US2788496A

US2788496A - Active transducer

Info

Publication number: US2788496A
Application number: US360301A
Authority: US
Inventors: John G Linvill
Original assignee: Bell Telephone Laboratories Inc
Current assignee: AT&T Corp
Priority date: 1953-06-08
Filing date: 1953-06-08
Publication date: 1957-04-09
Anticipated expiration: 1974-04-09
Also published as: DE1127401B; BE529361A; NL97559C; GB753085A; FR1102666A; NL187239B

Description

April 9, 1957 J. G. LlNVlLL 2,788,496

ACTIVE TRANSDUCER Filed June 8, 1953 2 Sheets-Sheet 1 F IG.

1 3 1' 2 1 9 d b 7 1 l I I 2 NEGATIVE PASSIVE PASSIVE NETWORK 7 b NETWORK [a F IG. 3

a b +0. 0, +0, 2 x Q .7 2 1 pa I i c x 20 I00 I000 l0,000

FREOUENCY- (.25.

F IG. 4

7 FIG. 7

NEGATIVE mpa'omvcs CONVERTER FIG. 8

3 M 7 T 1 I! II R NEGATIVE lMPEDANCL co/vvmrm 4 F IG. 6 2

/N VENTOR NEGATIVE J. G. L/NV/LL IMPEDANCE CONVERTER RL BY k WfW ATTORNEY April 9, 1957 J. G. LlNVlLL 2,788,496

ACTIVE TRANSDUCER Filed June 8, 1953 2 Sheets-Sheet 2 FIG/3 51 W 2 /7 I R6 5 AvAvAv 5 u u NEGATIVE s IMPEDANCE 5 CONVERTER C? L.

FIG. /4

FREOUENC Y- c. P. 5.

FIG.

m "J 9 E 0 b; w 2 .z e & v m 0 r:

I00 I000 |0,000 20,000 600 I000 I400 I800 FREOUENC Y- 0/? s. FREOUENCV- 0. PS.

F l6. l0-

l 3/ 30 3g a4 NEGATIVE NEGATIVE lMPDANCE AMP 'IMPEDANCE I 5; CONVERTER CONVERTER L J 1 0 Y J a 6 is 27 26 lNl ENTOR J G. LIN V/LL ATTORNEY nited States Patent "ice 2,788,496 it ACTIVE TRANSDUCER John G. Linvill, Whippany, N; 1., assignor to Bell Telephone Laboratories, Incorporated, New York, N. Y., a corporation of New York Application June 8, 1953, Serial No. 360,301

3 9 Claims. (c1. ass-so procedure for an active transducer having an unrestricted transmission characteristic. Other objects are to reduce the loss, or even provide a gain, in the transmission band of an active wave filter. Another object is to eliminate one type of reactor in a transducer without restrictingits transmission characteristic. Another object is to reduce the number of component reactors required. Further objects are to decrease the size and cost of a transducer which must meet particularly severe transmission requirements.

The network designer oftenencounters transmission requirements which he cannot meet economically in a passive transducer. For example, a wave filter which will transmit very low frequencies usually requires very large and expensive inductors. For this case, a passive transducer made up of only resistors and capacitors,

usually called anR-C network, .is attractive. However, for a given characteristic, an R-C filter introducesex cessive loss in the transmission band and requires many more elements than does a filter which includes both inductors and capacitors. These defects may be overcome by including an active element. One active transducer of this type is disclosed in United States Patent 2,549,065, issued April 17, 1951, to R. L. Dietzold. .The active element is a stabilized feedback amplifier. The

passive elements are resistors and only one type of reactor, either capacitive or inductive.

The present invention is directed to another type of active transducer with unrestricted transmission characteristic. The circuit comprises two passive networks and a negative impedance converter connected in tandem between them. The negative impedance converter, hereafter called simply a converter, has an impedance conversion ratio, designated M, which is negative. Thus, it presents at one pair of its terminals an impedance which is M times the impedance connected to its other terminal pair. Each of the passivenetw orks is made up of one or more resistors and one or more reactors. The reactors may include both inductors and capacitors, or they may be all of the same type. Usually, it is preferred to use only capacitors for the reactors. Resistors and capacitors are, in general, cheaper, smaller, and more nearly ideal elements than are inductors. The passive networks may be simple structures, either balanced or unbalanced. They may, for example, be lattice, ladder, bridged-T, or twin-T networks. 7 reactors required in the networks is no greater than that required in apassive transducer having a comparable transmission characteristic. 3

At their ends facing the converter, the passive networks have driving-point impedances oneof which is equal to -M times the other at oneor more preselected active transducer in accordance with the invention.

