Apparatus and method for stabilization of a thermistor temperature sensor

A thermistor temperature sensor having non-linear electrical response characteristics with temperature, a material constant, beta, that varies with temperature, and providing an output signal indicative of the temperature desired to be measured is utilized herein. A time dependent signal having a configuration that offsets the non-linear temperature dependence of the sensor output signal is generated and compared to the sensor output signal. Whenever the sensor output at least equals that of the offsetting signal, a comparator generates an output signal that, after appropriate delay, is indicative of the actual temperature. The material "constant" of the sensor is maintained substantially constant over at least a substantial range of temperatures. Additionally or alternatively, the sensor is energized in such manner as to substantially eliminate self-heating induced variations in the sensor output signal.

BACKGROUND OF THE INVENTION 
The present invention relates generally to temperature sensing devices. 
More particularly, the present invention relates to temperature monitoring 
systems employing temperature sensors having non-linear electrical 
response characteristics with temperature and circuitry for linearization 
and stabilization of these characteristics. More specifically, the present 
invention concerns apparatus and method for the linearization, 
stabilization or substantial elimination of other deleterious non-linear 
variations with temperature of systems utilizing thermistor temperature 
sensors. 
Thermistor temperature sensors have long been recognized for their utility 
in accurately and precisely monitoring a wide range of temperatures. The 
thermistor's usefulness as a temperature sensing device derives from its 
characteristic very large change in resistance with temperature. 
Unfortunately, this change is also highly non-linear, as evidenced by the 
following first order mathematical approximation of a thermistor's 
resistance with temperature: 
##EQU1## 
where R.sub.T represents the thermistor resistance at any absolute 
temperature T; R.sub.T.sbsb.o represents the thermistor resistance at 
absolute reference temperature T.sub.o ; and .beta. (also referred to as 
"beta") represents the thermistor material constant. Thus, in order to 
effectively employ a thermistor in an accurate temperature monitoring 
system, the circuitry utilized in association with the thermistor must be 
either capable of following this non-linearity or, as done in essentially 
all commercially practical devices, assume absolute linearity with 
temperature for each incremental temperature range. 
Accordingly, prior art devices have developed numerous linearization 
techniques to insure measurement accuracy. The majority of these 
techniques are based upon trading precision for accuracy (or, in other 
words in such cases, sensitivity for linearity), by placing additional 
resistances in series and/or parallel with the thermistor to reduce its 
rate of change of resistance with temperature [i.e., the curvature of the 
thermistor's Resistance-Temperature (or R-T) characteristic]. 
One prior art reference discloses a different linearization technique in 
which a time variant signal, whose characteristics are the inverse of 
those of a non-linear temperature sensor, is periodically generated and 
compared with that of the sensor output. Upon coincidence in the value of 
the two signals the comparator emits a short pulse of constant duration. 
Because the two signals are the inverse of each other, a measurement of 
the time from the start of the comparison to the time at which both 
signals have equal value--the so called "intersection time"--yields a 
linearized indication of the sensor's temperature. Although this technique 
does serve to achieve a highly linearized signal with temperature, when 
taken by itself such a device still suffers from serious loss of accuracy 
and precision in measurement, and requires unnecessary, additional, 
complex circuitry. 
One reason for these deficiencies results from the fact that the value of 
beta (the thermistor material constant) has been assumed to be absolutely 
independent of temperature in all devices employing non-linear, 
semiconductor temperature sensors of which I am aware. In fact, especially 
over a large temperature range, substantial variations in the value of 
beta occur, resulting in an appreciable loss of both accuracy and 
precision. 
A second deficiency in the performance of thermistor temperature sensing 
devices such as that described above stems from error induced by the heat 
generated within the thermistor itself as the signal current passes 
therethrough. One approach taken by others to mitigate the effects of 
self-heating has involved the intermittent operation of the thermistor 
sensor. But such periodic energizations reduces the frequency with which 
temperature samplings can be made by increasing the time necessary for 
each measurement, as well as necessitating further complex circuitry added 
to the system. 
This approach is not the only source of unnecessary complex, circuitry in 
the above offsetting signal linearization scheme. As previously explained, 
after the variable intersection time is reached, a coincidence pulse of 
constant duration is generated to terminate this period and initiate 
temperature computation and display. A "controller" must be provided to 
initiate the temperature computation and to subsequently begin a new 
measurement cycle. Moreover, the necessity to manually restart each such 
cycle severely limits the rapidity with which successive measurements can 
be made. 
