## Circuit Elemants Resistors Capacitors Operational Amplifiers Operational Transconductance Amplifiers

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The information in this chapter enables the engineer to design a wide variety of practical active filters for operation in the audio-frequency range, and, if operational transconductance amplifiers (OTAs) are used, even into the megahertz range. T...

The information in this chapter enables the engineer to design a wide variety of practical active filters for operation in the audio-frequency range, and, if operational transconductance amplifiers (OTAs) are used, even into the megahertz range. The equations presented permit the user to complete the design and arrive at a fairly comprehensive evaluation of the performance to be expected from the filter, without requiring complicated mathematics. Out of the countless different filters proposed in the technical literature, only those few circuits that have been proven to be practical, state-of-the-art designs are discussed in this chapter. Given the limited space available in a reference volume such as this, enough information can be provided only for the design of filters of relatively simple specifications; if system requirements are very stringent, the reader should consult the many excellent books (references 1-6) or papers referred to in the text.

The technology of hybrid and monolithic integrated circuits has profoundly influenced the design and imple-mentation of audio-frequency filters. Integration has allowed the realization of filters that are small in size, inexpensive, and mass-producible. During the past ten years, active R-C networks, typically comprising resis-tors, capacitors, and operational amplifiers, have been the primary means of integrated audio-filter implemen-tation. Active R-C filters have eliminated the need for the bulky, expensive inductors required in passive implementations, and tuning is simplified and involves the adjustment of only resistors. Also, tuning can be automated in manufacture, using commercial laser trimming systems. In addition, active R-C filters have provided opportunities for standardization and modu-larity that significantly simplify design and fabrication.

Switched-capacitor (SC) networks have allowed audio-frequency active filters to be realized with the metal-oxide-semiconductor (MOS), large-scale-inte-gration (LSI) technologies associated with digital net-works. Switched-capacitor filters typically contain ca-pacitors, FET switches, and operational amplifiers. The switches are operated by clock signals that are digitally derived from a stable frequency source such as a crystal-controlled oscillator. The characteristics of the filter are then determined by capacitor ratios and the clock frequency, both inherently precise and stable parameters. Hence, SC filters rarely require trimming. The most important attribute of SC filters is that their implementation in silicon is compatible with digital-circuit integration. Hence, digital and analog circuitry can coexist on the same LSI chip.

For applications at much higher frequencies, such as in the read/write channels of magnetic disk recording systems, active R-C filters based on operational amplifi-ers (op amps) prove insufficient because of their band-width limitations. Also, discrete passive L-C filters are no alternative because they are too large, too costly, and not compatible with integrated circuit (IC) technology. In these cases, the designer uses operational transcon-ductance amplifiers (OTAs) as active devices. It is quite possible to design OTAs to have a much higher band-width than op amps (up to several hundred megahertz) so that active filters in the radio-frequency range become possible. This latter technology is used mainly for integrated filters because discrete OTAs are not readily available, and because the filter performance depends unavoidably on OTA parameters which must somehow be tuned. Techniques for handling these problems are becoming available on ICs (see reference 5).

Integrated filters, especially for high-frequency appli-cations, must be designed in fully differential, balanced form because the filters will normally share an IC chip with digital circuitry so that ground and power-supply lines (ac ground) are noisy, due, for example, to digital switching transients. Referring a signal to ac ground will, therefore, likely result in a severely restricted dynamic range (low signal-to-noise ratio). This problem is greatly reduced by referring two differential signal lines with equal and opposite signal voltages ±v5/2 to each other: v, = vs/2 — (—vs/2) = vs. Noise voltages on power-supply and ground lines appear as common-mode signals and are rejected. Since OTA-based filters are intended mainly for IC implementation and for high frequencies, the corresponding examples below will be shown in differential form. Conversions from single-ended to differential form are quite straightforward (see reference 5).

Although the design of high-frequency integrated OTA-C filters is, in principle, quite simple, in practice numerous constraints must be considered which are beyond this introductory discussion. The presentation in this book should make the reader aware of the possibili-ties, but details are left to references 5 and 7-10. To design active filters, whether active R-C (based on op amps or OTAs) or SC, which are used extensively in instrumentation and communication systems, one must first understand what an active filter is and how its performance requirements are specified. An electric filter is a network that transforms an input signal in some specified way into a desired output signal. Although many applications exist where filter requirements are set in terms of time-domain specifications, the majority of filters are designed to satisfy certain frequency-domain criteria. Thus, as shown in Fig. 1, a filter is a two-port network with input voltage V1 and output voltage V2; the circuit response is described by a transfer function H(s) defined by