2,788,496 Patented Apr. 9, 1957 natural frequencies of the transducer. Thenatural frequencies are the complex frequencies at which the transfer impedance of the transducer is infinite. The transfer impedance of the transducer is unrestricted as to roots and poles in that-any physically realizable transmission characteristics may be provided. This is true even when the passive networks are restricted to reactors of only one type. As examples, lowpass, high-pass, and bandpass wave filtersare disclosed. The loss in the trans mission band may be reduced by the active portion of the transducer. Also, a gain in the band may be provided by proper design of t he converter or by inserting an amplifier in tandem between two active transducers.

The nature of the invention and its various objects, features, and advantages will appear more fully in the following detailed description of preferred embodiments illustrated in the accompanying drawing, of which Fig. l is a block diagram of an active transducer in accordance with the invention;

Fig. 2 represents the complex frequency plane it on which a re plotted the poles of the transfer impedance of a typical low-pass filter in accordance with the invention;-

Fig. 3 shows, on the complex frequency plane, the distribution of the zeroes and the poles of the difierence between the driving-point impedances of the passive networks at their ends facing the converter, for the low-pass filter example; e n

Fig 4 shows the circuitof the low-pass filter when the passive networks are unbalanced ladder structures made up of resistors and shunt capacitors; r i

Fig. 5 is atypical relativeresponseversus frequency characteristic obtainable with the low-pass filter of'Fig. 4; Fig. 6 shows the circuit of a high-pass filter in accordance with the invention, in which the passive networks are constituted by resistors and shunt inductors;

Figs. 7, 8, and 9 show, respectively, the pole-zero dis tribution, the network configuration, and a typical characteristic of a second high-pass filter in accordance with the invention;

Fig. 10 shows two active band-pass filters in accordance with the invention connected in tandem by an amplifier;

Fig. 11 shows the relative response characteristic of the filter of Fig. 10;

Fig. .12 shows a generalized lattice structure which may be used for the passive networks in Fig. 1; and

Figs. 13 and 14 show, respectively, the network configuration and the characteristic of a low-pass filter in accordance with the invention in which a twin-T network provides a peak of attenuation at a finite frequency.

Fig. 1 shows in block diagram an embodiment of an The transducer comprises two four-terminal passive networks 1 and 2 and an interposed negative impedance converter 3 connected in tandem between a pair of input terminals 5, 6. and a pair of output terminals 7, 8. A suitable source of signals, not shown, may be connected to the The total number of input terminals and a suitable load, not shown, may be connected to the output terminals. l

The converter 3 is an active four-pole network which presents at its input terminals 9, 10 an i impedance which, over the frequency range of interest, is equal to M times the impedance connected to its output terminals 11, 12. The converter 3 may be of thevacuum-tube type, examples of which are disclosed in the paper by J. L. Merrill, Jr., entitled, Theory of the negative impedance converter, in the Bell System Technical Journal, vol. XXX, No. 1, January 1951, pages 88 to 10 9. Preferably, however, the converter 3 is of the transistor type, suitable circuits of which are disclosed in the copending United States patent application of R. L. Wallace, Jr.,

and the present applicant, Serial No. 310,084, filed September 17, 1952, now Patent No. 2,726,370 issued December 6, 1955. A converter using one or more transistors is preferred because it may be designed to have more nearly ideal characteristics. Therefore, a transducer employing a converter of this type may be designed to meet a prescribed transmission characteristic within closer limits, and the characteristic will be more stable with time.

The converter 3 has a current transfer ratio Mi, which is the ratio of the input current'lato the output In, and a voltage transfer ratio Me, given by the ratio of the input voltage Ea to the output voltage Eb. One of these ratios is always negative. The ratio M1 is substantially unity if junction transistors are used in the converter. Its value may be approximately doubled by using point contact transistors, but at a sacrifice in the stability of the converter. This ratio may, of course, be extended by associating a transformer with the converter. As explained in the above-mentioned patent application, the magnitude of Me may be selected within wide limits.

The impedance conversion ratio M of the converter, which is the ratio of Me to Mt, may be given any negative value within a Wide range by choosing appropriate values of Me and Mi. A judicious choice of M may help in obtaining convenient values for the component impedance elements in the passive networks 1 and 2.