I have found several techniques and circuits for significantly improving 
both the accuracy and precision of temperature systems having sensors with 
non-linear characteristics. Generally, the present invention contemplates 
the utilization of a time exponential function to offset the temperature 
dependent exponential function characteristic of thermistor sensors 
described by equation (1). More specifically, I have found an extremely 
simple current division network which, when used in conjunction with 
thermistor sensor output signal amplifiers, substantially reduces or 
eliminates the aforementioned variations in the value of beta, regardless 
of temperature. 
Additionally, I have found that by impressing a particular feedback signal 
upon the thermistor sensor in place of the conventional constant voltage 
power source, the power dissipated by the thermistor itself may be fixed 
so as to substantially avoid the self-heating phenomena without having to 
periodically operate the same. Beyond this, by proper selection of this 
feedback signal, the thermistor output signal level may be compressed to 
such an extent that the linear response range of a temperature measuring 
system having a thermistor sensor may be substantially (and, at least 
theoretically, infinitely) expanded beyond that of all presently known 
systems. 
SUMMARY OF THE INVENTION 
It is, therefore, an object of the invention to provide an apparatus and 
method for the linearization, stabilization or substantial elimination of 
other deleterious non-linear variations with temperature of systems 
utilizing thermistor temperature sensors. 
It is another object of the invention to provide an apparatus and method, 
as above, without significantly reducing sensor sensitivity. 
It is still another object of the invention to provide an apparatus and 
method, as above, in which linearization is achieved by generation of a 
time dependent signal whose configuration offsets the non-linear 
temperature dependence of the sensor output signal and by subsequent 
comparison of such time dependent offsetting signal with the temperature 
dependent sensor output signal substantially without loss of accuracy or 
precision, and without the need for additional, complex circuitry. 
It is yet another object of the invention to provide an apparatus and 
method, as above, in which beta, the semiconductor material "constant," is 
maintained substantially constant over at least a substantial range of 
temperatures. 
It is a further object of the invention to provide an apparatus and method, 
as above, in which variations in the sensor output signal attributable to 
the self-heating phenomena are substantially eliminated. 
It is still a further object of the invention to provide an apparatus and 
method, as above, having a simple current division network which, when 
used in conjunction with a thermistor sensor output signal amplifier, 
substantially reduces or eliminates variations in beta with temperature. 
It is yet a further object of the invention to provide an apparatus and 
method, as above, having a particular feedback signal impressed upon the 
thermistor sensor for substantially eliminating variations arising from 
the self-heating phenomena and compressing the thermistor output signal 
level in such a manner that the linear response range of a temperature 
measuring system is substantially expanded beyond that of all presently 
known systems. 
These and other objects and advantages of the present invention over 
existing prior art forms will become more apparent and fully understood 
from the following description in conjunction with the accompanying 
drawings. 
In general, an apparatus and method embodying the concept of the present 
invention includes a temperature sensor having electrical response 
characteristics that vary non-linearly with temperature and providing an 
output signal indicative of the temperature desired to be measured. A time 
dependent signal having a configuration that offsets the non-linear 
temperature dependence of the sensor output signal is generated and 
compared to the sensor output signal. Whenever the sensor output at least 
equals that of the offsetting signal, a comparator generates an output 
signal. The material "constant" of the sensor is maintained substantially 
constant over at least a substantial range of temperatures. Additionally 
or alternatively, the sensor is energized in such manner as to 
substantially eliminate self-heating induced variations in the sensor 
output signal.

DESCRIPTION OF A PREFERRED EMBODIMENT 
FIG. 1 depicts an exemplary thermistor temperature sensing apparatus 
according to the concept of the present invention, which apparatus is 
generally indicated by the numeral 10. A fixed voltage E is impressed upon 
and energizes a thermistor 11 through a switch 12, described further 
hereinafter. In order to amplify the thermistor output signal to 
magnitudes acceptable for further processing, the output signal from the 
thermistor is received by the inverting input terminal (typically 
indicated in the drawings by a minus sign) of a conventional operational 
amplifier 14 after passing through first compensating resistor 15. A fixed 
voltage source V.sub.cc is connected through a second compensating 
resistor 16 to the inverting input terminal of operational amplifier 14, 
which operational amplifier 14 has its non-inverting input terminal 
(typically indicated in the drawings by a plus sign) connected to ground. 
