-15
`
`±tOV in ~ 411
`
`+15 (~
`—15 J
`
`+15
`
`417
`
`—15 +5
`OJ L
`
`22k
`
`N. op-amps with switches
`
`reset
`
`+~5
`
`15
`
`100k
`
`r
`
`O. FET gain control
`
`Q. integrator with reset
`
`+
`
`reset
`
`+15
`
`Vmnvo~
`
`~ load — V wntrol ~~ ~~~
`
`p, current source
`
`261
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTER
`~SCILLAT~RS
`
`With only the techniques of transistors and
`op-amps it is possible to delve into a num-
`ber of interesting areas of linear (as con-
`trasted with digital) circuitry. We believe
`that it is important to spend some time do-
`ing this now, in order to strengthen your
`understanding of some of these difFicult
`concepts (transistor behavior, feedback,
`op-amp limitations, etc.) before introduc-
`ing more new devices and techniques and
`getting into the large area of digital elec-
`tronics. In this chapter, therefore, we will
`treat briefly the areas of active filters and
`oscillators. Additional analog techniques
`are treated in Chapter 6 (voltage regula-
`tors and high-current design), Chapter 7
`(precision circuits and low noise), Chap-
`ter 13 (radiofrequency techniques), Chap-
`ter 14 (low-power design), and Chapter
`15 (measurements and signal processing).
`The first part of this chapter (active filters,
`Sections 5.01-5.11) describes techniques
`of a somewhat specialized nature, and it
`can be passed over in a first reading. How-
`ever, the latter part of this chapter (oscil-
`lators, Sections 5.12-5.19) describes tech-
`niques of broad utility and should not be
`omitted.
`
`ACTIVE FILTERS
`
`In Chapter 1 we began a discussion of fil-
`ters made from resistors and capacitors.
`Those simple RC filters produced gentle
`high-pass or low-pass gain characteristics,
`with a 6dB/octave. falloff well beyond
`the —3dB point. By cascading high-pass
`and low-pass filters, we showed how to
`obtain bandpass filters, again with gentle
`6dB/octave "skirts." Such filters are suf~i-
`cient for many purposes, especially if the
`signal being rejected by the filter is far
`removed in frequency from the desired
`signal passband. Some examples are by-
`passing of radiofrequency signals in audio
`circuits, "blocking" capacitors for elimina-
`tion of do levels, and separation of mod-
`ulation from a communications "carrier"
`(see Chapter 13).
`
`5.01 Frequency response with RC filters
`Often, however, filters with flatter pass-
`bands and steeper skirts are needed. This
`happens whenever signals must be filtered
`from other interfering signals nearby in
`
`263
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`i.o
`os
`o.s
`o.~
`0.6
`
`0a
`
` 0.5
`
`1 2
`frequency (Hz)
`
`3
`
`1 2
`normalized frequency
`
`3
`
`0.4
`.Q 0.3
`
`0.2
`
`0.1
`
`0
`
`A
`
`1.0
`o.s
`o.s
`o.~
`o.s
`0.5
`o.a
`— 0.3
`0.2
`o. i
`
`0
`
`a
`
`i.o
`
`~, o.i
`a
`
`0.01
`
`a
`
`o.00i
`
`o. i
`
`c
`
`1.0
`10
`normalized frequency (log scale)
`
`100
`
`Figure 5.2. Frequency responses of multisec-
`tion RCfilters. Graphs A and B are linear plots,
`whereas C is logarithmic. The filter responses
`in B and C have been normalized (or scaled)
`for 3dB attenuation at unit frequency.
`
`ACTIVE FILTERS AND OSCILLATORS
`264 Chapter 5
`
`frequency. The obvious next question is
`whether or not (by cascading a number
`~f idertieal 'c•:'-pass filters, gay) ~v~ ;,an
`generate an approximation to the ideal
`"brick-wall" low-pass frequency response,
`as in Figure 5.1.
`
`~
`
`Vogt
`V~~
`
`0
`
`fo
`
`Figure 5.1
`
`We know already that simple cascading
`won't work, since each section's input
`impedance will load the previous section
`seriously, degrading the response. But
`with buffers between each section (or by
`arranging to have each section of much
`higher impedance than the one preceding
`it), it would seem possible. Nonetheless,
`the answer is no. Cascaded RC filters do
`produce a steep ultimate falloff, but the
`"knee" of the curve of response versus
`frequency is not sharpened. We might
`restate this as "many soft knees do not
`a hard knee make." To make the point
`graphically, we have plotted some graphs
`of gain response (i.e., Vout/V;,,) versus
`frequency for low-pass filters constructed
`from 1, 2, 4, 8, 16, and 32 identical RC
`sections, perfectly buffered (Fig. 5.2).