It is obvious that a given impedance conversion ratio M may be obtained with an infinite number of combinations of the ratios Ml and Me. However, their choice determines the gain factor G, which is the product of M1 and Me. If the networks 1 and 2 and the ratio M all remain fixed, the power gain of the transducer from input to output is inversely proportional to the magnitude of G.

The theory of the design of a transducer of the type shown in Fig. 1 using such a converter will now be presented. The transfer functions of a four-terminal transducer made up of lumped impedance elements are rotational functions of the frequency 7". The analysis is simplifiedby the introduction of the parameter p, called the complex frequency and defined as where ois the real part, in is the imaginary part, and w is the radian frequency 2117. A fuller discussion of the concept of complex frequency and the complex frequency plane may be found, for example, in chapter II of the book Network Analysis and Feedback Amplifier Design, by H. W. Bode, published by D. Van Nostrand Company, New York, 1945 The transfer impedance Z'rof any four-terminal network may be expressed as the ratio of two polynomials in P giving The transfer impedance becomes infinite at the complex frequencies at which the denominator D(p) is zero. Therefore, these complex frequencies are the natural frequencies of the network. In passive R-C network-s, or R-L networks (comprising only resistors and inductors), these zeroes are restricted to the negative real axis of the complex frequency plane. This constraint seriously limits the quality of approximation to an ideal filter characteristic obtainable if N(p) and D(p) are polynomials of a limited degree. Active R-C or R-L networks, however, can have natural frequencies anywhere in the left-half plane, the same as passive networks comprising resistors, inductors, and capacitors.

In the ideal case, if the current transfer ratio Mt is unity, the current Ia entering the converter 3 is equal to the current Ib leaving it, that is,

lo=lb 3 Assuming a voltage transfer ratio Me of l, the input voltage Ea is the negative of the output voltage, that is,

output terminals 7, 8, which is the ratio of the output voltage E2 to the input current I1 when the output current I2 is zero, is given by In Equation 5, Z122. is the transfer impedance of the network 1, Z121; is the transfer impedance of the network 2, 2225 is the driving-point impedance of the network 1 at the terminals 9, 10, and Zllb is the driving-point impedance of the network 2- at the terminals 11, 12. The derivation of Equation 5 involves, as an intermediate step, the evaluation'of the input current In. to the converter 3. The load at the terminals 9, 10 of the network 1 is the input impedance Z: seen at the terminals 9, 10 of the converter 3, given by ZI=MZ11b where M is the impedance conversion ratio of the converter. When M is -1, as assumed here,

ZI=-Z11b Therefore, in accordance with network theory,

I Z I 1 12a 8 a 22Q 11b But since the current Ib flowing into the network 2 is equal to la, the output voltage E2 is It will be observed from Equation 11 that, in the present case whereMe and Mi are each of unit magnitude, the transducer obeys the law of reciprocity only in magnitude. The transfer impedance changes sign when the input and output terminal pairs are interchanged.

The driving-point impedance Z11 of the transducer at the input end is E Z 0 2 Z Z 11 I1 no 22a llb where E1 is the input voltage and Zlla. is the driving-point impedanceof the network 1 at the terminals 5, 6. The driving-point impedance Z22 at the output end of the transducer is where Z221; is the driving-point impedance of the network 2 at the terminals 7, 8.

In Equation 2, the zeroes of the numerator N(p) are associated with the structure of the network, not with its natural frequencies. For instance, ladder networks have zeroes of transfer impedance at frequencies where shunt elements become short circuits or series elements become open circuits, Thus, an R-C ladder-type network. made up of resistors and. three. shunt; capacitors will have three zeroes of transfer impedance at infinite frequency, where the capacitors are short circuits, irre spective of the values of the capacitors or the natural frequencies of the network. Lattice, bridged-T, and twin-T networks will have zeroes of transmission at frequencies where a bridge-like balance occurs, regardless of the location of the natural frequencies of the complete network.

A suggested procedure for designing an active transducer in accordance with the invention comprises three steps. First, the designer prescribes the desired transfer impedance, within a constant multiplier, in the form of Equation 2. Next, he selects the zeroes of the denominator D(p) as the natural frequencies of an active network of the type shown in Fig. 1. Then, by a method to be described below, he obtains for the networks 1 and 2 a pair of driving-point impedances Z2221, and Zllb which are consistent with these natural frequencies. Finally, he synthesizes networks which have the driving-point impedances Z223. and Zllb. The networks chosen must also be of a form which will provide the desired zeroes of transmission at the zeroes of the numerator N(p).