Operational amplifier 14 may be connected and operated in any of the 
well-known differential amplifier configurations with resistive feedback 
between the differential amplifier output terminal and its inverting input 
terminal 13 provided by resistor 17. Resistor 17 may, for convenience, be 
selected to have a value equal to whatever reference resistance 
R.sub.T.sbsb.o is desired. Again for convenience in implementation, the 
reference resistance R.sub.T.sbsb.o may, in turn, be selected to have a 
value equal to the thermistor's measured resistance at any intermediate 
temperature between the desired operational temperature extremities. 
The amplified thermistor output signal from operational amplifier 14, which 
will be hereinafter referred to as signal E.sub.1, is received by the 
non-inverting input terminal of any conventional voltage comparator 18, 
such as that commercially available from National Semiconductor Corp. of 
Santa Clara, Calif. and identified as Model No. LM 311. Comparator 18 
serves to continually compare the instantaneous voltage signal E.sub.1 
with the instantaneous voltage output of an RC network 20, hereinafter 
referred to as signal E.sub.c that is fed to the inverting input terminal 
of comparator 18. 
RC network 20 includes a capacitor 21 and a resistor 22, either or both of 
which may be variable so that the network's decaying time constant, .tau., 
may be varied as explained below. In order to intermittently recharge RC 
network 20, a fixed voltage source V having a magnitude well above the 
value of E.sub.1 associated with the maximum desired environmental 
temperature, is also connected, through an analog switch 23, to the 
inverting input terminal of comparator 18. Analog switch 23 may be any 
conventional switch having at least one pole and adequate switching speed, 
and capable of remote electrical control. One exemplary, suitable switch 
is Model No. MC 14066 manufactured by and commercially available through 
Motorola, Inc. of Phoenix, Arizona. 
Analog switch 23 is operated by a time delay circuit 24 having a fixed time 
delay t.sub.a, which circuit 24 receives the output signal from comparator 
18. Although essentially any time delay circuit capable of inducing delays 
of the magnitude indicated below would be acceptable for use herein, I 
have found National Semiconductor's Model No. LM 322 Precision Adjustable 
Time Delay, readily commercially available, to be conveniently compatible 
with the remaining elements in apparatus 10. 
Having described the fundamental elements with which the present invention 
functions, its operation now also may be delineated. Initially, the 
skilled artisan will appreciate that operational amplifier 14 generates 
variable voltage signal E.sub.1 in direct proportion to the thermistor's 
instantaneous output current signal, which current signal exponentially 
increases with temperature as its resistance decreases. When the 
thermistor is placed in the environment in which measurement of the actual 
(also referred to as ambient) temperature is desired, its output signal 
would, of course, be a function of this actual temperature. Thus, 
operational amplifier 14's output voltage E.sub.1 for any temperature T 
may be expressed as follows: 
##EQU2## 
where E is the absolute value of the constant thermistor supply voltage 
and T.sub.o is a reference temperature. 
Signal E.sub.1 is compared with the exponentially decaying voltage E.sub.c 
of RC network 20. RC network 20 has been found to be a simple and highly 
inexpensive manner of generating a time exponential function capable of 
accurately offsetting and thereby linearizing the temperature-dependent 
exponential function characteristic of thermistor sensors as stated in 
equation (1) above. Comparator 18 generates a single voltage pulse of 
short duration each time the voltage of signal E.sub.c, charged to a 
steady-state value V that is greater than the value of E.sub.1 for the 
maximum temperature desired, decays to equal the instantaneous voltage of 
signal E.sub.1 at a time t.sub.1. Thus, at time t.sub.1, 
EQU E.sub.1 =E.sub.c (3) 
Substituting equation (2) and the well-known equation representing a 
capacitor's voltage during discharge, 
##EQU3## 
where the time constant, .tau., equals the product of RC network's 
resistance and its capacitance. Solving for t.sub.1 
##EQU4## 
Equation (5) represents time at which signals E.sub.1 and E.sub.c are equal 
and comparator 18 generates an output pulse. By delaying the trailing 
logic transition of the output pulse from comparator 18 for a time period 
t.sub.a, where 
##EQU5## 
the frequency of the thusly extended pulses may be seen to be directly 
proportional to the absolute temperature sensed by the thermistors since 
##EQU6## 
By appropriately selecting .tau. and scaling t.sub.a the output of time 
delay circuit 24 may be made to constitute a pulse train whose frequency 
is equal to the actual measured temperature in any temperature scale 
desired. Of course, the frequency of this output pulse train may be 
determined and displayed or otherwise utilized in any conventional manner. 
