CA1045220A - Subnetworks for filter ladder networks - Google Patents

Abstract

Abstract of the Disclosure The invention concerns a one-port RC subnetwork for active filter ladder networks. The subnetwork includes a single pair of terminals, to one of which there is applied a reference potential, the subnetwork consisting of an operational amplifier having an input point and an output point. A first capacitor and a second capacitor are connected in series between the single pair of terminals and a feedback path from the output point of the amplifier with the parallel circuit of a first resistor and a third capacitor to the junction of the first and second capacitors. The first resistor and the third capacitor are dimensioned so that the feedback path has an effective time constant essentially equal to the time constant of the amplifier in the fall-off region of gain-frequency characteristic thereof in order to extend the effective band width of the subnetwork, so that in the equivalent circuit the series circuit of the subnetwork further comprises in series: a parallel connection of two elements each having a complex impedance l/bs.

Description

The invention relates to subnetworks for an active filter ladder network. Such networks use an inductor simultation technique employing a new resistive and capacitive components with active devices, such as amplifying elements, to avoid the use of inductive coils.
It is known to produce gyrator circuits which can be modified to give a substantially pure l/s2 (or pure s ie inductive) element. However, such circuits utilize more than one operational amplifier, generally an operational amplifier. Circuits providing 1/s2 elements are known, however, as it is also known that such elements are unstable, their use in ladder filter networks has not been pursued very far. Now a method ha~ been found of utilising the properties of simulated l/s2 elements in subnetworks suitable for stable use in a ladder filter network; such as a third order low-pa~s "pseudo-elliptic" filters.
It is an object of the present invention to provide a subnetwork having an input terminal and a reference terminal, which subnetwork is equivalent to a component having an impedance of the form l/as2 in series or in parallel between said terminals with a component having an impedance of the form l/sb, where a and b are constants and s is the complex frequency variable.
According to the invention there is provided a one-port RC subnetwork for active filter ladder networks comprising a single pair of terminals, to one of which there is applied a reference potential, said subnetwork consisting of an operational amplifier having an input point and an output point and in which a first capacitor and a second capacitor are connected in series between the single pair of terminals; and a feedback path from the output point of the amplifier with the parallel circuit o~
a first resistor and a third capacitor to the junction of the ;- , ~
~: , , ' ' lV4~'~ZO
first and second capacitors, the first resistor and the third .
capacitor being dimensioned such that the feedback path has an effective time constant essentially equal to the time constant of the amplifier in the fall-off region of gain-frequency characteristic thereof in order to extend the effective band width of the subnetwork, so that in the equivalent circuit the series circuit of the subnetwork further comprises in series: .
a parallel connection of two elements each having a complex impedance l~bs.
The invention will now be de~cribed, by way of example, with reference to the accompanying diagrammatic drawings in which:-FIGURE 1 shows a first subnetwork embodying the invention; :
FIGURE 2 shows the equivalent circuit of FIGURE l; : :
FIGURE 3 shows a second subnetwork embodying the invention;
FIGURE 4 show~ the equivalent network of FIGU~ES 3 and 6, FIGURE 5 shows a third subnetwork embodying the invention;
FIGURE 6 show-q a fourth subnetwork embodying the invention;
FIGURE 7 shows the equivalent network of FIGU~E 5; :
FIGURE 8 shows a third order low-pass filter incorporating the subnetwork of FIGURE l;
FIGURE 9 shows another third order lo.w-pass filter :
incorporating the subnetwork of FIGURE 5;
FIGURE 10 shows a further subnetwork;
FIGURE 11 shows a circuit similar in component configur-ation to that of FIGURE 5;
FIGUnE 12 shows the equivalent circuit of FIGURE. 11; ~ .
FIGURE 13 shows a modified form of subnetwork shown in FIGURES 5 and 11;

