New Approach to Understanding the Operation of Thermistors in Gas Chromatography David Buhl Lawrence Radiation Laboratory, University of California, Livermore, Calif, Optimum performance in a chromatography system can be obtained only if t h e operation of the thermistor detector i s completely understood. However, t h e traditional approach to analyzing a chromatography bridge has been to treat the sensing element as a variable resistor. While this method may be adequate for a hot wire detector, it fails to properly account for the negative resistance found in thermistor detectors. The present article is an attempt t o analyze the operation of a thermistor bridge using a negative resistance equivalent circuit. A calculation of signal and noise output is made using t h e equivalent circuit in a standard Wheatstone bridge. The signal in thiscase i s due to a change in dissipation constant and the noise is due to current and temperature fluctuations. The calculated noise level then determines the minimum detection threshold. A practical example is worked out which demonstrates that a temperature stability O C and a current stability of 50 p/m are of 4 X required for a 1.5-wV noise level. Several techniques for matching thermistors to reduce the noise level are considered. The analysis indicates the importance of having both well-matched bridge components and thermistors, and a stable power source and temperat u r e environment. The curves presented were plotted o n a computer using the response equations derived. The points on several of the curves represent experimental data used to support the theoretical results obtained.
THERMISTORS are widely used for high sensitivity measurements in gas chromatography. Usually very low noise levels are required in these measurement systems. In such a low noise application, it is important to understand the basic electrical behavior of a thermistor. This is necessary to optimize the performance of the thermistor in a chromatography system. Previous articles (1-4) have dealt with the problem of obtaining the response equations which describe the behavior of the thermistor. However, the fact that the thermistor is a negative resistance device has been often overlooked. In effect there is a positive feedback which at some point in the characteristic V-I curve causes the voltage across the thermistor to decrease with increasing current resulting in a negative resistance. This fact is important and will be utilized in the bridge response equations to be developed. The thermistor bridge used in chromatography is a basic Wheatstone bridge. Variations are made for the purpose of balancing and nulling (see Figure 1). In analyzing the operation of a thermistor bridge it has been customary to treat the thermistor as a variable resistor and solve the Wheatstone bridge equations in a linear fashion. This makes use of the
(1) V. J. Coates, H. J. Noebels, and I. S. Fagerson, “Gas Chromatography,” Academic Press, New York, 1958, pp 165-194. (2) S. Dal Nogare and R. S. Juvet, “Gas and Liquid Chromatography,” Wiley, New York, 1962, pp 189-214. (3) A. I. M. Keulemans, “Gas Chromatography,’’ Reinhold, New York, 1958, pp 84-91. (4) R. E. Walker and A. A. Westenberg, Rev. Sei. Instr., 28, 189 (1957).
partial derivative relationship between dissipation constant and thermistor resistance as follows: AR
dR A6 36
= -
where A6 = change in dissipation constant; AR = change in resistance of the thermistor. This approach is somewhat inadequate because it does not take into account the negative resistance exhibited by thermistors. Another related problem is that the optimum performance of the thermistor bridge is determined by maximizing the equations for the output voltage with respect to the bridge resistor R I . The result of such a calculation is to show that maximum output is obtained when the value of the bridge resistor R I is equal to the dc resistance of the thermistor (Rdc = VjI). Actually, as will be shown, the negative resistance of the thermistor produces a pole (V,,, = a) in the output response as a function of the bridge resistor R1, and thus there is no resistance which produces a maximum output (see Figure 13). These problems and others have prompted us to take a new approach to understanding the basic thermistor theory. In this article we will develop the response equations from the fundamental physics of the thermistor. We begin by first setting up a circuit model and equations to describe the thermistor behavior. This will give us a response equation, and also indicate sources of noise due to variations in undesired parameters. Note that since the thermistor output in chromatography is dependent on three variables-dissipation constant, current, and temperature-one of these can be considered a signal and the other two as noise. Later, the effects of the bridge resistors and other components on the output signal and noise will be considered. An additional noise contribution is due to the thermal noise inherent in a resistor. The level of this noise will be an ultimate limit to the sensitivity of any chromatographic system. EQUIVALENT CIRCUIT FOR A THERMISTOR
To tackle the problem of determining the thermistor model and the equations describing its behavior we first consider the basic equations of the thermistor.