It should be noted that by any of the modifications well known to the 
skilled artisan, time delay circuit 24 could be made to provide an output 
signal having any characteristic proportional to the actual measured 
(i.e., ambient) temperature, rendering apparatus 10 directly compatible 
with whatever output device or configuration is desired. 
Where, as here, the proportional characteristic is frequency, I have found 
it convenient to directly determine the frequency by use of a binary coded 
decimal (hereinafter, BCD) up/down presettable pulse counter 25, the 
output of which may be processed by BCD-to-7 segment decoder 26 so as to 
constitute a signal capable of being directly, digitally displayed by 
7-segment numeric display 27. 
Apparatus 10 provides means for substantially stabilizing variations in 
thermistor beta with temperature. FIG. 2 illustrates that portion of 
apparatus 10 utilized for beta stabilization and incorporates an 
equivalent circuit for the actual thermistor 11. The equivalent circuit 
includes an ideal constant-beta thermistor element 30 having resistance 
R.sub.T in parallel with a resistor 31 having resistence R.sub.p, both of 
which are in series with resistor 32 having resistance R.sub.S. The 
resultant effective resistance of this network is designated as "R" in 
FIG. 2 and, by definition, equals the actual (or measured) thermistor 
resistance at several preselected temperatures from a minimum of T.sub.min 
to a maximum at T.sub.max. By inspection of the equivalent circuit, the 
measured resistance of actual thermistor 11, R.sub.TH, is expressed as 
follows: 
##EQU7## 
From equation (1), beta may be expressed as 
##EQU8## 
where R.sub.T.sbsb.min is the effective thermistor network resistance at 
minimum temperature T.sub.min ; and R.sub.T.sbsb.max is the effective 
thermistor network resistance at maximum temperature T.sub.max. Finally, 
also from equation (1) 
##EQU9## 
where R.sub.T is the resistance of an ideal thermistor element at 
temperature T; and R.sub.T.sbsb.min is the resistance of an ideal 
thermistor element at minimum temperature T.sub.min. Because thermistor 
beta varies non-linearly with temperature, no single, unique values for 
R.sub.S and R.sub.P exist. Thus, in order to determine the single values 
for R.sub.S and R.sub.P that yield the least variation of beta throughout 
the desired temperature range, equations (8), (9) and (10) are preferably 
solved in the following manner. 
After selecting the desired temperature range, the actual resistance of 
thermistor 11 at both the minimum and maximum temperatures may be 
emperically ascertained and a single value each for R.sub.S and R.sub.P 
arbitrarily chosen. Next the effective thermistor network resistance is 
calculated for both the minimum and maximum temperatures from equation 
(8), which in turn permits calculation of a somewhat averaged value of 
beta from equation (9). Finally, equation (10) is solved for R.sub.T, the 
resistance of an ideal thermistor element, at two temperatures 
intermediate between the range of desired temperatures and these values 
compared to the effective thermistor network resistances found from 
equation (8). This process is repeated until those values of R.sub.S and 
R.sub.P that yield ideal and effective network thermistor resistances that 
are substantially equal are determined. 
Returning to FIG. 2, both parallel resistor 31 and series resistor 32 are 
imaginary resistances created to account for beta variations with 
temperature. However, because series resistor 32 will always have a 
negative resistance for a thermistor temperature sensor, the effects of 
series resistor 32 may be canceled out by the insertion of first 
compensation resistor 15, a positive (real) resistance of equal magnitude. 
The effects of parallel resistor 31 may be cenceled by the duplication of 
the current division of thermistor current components i.sub.T and i.sub.P 
between the ideal thermistor 30 and parallel resistor 31, respectively, 
with a similar current division between feedback resistor 17 and second 
compensation resistor 16. To accomplish this, second compensation resistor 
16 must be selected such that the current component i.sub.P through it 
(i.e., V.sub.cc /R.sub.16, where R.sub.16 is the resistance of resistor 
16), equals the current component through parallel resistor 31 (i.e., 
E/R.sub.P). Thus, the judicious selection of the values of resistors 15 
and 16 to equal those of series resistor 32, and parallel resistor 33 
multiplied by (V.sub.cc /E), respectively, substantially reduces 
thermistor beta variations with temperature. 