lV45ZZ0 FIGU~E 14 shows the equivalent circuit of FIGURE 13.
Referring now to FIGURE 1 the first subnetwork comprises a pair of input terminals 1 and 2 across which is coupled a pair of capacitors 3 and 4 in series to the junction between which there is connected a resistor 5, the other side of which is connected to the output of an output amplifier 6. The input ,of the amplifier is connected directly to the terminal 1.
FIGURE 2 shows the equivalent network of FIGURE 1 and comprises a l/s element 7 in series with a 1/s2 element 8.
The second subnetwork shown in FIGURE 3 comprises an amplifier 9 having its input and output connected by way of a resistor 10. The input to the amplifier 9 is also connected to the subnetwork input terminal 1 by way of a capacitor 11, The terminal 1 is also connected to the output of the amplifier 9 by way of a capacitor 12. The signals applied to the amplifier 9 are referenced relative to earth by means of a coupling line 13 to the earthed supply line 14 connected to the terminal 2, The equivalent circuit to FIGURE 3 as shown in FIGURE 4 and comprises an l/s element 15 in parallel with 1/s2 element 16.
FIGURE 5 shows a subnetwork which has an equivalent as shown in FIGU~E 7 and comprises an amplifier 17 connected to the input terminal 2 by way of a capacitor 19. The output from the amplifier 17 is also connected to the input terminal 1 by way of a capacitor 20, The equivalent network of the circuit of FIGURE `5 is shown in FIGURE ~ and comprises a l/s element 28 connected in series with a l/s2 element 29 which is al~o in parallel with a l/s element 30.
~ eferring now to FIGURE 6 this subnetwork comprises a differential amplifier 21 one of the inputs of which is connected . . , . . , . ~ -:
. , ~ , - .

to the input terminal 1 by way of a capacitor 22 and to the out-put terminal of the amplifier by way of a resistor 23, The other input terminal to the amplifier 21 is connected to the output terminal by way of a resistor 24 and by way of a capacitor 25 and a resistor 26 in parallel to the earthed line 27 connected to the input terminal 2.
Referring again to the subnetwork shown in FIGURE 1 and the e~uivalent network shown in FIGURE 2, the input impedance Zin of this network is given by: -(l/SCl) + (1/SC2) ,+ (l/S ClC2R3) in l+(l-k) (l/sC2R3) where: Cl and C2 are the values of capacitance of the capacitors 3 and 4 and R3 is~the reistance of the resistor 5; and assuming that the input impedance of the amplifier is sufficiently large to be negligible, the output impedance is sufficiently small to be negligible, and the voltage gain is k; s is the complex frequency variable. If now the gain k is exactly unity the input impedance is:-Zi =(l/SCl) + (1/SC2 ) + (1/S2ClC2R3) which has the general form Zd and as shown in FIGURE 2 the equivalent network input impedance is given by:-Zd = (l/sC4) + (l/s M5)where: C4 is the capacitance of the capacitor 7 and M5 is the value of the l/s2 element 8.
One use of the subnetwork having impedances of the above general form is shown in FIGURE 8. There are several particular properties of this network which make it especially useful in the construction of networks with low sensitivity.
If the amplifier is not 'ideal' the changes in Zd caused - lO~SZ2V
by departures from the ideal are largely negligible. Thus, if the input capacitance is not negligible it can be absorbed into the term C4, while if the output resistance is not negligible it can be absorbed into the resistance term R3 which should be as small as convenient. Furthermore, if gain k departs slightly from unity and in practice it may lie within the range 1 to 0.99 the first order effects are simply to change the value of C4 and M5 slightly, and to introduce into the impedance ~d a small term in 1/s3 which is negligible.
In the subnetwork shown in FIGURE 3 and the equivalent network of FIGURE 4 the input admittance Yin of this network is given by ~

- s/C6 + C7 + s C6C7Rg Yin =
I + sC6 R8/(l+A) where: C6 and C7 are the capacitances of the capacitors 11 and 12 and R8 is the resistance of the resistor 10; and where it has been assumed that the input admittance and the output impedance are negligible and the voltage gain -A. If now the voltage gain is very large (i.e. negligibly different from infinity), the input admittance is:-Yin s(C6 + C7) + s C6C7R8 which has the general form as related to FIGURE 4 and is givenby:

Ye sCg ~ s Mlo where: Cg is the capacitance of the capacitor 15 and Mlo is the value of the element 16. A filter network employing a subnetwork having the general form of admittance as set out above is shown in FIGUR~ 9.
~, ~

. . .
- ~

-` 16)45ZZO
The subnetwork of FT~uRE 3 like the subnetwork of FIGURE
1 also ha~ the property that the departure of the amplifier from the ideal ha~ a largely negligible effect on the admittance Ye~
thus the first order effect of the gain A being finite (but still large) is to alter slightly the value of Cg and Mlo, and to add ~a small admittance proportional to s3 which can be neglected.
This network is therefore especially suited to the construction of networks with low sensitivity, The resistance R8 f the resistor 10 should be made as large as convenient.
Referring now to FIGURE 8 the filter network include~ , a subnetwork substantially as shown in FIGURE 1 and is given the same reference numerals as in this Figure the other elements included in FIGURE 8 comprise an input resistor 28 and a capacitor -29 in series an output resistor 30 and a capacitor 31 in parallel a subnetwork input resistor 32 and an output amplifier 33, In operation the network of FIGURE 8 acts as a third order low-pass 'pseudo-elliptic' filter which with the element values of TABLE 1 ~below) has a passband ripple of ldB, cut-off frequency of 3,40 kHz, and stop-band discrimination of 30 dB, With the element values of TABLE 2 (below) the above parameters are maintained except that the passband ripple of ldB is now 0,1 dB, Resistor 28 = 85.28 Kohms Capacitor 29 - 1847 pF
Re~istor 30 = 144,9 Kohms Capacitor 3 = 12,220 pF
Resistor 32 - 20,00 Kohms Capacitor 4 = 12,220 pF
Resistor S = 191,5 Kohms Capacitor 31 = 468.1 pF

