Application Report SBOA093A – October 2001
Handbook Of Operational Amplifier Active RC Networks Bruce Carter and L.P. Huelsman ABSTRACT While in the process of reviewing Texas Instruments applications notes, including those from the recently acquired BurrBrown – I uncovered a couple of treasures, this handbook on active RC networks and one on op amp applications. These old publications, from 1966 and 1963, respectively, are some of the finest works on op amp theory that I have ever seen. Nevertheless, they contain some material that is hopelessly outdated. This includes everything from the state of the art of amplifier technology, to the parts referenced in the document – even to the symbol used for the op amp itself:
These numbers in the circles referred to pin numbers of old op amps, which were potted modules instead of integrated circuits. Many references to these numbers were made in the text, and these have been changed, of course. In revising this document, I chose to take a minimal approach to the material out of respect for the original author  L.P. Huelsman, leaving as much of the original material in tact as possible while making the document relevant to present day designers. I did clean up grammatical and spelling mistakes in the original. I even elected to leave the original RC stick figure illustrations, which have minimal technical content – but added to the readability of the document.
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Contents CHAPTER 1........................................................................................................................................... 6 Introduction..................................................................................................................................... 6 CHAPTER 2......................................................................................................................................... 11 The InfiniteGain SingleFeedback Circuit .................................................................................... 11 The Operational Amplifier ............................................................................................................. 11 The Basic Single Feedback Circuit ............................................................................................... 12 The Voltage Transfer Function ..................................................................................................... 13 The Passive Networks.................................................................................................................. 16 Network Design ............................................................................................................................ 18 Conclusions.................................................................................................................................. 20 CHAPTER 3......................................................................................................................................... 21 The InfiniteGain MultipleFeedback Circuit .................................................................................. 21 The Basic Multiple Feedback Circuit............................................................................................. 21 The Voltage Transfer Function ..................................................................................................... 21 Network Design ............................................................................................................................ 23 Conclusions.................................................................................................................................. 26 CHAPTER 4......................................................................................................................................... 27 The Controlled Source Circuit....................................................................................................... 27 The VoltageControlled Voltage Source........................................................................................ 27 Network Design ............................................................................................................................ 28 Other Realizations with VoltageControlled Voltage Sources........................................................ 33 Conclusions.................................................................................................................................. 34 CHAPTER 5......................................................................................................................................... 36 The NIC In Active RC Circuits....................................................................................................... 36 The NIC (NegativeImmittance Converter).................................................................................... 36 A Realization for the INIC ............................................................................................................. 38 Stability of the INIC....................................................................................................................... 39 The Basic INIC Circuit .................................................................................................................. 40 Network Design ............................................................................................................................ 41 Conclusions.................................................................................................................................. 44 CHAPTER 6......................................................................................................................................... 45 Another Active Device: The Gyrator............................................................................................. 45 Definition of a Gyrator................................................................................................................... 45 Properties of the Gyrator .............................................................................................................. 45 A Gyrator Realization ................................................................................................................... 46 Circuit Realizations....................................................................................................................... 47 Conclusions.................................................................................................................................. 48 CHAPTER 7......................................................................................................................................... 49 A Summary................................................................................................................................... 49 SECTION II .......................................................................................................................................... 51 Circuits ......................................................................................................................................... 51 Introduction................................................................................................................................... 51 APPENDIX A ....................................................................................................................................... 80 References................................................................................................................................... 80 Chapter 1 ..................................................................................................................................... 80
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Chapter 2...................................................................................................................................... 80 Chapter 3...................................................................................................................................... 80 Chapter 4...................................................................................................................................... 81 Chapter 5...................................................................................................................................... 81 Chapter 6...................................................................................................................................... 81 APPENDIX B ....................................................................................................................................... 82 Describing Active Filters ............................................................................................................... 82 Describing the Filter...................................................................................................................... 82 Optimizing the Circuit.................................................................................................................... 82 Limiting Specifications .................................................................................................................. 82 Conclusion.................................................................................................................................... 83 APPENDIX C ....................................................................................................................................... 84 Figures Figure 11. Model for an Ideal Operational Amplifier ........................................................................ 9 Figure 12. Circuit Symbol for an Operational Amplifier .................................................................. 9 Figure 21. Symbolic Representation of the Operational Amplifier ............................................... 11 Figure 22. OpenLoop Transfer Characteristics of the Operational Amplifier ............................. 11 Figure 23. Open Loop Frequency Characteristic of a Typical Operational Amplifier................... 12 Figure 24. Basic Single Feedback Operational Amplifier Circuit.................................................. 12 Figure 25. The Port Variables for Network A..................................................................................13 Figure 26. The Port Variables for Network B..................................................................................13 Figure 27. The Port Variables for the Basic SingleFeedback Circuit........................................... 14 Figure 28. Dual Summing SingleFeedback Circuit ....................................................................... 15 Figure 29. BridgedT RC Network ................................................................................................... 16 Figure 210. TwinT RC Network....................................................................................................... 17 Figure 211. Low Pass Network A .................................................................................................... 18 Figure 212. High Pass Network A.................................................................................................... 19 Figure 213. Single Zero – Single Pole Network A .......................................................................... 19 Figure 31. MultipleFeedback (MFB) Operational Amplifier Circuit .............................................. 21 Figure 32. Basic MultipleFeedback Circuit.................................................................................... 22 Figure 33. Low Pass MFB Filter ...................................................................................................... 23 Figure 34. High Pass MFB Active Filter .......................................................................................... 24 Figure 35. Band Pass MFB Active Filter ......................................................................................... 25 Figure 41. VCVS Circuit Model ........................................................................................................ 27 Figure 42. VCVS Circuit Symbol ..................................................................................................... 27 Figure 43. NonInverting Operational  Amplifier VCVS ................................................................ 28 Figure 44. VCVS Low Pass Active Filter ......................................................................................... 29 Figure 45. Operational Amplifier VCVS Low Pass Active Filter .................................................... 30 Figure 46. VCVS High Pass Active Filter ........................................................................................ 31 Figure 47. Operational Amplifier VCVS High Pass Active Filter ................................................... 31 Figure 48. VCVS Band Pass Active Filter ....................................................................................... 32 Figure 49. Inverting Operational Amplifier VCVS........................................................................... 33 Figure 410. Inverting VCVS Low Pass Filter...................................................................................33 Figure 411. Inverting VCVS High Pass Filter.................................................................................. 34 Figure 51. TwoPort Network With Load......................................................................................... 36 Figure 52. The Port Variables for a TwoPort Network .................................................................. 36 Figure 53. INIC Input Circuit ............................................................................................................ 37 Figure 54. INIC Output Circuit ......................................................................................................... 37
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Figure 55. Operational Amplifier Realization of the INIC............................................................... 38 Figure 56. Circuit Model of the Operational Amplifier Realization INIC ....................................... 39 Figure 57. Basic Voltage Transfer Circuit Using the INIC ............................................................. 40 Figure 58. INIC Low Pass Active Filter ........................................................................................... 41 Figure 59. INIC High Pass Active Filter........................................................................................... 42 Figure 510. INIC Band Pass Active Filter........................................................................................ 43 Figure 61. Gyrator Symbol .............................................................................................................. 45 Figure 62. The Input Admittance of a Terminated TwoPort Network .......................................... 46 Figure 63. Gyrator Realization Using Two INIC’s........................................................................... 47 Figure 64. Gyrator Band Pass Active Filter .................................................................................... 47 CIRCUIT 1: Single Feedback Low Pass ........................................................................................... 52 CIRCUIT 2: Single Feedback High Pass .......................................................................................... 54 CIRCUIT 3: Single Feedback Band Pass ......................................................................................... 56 CIRCUIT 4: Multiple Feedback Low Pass ......................................................................................... 59 CIRCUIT 5: Multiple Feedback High Pass........................................................................................ 62 CIRCUIT 6: Multiple Feedback Band Pass....................................................................................... 64 CIRCUIT 7: Controlled Source Low Pass ........................................................................................ 66 CIRCUIT 8: Controlled Source High Pass........................................................................................ 68 CIRCUIT 9: Controlled Source Band Pass....................................................................................... 71 CIRCUIT 10: INIC Low Pass .............................................................................................................. 73 CIRCUIT 11: INIC High Pass ............................................................................................................. 75 CIRCUIT 12: INIC Band Pass ............................................................................................................ 78
Table 1.
4
Tables Summary of the Advantages and Disadvantages of the Various Realization Techniques .................................................................................................................. 49
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ACTIVE RC NETWORK THEORY The subject of active RC networks is one that has attracted considerable attention in the past few years from network theorists. Many new active devices and many new techniques have been developed. Some of these techniques have been of great theoretical interest, but of little practical value. Others, however, offer great practicality and have great potential for everyday application. In writing this hand book, the goal has been to screen the large volume of literature on this subject, and present only those techniques that are of definite practical value to the working engineer. All of the realization schemes described in Chapters 2 through 5 have been proven on the bench, and full details on their implementation are given in the “circuits” section of this handbook. In addition, each of these techniques is described in the text, where some of the pertinent theoretical background is given. The reader who is interested in a more detailed theoretical treatment will find that the references listed in Appendix A will give him an excellent introduction into the considerable literature on this subject.
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CHAPTER 1 Introduction
This is a handbook on active RC networks. The first question about this subject that one might ask is, "What is an active RC network?” The answer is simple. It is collection of resistors, capacitors, and an active element (or elements). Viewed in another sense; it is a circuit without inductors. Why leave out inductors? There are many reasons. First of all, the inductor is a relatively large and heavy element. This is especially true at frequencies in the audio range and below.
Second, inductors generally have more dissipation associated with them than capacitors of similar size. In other words, commercially available inductors are not nearly as “ideal" as commercially available capacitors. If you have tried to use network synthesis techniques you have probably discovered that the dissipation (or resistance) associated with inductors can cause considerable difficulty. For these reasons (and a few others such as nonlinearity, saturation, and cost) more and more interest is being shown in circuit design techniques which avoid the use of inductors, namely active RC networks.
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Can active RC networks do everything that passive RLC networks can do? Yes, and more! They can have natural frequencies any place in the left half at the complex frequency (or "s") plane. They can function as oscillators, in other words they can have natural frequencies on the jω axis. They can provide transformation ratios just like the coupled coils of a transformer do (however they can’t provide the isolation). They can even provide perfect coupling and thus realize "ideal" transformers, which actual coupled coils cannot do. They can gyrate microfarads of capacitance into hundreds of henries of inductance, etc. There won't be space in this handbook to cover all of the things that active RC networks can do. Instead, we'll try to show you in detail how to use them to do some of your more common filtering tasks. If you are interested in more specialized applications, some references are given in Appendix A.
Natural Frequencies for Passive RC Circuits How does the tremendous capability of active RC networks come about? Certainly not from the passive elements, the resistors and capacitors. Taken by themselves these elements can produce natural frequencies only on the negative real axis of the complex frequency plane, a relatively uninteresting region for most filtering applications. Active RC networks, on the other hand, can have natural frequencies anywhere on the complex frequency plane. Right half plane natural frequencies, of course, are not useful because they signify unstable network behavior, so we'll just consider the usable active RC natural frequencies as being in the left half plane or on the jω axis. Since it is the "active” element that gives active RC networks their potential, let's briefly consider such elements in more detail.
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Stable Natural Frequencies for Active RC Circuits There are several types of active elements that can be used in active RC networks. First, there is the ideal voltage amplifier of high gain. By “high" here we mean a gain in the order of at least 60db. By "ideal" we mean infinite input impedance and zero output impedance. The operational amplifier is an example of such an active element. Second, there is the ideal voltage amplifier of low gain. By "low” here we mean a gain in the order of 20 db or less. Such an element is sometimes referred to as a controlled source.
Third, there is the NIC (negativeimmittance converter, also sometimes referred to as a negativeimpedance converter). This is a twoport device (a device with two sets of terminal pairs) with the property that impedance connected across one set of terminals appears as negative impedance at the other set of terminals. Fourth, there is the gyrator, a device that converts capacitance to inductance and vice versa.
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An interesting point to be noted here is that any of the last three types of active elements listed above can also be realized very simply and accurately with operational amplifiers. Thus, we see that the operational amplifier can be considered as a basic building black for constructing every type of active RC network. Many more details about the active elements introduced above will be given in the sections that follow. The networks that use operational amplifiers to realize these active elements will also be discussed. First, however, let us say a few things about the operational amplifier. The modern differential input operational amplifier may be simply modeled as an ideal voltage amplifier of very low output impedance (we'll assume that it is zero), very high input impedance (we'll assume that it is infinite), and very high gain, with the property that the output voltage is proportional to the difference in the voltages applied to the two input terminals. An equivalent circuit for such a model is shown in Figure 11.
Figure 11. Model for an Ideal Operational Amplifier The circuit symbol that will be used in future discussions is shown with the same terminal designation in Figure 12:
+ Figure 12. Circuit Symbol for an Operational Amplifier
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As a result of the properties of the operational amplifier, when it is inserted in circuit configurations, the voltage between the input terminals  and + in Figures 11 and 12 is driven to zero. Due to the high input impedance and zero voltage, the current into both of these terminals may be considered as zero. These two characteristics comprise the "virtual ground" concept that is a basic tool for analyzing operational amplifier circuits. For more detailed information on the properties and characteristics of operational amplifiers, you should consult the "Handbook of Operational Amplifier Applications", SBOA092, which is available from Texas Instruments. In the remainder of this handbook, we shall discuss in detail how the various types of active elements introduced above may be used to produce the most common types of network characteristics, namely, the lowpass, the highpass, and the bandpass characteristics. We shall see that each of the active elements has advantages and disadvantages in the different circuit configurations. So, without more delay, let us start our investigation of active RC networks, a world without inductors.
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CHAPTER 2 The InfiniteGain SingleFeedback Circuit The first active element that we shall consider for realizing active RC networks is the operational amplifier. In this chapter we shall investigate its use directly as an operational amplifier, in other words we shall not first modify it so that its characteristics approach those of some other active device. It may be helpful at this point to review briefly some of the characteristics of the operational amplifier. Those readers who are familiar with operational amplifiers may skip the next section without loss of continuity.
The Operational Amplifier 
+ E1 
+ E2 
+
+ Eo 
Figure 21. Symbolic Representation of the Operational Amplifier In Figure 21 we have shown a symbolic representation for an operational amplifier that defines the input voltages E1 and E2 and the output voltage Eo. In terms of these voltages we may plot a typical openloop DC transfer characteristics as shown in Figure 22.
Figure 22. OpenLoop Transfer Characteristics of the Operational Amplifier From this figure we see that the ““ terminal may be referred to as the "inverting" input terminal, while the “+” terminal may be referred to as the "noninverting" input terminal. In a typical operational amplifier the magnitude of Eo is near the power supply rails at saturation. The openloop DC gain of the amplifier shown in Figure 23 is 100000, so we see that the magnitude of Es, the differential input voltage that produces saturation, is only 100 µV. Therefore, openloop operation of an operational amplifier is not practical. The openloop frequency characteristic of a typical operational amplifier is shown in Figure 23.
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Figure 23. Open Loop Frequency Characteristic of a Typical Operational Amplifier The slope of the rolloff is 20 dB/decade. A compensation network determines the location of the break point. In most voltage feedback operational amplifiers, this network is integral with the operational amplifier circuitry; in uncompensated amplifiers, the designer must supply it external to the amplifier packaging. Stability considerations determine the proper choice of compensation network for a given circuit configuration; however, most operational amplifiers are compensated so as to provide adequate performance for the majority of circuit applications. For additional information on stability, compensation, or other general properties of operational amplifiers, the reader is referred to the "Handbook of Operational Amplifier Applications", SBOA092, published by Texas Instruments.
The Basic Single Feedback Circuit The basic circuit that will be considered in this chapter consists of two passive networks, which we will refer to as network A and network B, and an operational amplifier. Network A is connected between the input to the circuit and the input terminal of the operational amplifier Network B is used as a feedback network from the output to the input of the operational amplifier. The circuit is shown in Figure 24. Network B
+
Network A
+
+
E1
E2