In order to more readily demonstrate this, the values of R.sub.S and 
R.sub.P were calculated for temperatures T.sub.1 =100.degree. F.; T.sub.2 
=200.degree. F.; T.sub.3 =300.degree. F.; and T.sub.4 =400.degree. F. and 
were found to be -10.2 .OMEGA. and 760 K.OMEGA., respectively. Utilizing 
these temperatures and the actually measured thermistor resistances as 
supplied by Thermometrics, Inc. of Edison, New Jersey, Table 1 was 
compiled to illustrate those temperatures measured by apparatus 10 both 
with and without beta compensation circuitry for each twenty degree 
temperature increment from 100.degree. to 440.degree. F. 
TABLE I 
______________________________________ 
Measured- Measured- 
Actual Mfr's Value Uncompen- Compensated 
T (.degree. F.) 
R.sub.TH (.OMEGA.) 
sated T(.degree. F.) 
T(.degree. F.) 
______________________________________ 
100 59,851. 102.7 100.00 
120 37,980. 121.9 120.30 
140 24,746. 141.3 140.35 
160 16,519. 160.7 160.25 
180 11,277. 180.3 180.12 
200 7,860. 200.0 200.00 
220 5,580. 219.8 219.94 
240 4,034. 239.8 239.92 
260 2,967. 259.7 259.90 
280 2,215. 279.8 279.94 
300 1,678. 300.0 300.00 
320 1,289. 320.2 320.05 
340 1,003. 340.4 340.07 
360 789.9 360.7 360.06 
380 628.7 381.0 380.05 
400 505.6 401.4 400.00 
420 410.5 421.9 419.90 
440 336.3 442.4 439.70 
______________________________________ 
It should be noted that this beta stabilization technique may be 
successfully applied over all temperature ranges and with all 
semiconductor devices exhibiting a negative exponential resistance 
characteristic with temperature. Moreover, other circuit configurations 
could be employed whereby the resistances of R.sub.S and R.sub.P, being 
permitted to vary with temperature, would substantially reduce or 
eliminate beta variations with temperature over even greater ranges than 
that herein. However, the configuration of FIG. 2 is generally preferred 
because of its simplicity, relatively low cost, and excellent performance 
over ranges of temperatures usually more than sufficient for any 
particular application. 
Beta variation isn't the only factor adversely effecting devices employing 
thermistor temperature sensors. Again with reference to FIG. 1, in order 
to minimize the deleterious effects of thermistor self-heating, after 
passing through logic inverter 28 the output pulse train from time delay 
circuit 24 also operates analog switch 12, which switch may be similar in 
manufacture to switch 23. This results in the thermistor power supply E 
being disconnected during the delay or "off" portion of each complete 
cycle, t.sub.1 plus t.sub.a, in turn causing signal E.sub.1 to go to zero. 
As seen from the low temperature waveforms in FIG. 3 and the high 
temperature waveforms in FIG. 4, coordinated in time but not necessarily 
to scale, as the temperature becomes greater the duty cycle of the output 
signal from time delay circuit 24 decreases, thus causing the thermistor 
to operate for a proportionally shorter time (see waveform E.sub.1), 
reducing the average power dissipated and thereby substantially reducing 
self-heating induced errors. 
Pulsed operation of thermistor 11 will not, however, totally eliminate 
self-heating induced errors. Self-heating is, of course, defined as the 
power a device must dissipate. Thus, the power dissipated in any device, 
including a thermistor, at any particular temperature T may be expressed 
as follows: 
##EQU10## 
where E equals the voltage impressed across its terminals; and R.sub.T 
equals its resistance at temperature T. In apparatus 10, the thermistor 
voltage is held constant for all temperatures. As temperature increases, 
thermistor resistance decreases. Thus, notwithstanding pulsed operation, 
self-heating must still vary with temperature, permitting self-heating 
induced errors to also still vary with temperature. 
FIG. 6 discloses in simplified form a circuit for substantially eliminating 
self-heating induced variations in thermistor temperature sensing devices. 
In order to better understand the functioning of this circuit, the 
following discussion is deemed pertinent. 
When temperature T equals any preselected reference temperature T.sub.o, we 
shall define the output signal from thermistor operational amplifier 14, 
generally known as E.sub.1, to be equal to some voltage E.sub.o. 