`The first graph shows the effect of cas-
`cading several RC sections, each with its
`3dB point at unit frequency. As more
`sections are added, the overall 3dB point
`is pushed downward in frequency, as you
`could easily have predicted. To compare
`filter characteristics fairly, the rolloff fre-
`quencies of the individual sections should
`be adjusted so that the overall 3dB point
`is always at the same frequency. The other
`graphs in Figure 5.2, as well as the next few
`graphs in this chapter, are all "normalized"
`in frequency, meaning that the —3dB point
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTERS
`5.02 Ideal performance with LC filters
`
`265
`
`97.5
`
`605
`
`743
`
`583
`
`143
`
`605
`
`97.5
`
`5725
`
`Ok T
`
`4979
`16 16 ~ ~ 16
`5236
`
`5025
`16 m ~ 16
`15025
`
`5236
`16 T m 16
`14979
`
`5725
`16 ~ ,Or--~
`
`ma c0 ca
`
`~ m
`
`1 2 14 16 18 20 22 24
`frequency (kHz)
`
`Figure 5.3. An unusually good passive bandpass filter implemented from inductors and capacitors
`(inductances in mH, capacitances in pF). Bottom: Measured response of the filter circuit. [Based
`on Figs. 11 and 12 from Orchard, H. J., and Sheahan, D. F., IEEE Journal of Solid-State Circuits,
`Vol. SC-5, No. 3 (1970).]
`
`(or breakpoint, however defined) is at a fre-
`quency of 1 radian per second (or at 1 Hz).
`To determine the response of a filter whose
`breakpoint is set at some other frequency,
`simply multiply the values on the frequen-
`cy axis by the actual breakpoint frequency
`f~. In general, we will also stick to the
`log-log graph of frequency response when
`talking about filters, because it tells the
`most about the frequency response. It
`lets you see the approach to the ultimate
`rolloff slope, and it permits you to read
`off accurate values of attenuation. In this
`case (cascaded RC sections) the normal-
`ized graphs in Figures 5.2B and 5.2C dem-
`onstrate the soft knee characteristic of pas-
`sive RC filters.
`
`5.02 Ideal performance with LC filters
`As we pointed out in Chapter 1, filters
`made with inductors and capacitors can
`
`have very sharp responses. The parallel
`LC resonant circuit is an example. By
`including inductors in the design, it is pos-
`sible to create filters with any desired flat-
`ness of passband combined with sharpness
`of transition and steepness of falloff out-
`side the band. Figure 5.3 shows an exam-
`ple of a telephone filter and its character-
`istics.
`Obviously the inclusion of inductors in-
`to the design brings about some magic that
`cannot be performed without them. In
`the terminology of network analysis, that
`magic consists in the use of "off-axis poles."
`Even so, the complexity of the filter in-
`creases according to the required flatness
`of passband and steepness of falloff outside
`the band, accounting for the large number
`of components used in the preceding fil-
`ter. The transient response and phase-shift
`characteristics are also generally degraded
`as the amplitude response is improved to
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`and the gyrator. These devices can mimic
`the properties of inductors, while using
`viiiy i~.~i.~7ivij CLi~l~i I~UiJCLl.1LV1J 111 [LUU1L1V11 LV
`op-amps.
`Once you can do that, you can build in-
`ductorless filters with the ideal properties
`of any RLC filter, thus providing at least
`one way to make active filters.
`The NIC converts an impedance to its
`negative, whereas the gyrator converts an
`impedance to its inverse. The following ex-
`ercises will help you discover for yourself
`how that works out.
`
`EXERCISE 5.1
`Show that the circuit in Figure 5.4 is a negative-
`impedance converter, in particular that Z;n =
`-Z. Hint: Apply some input voltage V, and
`compute the input current I. Then take the ratio
`to find Z;n = V/I.
`
`z,~ _ -z
`
`Figure 5.4. Negative-impedance converter.
`
`~
`
`r 1,
`
`IL
`
`z
`TJ_~_
`
`R
`
`R
`
`N IC
`
`z
`z'" Z
`—i
`
`R
`
`NIC
`
`ACTIVE FILTERS AND OSCILLATORS
`266 Chapter 5
`
`approach the ideal brick-wall characteris-
`tic.
`Th ~ ~~~n ~h~~i~ of .f'.1 iii u u vui yuuui v ~.