To illustrate the application of the procedure outlined above, the design of several active wave filters of the form shown in Fig. 1 will now be described.- The first example is a low-pass filter which has a Butterworth characteristic, a cut-off frequency-1 fc of 1000 cycles per second, and an attenuation rising at the rate of 18 decibels per octave of frequency. In the form of Equation 2, the transfer impedance of such afilter. may be written as N (P) K 1m) r (if 21 21r +2 21 21rf where K is a numerical constant which determines the impedance level of the filter. The transfer impedance will have three poles, corresponding to the zeroes of D(p), and three zeroes. The. zeroes all fall at infinite frequency. The poles are known to fall on a semicircle in the left half of the complex frequency plane. Fig. 2

shows the poles Pa, $5., and Pb plotted on a complex frequency plane in which the real (or tr) axis is horizontal and the imaginary (or its) axis is vertical. The coordinates of these poles are Pa=21r500+j21r866 Fa=-21r500j21r866 (16) Pb= 21r1000+j0 (17) It is seen that the poles Pa and P: are conjugates and the pole Pb falls on the negative real axis.

The natural frequencies of the filter must occur at these complex frequencies. The circuit shown in Fig. 1 will have natural frequencies when the driving-point impedance Z2211 of the network 1 and the driving-point impedance Zllb of the network 2 are equal. At each of these frequencies, the sum Z of the impedance -Z1lb seen looking into the converter 3 at its input terminals 9, 10 and the impedance Z222. is zero, that is, when The zeroes of Z will, therefore, be at the complex fretmf where D(p) isithe denominator in Equation 14. selection of the'points 0'1, 02, and as is arbitrary as far as the transfer impedance Z21 of; the filter is concerned. Of course, none should coincide with the pole Pb. However, as explained below, the driving point impedances and othercharacteristics of the filter are influenced by the selection.

There remains, now, only the synthesis of the passive R-C networks 1 and 2. A suggested method involves, first, the expansion of Equation 19 in partial fractions. It is found that the residues are always real but may be positive or negative. The terms are divided into a first group with positive residues and a second group with negative residues. It is known that any function with simple poles on the negative real axis and positive real residues in those poles is the driving-point impedance of an R-C network. Therefore, the first group of terms is associated with the impedance Zen. of the network 1. The second group of terms is associated with the impedance Zllb of the network 2. Although the network 2 can only provide positive residues in the poles of Ziib, they will appear as negative residues when viewed from the input terminals 9, 10 of the converter 3. It is now assumed that the networks 1 and 2 will be ladder-type structures comprising resistors and shunt capacitors. The required values of the component elements are found by making a Cauer synthesis. I For the distributions of the critical frequencies Pa, Pa, Pb, 0,, (r and 0' shown in Fig. 3, the networks 1 and 2 will have the configurations shown in Fig. 4. The network 1 comprises three series resistors R1, R2, and R3 and. two shunt capacitors C and C2. The network 2 consists of the parallel combination of a resistor R4 and a capacitor C3. The fact that the forms shown for the networks 1 and 2 are correct can be understood from the following analysis. From the distribution of the poles of Z shown in Fig. 3, it can be observed that the residues in the poles at a, and a, are positive while the residue in the pole at a is negative. Therefore, the constant K, appearing in Equation 14, and the partial fractions associated with the poles at a, and a, are identified with the impedance Z222. of the network 1 and the partial fraction associated with the pole a, is identified with the impedanceZnb as seen through the converter 3. Accordingly, the network 1 will include two capacitors and the network 2 only one capacitor. The required values of the resistors R1, R2, R3, and R4, in ohms, are .600, 2040, 1200, and 1300, respectively, and the values of the capacitors C1, C2, and C3, in microfarads, are 0.384, 0.172, and 0.0820, respectively. It is assumed that the impedance conversion ratio M of the converter 3 is l. Fig. 5 shows a typical relative response characteristic obtainable with the low-pass filter of Fig. 4. It is assumed that the signal source of voltage E1 connected to the terminals 5, 6 has zero internal impedance. if the source has an equivalent series impedance R0, the series resistor R1 is replaced by a resistor of smaller value R1 given by R1'=R1R0 (20) In Fig. 5, the ratio of the output voltage E2 to the input voltage Ei, expressed in decibels, is plotted against the frequency in cycles per second, on a logarithmic frequency scale. The. response relative to that at zero frequency is shown. By properly choosing the gain factor G of the converter 3, there may be provided any gainwhich is consistent with the stability'of the con verter and the networks 1 and 2.