Substituting these values in equation (1), the actual thermistor 
resistance R.sub.T becomes equal to R.sub.T.sbsb.o. Thus, from equation 
(11), at temperature T.sub.o the power dissipitated by thermistor 11 is: 
##EQU11## 
In order to maintain power dissipation constant, the power dissipated by 
thermistor 11 at a particular temperature T.sub.o must equal the 
thermistor power dissipation at any temperature T. Therefore, equating 
equations (11) [with equation (1) substituted therein] and (12), 
##EQU12## 
Solving for thermistor voltage E at any temperature T we find 
##EQU13## 
Equation (14) relates a variable voltage E impressed upon thermistor 11 for 
any temperature T to a reference voltage E.sub.o at a reference 
temperature T.sub.o. For reasons which will become more evident 
hereinafter, it is necessary to relate the reference voltage E.sub.o to 
the thermistor operational amplifier 14 output signal E.sub.1 for any 
temperature T. However, first the relationship between signals E.sub.1 and 
E must be expressed. 
From the well known voltage gain of an ideal inverting operational 
amplifier having an input signal E applied through an input resistor 
R.sub.T and the output signal E.sub.1 fed back to the inverting input 
through a resistor R.sub.T.sbsb.o it is known that 
##EQU14## 
Inserting the values of E and R.sub.T from equations (14) and (1), 
respectively, 
##EQU15## 
Rearranging and equating equations (14) and (16), and solving the same for 
E yields 
##EQU16## 
Since FIG. 6 incorporates an inverting operational amplifier, 
##EQU17## 
Equation (18) indicates that permitting the voltage (E) impressed upon a 
thermistor to vary in inverse proportion to an amplified output signal 
(E.sub.1) of a thermistor (times some fixed constant E.sub.o.sup.2) will 
maintain power dissipation by the thermistor totally constant for all 
temperatures and completely eliminate self-heating variations in 
temperature measurements. More fundamentally, as seen from the .beta./2 
term in the exponent of equations (14) and (16), operating a thermistor so 
as to effectively obtain half the value of its material constant, beta, 
yields voltage outputs that are the square root of what they would have 
been for any given temperature T without such operation. It is this square 
root compression of the resulting thermistor voltage output that underlies 
the results achieved by this aspect of the present invention, permitting a 
substantial (and theoretically infinite) expansion in the range of 
linearization without the deleterious effects of self-heating. 
FIG. 6 presents a simplified circuit suitable for the implementation of 
self-heating elimination. Feedback transfer function 34 is introduced 
between the output of operational amplifier 14 and the input of thermistor 
11. Feedback transfer function 34 may comprise any circuit capable of 
generating the transfer function (-k/x) where x equals E.sub.1, and k is 
made, by definition, to equal E.sub.o.sup.2. 
FIG. 5 shows an apparatus according to the concept of the present invention 
incorporating both the beta stabilization of FIG. 2 and the square root 
compression of FIG. 6 and is generally designated by the numeral 35. Most 
of the elements and their interconnection in the circuit of FIG. 5 are 
substantially similar to that of FIG. 1. Of course, since self-heating 
variations are eliminated by the continual application of a particularly 
variable thermistor supply voltage E, thermistor's 11 supply voltage is no 
longer pulsed and analog switch 12 of FIG. 1 is eliminated, as is switch's 
12 control signal from time delay circuit 24. Because the output signal 
from feedback transfer function 34 is an analog signal whose value must be 
negative for impression upon thermistor 11, in the event such signal is 
not already negative, digital logic inverter 28 is replaced by an 
inverting amplifier 36, which may be a conventional amplifier having unity 
gain and resistive feedback (not shown). Fixed voltage source V.sub.cc 
also may be eliminated, and the second compensating resistor 16 connected 
directly to the output of feedback transfer function 34. 
Although it is believed to be well within the skills of the ordinary 
artisan to develop numerous circuits for effectuating feedback transfer 
function 34, all of which must be taken to be within the spirit of the 
present invention, FIG. 7 depicts one such exemplary circuit. Signal 
E.sub.1 is received, through input resistor 40 having resistance R.sub.1, 
by the inverting input terminal of the first of four conventional, 
inverting operational amplifiers 41, 42, 43, and 44, amplifiers 41, 43 and 
44 of which have their non-inverting input terminals connected to ground. 
A constant voltage V.sub.A is received, through input resistor 45 having 
resistance R.sub.3, by the inverting input of operational amplifier 44. 