`components (R, L, C) is a highly devel-
`oped subject, as typified by the authorita-
`tive handbook by Zverev (see chapter ref-
`erences at end of book). The only problem
`is that inductors as circuit elements fre-
`quently leave much to be desired. They are
`often bulky and expensive, and they de-
`part from the ideal by being "lossy," i.e., by
`having significant series resistance, as well
`as other "pathologies" such as nonlinear-
`ity, distributed winding capacitance, and
`susceptibility to magnetic pickup of inter-
`ference.
`What is needed is a way to make
`inductorless filters with the characteristics
`of ideal RLC filters.
`
`5.03 Enter active filters: an overview
`
`By using op-amps as part of the filter de-
`sign, it is possible to synthesize any RLC
`filter characteristic without using induc-
`tors. Such inductorless filters are known
`as active filters because of the inclusion of
`an active element (the amplifier).
`Active filters can be used to make low-
`pass, high-pass, bandpass, and band-reject
`filters, with a choice of filter types accord-
`ing to the important features of the re-
`sponse, e.g., maximal flatness of passband,
`steepness of skirts, or uniformity of time
`delay versus frequency (more on this short-
`ly). In addition, "all-pass filters" with flat
`amplitude response but tailored phase ver-
`sus frequency can be made (they're also
`known as "delay equalizers"), as well as the
`opposite - a filter with constant phase shift
`but tailored amplitude response.
`
`Figure 5.5
`
`q Negative-impedance converters and
`gyrators
`
`Two interesting circuit elements that
`should be mentioned in any overview are
`the negative-impedance converter (NIC)
`
`EXERCISE 52
`Show that the circuit in Figure 5.5 is a gyrator,
`in particular that Z;n = R2/Z. Hint: You can
`analyze it as a set of voltage dividers, beginning
`at the right.
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTERS
`5.04 Key filter performance criteria
`
`267
`
`The NIC therefore converts a capacitor
`to a "backward" inductor:
`
`Zc= 1~~wc~Zin=j~cvC'
`i.e., it is inductive in the sense of generat-
`ing acurrent that lags the applied voltage,
`but its impedance has the wrong frequency
`dependence (it goes down, instead of up,
`with increasing frequency). The gyrator,
`on the other hand, converts a capacitor to
`a true inductor:
`ZC = 1~jc~C --~ Zin = j~.vCRa
`i.e., an inductor with inductance L =
`CR2.
`The existence of the gyrator makes it
`intuitively reasonable that inductorless fil-
`ters can be built to mimic any filter us-
`ing inductors: Simply replace each induc-
`tor by a gyrated capacitor. The use of
`gyrators in just that manner is perfectly
`OK, and in fact the telephone filter illus-
`trated previously was built that way. In ad-
`dition to simple gyrator substitution into
`preexisting RLC designs, it is possible to
`synthesize many other filter configurations.
`The field of inductorless filter design is ex-
`tremely active, with new designs appearing
`in the journals every month.
`
`Sallen-and-Key filter
`
`Figure 5.6 shows an example of a simple
`and even partly intuitive filter. It is known
`as a Sallen-and-Key filter, after its inven-
`tors. The unity-gain amplifier can be an
`op-amp connected as a follower, or just an
`emitter follower. This particular filter is a
`2-pole high-pass filter. Note that it would
`be simply two cascaded RC high-pass fil-
`ters except for the fact that the bottom of
`the first resistor is bootstrapped by the out-
`put. It is easy to see that at very low fre-
`quencies it falls off just like a cascaded RC,
`since the output is essentially zero. As the
`output rises at increasing frequency, how-
`ever, the bootstrap action tends to reduce
`
`the attenuation, giving a sharper knee. Of
`course, such hand-waving cannot substi-
`tute for honest analysis, which luckily has
`already been done for a prodigious variety
`of nice filters. We will come back to active
`filter circuits in Section 5.06.
`
`c c
`input----I
`
`+~
`
`output
`
`R R
`
`Figure 5.6
`
`5.04 Key filter performance criteria
`There are some standard terms that keep
`appearing when we talk about filters and
`try to specify their performance. It is worth
`getting it all straight at the beginning.
`
`Frequency domain
`
`The most obvious characteristic of a filter
`is its gain versus frequency, typified by
`the sort of low-pass characteristic shown
`in Figure 5.7.