Figure 24. Basic Single Feedback Operational Amplifier Circuit It should be noted that the operational amplifier is used in an inverting configuration, i.e., with its noninverting input terminal (+) grounded. We shall call this circuit an infinitegain singlefeedback circuit since the operational amplifier that is the active element normally has very high gain, and since the feedback around it is made to a single point.
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To characterize the properties of the two passive networks, we shall use their y parameters. For network A, we may define voltage and current variables as shown in Figure 25. I2a
I1a +
Yija
E1a
+ E2a 

Figure 25. The Port Variables for Network A The relations between these variables and the y parameters of the network are:
I1a = y 11aE1a + y12aE 2a I 2a = y 12aE1a + y 22aE 2a
(1)
Similarly, for network B and the variables shown in Figure 26: I2b
I1b + E1b 
Yijb
+ E2b 
Figure 26. The Port Variables for Network B
I1b = y11bE1b + y12bE 2b I2b = y12bE1b + y 22bE 2b
( 2)
All of the voltage and current variables and the y parameters defined in equations (1) and (2) are functions of "s", the complex frequency variable.
The Voltage Transfer Function The basic network configuration for the infinitegain singlefeedback circuit has been redrawn in Figure 27 to indicate the variables of the two passive networks explicitly.
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I1b +
Yijb
E1b 
+ E2b 
I2a
I1a Yija
+

+
+
E1
E1a
E2a
E2




+
+
Figure 27. The Port Variables for the Basic SingleFeedback Circuit In Chapter 1, it was pointed out that due to the "virtual ground," the voltage between the inverting and noninverting terminals of the operational amplifier may be considered to be zero. Thus, the voltage E2a shown in Figure 27 is zero. From the second equation of (1), we see that under this condition I2a = y12aE1a . In addition, since E1a and E1 are equal, we may write:
I2a = y12aE1
(3)
Similarly, for network B, E1b is zero, and E2b=E2. Thus we see that:
I1b = y12bE 2
( 4)
The virtual ground concept also tells us that the current into terminal 1 of the operational amplifier is negligibly small. Thus we see that I2a = I1b. We may now combine equations (3) and (4) to obtain:
E 2 − y12a = E1 y12b
( 5)
This is the opencircuit voltage transfer function for the infinitegain singlefeedback active circuit configuration.
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Let us examine the voltage transfer function given in equation (5) more closely. If networks A and B are passive RC networks, their natural frequencies will be on the negative real axis of the complex frequency plane. Let us assume that both of the passive networks have the some natural frequencies; then the denominators of the functions y12a and y12b will cancel and the locations of these natural frequencies will not affect the voltage transfer function of the overall network. The poles of the voltage transfer function of the active network configuration will then be determined solely by the zeros of the transfer admittance y12b. Since a passive RC network can have the zeros of its transfer admittance anywhere on the complex frequency plane, this says that we can realize complex conjugate poles in our voltage transfer function. Such poles will, of course, be restricted to the left half of the complex frequency plane for reasons of stability. Similarly, the zeros of the voltage transfer function given in equation (5) will be determined by the zeros of y12a, and therefore we can realize any desired real or complex conjugate zeros in our voltage transfer function. Thus we see that an infinitegain singlefeedback active RC network configuration can be used to realize almost any desired polezero configuration. One other property of this circuit should be noted. Suppose that another network with transfer admittance y12c is also connected to the input terminal of the operational amplifier. The connection is shown in Figure 28, where the additional network is labeled as network C.
+
Network C
Network B
E1c 
+
Network A