H(s) = N(s)/D(s) = V2(s)1111(s) (Eq. 1)

where, in steady-state, s = jco is the frequency parameter, co = 271-f is the radian frequency (rad/s), f is the frequency in hertz (Hz). As indicated, H(s) is a ratio of two polynomials N(s)

and D(s). The roots of N(s) are the transmission zeros of the filter, i.e., points of infinite attenuation; the roots of D(s) are its poles. The transfer function is a complex quantity that may be expressed as

H(jco) = 11-1(i co)I exP [fri((0)] (Eq. 2)

where, 11/C/01 is the magnitude, 4(co) is the phase. Thus, to specify a transfer function completely, both magnitude and phase must be given at a sufficient number of frequency points. In many cases, the magni-tude response is the dominant specification with the phase response either loosely specified or unspecified. In this case, a minimum-phase filter is designed to meet the magnitude specification, and whatever phase the design provides is accepted. When both magnitude and phase are specified, one widely accepted design proce-dure is to design first a minimum-phase filter to meet the magnitude response, as previously mentioned, and then to design a tandem nonminimum-phase all-pass filter, which, when cascaded with the minimum-phase filter, meets the desired phase specification. This non-minimum-phase all-pass network is often referred to as a phase or delay equalizer. Filtering implies that certain frequency components of the input signal, those in the passband or passbands, are transmitted or passed to the output, whereas those in the stopband(s) are not transmitted. The most frequent-ly used method of identifying the location of passbands and stopbands on the frequency axis is by specifying, versus frequency, the magnitude characteristic via the loss curve in decibels (dB), defined as

a(w) = —20 log IH(jco)I = —20 log I V2/1/11 (Eq. 3) In the stopbands, where IV2 I << I Vi I IHUOI is small and the loss a is large, for example IH(ja))1< 0.01 or a > 40 dB. In the passbands, I V2I = I V1 I or even 1 V2 1 > 1 V1 1 , so that IH(jco)I = 1 (a = 0 dB) or IH(joo)I > 1 (a < 0 dB; i.e., the circuit provides gain, something an active filter can do, whereas a passive filter always provides a loss). If the phase response is of prime importance, then 4,(w) is specified directly in degrees or radians; alterna-tively, and perhaps more frequently, one prescribes the delay T(w) in seconds, defined as

= —(dIdco)[0(a))] (Eq. 4)

For best, distortion-free transmission, delay should be constant, T(w) = To; i.e., the phase should be linear, (kw) = —coTo, over the frequency range of interest. This is especially important when filtering pulse sig-nals, such as in read/write channels of magnetic disk drives. Some additional criteria of practical interest in active-filter design are sensitivity, dynamic range, noise, power dissipation, number and range of compo-nents, method of fabrication, and cost. All of these specifications place limitations and constraints on the acceptable design. In more cases than one would like, the specifications conflict so that engineering tradeoffs have to be made to resolve the conflict. In the following sections, the components used for active filters, some important design criteria, and sever-al state-of-the-art practical active filters will be dis-cussed in detail.

CIRCUIT ELEMENTS

Active filters are constructed from resistors, capaci-tors, and, usually, operational amplifiers (op amps) for low-frequency applications or operational transcon-ductance amplifiers (OTAs) for applications at high frequencies. A few comments will be made about these components, especially about the op amp because of its serious effect on filter performance.

Resistors*

Resistors used in active-filter design are carbon composition, metal or carbon film, thin or thick film, wirewound, and diffused. The selection depends on cost, on the technology used to implement the filter, and on filter requirements. Carbon composition resis-tors are the least expensive, but they have large toleranc-es and temperature coefficients. Further, tracking is not very good so that composition resistors should be used only for uncritical applications. Metal-film and wire-wound resistors are better than composition types in all respects, although more expensive, and are the most frequently used resistors in active-filter design today. Wirewound resistors have somewhat larger parasitics (L and C) than metal-film resistors and should not be used for high-frequency applications.

