Switched capacitor function generator

An analog function generator useful in providing a variety of functions for analog signal processing applications includes a pulse width modulator and a switched capacitor operational amplifier. Capacitors in the input of the operational amplifier and in the feedback loop of the operational amplifier are selectively switched by the output of the modulator to create output voltages of the amplifier that are polynomial, logarithmic or exponential functions of the input voltages to the amplifier and pulse width modulator.

This invention relates generally to an analog functional circuit for use in 
very large scale integrated circuits (VLSI) and more particularly the 
invention relates to switched capacitor circuits for pulse width 
modulation and generation of polynominal functions. 
Switched capacitor techniques are known for creating large effective 
resistance (R) to use with small capacitances (C) in low frequency analog 
VLSI circuits. A major application is in audio frequency filters which 
require large RC values. Having a large effective resistance permits use 
of small equivalent capacitance and, hence, space saving in VLSI circuits. 
As will be described further hereinbelow, the high effective resistance is 
obtained by providing a switched capacitor in the input of an operational 
amplifier and a switched capacitor in the feedback loop of the operational 
amplifier. This circuit is equivalent to a one-pole low pass filter having 
a large input resistance value. However, known prior art switched 
capacitor VLSI circuits cannot provide many of the functions that would be 
useful in low frequency applications. 
The present invention is directed to a functional building block using 
switched capacitor circuits. The building block comprises a pulse width 
modulator and a switched capacitor operational amplifier with the 
capacitors being selectively switched by the output of the modulator. 
Signal multiplication, voltage expansion, gain control, voltage division, 
variable pole filters, and compressors are some of the functions achieved 
with the functional building block. Functions available with the circuitry 
include x.y, x.sup.2, x/y, x, xy.sup.n, xy.sup.-n, log x, and e.sup.x 
where x and y are input wave forms creating the f(x,y) outputs. With these 
functions a wide range of analog applications can be realized. 
Accordingly, an object of the invention is an analog function generator for 
providing a family of low frequency VLSI circuits. 
A feature of the invention is a pulse width modulator for generating timing 
pulses for use in switched capacitor circuitry.

Referring now to the drawings, FIGS. 1a-1c illustrate the structure and 
operation of a switched capacitor circuit in accordance with the prior 
art. FIG. 1a illustrates schematically an amplifier A having a switched 
capacitor C.sub.I and a fixec capacitor C.sub.F in its feedback loop and a 
switched capacitor C connected to the input of the amplifier. FIG. 1b 
illustrates the switching signals, .phi..sub.1 and .phi..sub.2, which 
control the switches in the circuitry of FIG. 1a, and FIG. 1c is the 
equivalent one-pole filter of the circuit of FIG. 1a. 
In time period .phi..sub.1, the switches labeled .phi..sub.1 close. 
Capacitor C takes on charge q=XC (X is the input voltage waveform). During 
the same time period Capacitor C.sub.I is emptied. C.sub.F maintains its 
current charge, qf=vC.sub.F (v is the output voltage waveform). In time 
period .phi..sub.2, the switches labeled .phi..sub.2 close. Charge q 
discharges into amplifier junction "a". Also, a charge q=vC.sub.I flows 
into junction "a" as C.sub.I charges up to voltage v. The differential 
charge flows into capacitor C.sub.F 
EQU qf=q-q.sub.i, (1) 
This causes an incremental change in voltage out 
EQU .DELTA.v=q/C.sub.F =X(C/C.sub.F)-v(C.sub.I /C.sub.F) (2) 
The change occurs in time interval .DELTA.t=1/f. (This time is short 
compared with changes in either x or v.) The change of voltage out with 
time thus equals: 
EQU .DELTA.v/.DELTA.t=X(C.sub.f /C.sub.F)-v(C.sub.I f/C.sub.F) (3) 
With f large compared with variations in X and v, the equation can be 
written: 
EQU C.sub.F 9dv/dt)=)C.sub.F)X-(C.sub.I f)v (4) 
This is the same as the equation for the conventional amplifier shown in 
FIG. 1(c) if the component values are given by: 
EQU R.sub.I =1/C.sub.I f; R=1/Cf; C.sub.F =C.sub.F (5) 
This is a 1-pole low-pass filter with a gain and cutoff frequency given by: 
EQU G=R.sub.I /R=C/C.sub.I ; f.sub.3 dB =1/R.sub.1 C.sub.F =f(C.sub.I /C.sub.F) 
(6) 
For frequencies well below f.sub.3 dB, the output is given by 
EQU v=X)C/C.sub.I) (7) 
The cutoff frequency, f.sub.3 dB, can be made low by choosing the proper 
switching frequency, f, and ratio of capacitors C.sub.I C.sub.F. With this 
approach the C's can be made small enough for VLSI circiuts. 