`The passband is the region of frequen-
`cies that are relatively unattenuated by the
`filter. Most often the passband is con-
`sidered to extend to the —3dB point, but
`with certain filters (most notably the "equi-
`ripple" types) the end of the passband may
`be defined somewhat differently. Within
`the passband the response may show vari-
`ations or ripples, defining a ripple band, as
`shown. The cutoff frequency, f~, is the end
`of the passband. The response of the filter
`then drops off through a transition region
`(also colorfully known as the skirt of the fil-
`ter's response) to a stopband, the region of
`significant attenuation. The stopband may
`be defined by some minimum attenuation,
`e.g., 40dB.
`Along with the gain response, the other
`parameter of importance in the frequency
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTERS AND OSCILLATORS
`268 Chapter 5
`
`frequency (linear) —y
`
`1
`
`-~'o
`
`vE
`
`C
`
`passband
`
`apple
`} J band
`
`~ skirt
`~ N
`~'
`transition region
`~,
` N
`stopband
`a
`
`mO
`
`f~
`
`log frequency
`
`frequency (l inear)
`
`A
`B
`Figure 5.7. Filter characteristics versus frequency.
`
`terms for some undesirable properties of
`filters.
`
`a,~
`
`3~r
`
`z,~
`
`r
`
`~ a
`
`~, 1.0
`
`U ~
`
` ~.~
`
`~,
`v o.s
`
`o.a
`
`0.2
`
`v Q
`
`domain is the phase shift of the output
`signal relative to the input signal. In other
`words, we are interested in the complex
`response of the filter, which usually goes
`by the name of H(s), where s = jc.~, where
`H, s, and c~ all are complex. Phase is
`important because a signal entirely within
`the passband of a filter will emerge with
`its waveform distorted if the time delay of
`different frequencies in going through the
`filter is not constant. Constant time delay
`corresponds to a phase shift increasing
`linearly with frequency; hence the term
`linear-phase filter applied to a filter ideal
`in this respect. Figure 5.8 shows a typical
`graph of phase shift and amplitude for a
`low-pass filter that is definitely not a linear-
`phase filter. Graphs of phase shift versus
`frequency are best plotted on a linear-
`frequency axis.
`
`Time domain
`
`As with any ac circuit, filters can be
`described in terms of their time-domain
`properties: rise time, overshoot, ringing,
`and settling time. This is of particular
`importance where steps or pulses may be
`used. Figure 5.9 shows a typical low-
`pass-filter step response. Here, rise time
`is the time required to reach 90% of the
`final value, whereas settling time is the
`time required to get within some specified
`amount of the final value and stay there.
`Overshoot and ringing are self-explanatory
`
`0
`
`0
`
`2.0
`
`0.5
`1.0
`1.5
`normalized frequency
`(l inear scale)
`Figure 5.8. Phase and amplitude response
`for an 8-pole Chebyshev low-pass filter (2dB
`passband ripple).
`
`5.05 Filter types
`
`Suppose you want a Toes-pass filter with
`flat passband and sharp transition to the
`stopband. The ultimate rate of falloff,
`well into the stopband, will always be
`6ndB/octave, where n is the number of
`"poles." You need one capacitor (or
`inductor) for each pole, so the required
`ultimate rate of falloff of filter response
`determines, roughly, the complexity of the
`filter.
`Now, assume that you have decided
`to use a 6-pole low-pass filter. You are
`guaranteed an ultimate rolloff of 36dB/
`octave at high frequencies. It turns out
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTERS
`5.05 Filter types
`
`269
`
`Butterworth and Chebyshev filters
`
`The Butterworth filter produces the flattest
`passband response, at the expense of steep-
`ness in the transition region from passband
`to stopband. As you will see later, it also
`has poor phase characteristics. The ampli-
`tude response is given by
`
`1
`
`Vout
`[1 + (,f lf~)2~~ 2
`V n
`where n is the order of the filter (number
`of poles). Increasing the number of poles
`flattens the passband response and steep-
`ensthe stopband falloff, as shown in Figure
`5.10.
`
`~.o
`
`o.i
`
`E
`
`am
`
`.~ 0.01
`
`o.00i
`
`0.1
`
`1.0
`normalized frequency
`
`10
`
`Figure 5.10. Normalized low-pass Butterworth-
`filter response curves. Note the improved
`attenuation characteristics for the higher-order
`filters.
`
`The Butterworth filter trades off every-
`thing else for maximum flatness of re-
`sponse. It starts out extremely flat at zero
`frequency and bends over near the cut-
`of~' frequency f~ (f~ is usually the —3dB
`point).