VI = P = Q
=
6 (TT - Tu)
where Temperature of the thermistor bead Wall or gas temperature P = Power into the thermistor Q = Heat dissipated by the thermistor To = Temperature at which R Ois measured @ = Characteristic temperature of the thermistor 6 = Dissipation constant of the thermistor and its environment TT T,
= =
VOL. 40, NO. 4, APRIL 1968
0
715
One Sided B r i d g e
Wheatstone B r i d g e (b1
(4
Wheatstone Bridge with DC Balance (c)
Resistor
-@-
Thermistor
e
Power Source
Re c o r d e r
Dupont-E R L B r i d g e (d1 Figure 1. Thermistor bridge configurations
The first equation comes from the basic semiconductor nature of the thermistor and the fact that the conductivity of a semiconductor is proportional to the carrier density, which is in turn an exponential function of temperature. The second equation is a thermodynamic one and expresses the fact that the temperature drop from the bead to the wall is proportional to the heat dissipated, Using Equations 2 and 3 an expression for the voltage-current characteristic can be derived (see Appendix I). The simplest form of the equation is obtained by expressing the thermistor parameters as the characteristic powers Poand PI: (4)
Where :
P 716
=
VI
ANALYTICAL CHEMISTRY
PI = 6T,
6T,2 =
P
This equation is plotted (Figure 2) and shows the definite resistance (negative Slope) region of the thermistor. To show the close correlation of theory and experiment, the experimental results are plotted as points on the theoretical curve. We will now use this curve to develop a circuit model for the thermistor. The circuit shown in Figure 3a produces a V-Zcharacteristic shown in Figure 36. We have added two voltage generators onto the circuit to take care of the temperature and dissipation constant which are considered later, The V-Z characteristic produced is in certain respects identical to that of the thermistor (Figure 2) and over a small region of interest can be considered equal to it. This is the linear approximation we shall make to the thermistor, Note that this is entirely
Current (mA)
Figure 2.
V-I characteristics of the thermistor
different from the approximation usually made (Figure 4). For the purposes of the present small signal analysis we can drop the battery shown in the equivalent circuit, because it provides only a dc level. The series resistor RT represents the slope of the V-Zcurve, or the sensitivity to current change: R T =bV Z ~
=
PI
' 6 = const T~ = const
i
In gas chromatography we are attempting to measure a change in dissipation constant, so this can be considered the signal. The sensitivity to current and to temperature is the noise. The slope of the V-Z curve for the thermistor can be derived from the same original equations used to obtain the V-Zcurve itself and the form is given as (see Appendix 11):
T- +
(can be ignored in analysis)
(7) Equation 6, which is plotted in Figure 5 , is seen to be negative for currents above 2.5 mA. This is the negative resistance region referred to previously, and is important in the bridge gain to be discussed later on. For the sake of comparison, Equation 7 is also plotted, and shows the difference between the equivalent circuits of Figure 3 and Figure 4. The second noise variable to be considered is the temperature. It enters into Equation 3 as Tu,the wall temperature. The thermistor temperature TT is essentially a coupling variable between Equations 2 and 3, and we have no interest in it other than that. The temperature sensitivity for our linear model is just the partial derivative:
I Current
Figure 3a. Equivalent circuit for thermistor
Figure 36. Characteristic of Figure 3a VOL. 40, NO. 4, APRIL 1968
717
Figure 4a. Variable resistor equivalent circuit Current
Characteristic of Figure 40
Figure 4b.
6 Z
= =
const const
that the temperature appearing in their derivation is the thermistor temperature T T ,rather than the wall temperature Tu.) The change in the thermistor due to dissipation constant is obtained in the same way as previously:
This is obtained similar to the slope equation (Equation 6) and is derived in Appendix I11 (9)
8V 86
P6 = -
This sensitivity to temperature variation is plotted as a function of current in Figure 6 . Having considered the two noise sensitivities, we will now investigate the sensitivity to dissipation constant. (This is similar to the treatment of Sapoff and Oppenheim (5) except
In Appendix IV this is derived to give:
bV
negative r e s i s t a n c e re
I 0
I
I
1
I
2
I
I
3
I
I
4
1
I
5
I
1
6
Figure 5. Resistance plot for the thermistor ANALYTICAL CHEMISTRY
[(I
+ P / P ~+) P/P~I ~
(1 1)
I
C u r r e n t (mA)
718
( V/6)PIP0
as
c ( 6 = - =
( 5 ) M. Sapoff and R. M. Oppenheim, Proc. ZEEE., 51,1292 (1963).
-500
(10)
= const Tu = const
Z
I
I
7
I
1
8
I
9
1
0.1c
0. OE
-Y . -
0.06
>
-I
.-h 0. 04 Y
>
.A
.d c
t L?