Capacitors*

Of the numerous different types of capacitors avail-able, those commonly used in active filters are ceramic disc, Mylar, polystyrene, Teflon, and thin-film capaci-tors. Once again, the selection depends on factors such as cost, available range, tolerances, temperature coeffi

cients, and dissipation factor (loss). Ceramic and Mylar capacitors are the least expensive types and have the highest loss; they are used only for uncritical applica-tions. Teflon, thin-film, and especially polystyrene (or for small values, mica) capacitors are more expensive but have much lower dissipation factors and are there-fore better suited for critical filter designs. Of course, in integrated implementations, metal-oxide-semiconduc-tor (MOS) capacitors are being used. In setting filter parameters, apart from tolerances and temperature coefficient, the dissipation factor (DF) or quality factor (Q c) is of some importance. If loss is modeled by means of a resistor R, in parallel with capacitor C, Q, and DF are defined as

Q, = 1/DF = coCR, (Eq. 5)

where co is some critical frequency of interest, usually chosen at the passband edge. Also, it should be remembered that Q, is a strong function of temperature. Typical values of Q, range from less than 100 (ceramic) up to several thousand (polystyrene).

Operational Amplifiers Although, as was mentioned, IC filters preferably use OTAs, the active element used in the vast majority of all discrete active filters is the operational amplifier. The "op amp" is an integrated circuit with five or more terminals, three of which are used for handling the signal (Fig. 2): the inverting (V-) and noninverting (V+) input terminals, at which the input voltages are applied, and the output terminal (V,). The remaining terminals are for power supply and, in some models, for offset compensation and frequency compensation. The function of the op amp is described by

- V° = A(s)(V+ — V-)

(Eq. 6)

where A(s) is the open-loop gain. In the ideal op amp, A is assumed to be infinite (i.e., A —> co), and, furthermore, all input impedances are infinite and the output impedance is zero. Thus, the ideal op amp is an ideal voltage-controlled voltage source with infinite gain; the input currents into the inverting and noninverting input terminals are zero. Further, since A —+ 00 and V, must remain finite in practice, it follows that the input voltage V, = V+ — is zero; the operation is defined such that V, = 0 when V, = 0.

Fig. 2. Op-amp symbol.

To get a feel for their operation, op-amp circuits often are first analyzed or designed under the assumption of ideal amplifiers (A --> co); it should, however, be strongly emphasized at this point that, except for uncritical applications at very low frequencies (f < 1 kHz), the operation of an active filter will rarely be satisfactory in practice if its design is based on ideal op amps. The main reason is that the op-amp gain A(s) is a strong function of frequency. Specifically, for most op amps used in active-filter design, the gain decreases throughout the useful frequency range by 20 dB/ decade. This frequency response is required for stability reasons and is achieved by use of internal or external compensation capacitors. Thus, the most widely used and quite realistic op-amp model is

A(s) = Aocrl(s + = 0,1 (s + o) (Eq. 7)

where, s = jw (as defined previously), o- is the open-loop —3 dB frequency (usually :5- 27r • 10 Hz), Ao is the dc gain (usually > 100 dB), = Aoo- is the gain-bandwidth product (-= 27r • 1 MHz). In most practical applications, I sj >> a, so that, instead of Eq. 7, a commonly used model is

A(s) = co fl s (Eq. 8)

Analyzing an active filter with op amps represented by Eq. 7 or 8 rather than by a simple constant-gain model increases the degree of the describing network function (Eq. 1) by one for each amplifier used. Thus, the filter acquires parasitic poles and zeros in addition to suffering shifts of the nominal pole-zero locations. Stated differently, the two polynomials N(s) and D(s) in Eq. 1 change from their ideal form. It is for this reason that filters designed with the assumption of ideal amplifiers (co, —› 00) do not normally have a satisfactory frequency response but show a potentially rather large deviation from their nominal performance due to finite WI. Occasionally, a filter will behave unpredictably, possibly oscillate, in spite of a design based on the op-amp model of Eq. 7 or 8. The reason may often be found in the fact that, whereas the representation of Eq. 7 describes the op-amp gain very well, the phase shift is larger than indicated by Eq. 7. An adequate remedy is to multiply Eq. 7 by an "excess phase factor," exp(— joico2), where w2 is a normalizing frequency of value w2 = 3w,, and to use this augmented model in the analysis. Since for all practical purposes w2 is much larger than operating frequency co, it has been found convenient for keeping the algebra manageable to set exp(—s/w2) = 1 — s/w2; i.e., a very accurate op-amp

model for use in highly selective filters at fairly high frequencies is

A(s) = (wils)(1 — sic02) (Eq. 9)