i Fig. 6 shows the circuit of a second wavefilter'in accordance with the invention. This is a high-pass filter having a Butterworth characteristic and an attenuation which rises at the rate of 18 decibels per octave. The passive networks 1 and 2 are ladder-type RL structures, comprising resistors and shunt inductors. Their configurations are the same as those shown for the low-pass filter of Fig. 4 except that the three shunt capacitors C1, C2, and C3 are replaced, respectively, by the three shunt inductors L1, L2, and L3. Otherwise, the circuit of Fig. 6 is similar to that of Fig. 4. The component resistors and inductors required in the filter of Fig. 6 may be evaluated by the same procedure described above in connection with the filter of Fig. 4.

Figs. 7, S, and 9 relate to another high-pass wave filter in accordance with the invention. The filter has a Butterworth characteristic with an attenuation rising at the rate of 24 decibels .p'er octave. On the complex frequency plane of Fig. 7, the

points

14, 15, 16, and 17 marked by xs are poles of the transfer impedance Z21 and zeroes of the impedance Z. These points all fall on a semicircle 23 in the left half of the plane and have a uniform spacing S from the origin. They also have a uniform angular spacing U of 45 degrees on the semicircle 23, and the points 14 and 17 have equal angular spacings of U/2 from the for axis. Thus, the points 14 and 17 are conjugate, as are also the points and 16.

The

points

19, 20, 21, and 22 marked by +s are poles of Z. These poles all fall on the negative real axis and have spacings of S1, S2, and S3, as shown. The displacement of the nearest pole 22 from the origin is St. The impedance Z21 has a fourth-order zero at the origin, as shown by the circle 24-.

Fig. 8 is a schematic circuit of a filter having the distribution of poles and zeroes shown in Fig. 7. Each of the passive networks 1 and 2 is an RC ladder structure comprising two series capacitors and three shunt resistors. The procedure described in connection with Fig. 4 may be used to find the required values of these elements. Fig. 9 shows a relative response characteristic obtainable with the filter of Fig. 8 when the source E1 and the load Rn each have a high impedance compared to the resistance of the end resistor connected in parallel therewith.

In accordance with the invention, two or more active transducers may be connected in tandem and isolated from eachother by one or more amplifiers. As an example, Fig. 10 shows two band-pass wave filters 25 and 26 connected in tandem between input terminals 5, 6 and output terminals 7, 8 and an interposed amplifier 27, preferably of the transistor type. The filter 25 comprises two

passive networks

29 and 30 connected through a converter 31. In the filter 26, the corresponding units are designated 33, 34, and 35. Each of the

networks

29 and 33 is constituted by the series combination of a resistor and a capacitor in a series branch. Each of the

networks

30 and 34 is made up of the parallel combination of a resistor and a capacitor in a shunt branch.

Fig. 11 shows a typical relative response characteristic obtainable with the filter of Fig. 10. The midband frequency is located at 1000 cycles per second and the band width is approximately 200 cycles. Zeroes of tnan-smission occur at zero and infinite frequencies. By properly designing and adjusting the amplifier 27, considerable gain in the transmission band may be achieved, if desired.

In the filter circuits shown in Figs. 4, 6, 8, and 10, the zeroes of transmission occur only at zero or infinite frequency. With the RC and L-C ladder-type passive networks used in these filters, it is impossible to obtain a zero of transmission at a real frequency between zero and infinity. However, in accordance with the inven- 8 tion,-intermediate zeroes of transmission (peaks of attenuation) may be provided by using R-C or L-C lattice networks, or unbalanced equivalents thereof such as bridged-T or twin-T structures.

Fig. 12 shows the generalized circuit of a lattice structure which may be used for either or both of the passive networks 1 and 2 in Fig. 1. The lattice comprises two equal series impedances Za, Za and two equal diagonal impedances Zb, Zb connected between a pair of input terminals 37, 33 and a pair of

output terminals

39, 40. The synthesis of an active filter of the type shown in Fig. 1 using two such lattice networks to provide intermediate attenuation peaks proceeds as explained above in connection with Fig. 4 through the specification of the desired transfer impedanceZzi, the selection of the impedance Z, and the evaluation of the driving-point im pedances Z229. and Z1111. At this point, one has the driving-point impedances of the networks 1 and 2 and knows the frequencies at which these networks should introduce peak-s of attenuation.