E1a
+
+ E2


Figure 28. Dual Summing SingleFeedback Circuit The input voltages to networks A and C are E1a and E1c respectively. An analysis similar to the one made in the preceding paragraph shows that the output voltage E2 for this circuit is given by the relation:
éy ù y E 2 = − ê 12a E1a + 12c E1c y12b ë y12b
( 6)
Thus we see that the infinitegain singlefeedback circuit configuration can also be used for summing signals from separate sources. This can be done without any interaction occurring between the sources.
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The Passive Networks In general, most filter designs require the use of complex conjugate poles. To produce these by the active RC technique described in this chapter, we thus require passive networks that have transfer admittances with complex conjugate zeros. There are several such network configurations, of which the two most common ones are the bridgedT network and the twinT network. It is beyond the scope of this handbook to analyze such networks in detail. For completeness, however, we will present a simple design procedure for each type of network in this section. C1
G1
G2
C2
Figure 29. BridgedT RC Network An example of a bridgedT network is shown in Figure 29. The units for the elements of this network are farads for the capacitors and mhos (G=1/R) for the resistors. For a transfer admittance normalized to one radian/second, and of the form:
s 2 + αs + 1 − y12 = s+α
(7 )
The elements will have the following values:
C1 = 0
(8)
G1 = 2.5 − α G2 =
1
æ 1 ö çç α − G1 è C2 = G1G2 Such a network is not useful for producing zeros that lie close to the jω axis for small values of the constant α in the numerator of equation (7). A useful range of the constant a for this circuit is: ½<α<2
(9)
It should be noted that the usual frequency and impedance normalizations must be applied to this circuit to determine the actual element values. An example of a twinT network is shown in Figure 210.
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C1
C2
G1
G2
G3
C3
Figure 210. TwinT RC Network If the network elements are chosen such that:
A1 = C1 = G1 = (5 − α )
1+ α 2+α
A 2 = C2 = G2 =
A1 (A1 − 1)
A 3 = C 3 = G3 =
A 1A 2 (1 + α )
(10)
the transfer admittance of this network will be:
− y12 =
(s + 1)(s2 + αs + 1) (s + σ1 )(s + σ2 )
(11)
Note that in equations (10), the numerical values of the capacitors in farads and the resistors in mhos are equal. For convenience, these values have also been referred to as A1, A2, and A3. The constants σ1 and σ2 in equation (11) can be found by the relations:
σ1 =
A1 + A 2 A3
σ2 =
1 σ1
(12)
This network may be used for producing zeros that are as close to the jω, axis as desired, i.e., for as small a value of the constant α as may be required. Extremely small values of α, however, may lead to oscillations in the overall active circuit. It should be noted that the numerator of the transfer admittance of this network as given in equation (11) is of third degree with a negative real zero at –1. The poles of the transfer admittance, however, are located symmetrically about 1. This may be seen from the second equation of (12). It is easily shown that as the constant α approaches zero, σ1 and σ2 both approach unity. Thus, when α equals zero, cancellation occurs between one of the poles and the negative real zero in the numerator of the transfer admittance. Similarly, for small values of α, the numerator zero very nearly cancels one of the poles, and the transfer admittance may be assumed to have the form:
− y12 =
s 2 + αs + 1 s +1
(13)
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without significant loss of accuracy.
Network Design Now let us consider the application of the infinitegain singlefeedback active RC circuit to the realization of three common filtering applications — the low pass, high pass, and band pass cases. For the low pass network, the frequency normalized voltage transfer function is of the form:
E2 −H = 2 E1 s + αs + 1
(14)
where H is a positive real constant giving the magnitude of the pass band gain. A common choice for the constant α is 2 . This gives a maximally flat (sometimes called a Butterworth) frequency response characteristic. A bridgedT network of the type shown in Figure 29 may be used to produce the complex conjugate poles. The element values can be found from equations (8). The bridgedT network is used as network B of Figure 24. The transfer admittance for network A must have no finite zeros, but must have a single negative real pole. A network satisfying these requirements is shown in Figure 211. G1
G2
C
Figure 211. Low Pass Network A This network has the transfer admittance:
− y12 =
G1G2 C s + (G1 + G2 ) C
(15)
Since this network must have the same pole location as network B, the factor (G1 + G2)/C must be set equal to the constant α. The complete network, together with summary of the design procedure, and some sample element values is given in the circuit section of this handbook as Circuit No. 1. For the high pass network, the frequency normalized voltage transfer function will be of the form
− Hs2 E2 = 2 E1 s + αs + 1
(16)
where H is a positive real constant giving the magnitude of the gain in the pass band. The same bridgedT network which was used for the low pass case may be used for network B for this case, since the poles of the transfer function will normally have similar locations The transfer admittance for network A must now have two zeros at the origin and a single pole on the negative real axis. A network satisfying these requirements is shown in Figure 212.
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C1
C2
G
Figure 212. High Pass Network A It has the transfer admittance
− y12 =
s2C1C2 (C1 + C2 ) s + G (C1 + C2 )
(17)
The element values must be chosen so that the factor G/(C1 + C2) equals the constant α. The complete network, a design procedure, and a set of typical element values are given in the circuit section as Circuit No. 2. For a band pass network, the frequency normalized voltage transfer function will be of the form
E2 − Hs = 2 E1 s + αs + 1
(18)
The magnitude of the gain in the pass band for this function is H/α, where H is a positive real constant. Most band pass filter applications require that the constant α be small, i.e., that the poles be close to the jω axis. For this case the bridgedT network is not satisfactory, and it will usually be necessary to use a twinT network of the type shown in Figure 210 as network B. If we assume that the transfer admittance of the twinT network is of the form given in equation (13), then the transfer admittance of network A must have a single zero at the origin and a pole at 1. A network satisfying these requirements is shown in Figure 213. G
C
Figure 213. Single Zero – Single Pole Network A It has the transfer admittance:
− y12 =
sG s+G C
(19)
The factor G/C must be set equal to unity. The complete network, a design procedure, and a set of typical element values are given in the circuit section of this handbook as Circuit No. 3.
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It should be noted that, in any realization produced by the method outlined in this chapter, the gain constant of the complete network realization is easily adjusted by changing the impedance normalization of either of the component passive networks, This is easily seen from equation (5). For example, to raise the overall network gain, one may either lower the impedance level of network A (and thus raise the magnitude of y12a) or raise the impedance level of network B (and thus lower the magnitude of y12b). It should also be noted that all circuits described in this chapter produce a signal inversion in addition to the frequency dependent properties that have been noted.
Conclusions At this point we may make same conclusions regarding some of the properties of the infinitegain singlefeedback active RC circuit configuration described in this chapter. These conclusions will assist us in determining the relative merits of this configuration as compared to the other configurations that will be described in the chapters that follow. One of the major disadvantages of this active RC circuit configuration is the large number of passive elements that it requires. For example, in the band pass network described as Circuit No. 3, we see that eight elements are needed. Another difficulty is brought about by the fact that bridgedT and twinT networks must be used to produce the complex conjugate poles. This means that any adjustment or trimming of the pole locations will be difficult since the passive elements interact to a high degree in such networks. On the positive side is the fact that this configuration has its pole locations determined completely by the passive networks. Thus the pole locations will remain relatively stable and independent of changes in the active element. This is a considerable advantage when it is desired to design highQ networks, where the poles are located close to the jω axis, since even small pole displacements may produce instability in this case. Another advantage of this configuration is that the output impedance of the network is equal to the output impedance of the operational amplifier, which with high loop gain is very low. Thus, this circuit may be used to drive other networks, without the need for an isolating stage, and without appreciable change in the circuit characteristics due to loading. Yet another desirable feature is the capability of summing signals at the input.
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CHAPTER 3 The InfiniteGain MultipleFeedback Circuit In the preceding chapter an active RC network configuration described as an infinitegain singlefeedback circuit was presented. An operational amplifier was used as the active element, and a single feedback path was provided around it. This chapter another active RC network configuration will be presented. An operational amplifier will again be used as the active element; however, more than one feedback path will be provided around it. The advantages and disadvantages of the two approaches will be compared.
The Basic Multiple Feedback Circuit The basic circuit that will be described in this chapter consists of a number of twoterminal passive elements, interconnected so as to form feedback paths around an operational amplifier. The operational amplifier is used in an inverting configuration, with its noninverting input terminal grounded. The general circuit configuration is shown in Figure 31.

+
+ E1
+ E2

Legend:
= Two Terminal Passive Element
Figure 31. MultipleFeedback (MFB) Operational Amplifier Circuit We shall call this circuit an infinitegain multiplefeedback circuit. In applying this circuit to the realization of transfer functions, it is practical to restrict each of the passive twoterminal elements to a single resistor or a single capacitor. In addition, if we limit ourselves to the realization of a voltage transfer function with a single pair of complex conjugate poles, and with zeros located only at the origin of the complex frequency plane or at infinity, then a maximum of five elements is necessary. The low pass, high pass, and band pass cases are included in this class of transfer functions. The extension of the method to other cases will be clear from the discussion that follows.
The Voltage Transfer Function The basic circuit that may be used to realize voltage transfer functions with a single pair of complex conjugate poles and with zeros restricted to the origin or infinity is shown in Figure 32.
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I4
I5
Y4
I1
I3
Y1
Y3
Y5

+
I2
E1
+
+ Eo 
E2
Y2

+

Figure 32. Basic MultipleFeedback Circuit Each of the elements Yj shown in this figure represents a single resistor or a single capacitor. Reference currents Ij and an interior voltage Eo have been defined in the figure to aid in the analysis. From the figure we may write:
E1 =
1 I1 + Eo Y1
(1)
It is also obvious from Kirchoff’s law that: I1 = I2+ I3+ I4
(2)
The virtual ground imposed by the operational amplifier requires that the voltage across both Y2 and Y3 equal Eo. Similarly, the voltage across Y4 is the difference between Eo and E2. Thus we may write expressions for the branch currents as:
I2 = Y2Eo
(3)
I3 = Y3Eo
I4 = Y4 (Eo − E 2 ) The virtual ground also requires that I3 = l5, thus we may write: I3 = Y3Eo = Y5E2 = I5
(4)
If we substitute the relations of equations (2), (3), and (4) into equation (1), we obtain:
E2 − Y1Y3 = E1 Y5 (Y1 + Y2 + Y3 + Y4 ) + Y3 Y4
( 5)
This is the opencircuit voltage transfer function for the infinitegain multiplefeedback circuit shown in Figure 32. The elements Yj of this network may readily be chosen so as to obtain low pass, high pass, and band pass voltage transfer functions. This will be shown in the next section.
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Network Design Let us first consider the use of the infinitegain multiplefeedback configuration to realize a low pass network. It is desired to obtain a frequency normalized voltage transfer function of the form:
E2 −H = 2 E1 s + αs + 1
( 6)
where H is a positive real constant which specifies the gain in the passband, i.e., the dc gain. If we compare the above equation with equation (5), we see that in order to have the numerator not be a function of “s”, both of the elements Y1 and Y3 must be resistors. Similarly, in order to generate the s2 term in the denominator, Y5 must be a capacitor, as must either Y2 or Y4. Y4, however, must be a resistor; otherwise, it will not be possible to realize the constant term in the denominator (this must come from the product Y3Y4). Thus, we must make the following choices for the elements of the circuit shown in Figure 32:
Y1 = G1
(7)
Y2 = sC2 Y3 = G3 Y4 = G4 Y5 = sC5 The circuit with these elements is shown in Figure 33.
G4
C5
G1 
+ E1
G3
+
C2
+ E2


Figure 33. Low Pass MFB Filter The voltage transfer function for this circuit is:
− G1G3 E2 = 2 E1 s C2C5 + sC5 (G1 + G3 + G4 ) + G3G4
(8)
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It should be noted that this circuit produces a signal inversion, as will be true for all the circuits realized by this technique. The specific solutions for the element values in terms of the constants α and H may be found by equating corresponding coefficients in equations (6) and (8). Such a process leads to a simultaneous set of equations that are, unfortunately, nonlinear. The nature of the set of equations is such, however, that constraints may be applied to develop a set of solutions. Such a set of solutions, together with other design information, is given in the circuit section of this handbook as Circuit No. 4. It should be noted that although the solutions given have been found to give good experimental results, they are not unique; i.e., other sets of solutions also exist. The high pass network can be considered in a manner similar to the low pass network. The frequency normalized high pass voltage transfer function will be of the form:
− Hs2 E2 = 2 E1 s + αs + 1
(9 )
where H is a positive real constant which specifies the gain in the pass band, i.e. the high frequency gain. The network elements shown in Figure 32 must be chosen as follows:
Y1 = sC1
(10)
Y2 = G2 Y3 = sC3 Y4 = sC4 Y5 = G5 The resulting voltage transfer function may be expressed in terms of the network elements as:
− s2C1C3 E2 = 2 E1 s C3C4 + sG5 (C1 + C3 + C4 ) + G2G5
(11)
The circuit configuration is shown in Figure 34.
G5
C4 C1

+ C3 E1
+
G2

E2 
Figure 34. High Pass MFB Active Filter
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The same comments that were made with respect to solving for the element values in the low pass network also apply in this case. A set of design equations and a summary of other information on this circuit are given in the circuits section of this handbook as Circuit No.5. There are several configurations of five elements that can be used to realize a band pass network with a frequency normalized transfer function of the form:
− Hs E2 = 2 E1 s + αs + 1
(12)
where H is a positive real constant and H/α is the magnitude of gain in the pass band. One of the most practical configurations is the one defined by the following choice of elements:
Y1 = G1
(13)
Y2 = G2 Y3 = sC3 Y4 = sC4 Y5 = G5 For such a choice, the voltage transfer function may be written in terms of the elements as
− sG1C3 E2 = 2 E1 s C3C4 + sG5 (C3 + C4 ) + G5 (G1 + G2 )
(14)
The circuit configuration is shown in Figure 35.
G5
C4 G1

