120s IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL 46, NO 5, OCTOBER 1997

ooting Voltage-to-Frequency c
F. N. Trofimenkoff, Senior Member, IEEE, Farmarz Sabouri, Member, IEEE,
Jichang Qin, and J. W. Haslett, Senior Member, IEEE

Abstract-A clock-controlledvoltage-to-frequencyconverter in
which the output frequency is proportional to the square root
of the input voltage is described and a detailed error analysis
is presented. Accuracies of about 0.02% of full-scale for the
voltage input range from 1 mV to 10 V have been achieved using
c~mmercia~~y-available CMOS components in good agreement
with theoreticai expectations.
Index Terms-Analog-to-digital conversion, flow measurement,
high temperature, precision square-rooting.

I. INTRODUCTION
voltage-to-frequency converter (VFC) with an output
frequency proportional to the square root of the input
voltage is useful in processing signals from liquid or gas orifice
and venturi flow measurement devices. In such applications, I_-_______________.______________j

a frequency measurement would then determine flow rate and cLC€KCONllloLLED ONESHOT
a count totalization over a period of time would determine Fig. 1. Clock-controlled one-shot or fixed mark implementation of a
volume delivered in that time period. A clock-controlled VFC square-rooting VFC.
in which the conversion accuracy depends, to first order, only
on reference voltage and clock stability has been described in given as
the literature [l]. In the present work, this VFC is modified
to produce a circuit with output frequency proportional to the
square root of the input voltage and having the same stability
properties. Indeed, as will be shown later, if the input signal
is the output of a strain gauge bridge pressure transducer
powered by a reference voltage, the reference voltage is
ratioed out and only clock stability is important. Use of a
crystal-controlled clock then leads to an exceptionally stable The charge balance for one period of oscillation is accurate to
measurement system. within Vr,flT,IR4 C because the one-shot pulse is only called
on the system clock edge. This is equivalent to {he amount
of charge that can be delivered through R4 by during
II. BASICCIRCUITTOPOLOGY one system clock period, T,. A more detailed description of
The basic topology for the clock-controlled one-shot im- this VFC, including timing diagrams, measurement resolution
plementation of the square-rooting VFC is shown in Fig. 1. considerations, etc., can be found in [l].
In this implementation, the one-shot period, M , is defined by If switches S2, which are also a parallel pair of analog
M = N I T , where T, is the period of the system clock and transmission gates driven in anti-phase, are used as shown in
Nl is determined by selecting the appropriate output tap of the Fig. 1, the amplitude of the waveform averaged by the lowpass
oscillator-timer integrated circuit. The resistor R4 is connected filter (R3and C,)will be equal to T/;ef2 during the time A4 and
to Vrefl during the time M and to ground during the time zero during the time (T - M ) . Then I/,,,,, i.e., the average
(7' - M ) through the switches S 1, which are a parallel pair of value of this waveform for the case when R3C3 -+ 00,will
analog transmission gates driven in anti-phase. The period of be given by
oscillation, T , of VFC can be determined by writing a charge M
balance equation for C4 for one period of oscillation. This is Voave =r K e f 2

Manuscript received May 14, 1997. This work was supported by Nat- -
- V,Kef2
ural Science and Engineering Research Council of Canada under Grants (3)
Kef1 '
OGP0003382 and OGP0007776.
The authors are with the Department of Electrical and Computer Engi- Now, if Vrefl
is taken to be Voave,as is the case in the circuit
neering, University of Calgary, Calgary, Alta., Canada T2N 1N4 (e-mail: of Fig. 1,
trof @ enel.ucalgary.ca).
Publisher Item Identifier S 0018-9456(97)09206-1. (4)
0018-9456/97$10.00 0 1997 IEEE
TROFIMENKOFF et al.: SQUARE-ROOTING VOLTAGE-TO-FREQUENCY CONVERTER 1209

Vt ii
-I

Ihb
VY

output voltage (b)

0
Fig. 3. (a) Analysis of the effect of pulse shape and delays on converter
accuracy and (b) finite T = R3C3 on converter accuracy.