0.02
0
1
0
3
2
4 C u r r e n t (mA)
6
5
7
8
9
Figure 6. Temperature sensitivity of the thermistor
x
0
1
2
3
4
5
6
7
8
9
C u r r e n t , (mA)
Figure 7a. Thermistor sensitivity to change in dissipation constant VOL 40, NO. 4, APRIL 1968
719
12
10
0
1
2
3
4
5
6
7
C u r r e n t , (mA)
Figure 76. Thermistor sensitivity corrected for bridge loading effect
50
40
30
20
10
0
Current (mA)
Figure 8. Power dissipated in the thermistor
720
ANALYTICAL CHEMISTRY
8
9
Figure 9. Wheatstone bridge 2
This form is then plotted in Figure 7a. Note that for this particular thermistor the sensitivity is quite flat beyond 3.5 mA (f573. This was also noted experimentally. It should be noted that the above equations are plotted on a computer, using numerical methods to solve the implicit equation (Equation 4):
v =fo
( V , I>
-v
=
fi (I)
Figure 10. Wheatstone bridge with thermistor equivalent circuit Ri
RThevenin
(12) Thevenin
Once the thermistor voltage function has been determined it is a simple matter to obtain the power function P (Figure 8): P = VI =
Ifi (I)
( 1 3)
The equations for the elements of the equivalent circuit have been derived as functions of the thermistor power P. This has been done because of the implicit nature of the voltage function. The computer was then used to plot the equations as a function of current. Most of the derivations of the above equations resulted in forms which were dependent on the thermistor temperature TT. This was related to the wall temperature T , in Appendix V and then subsequently used to express the equations in terms of known parameters T,, 6, p , Ro, and TO. This gives an exact form of the equations since the thermistor temperature TT is not constant. The problem of determining A8 (the change in dissipation constant) is not completely resolved and further information can be found in several of the references given (6, 7). Since the relation of the change in dissipation constant to the change in gas composition is a very complicated one, the experimental data in Figure 7 has been normalized at 5 mA. The circuit model which we have derived for the thermistor (Figure 3a) is equivalent to the total differential: 3V dV = - d I 3I
dV + bV - a + ds bT as -
where each partial derivative is understood to be taken with the other two variables constant. The interpretation of this equation is that the effects of current, temperature, and dissipation constant are all independent and add linearly: dV+AV
dI+AI
d T - + A T d6+AS
2 R1 R r
RThevenin
=
Figure 11. Reduced network for current source bridge The reason for going to all the trouble of establishing an equivalent circuit is that the problem of handling the bridge and the other components in the system can then be solved by elementary circuit theory. WHEATSTONE BRIDGE OUTPUT EQUATION
The basic bridge is the standard Wheatstone shown in Figure 9 and the other bridges are variations on the Wheatstone, whose effect can be easily calculated. So to start off with we will see how the Wheatstone bridge and the equivalent circuit we have developed are analyzed. Note that the equivalent circuit applies for small signals and is only valid over a certain range. The bridge of Figure 10 can be considered as an electrical network. Terminals 1 and 2 are open-circuited if the dc supply is a current source and shorted if it is a voltage source. The superposition principle applies because this is a linear network, and we can first short out the p T source and determine Vodue to pa. Note that because the two sources are identical in their position in the circuit, their equations will be the same, so we treat only the p6 source. For a current power source, the network looks like Figure 11, when the appropriate voltage sources have been shorted out. The Thevenin equivalent circuit for this network at the output terminals of the bridge is also shown and its elements are
Therefore
(6) L. J. Schmauch and R. A. Dinerstein, ANAL.CHEM., 32, 343 (1960).
(7) B. D. Smith and W. W. Bowen, Ibid.,36,82 (1964).