+ C3 E1
+
G2
+ E2


Figure 35. Band Pass MFB Active Filter A set of nonlinear equations must be solved for the element values. The solutions and the circuit design information are given in the circuits section of this handbook as Circuit No. 6.
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Conclusions In a manner similar to that of the last chapter, we may make some conclusions regarding the characteristics of the infinitegain multiplefeedback circuit configuration. One advantage we see is that the number of elements is greatly reduced from the number required for the infinitegain singlefeedback circuits of the preceding chapter. For example, for the low pass and high pass circuits, a maximum of five elements is required, compared with seven for the singlefeedback circuit, As may be seen from the design equations given in the circuits section of the handbook, there are also cases in which one of the five elements may be eliminated in the band pass case, four or five elements are required rather than eight, a considerable savings. The same advantage of low output impedance which was pointed out for the singlefeedback circuit holds true for the multiplefeedback circuit, since the output impedance of the circuit is just the closed loop output impedance of the operational amplifier. Thus this circuit may be used to drive other circuits without degradation of performance due to loading effects. The multiplefeedback circuit, however, has same disadvantages that the singlefeedback circuit does not. For example, it is not possible to obtain high Q band pass realizations with the multiplefeedback configuration without resorting to large spreads of element values. Another disadvantage is the fact that since feedback is made to two points, there is no one single point in the circuit which can be used to sum separate signals as could be done in the singlefeedback configuration. Also, if it is desired to realize transfer functions with zeros other than at the origin or infinity, the networks and the design procedure for the multiplefeedback case become considerably more complicated. Finally, this approach, in general, cannot be used to achieve as large a value of gain constants as may be obtained by the singlefeedback configuration. Articles have appeared in the literature discussing the application of the infinitegain multiplefeedback circuit configuration to the realization of more complicated transfer functions. For the reader who wishes to pursue this topic further, some references are given in Appendix A.
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Application Report SBOA093A – October 2001
CHAPTER 4 The Controlled Source Circuit In the preceding chapters, some general properties of active RC circuits were introduced, and two types of active RC circuit configurations were analyzed. Both of these configurations required an active element with a high value of gain. Thus, they were referred to as “infinitegain” realizations, and an operational amplifier was used to provide the gain. In this chapter a quite different circuit configuration will be presented. It requires on active element with a relatively low value of gain, which we shall refer to as a controlled source. In general, a controlled source is an active network element which has an output voltage or current which is a function of some single input voltage or current, but is unaffected by any of the other voltages or currents in the network. There are four types of controlled sources: the voltagecontrolled voltage source, the currentcontrolled voltage source, the voltagecontrolled current source, and the currentcontrolled current source. Certain physical devices have characteristics that make them act in a manner similar to some of these sources. For example, a transistor acts somewhat like a currentcontrolled current source. For those who remember the “fire bottle” (tube) days of electronics, a pentode acted very nearly like a voltagecontrolled current source.
The VoltageControlled Voltage Source In our discussion of the use of controlled sources in active RC circuits, we will limit ourselves to a single type of source, an ideal voltage amplifier with infinite input impedance, zero output impedance, and an output voltage which is equal to the input voltage multiplied by same positive or negative constant, This device we shall refer to as a VCVS (voltagecontrolled voltage source). A circuit model for it is shown in Figure 41.
Figure 41. VCVS Circuit Model From this figure we see that E2 = KE1, where the constant K is usually referred to as the “gain.” It may be either positive or negative. The circuit symbol that we shall use for the VCVS is shown in Figure 42:
Figure 42. VCVS Circuit Symbol Now let us see how we can realize a VCVS with an operational amplifier. For the values of K that are positive and greater than unity, the circuit shown in Figure 43 may be used.
27
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+

+ I2
E1
Rb I1
E2
Ra


Figure 43. NonInverting Operational  Amplifier VCVS Note that in this circuit, the noninverting terminal of the operational amplifier is used as the input terminal. Using the virtual ground concept introduced in Chapter 1, we see that the inverting and noninverting inputs must be at the same potential. Since the voltage at the inverting input is equal to RaI1 the voltage at the noninverting input must be the same; therefore, E1 = RaI1. No current flows into the operational amplifier terminals; therefore, the currents I1 and I2 are equal. Thus E2 = (Ra + Rb)I1. Combining these relations we obtain:
E 2 Ra + Rb = E1 Ra
(1)
Since the input impedance at terminal 2 (as well as that at terminal 1) of the operational amplifier is very large, the circuit shown in Figure 43 may be considered as an ideal VCVS with a value of K equal to (Ra + Rb)/Ra. It should be noted that the idealness of this VCVS realization is relatively independent of the impedance levels that are chosen for Ra and Rb. For generalpurpose applications, values of Ra in the range of 100 kΩ will usually prove satisfactory.
Network Design The controlled source described in the last section may be used, in connection with passive RC networks, to obtain various network functions. The first such function that we shall consider is the transfer function for the low pass network. The frequency normalized voltage transfer function is:
E2 H = 2 E1 s + αs + 1
( 2)
where H is a positive real constant giving the value of the gain in the pass band. A network configuration that will produce this function is shown in Figure 44.
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SBOA093A C1
G1
G2 K
+ E1
+ E2
C2


Figure 44. VCVS Low Pass Active Filter In terms of the elements shown in this figure, we may write the voltage transfer function as:
E2 KG1G2 = 2 E1 s C1C2 + s(C2G1 + C2G2 + C1G2 − KC1G2 ) + G1G2
(3)
From the preceding equations we see that no signal inversion is introduced by this network configuration. The transfer function given in equation (3) points out some interesting properties of this circuit realization. First of all, the low frequency gain, the gain in the pass band, may be found by evaluating equation (3) in the limit as “s” approaches zero. It is readily seen that this gain is simply equal to K. In other words, the overall gain specified for the circuit directly determines the gain of the VCVS. Second, if it is desired to change the value of the cutoff frequency for this circuit, this may done by changing the values either of the resistors or the capacitors. Such changes will not affect the gain in the pass band. If, in addition, the changes are made in such away that the ratio of the two elements changed remains the same, i.e. if the same percentage change is made to each of them, then the relative shape of the magnitude and phase characteristics of the network will remain unchanged. Thus this filter has the property that its cutoff frequency can be readily shifted. The values of the network elements must be found by simultaneously equating the coefficients of equations (2) and (3) The resulting set of equations is nonlinear, but solutions for the element values in terms of the constants α=and H are easily found set of solutions, together with other design information for the circuit is given in the circuits section of the handbook as Circuit No. 7. The low pass circuit described above is sometimes modified to the configuration shown in Figure 45.
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SBOA093A C1
G1
G2 +
+

+
C2
E1
Rb
E2
Ra 

Figure 45. Operational Amplifier VCVS Low Pass Active Filter If this figure and Figure 44 are compared, it is easily seen that the right terminal of the capacitor C1 has been moved from the output of the operational amplifier (which is also the output of the controlled source) to the junction of the two resistors Ra and Rb whose values determine the gain of the controlled source. If the magnitude of the impedance of the capacitor C1 is much larger than the values of the resistors Ra and Rb, so that the capacitor does not load the resistor network, then the voltage transfer function for the circuit shown in Figure 45 is:
E2 KG1G2 = 2 E1 s C1C2 + s(C2G1 + C2G2 ) + G1G2
( 4)
The above equation illustrates one advantage of this configuration; i.e. none of the coefficients in the denominator are a function of K, the gain of the controlled source. Thus K may be varied, changing the gain of the network, without changing the frequency characteristics of the network. Such a change of gain may, of course, be accomplished by varying either Ra or Rb in Figure 45. It should be noted that this circuit also has a disadvantage. Since there is no subtraction of terms in the coefficient of the first degree term in the denominator of equation (4), it is not possible to realize transfer functions of the type given in equation (2) in which the constant a has very small values. The frequency normalized voltage transfer function for a high pass network is:
E2 Hs2 = 2 E1 s + αs + 1
(5)
A realization for such a transfer function using a VCVS as the active element is shown in Figure 46.
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SBOA093A G1 C1
C2 K
+ E1
+
G2
E2


Figure 46. VCVS High Pass Active Filter The voltage transfer function is:
E2 Ks 2C1C2 = 2 E1 s C1C2 + s(C1G2 + C2G2 + C2G1 − KC 2G1 ) + G1G2
( 6)
The same comments that were made with respect to the low pass network also apply here; namely, H is positive, and the pass band gain (in this case the high frequency gain) is equal to the gain K of the VCVS and is not a function of the passive elements of the network. Similarly, the cutoff frequency can be changed by changing the values of the resistors or the capacitors, and, if equal percentage changes of the elements are made, the relative shape of the frequency characteristics of the network will remain unchanged. A set of formulas for determining the values of the network elements, together with other design information, is given in the circuits section of the handbook as Circuit No. 8. In a manner similar to that which was done for the low pass network, the high pass configuration shown in Figure 46 may be modified to the configuration shown in 47, by moving the right terminal of the resistor labeled G1 from the output of the controlled source to the junction of the two resistors Ra and Rb. G1 C1
C2 +
+
E1

G2
+
Rb
E2
Ra 

Figure 47. Operational Amplifier VCVS High Pass Active Filter
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If the magnitude of the resistance of the element G1 is much larger than the values of the resistors Ra and Rb, so that the gaindetermining resistor network is not loaded, then the voltage transfer function far the circuit shown in Figure 47 is:
E2 Ks 2C1C2 = E1 s2C1C2 + s(C1G2 + C2G2 ) + G1G2
(7 )
Since none of the denominator coefficients in equation (7) are functions of K, the gain of the controlled source, this gain may be varied, changing the gain of the network without changing the frequency characteristics of the network. As in the law pass case is not possible to realize a voltage transfer function of the type given in equation (5) in which the constant α has very small values. The frequency normalized voltage transfer function for a band pass network is:
E2 Hs = 2 E1 s + αs + 1
(8)
where H is a positive real constant and H/α is the gain in the pass band. There are several network configurations using a VCVS as the active element that may be used to realize such a transfer function. One such configuration that has been found to give good experimental results is shown in Figure 48: G2
G1
C1 K
+ E1
G3
C2

+ E2 
Figure 48. VCVS Band Pass Active Filter The voltage transfer function in terms of the elements is:
E2 sKC1G1 = 2 E1 s C1C2 + s(C1G3 + C2G1 + C2G2 + C1G1 + C1G2 − KC1G2 ) + G3 (G1 + G2 )
( 9)
It should be noted that in the denominator, K, the gain of the controlled source, appears only in the coefficient of the firstdegree term. Thus, in high Q realizations, we may adjust the real part of the pole locations by varying K without appreciably affecting the resonant frequency. Since this adjustment may be made by varying the value of either of the resistors associated with the operational amplifier realization for the VCVS (see Figure 43), this is a very convenient means of adjusting the Q of the network. From equation (9) we see that such adjustments will also change the overall gain constant H, but this is usually a minor effect. The design information for this circuit is given in the circuits section of the handbook as Circuit No. 9.
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Other Realizations with VoltageControlled Voltage Sources Up to this point, in this chapter we have described active RC circuit configurations that have required a noninverting VCVS. There are also many network configurations which require the use of an inverting VCVS, i.e., one in which K, the gain constant, is negative. An operational amplifier and the circuit shown in Figure 49 may produce such a source: Rb
I2 Ra 
+
+
+
I1 E1
E2


Figure 49. Inverting Operational Amplifier VCVS Note that in this circuit the noninverting input of the operational amplifier is grounded. If we apply the Q virtual ground concept to this circuit we see that E1 = I1Ra. Similarly, I2 = I1 and therefore E2 = I1Rb. Thus we may write:
E 2 − Rb = E1 Ra
(10)
For large values of Ra, this circuit may be considered as an ideal VCVS with a value of K equal to Rb/Ra. Since the resistor Ra must be chosen large, this places a limit an the maximum gain value that K may have, as well as on the frequency range aver which the circuit will effectively model the ideal VCVS. Actual values of Ra to be used will necessarily depend on the impedance level of the rest of the circuit, but values of the general order of 1 kΩ to 1 MΩ are not uncommon. An example of the use of an inverting VCVS to realize a low pass transfer function as given in equation (2) is shown in Figure 410: G3
G1
G2 K
+ E1 
C1
C2
+ E2 
Figure 410. Inverting VCVS Low Pass Filter
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The voltage transfer function for this network is:
− K G1G2 E2 = 2 E1 s C1C2 + s[C2 (G1 + G2 + G3 ) + C1G2 ] + G2 (G1 + G3 + K G3 )
(11)
To avoid misinterpretation, the negative sign associated with the constant K has been written into the equation, and the gain is expressed as a magnitude. In a similar manner the inverting VCVS may be used to realize a high pass network function of the type given in equation (5). The circuit is shown in Figure 411: C3
C1
C2 K
+ E1
G1
G2