filter and the effects of bias currents and input offset voltage of
(b)
the buffer amplifier can be assessed for a very long filter time
constant by writing a charge balance for the filter capacitor
Fig. 2. Circuits for input bias current and input offset voltage analysis of the
integrator and the lowpass filter. For single-ended operation, finite CMRR is [Fig. 2(b)] to give
taken into account by adding Vs/(2CMRR) to Vosland VOsz.
_ - IZz(R3+ T h 2 )
-Vf
- - T
(8)
and the output frequency f will be given by Kef2

f=”p. NIT, Kef2 (5)

and
The effect of a finite R3C3 is considered in Section IV.
If a strain gauge bridge pressure transducer powered by
Vref2 is used to measure the pressure drop, its output, V,,
will be proportional to Kef2 and Vref2 will cancel in ( 5 ) to
produce a fully ratiometric measurement system with a long-
term stability determined only by the stability of the clock
In (6)-(9), the subscripts 1 and 2 refer to amplifier number;
producing T, and the properties of the pressure transducer.
CMRR is the common mode rejection ratio; I i and I; are the
amplifier input bias currents for the inverting and noninverting
111. ERRORANALYSIS
terminals; rhl and rll are switch resistances when R4 is
The effects of mismatches in the “on” resistances of the connected to Vrefl and ground, respectively; r h 2 and r12 are
switches S1 and of integrator operational amplifier bias cur- switch resistances when R3 is connected to Vref2 and ground,
rents, input offset voltage and common-mode rejection ratio respectively; and Vf is voltage across capacitor Cs.
must be considered in the above VFC. The effects of these Errors due to the sets of switches S1 and S2 and to imperfect
nonidealities can be included in writing the charge balance for lowpass filtering as indicated in Fig. 3 are given by the
C, [Fig. 2(a)] to give following expressions.
5’ + I&(& +rhl) A. Error Due to Switches SI and S2 [Fig. 3(a)]
-- Kef1 Kef1
(6)

where
T
1-
( 1--
2J Thl
R4
- rll
+ rz1
The frequency error in percent of full-scale is given by

VS -
_ v, where
5’= vt + vos, + CMRRl
2 - &R2.
~ (7)
Similarly, the effect of mismatches in the “on” resistance
of the switches that connect Vref2and ground to the low pass
1210 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 46, NO. 5 , OCTOBER 1997

0.04, , , , . ( . . ( , . , ' . , - ,- ,

Input Voltage (Y)

Fig. 5. Measured errors. Error in percent of full-scale

= 100 x [fmeas - fFS&7EZl/fFS.

B. Error Due to Finite Lowpass Filter Time

Constant r = R3C3 [Fig. 3(b)]
The frequency error in percent of full-scale is given by
o---O Temperature = 25 C

where
v,
--
(13)
As an example, suppose auto-zeroing amplifiers such as
the TS913A are used in a single-ended configuration with
Vs = 15 V. VOScan be as high as 15 pV, the CMRR can
0.001 0.010 0.100 1.000 10.000
be as low as 110 dB, and the bias currents can be as high as Input Voltage (V)
90 pA with a possible mismatch between I: and Ig of 20 pA
at 25 "C. If 4066 CMOS analog switches are used, rh - rl Fig. 6. Measured errors as a function of temperature. Error in percent of
full-scale = 100 x ISmeas- SFSJ-I/~FS.
can be as high as 50 R. If R3 = 1 M a , C, = 0.22 pF, and
At = -50 nS, a reasonable value for the 4066 switches, the
total error in the output frequency calculated using (6)-(13) is mode rejection ratios, and to ripple on Vrefl = V,,,,. The
given in Fig. 4. It appears that 0.02% of full-scale accuracy is ripple component of the error cannot, of course, be reduced
possible at low oscillator frequencies and large R3C3. without increasing the system response time to changes in V,.
As noted earlier, and as is evident from (5), the voltage-
to-frequency transfer function is independent to first order of
IV. EXPERIMENTAL
RESULTSAND CONCLUSIONS passive component values. Since the operational amplifiers
Results of measurements on the circuit of Fig. 1 are pre- are auto-zeroed, this transfer function should be practically
sented in Fig. 5. The clock frequency, fc, in each case was independent of temperature if Vref2 and Xc are fixed. Error
1.024 MHz with N I = 1024,Nl = 512,Nl = 256, and plots for f p s I 4 kJ3z operation at 0, 25, and SO OC are
Nl = 128 yielding full-scale frequencies, f p s , of 1, 2, 4, shown in Fig. 6 to confirm this property of the converter. In
and 8 kHz, respectively. No adjustments other than the setting generating Fig. 6, the amplifiers and switches were changed
of f c were required to achieve 0.02% of full-scale accuracy from those used to generate Fig. 5 to show that there is
in the input voltage range from I mV to 10 V for full-scale practically no sensitivity to those components as well.
frequencies of 1, 2, and 4 kHz. At a full-scale of 8 kHz, In conclusion, the clock-controlled square-rooting VFC is
the error at higher input voltages begins to increase to a capable of 0.02% of full-scale accuracy operation when con-
more substantial level. The calculated error curve shown in structed with commercially-available 18 V CMOS components
Fig. 4 is in good agreement in both shape and magnitude and 18 V auto-zeroed amplifiers. If a crystal-controlled VFC
with that which was measured. It can be shown that the is used as part of a flow measuring system where a strain
errors for (V,/V,,f2) + 1 are due primarily to pulse-shape- gauge bridge pressure transducer is operated ratiometrically.
related effects, whereas the errors for (&/Vref2) 4 0 are due the above level of performance can be expected from the
primarily to amplifier offset voltages, amplifier finite common- signal processing circuits.
TROFIMENKOFF et al.: SQUARE-ROOTING VOLTAGE-TO-FREQUENCY CONVERTER 1211

