individual transfer functions:
A cascade of identical unity-gain low-pass filters. (a) The block diagram. (b) The circuit.
Electronic Circuits, Tenth Edition J 20 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
⋯ =
*A cascade of n first-order filters produces an nth-order filter, having n poles in its transfer function and a final slope of 20n dB>dec in the transition band. • As the order of the low-pass filter is increased by adding prototype low- pass filters to the cascade, the cutoff frequency also changes.
frequency scaling = , ⁄ = 2 1,
2 1. 15.4 Higher Order Op Amp Filters
For example, Let’s compute the cutoff frequency of a fourth-order unity-gain low- pass filter constructed from a cascade of four prototype low-pass filters: 2 1 0.435 ⁄ .
scaling by k f = ωc / 0.435 to place the cutoff frequency.
Electronic Circuits, Tenth Edition J 22 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
• A serious shortcoming of cascading identical low-pass filters The gain of the filter is not constant between zero and the cutoff frequency
ωc. An ideal low-pass filter, the passband magnitude is 1 for all frequencies below the cutoff frequency.
But, the magnitude is less than 1 (0 dB) for frequencies much less than the cutoff frequency. Magnitude of the transfer function for a unity-gain low-pass nth-order cascade
As the denominator Magnitude = , becomes larger than 1, so the magnitude becomes smaller = than 1 ⁄
Electronic Circuits, Tenth Edition J 23 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
• shortcoming: The gain of the filter is not constant between zero and the cutoff frequency. • Unity-gain Butterworth low-pass filter has a transfer function whose magnitude = ⁄ Properties
1. The cutoff frequency is ωc rad/s for all values of n. 2. If n is large enough, the denominator is always close to unity when
3. In the expression for H(jω) the exponent of is always even. Given an equation for the magnitude of the transfer function, how do we find H(s) ? The derivation for H(s) is greatly simplified by using a prototype filter.
Electronic Circuits, Tenth Edition J 24 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
To find H(s)
Because s = jω Now observe that s2 = - ω2
1 1 1 1 1 1 1
Electronic Circuits, Tenth Edition J 25 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
• The procedure for finding H(s) for a given value of n
1. Find the roots of the polynomial 1 1 2 0
2. Assign the left-half plane roots to H(s) and the right half plane roots to H(-s)
3. Combine terms in the denominator of H(s) to form first- and second-order factors.
Electronic Circuits, Tenth Edition J 26 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
Butterworth Filter Circuit
A circuit that provides the second-order transfer function for the Butterworth filter cascade.
• s-domain nodal equations at the noninverting terminal of the op amp and at
the node labeled
0 2 2 2 1 1 2 2 0 15.4 Higher Order Op Amp Filters
Transfer function for the circuit 1 2 1 Finally, set R = 1 1 2 1 The form required for the second-order circuit in the Butterworth cascade. Transfer function of the form 1 2 1 1 Choosing capacitor values and 1 15.4 Higher Order Op Amp Filters The Order of a Butterworth Filter
• The higher the order of the Butterworth filter, the closer the magnitude characteristic comes to that of an ideal low-pass filter. • At the same time, as the order increases, the number of circuit components increases. • It follows then that a fundamental problem in the design of a filter is to determine the smallest value of n that will meet the filtering specifications. The filtering specifications are usually given in terms of the abruptness
of the transition region (Ap, ωp, As, and ωs) 15.4 Higher Order Op Amp Filters
For the Butterworth filter, 1 20log 1
. 10 10log 1 . 10 1 20log 1 10log 1 . 10 1 log ⁄ 10 . 1 log ⁄
If ωp is the cutoff frequency ⁄
One further simplification the filtering specification 10 . ≫ 1 15.4 Higher Order Op Amp Filters
. 10
log 0.05
a good approximation for the calculation of
. ⁄
Electronic Circuits, Tenth Edition J 31 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
Butterworth High-Pass, Bandpass, and Bandreject Filters
• nth-order Butterworth high-pass filter
A second-order Butterworth high-pass filter circuit. To produce the second-order factors in the Butterworth polynomial
2 2 1 1
Electronic Circuits, Tenth Edition J 32 ames W. Nilsson | Susan A. Riedel 15.4 Higher Order Op Amp Filters
Transfer function
Setting 1
2 1 1
Electronic Circuits, Tenth Edition J 33 ames W. Nilsson | Susan A. Riedel 15.5 Narrowband Bandpass & Bandreject Filters
• The cascade and parallel component designs for synthesizing bandpass and bandreject filters from simpler low-pass and high-pass filters only broadband, or low-Q, filters will result. • If a high-Q, or narrowband, bandpass, or bandreject filter is needed, the cascade or parallel combination will not work..
An active high-Q bandpass filter. The Bode magnitude plot for the high-Q bandpass filter
Electronic Circuits, Tenth Edition J 34 ames W. Nilsson | Susan A. Riedel 15.5 Narrowband Bandpass & Bandreject Filters
transfer function 2 2 0.5 The largest quality factor we can achieve with discrete real poles arises when the cutoff frequencies, and thus the pole locations, are the same. The bandwidth and center frequency directly:
2
To build active filters with high quality factor values 1/ 15.5 Narrowband Bandpass & Bandreject Filters
At the node labeled a,
1 1/ 1/ 2
1 2 ⁄ Transfer function
2 1 where 1 2 1‖ 2 1 2
Electronic Circuits, Tenth Edition J 36 ames W. Nilsson | Susan A. Riedel 15.5 Narrowband Bandpass & Bandreject Filters
Standard form of the transfer function for a bandpass filter
The values of the resistors, which will achieve a specified center
frequency (ω0), quality factor (Q), and passband gain (K): ; 1 ; 1
Expressions for R1, R2, and R3 1 / 2 2 / 2 3 2
Electronic Circuits, Tenth Edition J 37 ames W. Nilsson | Susan A. Riedel 15.5 Narrowband Bandpass & Bandreject Filters
• A high-Q active band reject filter.
twin-T notch filter
The parallel implementation of a band reject filter that combines lowpass and high-pass filter components with a summing amplifier has the same low-Q restriction as the cascaded bandpass filter. 15.5 Narrowband Bandpass & Bandreject Filters
Summing the currents away from node a: 2 0 Summing the currents away from node b 2 0 Summing the currents away from the noninverting input terminal of the top op amp 0
1 4 1 1
Transfer function: 1 4 1 1 15.5 Narrowband Bandpass & Bandreject Filters
Standard form for the transfer function of a band reject filter:
1 4 1
One parameter is chosen arbitrarily; it is usually the capacitor value because this value typically provides the fewest commercially available options
1 Once C is chosen, 1 1
Electronic Circuits, Tenth Edition J 40 ames W. Nilsson | Susan A. Riedel End of Ch.15
Electronic Circuits, Tenth Edition J 41 ames W. Nilsson | Susan A. Riedel