The driving-point impedance Zn at either end of the lattice of Fig. 12 is D 2 B p) and the transfer impedance Zr in either direction is Z I) Z a T p Z 23 It is already known that the driving-point impedance of the network 1 will be Z223. and the driving-point impedance of the network 2 will be 2111;. When one substitutes expressions for the driving-point and the transfer impedances for the lattices, as given in

Equations

22 and 23, into an equation of the form of 5, the numerators of the expressions for Z122. and ZlZb are found to include constant multiplier-s and all of the factors of N(p) but no others. This is so because the drivingpoint and the transfer impedances of a lattice have the same denominator B(p). These factors of N(p) are divided in conjugate pairs between the numerators of Z12a and Z121) in any way which makes each numerator of no higher degree than the associated denominator. One thus obtains an expression for the transfer impedance of each of the lattices 1 and 2 in the form of Equation 23 in which the factors of the numerator T(p) are specified but a constant multiplier is yet to be determined. Expressions for the impedances Za and Zb of each of the lattices 1 and 2 may now be found by adding Zn to Zr to obtain Zb and subtracting Zr from Zn to get Za. Now, if the i-mpedances Za and Zb are to be R-C structures, one selects the largest permissible values for the constant multipliers. Finally, he determines the configurations of these branches and the required values of the component resistors and capacitors to provide the impedances Za and Zb. The synthesis of an RC lattice network from a specified drivingpoint impedance and a transfer impedance specified within a constant multiplier is described in greater detail, for example, in the paper by J. L. Bower and P. F. Ordung entitled, The synthesis of resistor-capacitor networks, in the Proceedings of the I. R. B, vol. 38, No. 3, March 1950, pages 263 to 269.

It is often desirable to convert the lattices thus found for the networks 1 and 2 into equivalent unbalanced structures. However, it is usually difiicult, if not impossible, to predict the configuration of the equivalent unbalanced network from the form of the lattice. When it is desired to obtain, for one of the networks 1 and 2, a particular type of unbalanced structure, one may use a modified method which avoids first finding a lattice and then seeking the unbalanced equivalent. This method, which is applicable to either R-C or -RL networks, involves exchanging part of the inherent latitude inthe choice of poles of the impedance Z for a constraint in the form of one of the networks 1 and 2.

19 There will now be presented, as an example of this modified method, an active low-pass wave filterg-cf the type shown in Fig. 1 in which an unbalanced, twin-T, R-C network is used to provide a peak of attenuation at a finite frequency. As shown in Fig. 13, the passive network 1 comprises a resistor R5 in series with a parallel twin-T structure. One of the Ts is constituted by the two equal series resistors R6, R6 and an interposed shunt capacitor C4. The other is made up of the equal series capacitors C5, C5 and the interposed shunt resistor R7. The network 2 is an R-C ladder structure with a series resistor, a shunt resistor, and two shunt capacitors which will provide a 12-decibel per octave attenuation rate at the higher frequencies. Fig. 14 gives the relative response characteristic of the filter. The band cuts off at 1000 cycles per secohd. The attenuation peak occurs at 2000 cycles.

As in the previous examples, the first step in designing the circuit of Fig. 13 is to select the poles and zeroes of the transfer impedance Z21 of the filter to provide an acceptable characteristic, that is, one having a reasonably fiat pass band from zero to 1000cycles, an attenuation peak at 2000 cycles, and a high-frequency attenuation rate of 12 decibels per octave. The method described in my paper entitledyThe approximation with rational functions of prescribed magnitude and phase characteristics, in the Proceedings of the I. R. E., vol. 40, No. 6, June 1952, pages 711 to 721, may be followed in determining the locations of the poles and zeroes.

In the network 1, the initial resistor R5 .is selected in accordance with the impedance of the voltage source E1. The twin-T structure is designed to have reasonable values for the component elements Re, R6, R7, C4, C5, and C5 and to provide the attenuation peak at 2000 cycles. A method of designing such a twin-T network is presented, for example, in United States Patent 2,106,785, to H. W. Augustadt, issued February 1,' 1938.