The outputs of operational amplifiers 41, 44, signals V.sub.1 and V.sub.B, 
respectively, are each fed back to their respective amplifier inverting 
inputs through monolithic transistors 46, 47 having constants K.sub.1 and 
K.sub.3, respectively, defined below. Signal V.sub.1 is received by the 
inverting input of operational amplifier 42 after passing through input 
resistor 48 having resistance R. Signal V.sub.B is directly received by 
the noninverting input of operational amplifier 42. 
The output of operational amplifier 42, signal V.sub.2, is fed back to its 
inverting input through resistor 49 having the same resistance R as 
resistor 48. Additionally, the output of operational amplifier 42 passes 
through transistor 50, monolithic with transistors 46 and 47, having 
constant K.sub.2 and is subsequently received by the inverting input of 
operational amplifier 43. Resistor 51 having resistance R.sub.2 provides 
feedback of the output signal from operational amplifier 43, hereinafter 
demonstrated to be the desired signal E, to its inverting input. 
Because transistors 46, 47 and 50 are monolithic, and assuming that 
operational amplifiers 41, 42, 43 and 44 may be considered to be ideal, it 
is well known that signals V.sub.1, V.sub.B and E may be expressed as 
follows: 
##EQU18## 
where k equals Boltzmann's Constant; T equals the ambient absolute 
temperature; q equals the charge of an electron; K.sub.1, K.sub.2 and 
K.sub.3 equals constants of transistors 46, 50, and 47, respectively, 
based upon that monolithic transistor's emitter reverse saturation 
current, current gain in the common base configuration utilized herein; 
and G equals the gap potential of the transistor's basic semiconductor 
material, generally silicon. 
From the voltage gain relationship for ideal operational amplifiers signal 
V.sub.2 may be expressed as follows: 
##EQU19## 
Substituting V.sub.1 and V.sub.B from equations (19) and (20), 
respectively, yields 
##EQU20## 
Finally, substituting equation (23) into equation (21) and solving for E 
yields 
##EQU21## 
where K' is defined as 
##EQU22## 
Thus, by merely selecting the values of the constants in equation (25) 
such that K' equals E.sub.o.sup.2, equation (18) may be properly 
implemented. 
From the discussion hereinabove, the operation of the elements within 
feedback transfer function 34 of FIG. 7 should now be more readily 
apparent. Initially, in order to obtain the reciprocal of signal E.sub.1, 
E.sub.1 is logarithmically amplified by operational amplifier 41 in 
conjunction with transistor 46. In order to eliminate the distortion in 
signal E.sub.1 introduced by reason of variations in the operating 
temperature of apparatus 35 itself [as represented by the term (kT/q) in 
equation (19)], a constant voltage V.sub.A is introduced and 
logarithmically amplified by operational amplifier 44 in conjunction with 
transistor 47. The logarithmic output signals V.sub.1 and V.sub.B from 
operational amplifiers 41 and 44, respectively, are summed and the 
difference therebetween amplified by operational amplifier 42, and the 
antilog taken thereof by operational amplifier 43 in conjunction with 
transistor 50. The value of feedback resistor 51 may be selected so as to 
adjust the absolute value of signal E to the desired suitable operational 
value. Thus, the output of operational amplifier 43 is a signal E 
inversely proportional to the thermistor amplifier output signal E.sub.1 
and independent of the operating temperature of apparatus 35. 
It should be emphasized that the circuit described immediately above does 
not eliminate the self-heating phenomena in thermistor temperature 
sensors: it merely eliminates all self-heating induced errors by 
effectively eliminating variations induced by self-heating over a 
substantial, and theoretically infinite, range of temperatures. In other 
words, this circuit fixes the formerly variable self-heating induced 
offset signal to a substantially constant value. Once initially accounted 
for, this constant offset signal may be thereafter ignored without 
suffering adverse consequences. Moreover, selection of the above constants 
may be utilized to directly determine the value of the constant offset 
signal. 
Inasmuch as the present invention is subject to many variations, 
modifications and changes in detail, a number of which have been expressly 
stated herein, it is intended that all matter described throughout this 
entire specification or shown in the accompanying drawings be interpreted 
as illustrative and not in a limiting sense. It should thus be evident 
that a device constructed according to the concept of the present 
invention, and reasonably equivalent thereto, will accomplish the objects 
of the present invention and otherwise substantially improve the 
thermistor temperature sensing art.