+ E2 
Figure 411. Inverting VCVS High Pass Filter The voltage transfer function for this circuit is:
− K s2C1C2 E2 = E1 s2C2 (C1 + C3 + K C3 ) + s[C2G1 + G2 (C1 + C2 + C3 )] + G1G2
(12)
The gain K has been shown as a magnitude in the same manner that was done in equation (11). It is also theoretically possible to use the inverting VCVS to realize a band pass transfer function. However, the value of the gain that is required for the source far even a moderately high value of Q is usually excessively high. Thus, such an application is of only limited value.
Conclusions The active RC circuits using controlled sources which hove been described in this chapter have several advantages and disadvantages when compared to the circuits of the preceding chapters. First, a new and different variable appears in the transfer function equations, namely K, the gain of the controlled source. In the infinitegain realizations given in Chapters 2 and 3, the network functions were almost completely unaffected by changes in the gain. In the realizations of this chapter, however, not only is there a strong dependence on the gain of the source, but also this effect may actually be used to vary the properties of the network. Thus, the presence of the gain as a variable is both an advantage and a disadvantage, and the potential user of these circuits must be aware of both.
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The high and low pass networks that use the controlled source present another new and useful characteristic; i.e., the pass band gain is independent of the element values. This has considerable application for the case where the cutoff frequencies are to be varied without changing the gain It should also be noted that the circuits presented in this chapter have the same low output impedance that the infinitegain realizations had. Thus, they may be used to drive other circuits without using isolating amplifier stages between them and the circuits that follow.
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Application Report SBOA093A – October 2001
CHAPTER 5 The NIC In Active RC Circuits In the first chapter of this handbook, the concept of an active device called an NIC (negativeimmittance converter) was introduced. It was pointed out that an NIC could be used in connection with resistors and capacitors to realize a wide range of network functions. In this chapter we shall explore some of the properties of this device and see in detail how it can be used in active RC circuits.
The NIC (NegativeImmittance Converter) Basically, the NIC is a twoport device that has the property that the impedance seen at either of its ports is the negative of the impedance connected to the other port. This “negative” action can come about in either of two ways. As a first way, the NIC can invert the direction of current flow with respect to that which would normally occur in a passive network, without disturbing the relative polarity of the input and output voltages. For example, consider a twoport network with a load impedance Z as shown in Figure 51:
Figure 51. TwoPort Network With Load If a current IZ flows out of port 2 as shown, then we would expect that a current I1 would flow into port 1 (assuming that the output voltage EZ and the input voltage E1 have the same polarity). If the twoport device somehow inverts one of these currents, then we have the situation wherein the Application of voltage E1 to port 1 causes a current to flow in a direction opposite to that shown for l1; i.e., it opposes the applied voltage. In other words, the input impedance is negative. More formally, such a device can be defined in terms of the twopart variables shown in Figure 52:
Figure 52. The Port Variables for a TwoPort Network and the equations:
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E1 = E 2
(1)
I1 = KI2 These equations may be said to define an ideal currentinversion negativeimmittance converter or INIC for short. The constant K is usually referred to as the “gain” of the INIC. In the next section we shall show that a single differentialinput operational amplifier can easily realize such a device. Let us investigate some properties of the INIC. First of all, consider the case where on impedance Z2 is connected across the terminals of port 2 as shown in Figure 53.
Figure 53. INIC Input Circuit The variables of port 2 are then constrained by the relation E2 = Z2I2. Substituting this relation into equations (1) we see that:
ZIN =
E1 Z =− 2 I1 K
( 2)
Thus the input impedance at port 1 is 1/K times the negative of the impedance connected across port 2. Thus we see that changing the gain K of the INIC easily varies the magnitude of this negative impedance. Similarly, if an impedance Z1 is connected across port 1 of the INIC as shown in Figure 54:
Figure 54. INIC Output Circuit then the impedance ZOUT seen at port 2 may be shown to be:
ZOUT =
E2 = −KZ1 I2
(3)
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It should be noted that again negative impedance is produced, but that in this case the gain constant K has the opposite effect to the one it had in equation (2). For the case where K = 1, the INIC will theoretically give the same results in either direction. Practically, however, stability considerations based on the network configuration in which the INIC is used usually do not permit interchanging ports 1 and 2 of the INIC, even for the unity gain case. More will be said about this when discussing the realization of the INIC. A second way in which the “negative” action of an NIC can be brought about is by inverting the voltage while keeping the direction of current flow through the twoport device unchanged. In terms of the variables shown in Figure 52, this type of NIC is defined by the relations:
I1 = −I2
( 4)
E1 = −KE 2 Such a device may be referred to as an ideal voltageinversion negativeimmittance converter or VNIC for short. The constant K is called the “gain” of the VNIC. Space does not permit developing some of the other properties of the NIC such as power relationships, impedance transformations, etc. The reader who is interested in learning more about this device should consult the references listed in Appendix A.
A Realization for the INIC An INIC (idealcurrentinversion negative immittance converter) may be realized by using a differentialinput operational amplifier as the active element. The circuit is shown in Figure 55. R1 I1 
+
+ R2 E1
I2 + E2