`In most applications, all that really mat-
`ters is that the wiggles in the passband re-
`sponse be kept less than some amount, say
`1 dB. The Chebyshev filter responds to this
`reality by allowing some ripples through-
`out the passband, with greatly improved
`
`~ s~i
`overshoot
`
`settle to 5%
`
`VF
`90%
`
`>°
`
`ringing
`
`tr (5°/al
`
`time ~
`
`Figure 5.9
`
`that the filter design can now be optimized
`for maximum flatness of passband re-
`sponse, at the expense of a slow transition
`from passband to stopband. Alternatively,
`by allowing some ripple in the passband
`characteristic, the transition from pass-
`band to stopband can be steepened con-
`siderably. A third criterion that may be
`important is the ability of the filter to pass
`signals within the passband without distor-
`tion of their waveforms caused by phase
`shifts. You may also care about rise time,
`overshoot, and settling time.
`There are filter designs available to opti-
`mize each of these characteristics, or com-
`binations of them. In fact, rational filter
`selection will not be carried out as just de-
`scribed; rather, it normally begins with a
`set of requirements on passband flatness,
`attenuation at some frequency outside the
`passband, and whatever else matters. You
`will then choose the best design for the
`job, using the number of poles necessary
`to meet the requirements. In the next few
`sections we will introduce the three popu-
`lar favorites, the Butterworth filter (max-
`imally flat passband), the Chebyshev fil-
`ter (steepest transition from passband to
`stopband), and the Bessel filter (maximally
`flat time delay). Each of these filter re-
`sponses can be produced with a variety of
`dift'erent filter circuits, some of which we
`will discuss later. They are all available
`in low-pass, high-pass, and bandpass ver-
`sions.
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTERS AND OSCILLATORS
`270 Chapter 5
`
`sharpness of the knee. A Chebyshev filter
`is specified in terms of its number of poles
`and passband ripple. By aiiowing greater
`passband ripple, you get a sharper knee.
`The amplitude is given by
`
`Vout
`
`1
`
`V n
`
`~l + EZCinlf ~fc~~ 2
`where Cn is the Chebyshev polynomial
`of the first kind of degree n, and e is
`a constant that sets the passband ripple.
`Like the Butterworth, the Chebyshev has
`phase characteristics that are less than
`ideal.
`
`Figure 5.11 presents graphs comparing
`the responses of Chebyshev and Butter-
`worth 6-pole iow-pass filters. As you can
`see, they're both tremendous improve-
`ments over a 6-pole RC filter.
`Actually, the Butterworth, with its max-
`imally flat passband, is not as attractive
`as it might appear, since you are always
`accepting some variation in passband re-
`sponse anyway (with the Butterworth it
`is a gradual rolloff near f~, whereas with
`the Chebyshev it is a set of ripples spread
`throughout the passband). Furthermore,
`active filters constructed with components
`of finite tolerance will deviate from the
`predicted response, which means that a
`real Butterworth filter will exhibit some
`passband ripple anyway. The graph in Fig-
`ure 5.12 illustrates the ef~'ects of worst-case
`variations in resistor and capacitor values
`on filter response.
`
`i .o
`
`o. i
`
`0.01
`
`0a
`a~
`
`mv
`
`aEA
`
`0.001
`
`0.1
`
`A
`
`1.0
`
`0.9
`
`?~ 0.8
`
`0 0.7
`o.s
`°a 0.5
`
`m~,
`
` 0.4
`
`frequency (linear) —~
`
`+5
`
`0
`
`10
`
`~ _5
`
`c m
`
` —10
`
`—15
`
`—20
`
`—25
`
`1.0
`normalized frequency
`
`Figure 5.12. The effect of component tolerance
`on active filter performance.
`
`Viewed in this light, the Chebyshev is
`a very rational filter design. It is some-
`times called an equiripple filter: It man-
`ages to improve the situation in the transi-
`tion region by spreading equal-size ripples
`throughout the passband, the number of
`ripples increasing with the order of the fil-
`ter. Even with rather small ripples (as little
`as 0.1 dB) the Chebyshev filter o~'ers con-
`siderably improved sharpness of the knee
`
`-' 0.3
`.Q
`~ o.z
`o. i
`
`0
`
`1 2
`normalized frequency
`
`3
`
`B F
`
`igure 5.11. Comparison of some common
`6-pole low-pass filters. The same filters are
`plotted on both linear and logarithmic scales.
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTERS
`5.05 Filter types
`
`271
`
`mar
`Gvas.
`m ir
`
`Gs~oP
`
`dmUN mO C m
`
`~cutof~
`
`fs~op
`
`frequency (log scale)
`
`>
`
`Figure 5.13. Specifying filter fre-
`quency response parameters.