It is assumed that any residual resistance in the switches show is small, 
so that 
EQU .sub.res =R.sub.res *C&lt;&lt;.phi..sub.1 or .phi..sub.2 (8) 
That is, the charge and discharge of C.sub.i and C is very fast compared 
with the switching periods. 
The low-pass filter illustrated is only one simple embodiment of 
switch-capacitor filter technology. In the general switch capacitor 
applications, multiple "resistors", switched capacitor "resistors" and 
normal capacitors are used in different circuit configurations to creat 
filters with "poles" and "zeros" in different locations. 
FIGS. 2a-2c illustrate a switched capacitor function generator in 
accordance with one embodiment of the present invention. FIG. 2a is a 
schematic of a pulse width modulator in which an output pulse, 
.phi..sub.t, is generated in response to the closing of the input switch 
by the clock signal (f) and comparing the charge generated on capacitor 
C.sub.T with a voltage v.sub.y. The generated pulse width is obtained from 
the NOR gate which is connected to receive the output of the comparator, 
CP, and the clock signal. FIG. 2b is a plot of the clock signals 
.phi..sub.1, .phi..sub.2, and .phi..sub.t ; and FIG. 2c is a schematic of 
a switched capacitor circuit which is operated by the clock signals of 
FIG. 2b. 
Referring to FIG. 2a, a pulse starts from clock (f) with the voltage across 
C.sub.T equal to "0". At the start of the clock pulse, charge flows from 
V.sub.C through R.sub.T, charging C.sub.T at an exponential rate. The 
comparator circuit, CP, senses when the voltage on C.sub.T has risen to 
equal the input voltage, V.sub.Y. The pulse end is then triggered by the 
comparator. The capacitor C.sub.T is discharged and held at zero volts 
until the next clock pulse (f). 
This circuit generates a pulse .phi..sub.t with repetition rate, f, 
starting at the same time as .phi..sub.1 and having a length t.sub.y given 
by 
EQU t.sub.y =-R.sub.T C.sub.T ln (1-y/V.sub.C) (9) 
In FIG. 2(c) the switching waveforms are used to charge a switched 
capacitor, C, in series with a resistor R for time period t.sub.y. 
The charge, q, that flow into C during this time is thus given by 
EQU q=XC(1-e.sup.-t.sbsp.y.sup./R.sbsp.C) (10) 
EQU q=XC(1-e.sup.+(R.sbsp.T.sup.C.sbsp.T.sup.)/RC(ln(1-y/V.sbsp.C.sup.))) (11) 
The remainder of the circuit is identical to the switch capacitor circuit 
described in FIG. 1a. Thus, the performance is the same if the value C' is 
substituted for by C where 
EQU C'=C(1-e.sup.+(R.sbsp.T.sup.C.sbsp.T.sup.)/RC(ln(1-y/V.sbsp.C.sup.)) (12) 
This can be rewritten using the relation, e.sup.aln(b) =b.sup.a. 