Figure 55. Operational Amplifier Realization of the INIC
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We may analyze this circuit by means of the virtual ground concept introduced in Chapter 1. This concept tells us that the voltage between the inverting and noninverting inputs of the operational amplifier is zero; thus we see that the voltages at the two ports of the overall network shown in Figure 55 are equal. Similarly, we know that no current flows into either of the amplifier inputs. Since the voltage across the two resistors and R2 must be the same, we see that I1R1 = I2R2; in other words, the ratio of the currents at the two ports is determined by the ratio of the resistors. We may write the above relationships as:
E1 = E 2 I1 =
( 5)
R2 I2 R1
These are the same relations given in equations (1), with K = R2/R1. Thus we see that the circuit shown in Figure 55 has the properties of an INIC, and that the gain constant K may be easily adjusted by changing the values of either of the resistors R1 or R2.
Stability of the INIC In the discussion given above, it was assumed that the voltage between the input terminals of the operational amplifier was zero. This assumption greatly simplified the analysis of the circuit. Actually, there will always be a small voltage present between these terminals. In the INIC circuit this small voltage becomes a significant factor in determining whether the circuit will be stable. To see this, consider the case where a resistor Ra is connected across port 1 of the INIC shown in Figure 55, and another resistor is connected across port 2. The circuit may be redrawn using the model for the operational amplifier shown in Figure 11 of Chapter 1. The result is shown in Figure 56.
Figure 56. Circuit Model of the Operational Amplifier Realization INIC The constant K shown in this figure represents the gain of the operational amplifier (not the gain of the INIC). In order for this circuit to be stable, the voltage E2  E1, even though it is small, must never be positive. It if should go positive, the feedback provided by the resistor networks will drive the operational amplifier into saturation. Therefore, the condition for stability is:
E1 ≥ E 2
( 6)
The voltages E1 and E2, however, if the currents I1 and I2 are zero, may be expressed in terms of the four resistors shown in Figure 56. Thus the inequality given in equation (6) may also be expressed as:
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Ra Rb ≥ R1 + Ra R 2 + Rb
(7)
This may be reduced to:
RaR 2 ≥ RbR1
(8)
For the case where R1 and R2 are equal, corresponding with an INIC gain of unity, we see that:
R a ≥ Rb
(9 )
is a necessary condition for stable operation of the INIC. To indicate this, it is customary to refer to port 1 as the opencircuitstable (OCS) port of this INIC realization, and port 2 as the shortcircuit stable (SCS) part. Similar restrictions may be developed for the situation where the INIC is embedded in networks containing capacitors as well as resistors, to ensure that the operational amplifier is not driven into saturation.
The Basic INIC Circuit There are several methods that have been proposed whereby a voltage transfer function may be realized by passive networks and an INIC. We shall discuss only one of these here, some others may be found in the references listed in Appendix A. The basic circuit configuration consists of two passive RC networks labeled A and B, and an INIC, and is shown in Figure 57.
Figure 57. Basic Voltage Transfer Circuit Using the INIC First let us consider the cascade connection of the B network and the INIC. If we describe the properties of the B network by its y parameters, then the effect of the INIC is to invert the current at the output port of network B and also multiply it by a constant. The y parameters of the cascaded connection of the B network and the INIC may thus be shown to be:
y11 = y11b y12 = y12b y 21 = −Ky12b
(10)
y 22 = −Ky 22b
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where y11b, y12b, and y22b are the y parameters of the B network by itself, and K is the gain of the INIC. Note that for the cascade connection, y12 is not equal to y21 as would be the case for a passive network. When the A network is connected in parallel, its y parameters add to those given in equations (10). Thus, the y parameters for the overall network are:
y11 = y11a + y11b y12 = y12a + y12b y 21 = y12a − Ky12b
(11)
y 22 = y 22a − Ky 22b Since the opencircuit voltage transfer function for an arbitrary network is simply the ratio y21/y22, we may write the voltage transfer function of the basic circuit as:
E 2 − y12a + Ky12b = E1 y 22a − Ky 22b
(12)
In the references, some specific rules are developed for determining the parameters of the component passive networks A and B in such a way that they can be realized by passive RC networks for a given desired voltage transfer function. We shall not develop these rules here. In the next section, however, we shall show the results of applying such developments to obtain the some three types of transfer functions that we realized in earlier chapters.
Network Design The basic circuit configuration described in the preceding section may be used to obtain various networks. The first network that we shall consider here is the low pass network having a frequencynormalized voltage transfer function of the form:
E2 H = 2 E1 s + αs + 1
(13)
A network configuration that realizes this voltage transfer function is shown in Figure 58.
Figure 58. INIC Low Pass Active Filter The elements C2 and G3 form the A network of Figure 57. For this network we have the following y12 and y22 parameters:
y12a = −G3 y 22a = sC2 + G3
(14)
Handbook Of Operational Amplifier Active RC Networks
41
SBOA093A
The elements C1, G1, and G2 shown in the figure form network B. This network has the y parameters:
y12b = y 22b
− sC1G1 sC1 + G1
sC1 (G1 + G2 ) + G1G2 = sC1 + G1
(15)
Substituting the expressions from equations (14) and (15) into equation (12), and rearranging terms, we find that the voltage transfer for the circuit shown in Figure 58 is:
E2 sC1 (G3 − KG1 ) + G1G3 = 2 E1 s C1C2 + s[C2G1 − KC1G2 + C1 (G3 − KG1 )] + G1 (G3 − KG2 )
(16)
If we constrain the elements of this network so that G3 equals KG1, then equation (16) reduces to:
E2 G1G3 = 2 E1 s C1C2 + s(C2G1 − KC1G2 ) + G1 (G3 − KG2 )
(17 )
If we compare this equation with equation (13) we see that the constant H is positive; no signal inversion is provided by this circuit. Equations relating the values of the network elements to the constants α and H of equation (13) have been tabulated and, together with other design information on this circuit, they are presented in the circuits section of this handbook as Circuit No. 10. In a similar fashion we may use passive RC networks and an INIC to realize a high pass filter. Such a network will have a transfer function of the form:
E2 s2H = 2 E1 s + αs + 1
(18)
A network configuration that realizes this function is shown in Figure 59.
Figure 59. INIC High Pass Active Filter The elements C2 and G2 shown in this figure constitute network A of Figure 57. The y parameters for this network are:
y12a = −sC2
(19)
y 22a = sC2 + G2
42
Handbook Of Operational Amplifier Active RC Networks
SBOA093A
The elements C1 and G1 constitute network B. Its y parameters are:
y 22b = − y12b =
sC1G1 sC1 + G1
( 20)
If we substitute the expressions from equations (19) and (20) into equation (12), after rearranging terms, we find that the voltage transfer function for the overall network is
E2 s2C1C2 + sG1 (C2 − KC1 ) = 2 E1 s C1C2 + s[C1G2 + G1 (C2 − KC1 )] + G1G2
( 21)
If we constrain the elements of this network so that C2 equals KC1, then equation (21) reduces to:
E2 s2C1C2 = 2 E1 s C1C2 + sC1G2 + G1G2
( 22)
We see that the constant H of equation (18) is positive. The design information for this circuit is given in the circuits section of the handbook as Circuit No. 11. As a final example of circuit design using an INIC, consider the band pass network with a frequency normalized voltage transfer function of the form: E2 − sH = 2 E1 s + αs + 1
(23)
A circuit that realizes this function is shown in Figure 510.
Figure 510. INIC Band Pass Active Filter The elements C2 and G2 comprise network A of Figure 57. The y parameters are: y12 a = 0 y22 a = sC2 + G2
(24)
The elements C1 and G1 comprise network B. Its y parameters are: y22b = − y12b =
sC1G1 sC1 + G1
(25)
The overall voltage transfer function for the circuit may be found by inserting the relations of (24) and (25) into equation (12). After rearranging terms we obtain:
− sKC1G1 E2 = 2 E1 s C1C2 + s(C1G2 + C2G1 − KC1G1 ) + G1G2
( 26)
Handbook Of Operational Amplifier Active RC Networks
43
SBOA093A
Note that the constant H of equation (23) is positive for this circuit, and that the circuit provides a signal inversion. The voltage transfer function given in equation (26) points out some of the interesting properties that the INIC realization of the band pass network possesses. In the denominator, it should be noted that the INIC gain constant appears only in the coefficient of the firstdegree term. Thus, in high Q realizations, it is possible to adjust the real part of the pole positions, i.e., the Q of the circuit, without significantly changing the magnitude of the pole positions, i.e., without significantly changing the resonant frequency. At a given frequency, therefore, we can control the bandwidth of the network by changing the gain of the INIC, i.e., by varying the values of either of the resistors associated with the operational amplifier INIC realization. A second interesting feature of this network is that by choosing a nonunity INIC gain constant, it is possible to design the network so that both resistive elements have the same value, and both capacitive elements have the same value. Thus, the problem of obtaining accurately specified passive element values is considerably minimized. Finally, if both resistors have the same value and are varied the same amount, the resonant frequency of the network is changed, although the Q of the network remains invariant. The same is true if the values of the capacitors are changed in the indicated manner. Thus we have a circuit where a single resistor may be changed to vary the bandwidth, and a pair of resistors may be changed to vary the resonant frequency, and the two effects do not interact. The design equations and other information for this circuit are given in the circuits section of the handbook as Circuit No. 12.
Conclusions In this chapter, an entirely new “breed” of circuits has been presented, namely, circuits that use an INIC as the active element. The INIC can, of course, also be used to produce single negativevalued elements which in turn con be used to compensate for dissipation, to reduce input capacitance, etc. We have used the INIC as an integral portion of circuits realizing low pass, high pass, and band pass voltage transfer functions. The band pass circuit is an especially attractive one because of the minimum number of elements used, the fact that all elements of a given kind have the some value, and the ease with which the Q and the resonant frequency of the network may be adjusted. It should be noted that all the circuits given in this chapter share a common disadvantage; namely, their output impedance is not zero. Therefore, if such networks are cascaded, suitable isolating stages must be used to separate them. The advantage of ease of adjustment provided by some of these networks must therefore be weighed against the disadvantage of the requirement for the extra circuitry involved in isolating the filtering stages. A more detailed comparison of the advantages and disadvantages of the various realization schemes that have been presented in Chapters 2 through 5 are given in Chapter 7.
44
Handbook Of Operational Amplifier Active RC Networks
Application Report SBOA093A – October 2001
CHAPTER 6 Another Active Device: The Gyrator In addition to the active elements that have been introduced in the preceding chapters, there is another one that deserves mention. It is called a gyrator. In this chapter we shall give a short introduction to the properties and potential uses of this element.
Definition of a Gyrator A gyrator is a nonreciprocal twoport device defined by the equations:
I1 = GE 2
(1)
I2 = −GE1 This twoport device is usually considered to have a common ground, and, in this case, the gyrator is represented by the symbol shown in Figure 61.
Figure 61. Gyrator Symbol The constant G is called the gyration conductance. The reference arrow drawn inside the circle in the figure indicates that the gyration action from terminal 1 to terminal 2 with terminal 3 common (as shown) is the same as that which would occur from terminal 2 to terminal 3 if terminal 1 was used as the common terminal. Similarly, it is the same as that which would occur from terminal 3 to terminal 1 if terminal 2 were used as the common terminal. We shall see what this gyration action consists of in the paragraphs that follow.
Properties of the Gyrator From equations (1) we see that the y parameters of a gyrator are:
y11 = 0
( 2)
y12 = G y 21 = −G y 22 = 0 If we calculate the input admittance YlN of a twoport network, defined by its y parameters, when an admittance Y2 is connected across the terminals of port 2, as shown in Figure 62:
45
SBOA093A
Figure 62. The Input Admittance of a Terminated TwoPort Network we obtain:
YIN = y11
y12 y 21 y 22 + Y2
(3)
For the case where the twoport is a gyrator we may substitute the y parameters from equations (2) into the expression in equation (3). Thus we obtain:
YIN =
G2 Y2
( 4)
This is the input admittance of a gyrator terminated in admittance Y2. Equation (4) tells us that if Y2 is a capacitor, then, at part 1 of the gyrator we see a twoterminal behavior exactly paralleling that of an inductor. In other words, a gyrator can be used to “gyrate” a capacitor into an inductor. Thus, with resistors, capacitors, and gyrators, we can achieve any network realization that can be achieved with resistors, capacitors, and inductors. If we can obtain a gyrator with a small value of G, then we can gyrate very small capacitors into very large inductors, a most useful feat! One other property of the gyrator may be of interest. This concerns the relationships at the terminals of the gyrator. For any twoport device, the total instantaneous power consumed by the device is:
p(t ) = e1 (t )i1 (t ) + e 2 (t )i2 (t )
(5)
Substituting relations equivalent to those given in equation (1), but in terms of functions of time, into the above equation, we see that
p(t ) = e1 (t )i1 (t ) − e1 (t )i1 (t ) = 0
( 6)
Thus we see that the gyrator neither adds energy to the circuit in which it is used, nor consumes it. As such, its terminal properties are those of a lossless passive network component. We shall see, however, that its realization inevitably requires the use of active elements.
A Gyrator Realization There are several ways of realizing a gyrator. One of the methods uses two INICs. Consider the circuit shown in Figure 63.
46
Handbook Of Operational Amplifier Active RC Networks
SBOA093A
Figure 63. Gyrator Realization Using Two INIC’s It is easily shown that this circuit has the y parameters given in equation (2), and thus functions as a gyrator. The values of the resistors (in mhos) shown in the figure determine the value of the gyration conductance. References to some other methods for realizing gyrators are given in Appendix A.
Circuit Realizations There are considerably fewer results available in the literature regarding the use of gyrators and RC circuits for the realization of transfer functions than there are for any of the other classes of networks that have been discussed in this handbook. Since the state of the art is relatively new, and since the active elements are considerably more complicated than any which have been discussed in the previous chapters, we shall not present general circuits for the realization of low pass, high pass and band pass circuits as was done in those chapters. To give an example of one form that such realizations may take, however, consider the circuit shown in Figure 64.
Figure 64. Gyrator Band Pass Active Filter The voltage transfer function for this circuit is:
E2 s = 2 E1 s + 0.2s + 1.01
(7)
Thus, the circuit realizes a band pass voltage transfer function.
Handbook Of Operational Amplifier Active RC Networks
47
SBOA093A
Conclusions Due to the complicated nature of the realizations for the gyrator, this network element has not achieved a wide usage at this time. In addition to the disadvantage of complexity, realizations that use it as the active element have two other disadvantages in that their output impedance will not be zero, and they will be capable only of the gain that a passive RLC circuit is capable of. Despite these disadvantages, the gyrator also has some potential advantages. First, its lossless nature provides a theoretical bar to circuit instability, since if no power is being supplied to the circuit, instability cannot occur. Second, since capacitors in general have a higher quality factor (lower dissipation) than inductors, gyration of a capacitor may produce a better inductor than those that are readily available. Finally, the possibility of using a gyrator for impedance multiplication implies the ability to realize very low frequency circuits without the need of relatively largevalued reactive elements.
48
Handbook Of Operational Amplifier Active RC Networks
SBOA093A
CHAPTER 7 A Summary In Chapters 2 through 5 of this handbook four different active RC synthesis techniques were presented. These were the infinitegain singlefeedback technique, the infinitegain multiplefeedback technique, a technique using controlled sources, and a technique using negativeimmittance converters. The various methods were all applied to the realization of low pass, high pass, and band pass voltage transfer functions. At this point the reader may well ask, “If I want to realize a low pass function (or a high pass or a band pass one) which of the methods is best?’ The answer to the question, of course, depends on how the word “best” is defined If “best” means fewest elements, then the infinitegain single feedback technique is certainly eliminated, If the network is to be cascaded with other networks, then the NIC approach is probably not a good one. Thus, the answer to such a question depends on the details of the application, which will vary considerably from one situation to another. The purpose of this handbook has been to give the prospective user several different techniques for each of the filter realizations, in order to permit him flexibility in selecting the technique that more nearly meets his specific application. To provide a further guide to such a choice, some of the advantages and disadvantages of the various realization techniques are summarized in Table I. A study of this table will provide a good review of the material that has gone before. Table 1.
Summary of the Advantages and Disadvantages of the Various Realization Techniques Realization Technique Property
InfiniteGain SingleFeedback
InfiniteGain MultipleFeedback
Controlled Source
NegativeImmittance Converter
Minimal number of Network elements

+
+
+
Ease of adjustment Of characteristics

0
0
+
Stability of characteristics
+
+


Low output Impedance
+
+
+

Presence of Summing input
+



Relatively high Gain available
+

+
+
Low spread of Element values
+

+
+
HighQ realization possible
+

+
+
+ Indicates the realization is superior for the indicated property 0 Indicates the realization is average for the indicated property  Indicates the realization is inferior for the indicated property
Handbook Of Operational Amplifier Active RC Networks
49
SBOA093A
This concludes our introduction to the wonderful world of active RC networks, a world without inductors. It is hoped that the reader will find the techniques that have been presented in this handbook useful, and that he will be able to apply them to his own filtering problems. Needless to say, the Applications Engineering Department of Texas Instruments welcomes any questions or comments that the reader might have on the material of this booklet or on any other operational amplifier application. Feel free to call us at any time.
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Handbook Of Operational Amplifier Active RC Networks
SBOA093A
SECTION II Circuits Introduction All of the examples shown in this section have been proven on the bench with the results shown. In addition, the transfer functions are given along with methods for determining component values and comments on the nominal range of values. We trust that these “hints and kinks” will enable you to readily modify the circuits given for your application. Two possible points of confusion, however, deserve mention. In the theory section we have, for simplicity, worked with frequency normalized transfer functions ωo (= 2πfo ) and conductance
G (= 1 R ) . In moving on to the real world of circuits, we find it convenient to “unnormalize” the transfer functions fo (= ωo 2π ) and deal with resistance R (= 1 G) . Have made this shift explicit,
we trust it will not create problems. Please note: •=
Since the design formulas give correct values to within an arbitrary constant, an impedance denormalization constant, k, is included and is to be chosen for convenience.
•=
The original edition of this handbook had part numbers of op amps listed on the schematics. While these part numbers were meaningful in 1966 when the original edition went to press – they are meaningless today.
•=
The state of the art in op amps has progressed, making these relatively low frequency examples somewhat dated. They are left “asis”, because they illustrate circuit design techniques – not state of the art performance levels.
It is our hope that the circuits presented will trigger the idea that develops into your circuit. Sharing your circuit with us will enable us to share it with other engineers. Similarly, we would welcome the opportunity to share our experience and the latest advancements in network theory and amplifiers with you.
Handbook Of Operational Amplifier Active RC Networks
51
SBOA093A
CIRCUIT 1: Single Feedback Low Pass Network B C1b R1b
R2b C2b
Network A Ra
Ra 
+
+
Ca
E1
+ E2