`
`as compared with the Butterworth. To
`make the improvement quantitative, sup-
`pose that you need a filter with flatness to
`0.1 dB within the passband and 20dB at-
`tenuation at a frequency 25% beyond the
`top of the passband. By actual calculation,
`that will require a 19-pole Butterworth, but
`only an 8-pole Chebyshev.
`The idea of accepting some passband
`ripple in exchange for improved steep-
`ness in the transition region, as in the equi-
`ripple Chebyshev filter, is carried to its log-
`ical limit in the so-called elliptic (or Cauer)
`filter by trading ripple in both passband
`and stopband for an even steeper tran-
`sition region than that of the Chebyshev
`filter. With computer-aided design tech-
`niques, the design of elliptic filters is as
`straightforward as for the classic Butter-
`worth and Chebyshev filters.
`Figure 5.13 shows how you specify fil-
`ter frequency response graphically. In this
`case (a low-pass filter) you indicate the al-
`lowable range of filter gain (i.e., the ripple)
`in the passband, the minimum frequen-
`cy at which the response leaves the pass-
`band, the maximum frequency at which
`the response enters the stopband, and
`the minimum attenuation in the stop-
`band.
`
`Besse) filter
`
`As we hinted earlier, the amplitude re-
`sponse of a filter does not tell the whole
`story. A filter characterized by a flat ampli-
`tude response may have large phase shifts.
`The result is that a signal in the passband
`will suffer distortion of its waveform. In
`situations where the shape of the wave-
`form is paramount, alinear-phase filter
`(or constant-time-delay filter) is desirable.
`A filter whose phase shift varies linearly
`with frequency is equivalent to a constant
`time delay for signals within the passband,
`i.e., the waveform is not distorted. The
`Besse) filter (also called the Thomson filter)
`had maximally flat time delay within its
`passband, in analogy with the Butterworth,
`which has maximally flat amplitude re-
`sponse. To see the kind of improvement in
`time-domain performance you get with the
`Besse) filter, look at Figure 5.14 fora com-
`parison of time delay versus normalized
`frequency for 6-pole Besse) and Butter-
`worth low-pass filters. The poor time-delay
`performance of the Butterworth gives rise
`to effects such as overshoot when driven
`with pulse signals. On the other hand, the
`price you pay for the Bessel's constancy
`of time delay is an amplitude response
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`Filter comparison
`In spite of the preceding comments about
`the Bessel filter's transient response, it still
`has vastly superior properties in the time
`domain, as compared with the Butterworth
`and Chebyshev. The Chebyshev, with its
`highly desirable amplitude-versus-frequen-
`cy characteristics, actually has the poor-
`est time-domain performance of the three.
`The Butterworth is in between in both fre-
`quency and time-domain properties. Table
`5.1 and Figure 5.15 give more information
`about time-domain performance for these
`three kinds of filters to complement the
`frequency-domain graphs presented earlier.
`They make it clear that the Bessel is a very
`desirable filter where performance in the
`time domain is important.
`
`0.6% overshoot
`
`i.o -
`
`a ~ 6-pole Chebyshev (0.5dB ripple)
`
`d ~
`
` 6-pole Butterworth
`0.5
`a ~~6-pole Bessel
`E
`
`0.5
`
`1.0
`
`2.0
`
`2.5
`
`3.0
`
`1.5
`time (S)
`Figure 5.15. Step-response comparison for 6-
`pole low-pass filters normalized for 3dB atten-
`uation at 1 Hz.
`
`ACTIVE FILTER CIRCUITS
`
`A lot of ingenuity has been used in invent-
`ing clever active circuits, each of which
`can be used to generate response functions
`such as the Butterworth, Chebyshev, etc.
`You might wonder why the world needs
`more than one active filter circuit. The
`reason is that various circuit realizations
`excel in one or another desirable property,
`so there is no all-around best circuit.
`Some of the features to look for in active
`filters are (a) small numbers of parts, both
`
`ACTIVE FILTERS AND OSCILLATORS
`272 Chapter 5
`
`with even less steepness than that of
`the Butterworth in the transition region
`between passbanci and stopband.
`
`0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
`
`frequency (radians/s or w)
`
`Figure 5.14. Comparison of time delays for
`6-pole Bessel and Butterworth low-pass filters.
`The excellent time-domain performance of
`the Bessel filter minimizes waveform distor-
`tion.