EQU C'=C(1-(1-y/V.sub.C).sup.(R.sbsp.T.sup.C.sbsp.T.sup./RC)) (13) 
Many different functions can be developed with this relationship. The first 
family of function generators evolves from setting the two time constants 
RC and R.sub.T C.sub.T equal to each other: 
For 
EQU RC=R.sub.T C.sub.T 
EQU C'=y*C/V.sub.C (14) 
This is a straight multiplier with gain and bandwidth 
EQU G=C'/C.sub.I =(y/V.sub.C)*(C/C.sub.I) (15) 
EQU F.sub.3 dB =f*C.sub.I /C.sub.F (16) 
The output, v, and two inputs, y and x, are given by 
EQU v=x*y(C/C.sub.I V.sub.C): MULTIPLIER (17) 
The above is a 2-quadrant multiplier; that is, the value of "y" must be 
positive because the time interval, t.sub.y, cannot take on negative 
values. If negative values of "y" are anticipated, a simple way to create 
a 4-quadrant multiplier is to use a zero-crossing detector (ZCD) and two 
inverting amplifiers, as shown in FIG. 3. 
A square-law voltage expandor is formed by connecting the same signal to 
both inputs of the multiplier (2 or 4 quadrants depending on the range of 
input voltage). 
FIGS. 4a and 4b show that in this case one inverter is saved by rectifying 
x before input to a 2-quadrant multiplier. 
A circuit to obtain the time average peaks of a waveform is shown in FIG. 
5a. This is used in a voice processing to vary gain at the rate of power 
changes in voiced syllables. A square-law syllabic expandor using this 
circuit is shown in FIG. 5b. 
The basic 2-quadrant multiplier can be used in a wide range of gain control 
applications where the input y in FIG. 2 is from a feedback sensing 
element. Normally, the sign of y in such applications is positive to the 
2-quadrant multiplier can be used. Applications include tape recorders and 
playback, AM radios, "Dolby" circuits, and mobile radio. 
Another basic function (divider) circuit in accordance with the invention 
is illustrated in FIGS. 6a-6c. The time circuit is the same as shown in 
FIG. 2a. Now, however, the charging capacitor in FIG. 6c which is being 
controlled is C.sub.I rather than C. The charge, q.sub.I, is then given by 
##EQU1## 
The relationships are the same as the circuit of FIG. 1 if C.sub.I ' is 
substituted for C.sub.I where 
EQU C'=C.sub.I 
(1-(1-y/V.sub.C).sup.(R.sbsp.T.sup.C.sbsp.T.sup.)/R.sbsp.I.sup.C.sbsp.I.su 
p.)) (19) 
The simple application is when R.sub.T C.sub.T and R.sub.I C.sub.I are 
matched. Then the value of C.sub.I is 
EQU C.sub.I =C.sub.I y/V.sub.C (20) 
With these values the circuit of FIG. 6 has gain bandwidth and transfer 
functions given by 
EQU G=C/C.sub.I V.sub.C /y; f.sub.3 dB =f*C.sub.I /C.sub.F (y/V.sub.C) (21) 
EQU v=x/y(V.sub.C C/C.sub.I): DIVIDER (22) 
The divider has the limitation that the cutoff frequency, f.sub.3 dB, 
varies with the input voltage. It also has the mathematical limitation of 
all dividers that division by zero implies infinite output voltage, v. The 
circuit saturates for small y and, therefore, would not be used for y that 
would change sign. It is useful as a 2-quadrant divider as long as the 
output desired can be limited. 
FIG. 7 illustrates the 2-quadrant divider used as a syllabic voice 
compressor. It will be noted that the actual relationship between Y, X and 
V is a feedback function whose stability depends on the peak-amplitude 
comparator () time constant. 