In most applications, however, the use of Eq. 8 leads to entirely satisfactory results. Further op-amp characteristics of some concern to filter designers are slew rate and dc offset voltage. Slew rate SR, given in volts per microsecond, refers to the maximum•rate of change of a signal voltage that the amplifier can maintain at its output. Violating slew-rate limitations results in gross signal and/or transfer-function distortion and should be avoided. Thus, if vo(t) = Vo sin cot is the amplifier output voltage, one needs to observe V, < SR/w. For example, if SR = 0.7 V/pts (a typical value) and the signal frequency is 15 kHz, the signal amplitude must satisfy V, < 7.2 V. There are two main reasons* for offset voltage. One is the need for dc input bias currents into the input stage of bipolar op amps (there is no input bias current in MOS op amps.); the second is imbalances in the input stage. To be able to provide the input bias, dc paths must exist from the inverting and noninverting input terminals to ground. To minimize offset, the resistances seen from these two terminals back into the network ought to be equal; then the voltage drops caused by the two bias currents at the op-amp inputs are equal and the direct differential input voltage is zero, resulting in zero contribution to offset. In practice, things are not quite so simple because due to imbalances in the op-amp input stage the bias currents are not exactly equal, causing a finite differential direct input voltage V. This voltage is multiplied by the dc closed-loop gain (see below), resulting in an output offset voltage that, however, frequently can be reduced to zero by means of a potentiometer connected to the offset-adjust terminals of the op amp. All amplifier parameters, especially the important terms Ao and cot, are strong functions of bias voltages and temperature, and in general are not well determined or predictable from unit to unit. Therefore, one strives to minimize filter dependence on these parameters and always uses op amps in closed-loop feedback configura-tions (Fig. 3) where the gain dependence on device parameters is reduced. Straightforward analysis, re-membering that op-amp input currents are zero and that V- = —V2/A, yields

V2/V1 = —(Z2/Zi )/[1 + (1 + Z2/Zi)/A] (Eq. 10) for Fig. 3A and V2/Vi = +(1 + Z2/21)41 + (1 + Z2/Zi)/A] (Eq. 11)

* References 3 and 5.

(A) Inverting gain.

(B) Noninverting gain

Fig. 3. Closed-loop feedback configurations.

for Fig. 3B. In both circuits, R is chosen to equal 1/[Y1(0) + Y2(0)] in line with the above discussion about offset minimization so that both op-amp inputs see the same dc impedance back into the circuit. Note, though, that resistor R does not affect the signal gain because no signal current flows through it. Thus, for simplicity, in the remaining discussion in this chapter, R will be neglected; i.e., R = 0 is assumed. For ideal amplifiers, or, in practice, in the frequency range where I A(*)1 >> I 1 + Z2( jco)/Z (jco)1 , Eqs . 10 and 11 reduce to

V2/Vi = —Z2(s)/Zi(s) (Eq. 12)

V2/V1 = 1 + Z2(s)/Zi (s) (Eq. 13)

respectively. The closed-loop gain functions are then independent of amplifier parameters, as desired, and are determined only by presumably accurately adjust-able and stable external impedances. The quantities Z1 (s) and Z2(s) are finally chosen to yield the desired frequency dependence of the gain. For example, setting Z1 = R1 and Z2 = KR results, of course, in the well-known amplifiers of inverting gain —K (Fig. 3A) and noninverting gain 1 + K (Fig. 3B). More will be said later about these important building blocks.

Operational Transconductance Amplifiers An operational transconductance amplifier (OTA) is a voltage-to-current converter described by the conversion parameter gm, the transconductance. Its circuit symbol is shown in Fig. 4A, and its function is given by

to = gm(V+ - V-) (Eq. 14)

Naturally, at very high frequencies the gain gm of a transconductance decreases because of parasitic poles. Just as in op amps, this decrease can be modeled by

gm(s) = g°,0/(s + c•) (Eq. 15)

but in an OTA, the pole o is at far higher frequencies (several tens to a few hundred megahertz) so that the effect may be neglected for most applications. Equation 14 indicates that an OTA generates an output current I° proportional to the differential input voltage V+ - V-. Ideally, the input and output impedances and the bandwidth of an OTA are infinite; in practice, for well designed circuits, the input impedance is greater than 108 11, the output impedance is greater than 105 fl, and the bandwidth is several hundred megahertz or, depend-ing on technology, reaches the gigahertz range. From these few numbers it can be seen that, in contrast to op amps, OTAs can be treated as almost ideal components for most active-filter design tasks. Only at the highest frequencies must OTA nonidealities, in particular the parasitic pole o, be taken into account. To introduce the appropriate differential symbols for later use: Fig. 4B shows an OTA with differential output terminals, and Fig. 4C shows an OTA with multiple differential inputs (here two) and differential outputs realizing 4+ = Jo = gm f(Ii1+ — 1/1—) — (72+ — v2—)] (Eq. 16)

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