E2 − Hωo = 2 E1 s + αωos + ωo 2 2
Choose : α ( = 2 for " max imally flat" ' Butterworth' response Let : b = (2.5 − α )
52
ωo = 2πfo Ao = H
Calculate : Ca =
4H k α 2 2πfo
C1b =
k 2πfo
C 2b =
b2 k ⋅ αb − 1 2πfo
Handbook Of Operational Amplifier Active RC Networks
α ( two such resistors) 2Hk 1 1 R1b = ⋅ b k 1ö 1 æ R 2b = ç α − ⋅ b k è Ra =
SBOA093A 0.00796 µF
92.1 kΩ Ω
707 Ω
0.0175 µF
707 Ω 
+ E1
49.3 kΩ Ω
+
1.59 µF
+ E2


EXAMPLE
Want : fo = 200 Hz
Calculate : Ca = 1.59 µF
A o = 100 (40dB )
C1b = 0.00796 µF
α= 2
Ra (2 ) = 707 Ω
k = 10−5
C2b = 0.0175 µF
R1b = 92.1 kΩ R 2b = 49.3 kΩ
Handbook Of Operational Amplifier Active RC Networks
53
SBOA093A
CIRCUIT 2: Single Feedback High Pass Network B C1b R1b
R2b C2b
Network A C1a
C2a 
+
+ E1
+
Ra
E2

E2 − Hs2 = 2 E1 s + αωos + ωo 2

ωo = 2πfo Ao = H
Choose : α ( = 2 for " max imally flat" response, 40 dB / decade rolloff ) α > 2, slower rolloff α < 2, peaking, faster rolloff Let : b = (2.5 − α )
54
Calculate : C1a = C2a = 2H
k 2πfo
C1b =
k 2πfo
C 2b =
b2 k αb − 1 2πfo
Handbook Of Operational Amplifier Active RC Networks
1 1 ⋅ 4Hα k 1 1 R1b = ⋅ b k 1ö 1 æ R 2b = ç α − ⋅ b k è Ra =
SBOA093A 0.0796 µF
92.1 kΩ
1.59 µF
0.175 µF
1.59 µF 
+
+ E1
92.1 kΩ
1.77 kΩ
+ E2


EXAMPLE
Want : fo = 20Hz
A o = 10 (20 dB )
α= 2 Choose : k = 10− 5
Calculate : C1a = C2a = 1.592 µF C1b = 0.0796 µF Ra = 1.77 kΩ R1b = 92.1 kΩ R 2b = 49.3 kΩ
Handbook Of Operational Amplifier Active RC Networks
55
SBOA093A
CIRCUIT 3: Single Feedback Band Pass Network B R1b C1b
R3b
R2b C2b
C3b
Network A Ra
Ca 
+
+
E2


− Hωos E2 = 2 E1 s + αωos + ωo 2
56
+
E1
ωo = 2πfo Ao = H α Q =1 α
Handbook Of Operational Amplifier Active RC Networks
SBOA093A
Let : α =1 Q
1+ α b = (2.5 − α ) 2+α H = Ao Q
Calculate : k Ca = H 2πfo C1b = b
1 1 ⋅ H k 1 1 R1b = ⋅ b k (b − 1) ⋅ 1 R 2b = b k (b − 1)(α + 1) ⋅ 1 R3b = b2 k Ra =
k 2πfo
C 2b =
b k ⋅ b − 1 2πfo
C3b =
b2 k ⋅ (b − 1)(1 + α ) 2πfo
Handbook Of Operational Amplifier Active RC Networks
57
SBOA093A 2.04 kΩ Ω
7.96 kΩ Ω
0.200 µF
1.79 kΩ Ω 10 kΩ Ω +
0.778 µF
0.888 µF
0.159 µF +
+
E1
E2


EXAMPLE
Want : fo = 100 Hz
A o = 10 (20 dB )
Q = 10
Calculate : Ca = 0.159 µF C1b = 0.200 µF C2b = 0.778 µF C3b = 0.888 µF Ra = 10 kΩ
Choose : k = 10
−4
R1b = 7.96 kΩ R 2b = 2.04 kΩ R3b = 1.79 kΩ
58
Handbook Of Operational Amplifier Active RC Networks
SBOA093A
CIRCUIT 4: Multiple Feedback Low Pass
R4
C5
R1
R3 
+
+
+ E2
C2
E1 

Ao = H
E2 − Hωo = 2 E1 s + αωos + ωo 2 2
ωo = 2πfo
TRANSFER FUNCTION
E2 − 1 R1R3 = 2 E1 s C2C5 + sC5 (1 R1 + 1 R3 + 1 R 4 ) + 1 R3R 4
Handbook Of Operational Amplifier Active RC Networks
59
SBOA093A
Calculate : Choose : k C= , (defines k ) 2πfo H = Ao α ( = 2 for " max imally flat" response, 40 dB / decade rolloff )
60
Handbook Of Operational Amplifier Active RC Networks
C5 = C = C2 =
k 2πfo
4 (H + 1) k 2 2πfo α
α 2Hk α R3 = 2(H + 1)k α R4 = = HR1 2k R1 =
SBOA093A
11.25 kΩ 1125 Ω
0.1 µF 1020 Ω 
+ E1
+
2.2 µF
+ E2


EXAMPLE
Want : fo = 100 Hz
Calculate :
A o = 10 (20 dB )
k = 6.28 × 10− 5 C5 = 0.1 µF
α= 2
C2 = 2.2 µF R1 = 1125 Ω
Choose : C = 0.1 µF
R3 = 1020 Ω R 4 = 11.25 kΩ
Handbook Of Operational Amplifier Active RC Networks
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SBOA093A
CIRCUIT 5: Multiple Feedback High Pass
C4 C1
R5 C3 
+
+ E1
+ E2
R2

E2 − Hs2 = 2 E1 s + αωos + ωo 2

ωo = 2πfo Ao = H
TRANSFER FUNCTION
E2 − s2C1C3 = 2 E1 s C3C4 + s(C1 + C3 + C4 ) R5 + 1 R 2R5 Choose : k C= , (defines k ) 2πfo
Calculate :
H = Ao
C4 = C H
α ( = 2 for " max imally flat" response, 40 dB / decade rolloff )
R2 =
C1 = C3 =
R5 =
62
Handbook Of Operational Amplifier Active RC Networks
k =C 2πfo
α k (2 + 1 H)
H(2 + 1 H) αk
SBOA093A
10 µF 10 µF
338 kΩ
10 µF 
+
+ E1
75.2 kΩ
+ E2


EXAMPLE
Want : fo = 0.1 Hz
A o = 1 (0 dB )
α= 2 Choose : C = 10 µF
Calculate : k = 6.28 × 10− 6 C1 = C3 = 10 µF C4 = 10 µF R 2 = 75.2 kΩ R5 = 338 kΩ
Handbook Of Operational Amplifier Active RC Networks
63
SBOA093A
CIRCUIT 6: Multiple Feedback Band Pass
C4
R5 C3
R1

+
+ E1
R2
E2

E2 − Hωos = 2 E1 s + αωos + ωo 2

Ao = H α Q =1 α ωo = 2πfo
TRANSFER FUNCTION
E2 − sC3 R1 = 2 E1 s C3C4 + s ⋅ 1 R5 (C3 + C4 ) + 1 R5 (1 R 4 + 1 R 2 ) Calculate :
Choose : k C= , (defines k ) 2πfo α =1 Q H = Ao Q
64
+
C3 = C 4 =
k =C 2πfo
R1 = 1 Hk 1 (2Q − H)k 2Q R5 = k R2 =
Handbook Of Operational Amplifier Active RC Networks
SBOA093A
10 µF 10 kΩ
200 kΩ
10 µF 
+
+ E1
527 Ω
+ E2


EXAMPLE
Want : fo = 1.6 Hz Q = 10 A o = 10 (20 dB )
Calculate : k = 10− 4 C3 = C4 = 10 µF H =1 R1 = 10 kΩ
Choose : C = 10 µF
R 2 = 527 Ω R5 = 200 kΩ
Handbook Of Operational Amplifier Active RC Networks
65
SBOA093A
CIRCUIT 7: Controlled Source Low Pass
Controlled Source
C1
+
+
R1
R2

+ (K1)R
C2
E1
E2 R


E2 Hωo = 2 E1 s + αωos + ωo 2 2
Ao = H = K ωo = 2πfo
TRANSFER FUNCTION
E2 K R1R 2 = 2 E1 s C1C2 + s[C2 R1 + C2 R 2 + C1 R 2 (1 − K )] + 1 R1R 2
Choose : k C1 = defines k 2πfo K = Ao α ( = 2 for " max imally flat" response ) R and (K − 1)R for appropriate speed and power consumptio n
66
Handbook Of Operational Amplifier Active RC Networks
Calculate : a2 + (K − 1) 4 mk C2 = mC1 = 2πfo
m=
2 αk α R2 = 2mk R1 =
SBOA093A
0.10 µF
+
+ 75 kΩ Ω
E1
3.94 kΩ Ω

+ 90 kΩ Ω
0.95 µF
E2 10 kΩ Ω


EXAMPLE
Want : fo = 30 Hz α= 2 C1 = 0.1 µF
A o = 10 (20 dB )
Calculate : m = 9. 5 k = 1.89 × 10−5 C2 = 0.95 µF R1 = 75 kΩ R 2 = 3.94 kΩ K = 10
Handbook Of Operational Amplifier Active RC Networks
67
SBOA093A
CIRCUIT 8: Controlled Source High Pass
Controlled Source
R1 C1
C2 +
+

+ (K1)R
E1
R2
E2 R

E2 Hs2 = 2 E1 s + αωos + ωo 2
68

Ao = K ωo = 2πfo
Handbook Of Operational Amplifier Active RC Networks
SBOA093A
Choose :
(
α = 2 for " max imally flat" , ' Butterwort h' response k (defines k ) C1 = 2πfo
)
C2 = C1 Calculate : α + α 2 + 8(K − 1) 4k 4 1 ⋅ R2 = α + α 2 = 8(K − 1) k
R1 =
Handbook Of Operational Amplifier Active RC Networks
69
SBOA093A
Controlled Source
39.3 kΩ Ω
+
+ 0.1 µF
0.1 µF

+ (K1)R
E1
717 Ω
E2 R


EXAMPLE
Want : fo = 300 Hz
70
Calculate : k = 1.884 × 10 − 4
α= 2 A o = 100 (40 dB )
R1 = 39.3 kΩ
C1 = 0.1 µF = C2
K = 100
R 2 = 717 Ω
Handbook Of Operational Amplifier Active RC Networks
SBOA093A
CIRCUIT 9: Controlled Source Band Pass
Controlled Source
R2 R1
C1 +
+