REFERENCES Jichang Qin was born in Beijing, China, in 1951.

He received the B.S. and M.S. degrees from the
[l] F. N. Trofimenkoff, C. 0. Li, and D. J. Paslawski, “Clock-controlled Beijmg Institute of Chemical Technology in 1982
voltage-to-frequency converter,” U.S. Patent 4 847 620, July 11, 1989; and 1984, respectively. In 1988, he received the
Canadian Patent 1288 165, Aug. 1991. M.S. degree in electrical and computer engineering
from the University of Calgary, Calgary, Alta.,
Canada, in 1991.
He was a Lecturer at the Beijing Institute of
Chemical Technology from 1985 to 1988 and
F. N. Trofimenkoff (M’63-SM’69) was born in worked on the development of semiconductor
Veregin, Sask., Canada, on August 10, 1934. He pressure transducers He has worked in the oil
received the B.E. degree in engineering physics and gas industry and is currently an Electronics Designer with McAllister
and the M.Sc. degree in physics, both from the Petroleum Services Ltd., Calgary.
University of Saskatchewan, Saskatoon, in 1957 and
1959, respectively. He was awarded an Athlone
Fellowship in 1959 and received the PhD. de-
gree in electrical engineering (semiconductor device
physics) from the University of London, Imperial J. W. Haslett (M’64-SM’79) was bom in
College of Science and Technology, London, U.K., Saskatchewan, Canada, on September 27, 1944. He
in 1962. received the B Sc. degree in electrical engineering
From 1957 to 1959, he worked on instrumentation for accurate humidity from the University of Saskatchewan, Saskatoon,
measurement in the Division of Building Research, National Research Council Sask., Canada, in 1966 and the M.Sc. and Ph.D
of Canada, and from 1962 to 1966 he was an Assistant Professor of Electrical degrees in electrical engineering from the University
Engineering at the University of Saskatchewan. In 1966, he joined the of Calgary, Calgary, Alta., Canada, in 1968 and
Electrical Engineering Department, University of Calgary, Calgary, A h . , 1970, respectively.
Canada. His current research interests are in the circuits and devices area In 1970, he joined the Department of Electrical
and in instrumentation related to the petroleum industry. Engineering, University of Calgary, and served as
Dr Trofimenkoff is a member of the Association of Professional Engi- Head from 1986 to 1997. His current research
neers, Geologists and Geophysicists of Alberta, the Engineering Institute of interests include optical imaging systems for spacecraft applications,
Canada, the Canadian Association of Physicists, and the American Society instrumentation systems related to drill stem testing of oil and gas wells,
for Engineering Education. high-temperature semiconductor device behavior, and the design of analog
and digital VLSI circuits
Dr. Haslett is a member of the Association of Professional Engineers,
Geologists and Geophysicists of Alberta, the Canadian Astronomical Society,
the Canadian Society of Exploration Geophysicists, and the American Society
Farmarz Sabouri (M’95) was born in Tehran, of Engineering Education.
Iran, in 1965. He received the B.Sc. degree in
electrical engineering in 1987 from Ferdowsi Uni-
versity, Mashad, Iran, the M.Sc. degree in electrical
engineering in 1990 from the Sharif University of
Technology, Tehran, and the Ph.D. degree in elec-
trical engineering from the University of Calgary,
Calgary, Alta., Canada, in 1996.
From 1990 to 1992, he was a Part-Time Instructor
at Ferdowsi University, and a Research Engineer at
the Sajad Electrical Research Center and the Trakho
Ballast Manufacturing Co., Mashad. He is currently with Analog Devices,
Somerset, NJ, where he designs high-speed analog-to-digital converters. His
current research interests include high-speed data converters and switched
capacitor circuits.