:
. ~, . :
- .
. . .:

4SZZO :

Resistor 28 = 63.62 Kohms Capacitor 29 - 1246 pF
Resistor 30 - 73.89 Kohms Capacitor 3 = 11.95 nF
Resistor 32 = 10.52 I~ohms Capacitor 4 = 11.95 nF
Resistor 5 = 195.8 Kohms Capacitor 31 = 468.1 pF
Referring now to FIGURE 9 which includes a subnetwork similar to that shown in FIGURE 3 and is therefore given the same reference numerals as similar integers in FIGURE 3 the filter also includes input resistor 34 and capacitor 35 feeding the network 36 and output resistor 38 and capacitor 39 feeding the amplifier 37.
The network of FIGU~E 9 acts as a third order low-pass Tchebychev filter, which with the element values of Table 3 has a pass band ripple of ldB and cut-off frequency of 3.4 kHz.

Resistor 34 = 1.303 Kohms Capacitor 35 = 133.86 nF
Resistor 24 = 144.3 Kohms Capacitor 22 = 4.00 nF
Resistor 38 = 4.197 Kohms Capacitor 23 = 4.00 nF
Capacitor 39 = 20.0 nF
Referring now to FIGURE 10, the subnetwork is connected to an earth line 40 and comprises a differential amplifier 41, of which the non-inverting input 42 is connected to an input terminal 43, and the inverting input terminal 44 is connected by way of a resistor 45 to earth. The output 46 of the ampliier 41 is connected by way of a first capacitor 47 to the inpnt 42 and by way of a second capacitor 48 to the input 44. As shown in Figure 2 the e~uivalent circuit comprises an impedance proportional to l~s in series with a capacitance.
Assuming the resistor 45 has a resistive impedance e~ual to Rl, and the capacitors 47 and 48 have capacitive impedance . .
, .

proportional to Cl and C2 respectively, then the l/52 element 9 as shown in Figure 2 has an impedance proportional to ClC2Rl.
This element 8 is in series with a capacitor 7 having an impedance proportional to Cl/2. The differential amplifier 41 has a high input impedance and low ou~put impedance and is arranged to provide unity gain i.e. the output signal is e~ual to the difference between the input signals. The impedance to ground measured from the terminal 43 is given by:-Zin = 1 + 2 S ClC2Rl sCl , Where:
s is the complex frequency variable.
If the input terminals of the amplifier are interchanged,then the admittance to ground measured from terminal 43 is given by:-Yin = s2RlClC2 ~ 2sC

which has the equivalent circuit shown in Fig. 4.
Referring now to Figures 11 and 12 the subnetwork shownin Figure 11 includes an amplifier 49 having unity gain, the input of which is connected to the junction between the resistor 50 and a capacitor 51 connected in series between an input terminal 52 and earth. The output of the amplifier 44 is connected by way of a capacitor 53 to the input terminal 52.
Assuming that the elements of the circuits have the following impedance values:-Resistor 50 = R

Capacitor 53 = Cl Capacitor 51 = C2 then the circuit as shown in Figure 12 includes a resistor 50' having an impedance proportional to the resistor 50 in parallel with a capacitor 53' having a capacitive impedance proportional .
to the capacitor 53 and an 1/s2 element 54 in parallel with a -~
capacitor 51' having a capacitive impedance proportional to the capacitor 51. The l/s element 54 has an impedance proportional to ClC2Rl. The amplifier 49 has a high input impedance and .
low output impedance and in the input frequency range where ~ Cl)2 is less than ~12, The circuit provides an equivalent circuit as shown in Figure 12 comprising a l/s element with a . .
parasitic shunt capacitor in series with a further parasitic capacitor and a resistor. The impedance to ground measured :
from the terminal 52 is given by Zin = 1 1 l+sRlCl sC ~s2C C ~
It will be appreciated that the circuit shown in Figure 11 is similar in layout to that shown in Figure 5 and described above, however the equivalent circuit contains a resistive element in parallel with the input series capacitive element.
Referring now to FIGU~E 13 it will be noted that this ;
Figure is similar to FIGU~E 1 with the addition of a capacitor 55 in parallel with the resistor 5. The other components are s given the same reference numerals as in FIGURE 1. It will be noted that with the addition of this further capacitor from the output of the amplifier 6 to the junction between the capacitors 3 and 4 the equivalent circuit, shown in FIGU~E 14, differs considerably from the equivalent circuit shown in :
FIGURE 2 if the components in FIGU~E 13 are proportioned in a g _ - . - . . , - -.