`
`There are numerous filter designs that
`attempt to improve on the Bessel's good
`time-domain performance by compromis-
`ing some of the constancy of time delay for
`improved rise time and amplitude-versus-
`frequency characteristics. The Gaussian
`filter has phase characteristics nearly as
`good as those of the Bessel, with improved
`step response. In another class there are in-
`teresting filters that allow uniform ripples
`in the passband time delay (in analogy with
`the Chebyshev's ripples in its amplitude re-
`sponse) and yield approximately constant
`time delays even for signals well into the
`stopband. Another approach to the prob-
`lem of getting filters with uniform time de-
`lays is to use all-pass filters, also known
`as delay equalizers. These have constant
`amplitude response with frequency, with
`a phase shift that can be tailored to in-
`dividual requirements. Thus, they can be
`used to improve the time-delay constancy
`of any filter, including Butterworth and
`Chebyshev types.
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTER CIRCUITS
`5.06 VCVS circuits
`
`273
`
`TABLE 5.1. TIME-DOMAIN PERFORMANCE COMPARISON FOR LOW-PASS FILTERSa
`Stopband attenuation
`
`Settling time
`
`Type
`
`f3de
`(Hz)
`
`Poles
`
`Bessel
`(-3.Od6 at
`f~ = 1.OHz)
`
`Butterworth
`(-3.OdB at
`f~ = 1.OHz)
`
`Chebyshev
`0.5d6 ripple
`(-0.5dB at
`f~ = 1.OHz)
`
`Chebyshev
`2.OdB ripple
`(-2.Od6 at
`f~ = 1.OHz)
`
`1.0
`1.0
`1.0
`1.0
`
`1.0
`1.0
`1.0
`1.0
`
`1.39
`1.09
`1.04
`1.02
`
`1.07
`1.02
`1.01
`1.01
`
`2
`4
`6
`8
`
`2
`4
`6
`8
`
`2
`4
`6
`8
`
`2
`4
`6
`8
`
`Step
`Over-
`rise time
`(0 to 90%) shoot
`(%)
`(s)
`
`to 1% to 0.1%
`(s)
`(s)
`
`f = 2f~
`(dB)
`
`f = 10f~
`(dB)
`
`0.4
`0.5
`0.6
`0.7
`
`0.4
`0.6
`0.9
`1.1
`
`0.4
`0.7
`1.1
`1.4
`
`0.4
`0.7
`1.1
`1.4
`
`0.4
`0.8
`0.6
`0.3
`
`4
`11
`14
`16
`
`11
`18
`21
`23
`
`21
`28
`32
`34
`
`0.6
`0.7
`0.7
`0.8
`
`0.8
`1.0
`1.3
`1.6
`
`1.1
`3.0
`5.9
`8.4
`
`1.6
`4.8
`8.2
`11.6
`
`1.1
`1.2
`1.2
`1.2
`
`1.7
`2.8
`3.9
`5.1
`
`1.6
`5.4
`10.4
`16.4
`
`2.7
`8.4
`16.3
`24.8
`
`10
`13
`14
`14
`
`12
`24
`36
`48
`
`8
`31
`54
`76
`
`15
`37
`60
`83
`
`36
`66
`92
`114
`
`40
`80
`120
`160
`
`37
`89
`141
`193
`
`44
`96
`148
`200
`
`~a~ a design procedure for these filters is presented in Section 5.07.
`
`active and passive, (b) ease of adjustability,
`(c) small spread of parts values, especially
`the capacitor values, (d) undemanding use
`of the op-amp, especially requirements on
`slew rate, bandwidth, and output imped-
`ance, (e) the ability to make high-Q fil-
`ters, and (fl sensitivity of filter characteris-
`tics to component values and op-amp gain
`(in particular, the gain-bandwidth product,
`fT). In many ways the last feature is one of
`the most important. A filter that requires
`parts of high precision is difficult to ad-
`just, and it will drift as the components
`age; in addition, there is the nuisance that
`it requires components of good initial ac-
`curacy. The VCVS circuit probably owes
`most of its popularity to its simplicity and
`its low parts count, but it suffers from high
`sensitivity to component variations. By
`comparison, recent interest in more com-
`plicated filter realizations is motivated by
`the benefits of insensitivity of filter prop-
`erties to small component variability.
`
`In this section we will present several
`circuits for low-pass, high-pass, and band-
`pass active filters. We will begin with the
`popular VCVS, or controlled-source type,
`then show the state-variable designs avail-
`able as integrated circuits from several
`manufacturers, and finally mention the
`twin-T sharp rejection filter and some in-
`teresting new directions in switched-
`capacitor realizations.