+ (K1)R
R3
E1
C2
E2 R


E2 Hωo s = 2 E1 s + αωos + ωo 2
Ao = H α
H=K
Q =1 α
ωo = 2πfo
TRANSFER FUNCTION
E2 = E1
Ks
é C C + C2 C2 C1 ù (1 − K ) + 1 + + s2C1C2 + s ê 1 + 1 R1 R2 R2 R3 ë R3
Choose : C1 =
C1 R1
k (defines k ) 2πfo
R and (K − 1R ) for desired op amp speed and power
Calculate : 1 k C2 = C1 = 2 4πfo R1 =
2 k
æ R + R2 ö ⋅ çç 1 è R1R 2
2 R1 = 3k 3 4 R3 = = 2R1 k 1æ 1ö K = ç 6.5 − Q 3è R2 =
Handbook Of Operational Amplifier Active RC Networks
71
SBOA093A
17.7 kΩ Ω 53 kΩ Ω
0.02 µF +
+
E1
+
106 kΩ Ω
53 kΩ Ω
0.01 µF
E2 47 kΩ Ω


EXAMPLE
Want : fo = 300 Hz Q = 10 C1 = 0.02 µF
72
Calculate : K = 2.133 A o = 21.33 ( 26.6 dB ) C1 = 0.02 µF C2 = 0.01 µF
Handbook Of Operational Amplifier Active RC Networks
k = 37.8 × 10−6 R1 = 53 kΩ R 2 = 17.7 kΩ R3 = 106 kΩ
SBOA093A
CIRCUIT 10: INIC Low Pass INIC R3 R1
KR
C1 
+
E1
+
+ R
R2
E2 C2


E2 Hωo = 2 E1 s + αωos + ωo 2 2
ωo = 2πfo
Choose : α ( = 2 for " max imally flat" response )
Calculate : α α2 + + (A o − 1) 2 4 k C2 = (defines k ) 2πfo
b= Let : K =1 R1 = R3
C1 =
Ao k A ⋅ = 2o C2 2 b 2πfo b
R1 = R3 = R2 =
b A ok
b (A o − k )k
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15.1 kΩ Ω 15.1 kΩ Ω
VCC
(10 kΩ Ω)
0.696 µF 
+
E1
VCC
E2 1.0 µF 
EXAMPLE
α= 2 A o = 10 (20 dB ) C2 = 1 µF
74
+ (10 kΩ Ω)
16.8 kΩ Ω

Want : fo = 4 Hz
VCC
+
Calculate : b = 3.79 k = 2.51 × 10− 5 C1 = 0.696 µF R1 = R3 = 15.1 kΩ R 2 = 16.8 kΩ
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SBOA093A
CIRCUIT 11: INIC High Pass INIC C2 KR +
R1
C1
+
+ R
E1
E2 R2


E2 s2 = 2 E1 s + αωos + ωo 2
ωo = 2πfo
TRANSFER FUNCTION
E2 s2C1C2 + s (C2 − KC1 ) R1 = 2 E1 s C1C2 + s[C1 R 2 + (C2 − KC1 ) R1 ] + 1 R1R 2 Note: Only unitygain possible with this configuration
(
Choose : α = 2 for " max imally flat" ' Butterwort h' response
)
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Calculate : C1 = C2 = α R1 = k
76
k (defines k ) 2πfo
R2 = K =1
Handbook Of Operational Amplifier Active RC Networks
1 αk
SBOA093A
0.1 µF 10 kΩ Ω, 0.1% +
0.1 µF
14.1 kΩ Ω
+
+ 10 kΩ Ω, 0.1%
E1
E2 7.02 kΩ Ω 

EXAMPLE
Want : fo = 160 Hz α= 2 C1 = 0.1 µF = C2
Calculate : k = 1.007 × 10− 4 R1 = 14.1 kΩ R 2 = 7.02 kΩ
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CIRCUIT 12: INIC Band Pass INIC
KR +
C1
R1
+ R
E1 + C2
R2 
E2 
ωo = 2πfo
− Hω0s E2 = 2 E1 s + αωos + ωo 2
Ao = H α Q =1 α
TRANSFER FUNCTION
E2 − Ks C1 R1 = 2 E1 s C1C2 + s(C2 R1 + C1 R 2 − K C1 R1 ) + 1 R1R 2
Choose :
Note :
k (defines k ) C1 = C2 = 2fo
K Ao = 2 −K 1 Q= 2−K
1 R1 = R 2 = k
fo =
1 2πC1R1
K = 2 −1 Q
Note: the choice of R in the INIC is relatively arbitrary, but it should be near 10 kΩ for best results.
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Handbook Of Operational Amplifier Active RC Networks
SBOA093A INIC
16.67 kΩ +
19.9 kΩ
0.1 µF
+ 8.33 kΩ
E1 19.9 kΩ 
+ 0.1 µF E 2 
EXAMPLE
Want : Q = 100 fo = 80 Hz C1 = 0.1 µF
Calculate : K = 1.99 A o = 199 (46 dB ) C1 = C2 = 0.1 µF k = 50.2 × 10− 6 R1 = R 2 = 19.9 kΩ
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APPENDIX A References The literature has many references to the topics in the field of active RC circuits that have been introduced in this handbook. Some of these are given in the list that follows. No attempt has been made to make this an all inclusive list; rather it is presented to give the interested reader a starting place in his pursuit of more de tailed information on this subject. The references are arranged by chapter.
Chapter 1 A general theoretical treatment of the subject of active networks, together with many additional realization techniques may be found in the book by K. L. Su, Active Network Synthesis, McGrawHill Book Co., Inc., New York, 1965
Chapter 2 General design formulas for several types of bridgedT and twinT networks, as well as a general discussion of the synthesis of passive RC networks may be found in the book by N. Balabanian, Network Synthesis, Chap. 7, PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1958. A more detailed presentation of several cases, together with several design charts may be found in Appendix VI of the book by C. J. Savant, Jr., Control System Design, 2nd Ed., McGrawHill Book Co., Inc., 1964 A simplified method for the design of twinT networks with transmission zeros in the righthalf of the complex frequency plane may be found in the article by B. A Shenoi, “A New Technique for TwinT RC Network Synthesis”, IEEE Transactions on Circuit Theory, Vol. CT11, No. 3, pp. 435436, Sept. 1964 A discussion of the adjustment problem in twinT networks may be found in the article by K. Posel, “A New Treatment of the RC ParallelT Network”, Proceedings of the Institution of Electrical Engineer’s (England), Vol. 110, No. 1, pp. 126—138, Jan. 1963.
Chapter 3 A tabular method which may be used to determine the voltage transfer function of a network using a fairly arbitrary feedback network is given in the article by A. G. J. Holt and J. Sewell, “Table for the Voltage Transfer Functions of SingleAmplifier DoubleLadder Feedback Systems,” Electronics Letters (published by the Institution of Electrical Engineers, England), Vol. 1, No.3, pp. 7071, May 1965 A design method for a third order filter using a single operational amplifier is given in the article by L. K. Wadhwa, “Simulation of ThirdOrder Systems with DoubleLead Using One Operational Amplifier,” Proceedings of the IRE, Vol. 50, No. 6, pp. 15381539, June 1962. Articles on similar filters with different zeros may be found in the February and April Proceedings issues of the same year.
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Chapter 4 The classic article in this area is the one by R. P. SalIen and E. L. Key, “A Practical Method of Designing RC Active Filters,” IRE Transactions on Circuit Theory, Vol. CT2, No. 1, pp. 7485, March 1955. Some additional circuits are given in the article by N. Balabanian and B. Patel, “Active Realization of Complex Zeros,” IRE Transactions on Circuit Theory, Vol. CT10, No.2, pp. 299300, June 1963
Chapter 5 The properties of a negativeimmittance converter as a twoport device were defined by A. I. Larky, “NegativeImpedance Converters,” IRE Transactions on Circuit Theory, Vol. CT4, No. 3, pp. 124131, September 1957. In addition to the circuit for an NIC given in the Larky paper, several other circuit realizations have appeared in the literature. See, for example, D. P. Franklin, “DirectCoupled NegativeImpedance Converter,” Electronics Letters, Vol. 1, No. 1, p. 1, March 1965. Power and impedance transformation properties are discussed by L. P. Huelsman, “A Fundamental Classification of NegativeImmittance Converters,” 1965 IEEE lnternational Convention Record Part 7, pp. 113118, March 1965. The basic circuit for the realization of a voltage transfer with an INIC presented in this chapter is described in more detail by T. Yanagisawa, ‘RC Active Networks Using Current Inversion Type Negative Impedance Converters,” IRE Transactions on Circuit Theory, Vol. CT4, No. 3, pp. 140144, Sept. 1957. Another basic approach that uses VNICs is given by J. G. Linvill, ‘RC Active Filters,” Proceedings of the IRE, Vol. 42, No. 3, pp. 555564, March 1954.
Chapter 6 Books which cover the topics of this chapter are the ones by Su (see Chapter reference) and L. P. Huelsman, Circuits, Matrices, and Linear Vector Spaces, McGrawHill Book Co., Inc., New York, 1962.
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APPENDIX B Describing Active Filters This handbook is addressed primarily to the design of active RC circuits employing operational amplifiers as the active elements. While we have seen that the operational amplifier is extremely versatile in active filter applications, it nonetheless holds true that for every requirement there is one operational amplifier that provides the optimum performance and value. It is, therefore, important far the user to work closely with a reputable manufacturer of operational amplifiers to gain the flexibility of being able to tailor an amplifier to the filter requirement. To fully utilize this service, it becomes necessary for the user to describe his filter requirement.
Describing the Filter There are many ways to describe a particular active filter. If you have evaluated a circuit from this handbook, referencing the page number along with the amplifier and circuit values employed provides a good starting point for the manufacturer. Any problems or limitations encountered should also be noted. The shape of the transfer function should be described by function (low pass, band pass, band rejection, high pass), passband gain, frequency response (3 dB points or accuracy over range of interest), rolloff characteristics (dB/octave or dB/ decode or rejection at specific frequencies), and type (Butterworthmaximally flat, Bessellinear phase, Tchebyscheffequal ripple, etc.). Alternatively, the filter may be described in terms of the basic function of the filter and overall system, a polezero plot, or a plot of the desired response versus frequency. Usually some combination of the above method is best used to describe a particular filter.
Optimizing the Circuit Given a complete description of the active filter requirement, the amplifier manufacturer can begin to make some very interesting design tradeoffs. In some band pass and high pass circuits it may be possible to relax the DC voltage and current offset and stability requirements that normally add to the cost of operational amplifiers. In certain filter circuits, the RC phase compensation network inside the operational amplifier may be modified to allow higher frequency performance. Thus, it is often desirable to wed the active amplifier and passive filter elements into a single committed active filter module. The resulting optimized circuit usually provides higher performance at lower overall cost than either a filter employing “standard” amplifiers a “standard” active filter that must be customized at least in terms of cutoff frequencies for your application.
Limiting Specifications The final filter design can be completely described by electrical, mechanical, and environmental specifications. In the design stage, however, it is important to indicate the critical or limiting specifications so as not to unduly restrict the design add to the design and product cost, and to help resolve any conflicting specifications design tradeoffs.
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Conclusion A reputable manufacturer of operational amplifiers with experience and competence in active filters offers valuable services to the user. Whether the service takes the form of circuit consultation, amplifier recommendation, special amplifier design, special filter design, or the supplying of a standard filter, the user must furnish a complete description of his filter requirements.
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APPENDIX C
84
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