Figure 12 shows a similar circuit for the case of a voltage power source. Note that the Thevenin equivalent circuit has the VOL. 40, NO. 4, APRIL 1968
721
RT
RThevenin
L 2 RlRy
%hevanin = R
T
Figure 12. Reduced network for voltage source bridge
same elements as those above for the current source. Thus the type of source does not affect the output of the bridge. Later, however, we will show that the source stability is very important in determining the noise level of the bridge. To carry the analysis one step further we consider the detector or recorder connected to the output of the bridge. If the recorder has a finite input impedance, the voltage appearing at the recorder input terminals will be just: VR =
RR
RR
+ RTheveoin VThevenin
Combining this with Equation 17 gives
6.E .+ -
8 4-
2-
722
ANALYTICAL CHEMISTRY
(19)
Thus the final voltage read by the recorder is related to the B ~ generator A ~ in the thermistor equivalent circuit by two voltage-divider factors. These factors are somewhat different because RT is negative for currents in the self-heating region and RThevenin is also negative as long as Rl > RTI. Thus both these factors are greater than one and can be very large ( VR-+ 03) for certain values of RI or RR. A plot of this gain is made in Figure 13 as a function of Rn/RTheve,inwhich shows the point made at the beginning, that there is no maximum gain. This plot has been examined experimentally and the pole-like behavior is very evident. As a useful experimental tool in providing increased system sensitivity, this method of loading the bridge tends to slow down considerably the time constant of the system and therefore may not be of much use practically. It does, however, indicate that if gain can be obtained externally (through an amplifier), a much more stable arrangement results if:
1
R1 >> RT and R R >> RThevenin (21) Figures 7a and 76 show two curves of thermistor response as a function of current. The second curve (Figure 7b) has been corrected for bridge loading in Equation 20. It can be seen from the normalized experimental points and by comparing the two curves that this loading factor is fairly significant at low currents. SIGNAL AND NOISE SENSITIVITY FOR A WHEATSTONE BRIDGE
The first bridge we will look at is the ordinary Wheatstone bridge (Figures l b and 10). It is meant mostly for illustration because the practical bridge circuits almost always contain some sort of balance controls (Figure IC). The sensitivity
of the bridge output to temperature fluctuations is represented by the voltage sources p T , and p T 2 . Similarly the resistors R T 1 and RT2represent the sensitivity to current. If the effects of bridge loading are ignored for the present, the noise output of the bridge is
1
Current Balance
In the same way the signal output of the bridge for the assumption used in the above equation (RL>> R T ) AVsignal
=
p~(aA6
(23)
Note that Equations 22 and 23 are quite similar to Equation 16, the difference being that the noise generators are on both sides of the bridge and so appear as differences or mismatch of sensitivities in the equations. The attempt to match the sensitivities and thus make the bridge insensitive to temperature fluctuations is an important nulling device used in the Du Pont-ERL bridge to be discussed later. To minimize the noise one can either minimize the temperature and current fluctuations themselves, or attempt to make the bridge insensitive to them by the nulling mentioned above. When we are dealing with very small changes in the dissipation constant (A6) or equivalently, with very dilute samples in the gas stream, the problem of noise becomes very acute and one must resort to one of the methods mentioned above, or preferably both. To get a better feel for the problem we will show an example which illustrates the order of magnitudes involved. The thermistor whose parameters were used to plot the curves in Figures 2 and 5 through 8 is one of a pair which was used to check experimentally some of the theory presented here. We can assume a Wheatstone bridge with a matched pair of these thermistors and proceed to calculate the noise contributions due to current and temperature fluctuations. The sensitivity to dissipation constant change, or signal, can also be obtained directly from Equation 11. Looking at this backwards, one can start by specifying the A6 due to the smallest sample one would want to detect. Then getting the voltage output of the bridge for this A6 one could specify the maximum tolerable noise in the system as AVnoise I ' 1 2 AVaignsl
(24)
Using this noise level, one can then easily calculate the balance requirements on the bridge and the stability requirements on the temperature of the system. Suppose, for example, that the thermistor mentioned above were operated somewhere near its peak sensitivity (Figure 7), approximately 5 mA. The sensitivity pais about 10 V/mW/' C. If one then requires a sensitivity to a A6 of 3 x 10-7 mW/" C (a dissipation constant change of about 1 p/m), this represents an output signal voltage of AVsignai= paA6
=
3 pV
(25)
Thus a noise level of less than 1.5 pV is required. Looking at the current noise first, we find from Equation 22 AIs AVnoise = (RTI- R T Z ) 2
or AVnoise =
(
Figure 14. Du Pont-ERL Bridge
This equation can be used to determine the bridge nulling and current stability necessary to achieve the required noise level. From Figure 5 we get an approximate value for R T 2 .
Is = 1 0 m A AV,,~,,
=
Ri-2
(" - ) ((RTIRLIRTz) (y)< RTZ
T IR T 2
-200 fl
2oo
lo-'>
< 1.5 pV
1.5 X lov6
Thus the product of the mismatch in the thermistor resistances (RT) and the stability of the current source must be less than 1.5 X (approximately 1 p/m). The matched thermistors we used in some experimental work were about 3 % matched in resistance. This then determines the stability needed in the current source.
5(0.03) < 1.5 X
10-6
IS
AIs
IS
(28)