Next, the driving-point impedance Z22. of the net- Work 1 at its end facing the converter 3 is expressed in the form of a partial fraction expansion. Then, the driving-point impedance Z1111 of the network 2 at its end facing the converter 3 is determined. This can be done through a consideration of the expression for Z, which, as given by Equation 18, is the difference between Z22a and Z111). One knows certain factors of Z at this point. Its numerator has a constant multiplier K1 and includes the factors of D(p). Its denominator includes as factors the denominator of Z229. and the denominator of Zub, as yet undetermined but known to be of the form (p-l-a) (p+b), Where a and b are as yet unknown. One writes an expression for Z, employing the known factors and putting the unknown factors in literal form, and. expands this expression in partial fractions. By known mathematical methods, the unknown constants K1, a, and b are selected consistent with the requirements that the residues in the poles of Z2211 must be positive and of the values already known and that the residues in the remaining poles, associated with Z11b, must be negative. With the constants K1, a, and b determined, one may write an explicit expression for Z. The impedance Zllb is found by substracting Z from Z22a- With Ziib thus determined, the final step is to synthesize an R-C ladder structure with shunt capacitors to obtain the network 2 shown in Fig. 13.

The synthesis procedure described above in connection with Fig. 4 results in a filter having a desired prescribed transmission characteristic. However, it does not lead to unique structures for the passive networks 1 and 2 because of the permissible latitude in the choice of the poles of the impedance Z. An infinite number of embodiments of the networks 1 and 2 may be found which will provide the desired transmission characteristic for the filter as a whole. The different filters will, however, differ in their driving-point impedances and also in the effects on the .transmissioh characteristic of imperfec' tions in the converter 3. w r I In some applications, it is important to be able to prescribe the driving-point impedance of the filter at one or both of the pairs of terminals 5 6 and 7-$. Part of the latitude in the choice of the poles of Z may be used to obtain amore desirable driving-point impedance at one or both of the ends of the filter. Of course, any desired impedance transformation Within the transducer may be provided by properly choosing the impedance conversion ratio M of the converter 3. r

As seen in Equation 6, the input impedance Zr of the converter 3 depends upon the impedance conversion ratio M. In a practical converter, the ratio M changes with time, temperature, or other environmental conditions, causing a corresponding change in Z1.- Equation 5 shows that a change in 'ZI from the value Z11b will not affect the zeroes of the transfer impedance Z21 of the filter but will change the poles. A change in the locations of these poles will-cause a change in the transmission characteristic of the filter. For example, moving a pole such as 14 in Fig.7 toward the its axis to a new position which is a fractional part H of its original distance from this axis will cause a maximum change I in the response characteristic of the filter given in decibels by A mathematical analysis shows that, for a given change in the ratio M, the resu1ting'shift.in pole location and consequent change: T in the filter characteristic are dependent upon the distributionof the poles of the impedance Z on the negative real complex frequency axis o'. From the standpoint of drift of the filter characteristic, a .good practical rule to apply is as follows: Assuming that Z has W poles, one is spaced from the origin by a distance Q which is between half and triple the average displacement of the zeroes of Z from the origin, another is located not more than Q/W from the origin, and the remaining poles are spaced approximately uniformly between these two. A somewhat involved mathematical analysis, here omitted, shows that in general the drift will be less with such a distribution of poles than if eitherof the end poles falls outside of the limits given, even though the spacing between poles is kept uniform. The analysis further shows that a nonuniform spacing, especially one in which adjacent poles have a spacing much less than the average, aggravates the drift problem. In Fig. 7, it is seen that the distribution of the four

poles

19, 20, 21, and 22 of the impedance Z falls Within the rule.. The distance Q from the origin of the most remote pole 19 is greater than 8/2 but not more than 38, where S is the radius of the semicircle 23 on which all of the

zeroes

14, 15, 16, and 17 of Z fall. Also, the pole spacings S1, S2, and S3 are: approximately equal.

It is to be understood that the above-described arrangements are illustrative of the application of the principles of the invention. Numerous other arrangements may be devised by those skilled in the art without departing from the. spirit and scope of the invention.

What is claimed is:

1. An active transducer. comprising two passive networks and a negative impedance converter connected in tandem therebetween, said networks having at their ends "facing said converter driving-point impedances which, at

a prescribed natural frequency of said transducer, are related to each other by a numerical factor equal in magnitude to the impedance conversion ratio of said converter.

2. A transducer in accordance with claim 1 having a transfer impedance with unrestricted zeroes and poles, said networks comprisingresistors and only a single type of reactor.

3. A transducer in accordance with claim 2 in which said single type of reactor is capacitive.

4. A transducer in accordance with claim 2 in which said single type of reactor is inductive.

5. A transducer in accordance with claim 1 in which said impedance conversion ratio is approximately 1.

6. A transducer in accordance with claim 1 in which said impedance conversion ratio has a magnitude greater than unity.

7. A transducer in accordance with claim 1 said converter is of the vacuum-tube type.

8. A transducer in accordance with claim 1 in which said converter is of the transistor type.

9. A transducer in accordance with claim 1 having the transmission characteristic of a low-pass filter.

10. A transducer in accordance with claim 9 in whicl said transmission characteristic has a peak of attenuation at a finite frequency other than zero.