`
`5.06 VCVS circuits
`
`voltage-source
`The voltage-controlled
`(VCVS) filter, also known simply as a
`controlled-source filter, is a variation of the
`Sallen-and-Key circuit shown earlier. It re-
`places the unity-gain follower with a non-
`inverting amplifier of gain greater than 1.
`Figure 5.16 shows the circuits for low-pass,
`high-pass, and bandpass realizations. The
`resistors at the outputs of the op-amps
`create a noninverting voltage amplifier
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTERS AND OSCILLATORS
`274 Chapter 5
`
`c,
`
`(dc-coupled)
`
`low pass filter
`
`R ~
`
`c.
`
`R ~
`
`high-pass filter
`
`R z
`
`cascaded to generate higher-order filters.
`When that is done, the individual filter sec-
`tions are, in general, not identical. In fact,
`each section represents a quadratic poly-
`nomial factor of the nth-order polynomial
`describing the overall filter.
`There are design equations and tables in
`most standard filter handbooks for all the
`standard filter responses, usually including
`separate tables for each of a number of
`ripple amplitudes for Chebyshev filters.
`In the next section we will present an
`easy-to-use design table for VCVS filters
`of Butterworth, Bessel, and Chebyshev
`responses (O.SdB and 2dB passband ripple
`for Chebyshev filters) for use as low-pass
`or high-pass filters. Bandpass and band-
`reject filters can be easily made from
`combinations of these.
`
`5.07 VCVS filter design using our
`simplified table
`To use Table 5.2, begin by deciding which
`filter response you need. As we mentioned
`earlier, the Butterworth may be attractive
`if maximum flatness of passband is de-
`sired, the Chebyshev gives the fastest roll-
`off from passband to stopband (at the
`
`TABLE 5.2. VCVS LOW-PASS FILTERS
`
`y Butter-
`°' worth
`a° K
`
`Bessel
`
`Chebyshev
`(0.5dB)
`
`Chebyshev
`(2.OdB)
`
`f„
`
`K
`
`}~
`
`K
`
`f„
`
`K
`
`bandpass filter
`
`2 1.586 1.272 1.268 1.231 1.842 0.907 2.114
`
`Figure 5.16. VCVS active filter circuits.
`
`of voltage gain K, with the remaining
`Rs and Cs contributing the frequency re-
`sponse properties for the filter. These are
`2-pole filters, and they can be Butterworth,
`Bessel, etc., by suitable choice of compo-
`nent values, as we will show later. Any
`number of VCVS 2-pole sections may be
`
`4 1.152 1.432 1.084 0.597 1.582 0.471 1.924
`2.235 1.606 1.759 1.031 2.660 0.964 2.782
`
`6 1.068 1.607 1.040 0.396 1.537 0.316 1.891
`1.586 1.692 1.364 0.768 2.448 0.730 2.648
`2.483 1.908 2.023 1.011 2.846 0.983 2.904
`
`8 1.038 1.781 1.024 0.297 1.522 0.238 1.879
`1.337 1.835 1.213 0.599 2.379 0.572 2.605
`1.889 1.956 1.593 0.861 2.711 0.842 2.821
`2.610 2.192 2.184 1.006 2.913 0.990 2.946
`
`Petitioner Intel Corp., Ex. 1037
`IPR2023-00783
`
`

`

`ACTIVE FILTER CIRCUITS
`5.07 VCVS filter design using our simplified table
`
`275
`
`expense of some ripple in the passband),
`and the Bessel provides the best phase char-
`acteristics, i.e., constant signal delay in
`the passband, with correspondingly good
`step response. The frequency responses for
`all types are shown in the accompanying
`graphs (Fig. 5.17).
`To construct an n-pole filter (n is an
`even number), you will need to cascade
`n/2 VCVS sections. Only even-order
`filters are shown, since an odd-order filter
`requires as many op-amps as the next
`higher-order filter. Within each section,
`Rl = R2 = R, and Cl = C2 = C. As is
`usual in op-amp circuits, R will typically
`be chosen in the range lOk to 100k. (It is
`best to avoid small resistor values, because
`the rising open-loop output impedance of
`the op-amp at high frequencies adds to
`the resistor values and upsets calculations.)
`Then all you need to do is set the gain, K,
`of each stage according to the table entries.
`For an n-pole filter there are n/2 entries,
`one for each section.
`
`Butterworth low-pass filters
`If the filter is a Butterworth, all sections
`have the same values of R and C, given
`simply by RC = 1/2~rf~, where f~ is the
`desired —3dB frequency of the entire filter.
`To make a 6-pole low-pass Butterworth
`filter, for example, you cascade three of the
`low-pass sections shown previously, with
`g