--` 1~J4SZ2Q
predetermined way which will now be described.
The admittance of the circuit of FIGURE 1 is given by:-S2clc2 + ~l-k) sclG3 Y.n =
S~Cl+ C2) + G3 where k is the actual gain of the amplifier 6. It is evident that if k is substantially equal to unity, the input impedance has the required form and consists of a capacitance in series with an impedance proportional to l/s2.
In practice k is frequency dependent. ~or amplifiers having a single dominant pole, with unity gain frequency given by:-sl = l/Tthe frequency dependence can be represented by the expre~sion (l/k) = 1 + (l/Ko) + sTl :
where Ko is the low-frequency open-loop gain of the amplifier, normally of the order of 104. The second term in the numerator can thus be written k((l/Ko) + sTl)sClG3 . .
The circuit shown in FIGURE 13 is obtained by replacing :
G3 by a parallel combination of the capacitor 55 having a capacitance C55 and the resistor 5 having a resistance R5 as shown in FIGURE 13, where:-G5 + sC4 = G5(1/k)/(1+(l/Ko)) = G (l+sTl/(l+(l/Ko))) The second term of this becomes ( (l/Ko) + STi ) SClG5/(1+(1/Ko) ) = S2ClC4 + SClG5/(l+Ko) The numerator of Yin is now given by s2Cl (C2+ C~,) + SClG5/ ( l+Ko) ~ ~ ~
The only "non-wanted" term is the term involving K

and since Ko is often of the order of 10 or 105 for readily ,. . ~ : .

)45Z20 available amplifiers, this term is negligible for practical purposes. Furthermore its effect decreases as the frequency increases, in contrast with the discrepancies which arise in the uncompensated circuit as shown in FIGURE 1.
The impedance of ~IGURE 13 is therefore given for practical purpo~es, by:-Z = (Cl + C2 + C~in slcl(C2 + C4) S2Cl(C2 + C4) Rs That is, as shown in FIGURE 14, a capacitor 56 having a capacitance Cl equal to the capacitor 3 in sexies with the parallel combination with two capacitors 57 and 58 having capacitancies C2 C55 equal to the capacitancies of the capacitors 4 and 55 respectively, in series with a "1/s2" element 59 having a value equal to Cl (C2 + C55) R5.

This result holds in practice over the frequency range :
where the amplifier gain is following a single-dominant-pole behaviour at frequencies substantially above the pole frequency in other word~ the range where the gain is falling off at 6dB
per octave. For readily available integrated circuit operatianal amplifiers, this range usually extends from 100HZ to lOOkHz, the pole frequency being of the order of 5Hz and the unity-gain frequency being of the order of lMHz the result is achieved, by arranging for the time constant of the parallel combination of the resistor 5 and the capacitor 55 to equal the effective time constant associated with the 6dB per octave fall-off region in the gain-frequency response of the amplifier i.e. where:-R5C4 = Tl/~l+(l/Ko)) Tl .; .: . . .
.. . . ..

1~45Z20 The circuit of FIGURE 13 allows a cheap amplifier ofmoderate band width to be used in a precision filter circuit where otherwise a more expensive amplifier of wide band width would be required, especially in the higher frequency range of frequencies from 1 OOHZ to lOOkHz. The circuit may be used to detect the audio frequency tones used in a telephone system to transmit the dialling digits. .

Claims

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. A one-port RC subnetwork for active filter ladder networks comprising a single pair of terminals, to one of which there is applied a reference potential, said subnetwork consisting of an operational amplifier having an input point and an output point and in which a first capacitor and a second capacitor are connected in series between the single pair of terminals; and a feedback path from the output point of the amplifier with the parallel circuit of a first resistor and a third capacitor to the junction of the first and second capacitors, the first resistor and the third capacitor being dimensioned such that the feedback path has an effective time constant essentially equal to the time constant of the amplifier in the fall-off region of gain-frequency characteristic thereof in order to extend the effective band width of the subnetwork, so that in the equivalent circuit the series circuit of the subnetwork further comprises in series: a parallel connection of two elements each having a complex impedance 1/bs.