The performance of the circuit of FiG. 7 is better understood as action on 
a sine wave. If X is a waveform, (A sin wt), the output is (.sqroot.A sin 
wt). The input value for Y is .sqroot.A (the circuit gives an output 
equal to the peak input voltage). The input waveform is, thus divided by a 
constant .sqroot.A and becomes 
EQU v=(A sin wt)/.sqroot.A=.sqroot.A sin wt (23) 
This circuit is the standard syllabic compressor used today except for use 
of the switch-capacitor invention to realize the required power law. In 
this circuit the value of C.sub.F is choosen so that the bandpass 
variation with the output voltage is not bothersome. 
Another basic function generator in accordance with the invention is shown 
in FIGS. 8a-8c. Here both switched capacitors in FIG. 8c are controlled. 
The effective capacitor values are still given by the equation (12) and 
(13), above. With these values the gain is given by 
##EQU2## 
If the three time constants are equated, the two terms including y cancel 
the gain becomes constant. 
EQU G=C/C.sub.I ; if R.sub.T C.sub.T =RC=R.sub.I C.sub.I 
The cutoff frequency depends on C.sub.I, but not C'. 
EQU f.sub.3 dB =f*C.sub.I /C.sub.F 
*(1-(1-y/V.sub.C).sup.(R.sbsp.I.sup.C.sbsp.I.sup./R.sbsp.I.sup.C.sbsp.I.su 
p.)) 
If the time constants are equated, the result is simple: 
EQU f.sub.3 dB =f*C.sub.I /C.sub.F *(y/V.sub.C); if R.sub.T C.sub.T =R.sub.I 
C.sub.I 
The variable filter element has a constant gain and a law-pass 3 db cutoff 
frequency that is a linear function of control voltage, y. 
The gain control and the variable pole filter described above are just two 
examples of filters whose characteristics are linearly controlled by 
voltage. In any switched capacitor filter, an R.sub.n can be added to any 
or all C.sub.n such that R.sub.n C.sub.n =R.sub.T C.sub.T. The poles can 
be varied by a control voltage, y. This capability can be of use in 
adaptive filtering applications. 
In the applications described above, the relationships are simplified by 
equating time constants. Other ratios of time constants create polynomial 
relationships that have other applications. The functions are as given 
below for the multiplier module of FIG. 2. 
EQU v=C/C.sub.I *(1-(1-y').sup.n)X (25) 
where 
EQU y'=y/V.sub.C 
EQU n=R.sub.T C.sub.T /RC 
The circuit of FIG. 2c, a time circuit R.sub.T C.sub.T, charges 
exponentially. This circuit is easily modified to generate a current 
V.sub.C /R.sub.T that does not vary with charging of C.sub.T. Then the 
time is given by 
EQU ty=y*R.sub.T C.sub.T /V.sub.C (26) 
The charge is given by 
EQU q=XC(1-e.sup.-t.sbsp.y.sup./RC)=XC(1-e.sup.(R.sbsp.T.sup.C.sbsp.T.sup.)/(RC 
)*Y) 
EQU C'=C(1-e.sup.-(R.sbsp.T.sup.C.sbsp.T.sup.)/(RC)*Y (27) 
The gain is given by 
EQU G=C/C.sub.I *(1-e.sup.-y); if R.sub.T C.sub.T /RC=1 
EQU v=aX(1-e.sup.-y); a=C/C.sub.1 (28) 
In a similar way the input X in FIG. 2 can be made a current generator with 
current i-X/R. In this case the function becomes 
EQU v=-aX ln (1-Y/V.sub.C); where A=C.sub.T R.sub.T /C.sub.I R (29) 
There have been described several embodiments of an analog function 
generator for providing a family of low frequency VLSI circuits which have 
heretofore been unavailable using switched capacitor building blocks. As 
is evident from the description, many functions can be implemented through 
simple variations in the placement and control of the capacitors. Thus, 
while the invention has been described with reference to specific 
embodiments, the description is illustrative of the invention and is not 
to be construed as limiting the invention. Various modifications and 
applications may occur to those skilled in the art without departing from 
the true spirit and scope of the invention as defined by the appended 
claims.