11. A transducer in accordance with claim 1 having the transmission characteristic of a band-pass filter.

12. A transducer in accordance with claim 1 having the transmission characteristic of a high-pass filter.

13. A transducer in accordance with claim 1 having the transmission characteristic of a wave filter with a peak of attenuation at a finite frequency other than zero.

14. A transducer in accordance With claim 1 in which one of said passive networks is a twin-T structure.

15. A transducer in accordance with claim 14 in which said twin-T structure has a peak of attenuation at a finite frequency other than zero.

16. A transducer in accordance with claim 14 in which said twin-T structure comprises only resistors and capacitors.

17. A transducer in accordance with claim 14 in which said twin-T structure comprises two T-networks connected in parallel, one of said T-networks including two series resistors and an interposed shunt capacitor and in which the other of said T-networks including two series capacitors and an interposed shunt resistor.

18. A transducer in accordance with claim 1 in which one of said passive networks is a ladder-type structure.

19. A transducer in accordance with claim 18 in which said ladder-type structure comprises only resistors and capacitors.

20. A transducer in accordance with claim 18 in which said ladder-type structure comprises only resistors and inductors.

21. A transducer in accordance with claim 18 in which said ladder-type structure comprises a series resistor and a shunt capacitor.

22. A transducer in accordance with claim 18 in which said ladder-type structure comprises a series resistor and a shunt inductor.

23. A transducer in accordance with claim 18 in which said ladder-type structure comprises a series capacitor and a shunt resistor.

24. A transducer in accordance with claim 1 in which each of said passive networks is a ladder-type structure.

25. In combination, two transducers in accordance with claim 1 and an amplifier connected in tandem therebetween.

26. A transducer in accordance with claim 1 in which the difference between said driving-point impedances is an impedance having a plurality of zeroes and a plurality of poles, said poles being W in number, all of said poles being located on the negative real axis of the complex frequency plane, one of said poles being spaced from theorigin by a distance Q which is between half and triple the average displacement of said zeroes from said origin, a, second of said poles being located not more than Q/ W from said origin, and the remaining poles being spaced approximately uniformly between said first and said second poles.

, 27. A transducer in accordance with claim 26 in which said zeroes fall approximately on a semicircle in the left half of said plane.

28. A transducer in accordance with claim 27 in which said zeroes are approximately equally spaced on said semicircle.

29. A transducer in accordance with claim 28 in which two of said zeroes are conjugate.

36. A transducer in accordance with claim 1 in which said impedances are also related to each other by said ratio at a second prescribed natural frequency of the transducer.

31. A transducer in accordance with claim 1 in which said impedances are also related to each other by said ratio at a plurality of other prescribed natural frequencies of the transducer.

32. A transducer comprising two passive networks and an interposed negative impedance converter connected in tandem, said converter having an impedance conversion ratio approximately equal to -1, said networks comprising resistors and only a single type of reactor, and said networks having at their ends facing said converter driving-point impedances which are substantially equal at a preselected natural frequency of the transducer.

33. A transducer in accordance with claim 32 in which said single type of reactor is capacitive.

34. A transducer in accordance with claim 32 in which said single type of reactor is inductive.

35. A transducer in accordance with claim 32 in which one of said networks is of the ladder type.

36. A transducer in accordance with claim 32 in which each of said networks is of the ladder type.

37. A transducer in accordance with claim 32 in which said impedances are also substantially equal at a second preselected natural frequency of the transducer.

38. A transducer in accordance with claim 32 in which said impedances are also substantially equal at a plurality of other preselected natural frequencies of the transducer.

39. A transducer in accordance with claim 32 in which the difference between said driving-point impedances is an impedance having a plurality of zeroes and a plurality of poles, said poles being W in number, all of said poles being located on the negative real axis of the complex frequency plane, one of said poles being spaced from the origin by a distance Q which is between half and triple the average displacement of said zeroes from said origin, a second of said poles being located not more than Q/W from said origin, and the remaining poles being spaced approximately uniformly between said first and said second poles.

References Cited in the tile of this patent UNITED STATES PATENTS 2,093,665 Tellegen Sept. 21, 1937 2,197,348 Roberts Apr. 16, 1940 2,243,440 Roberts May 27, 1941 2,549,065 Dietzold Apr. 17, 1951