EN Tempco article (very long)

[gstopp at fibermux.com]

     Hi DIY list,
     
     Here's the text from EN #95, starting on page 15. It's from a 
     "reader's questions" type of section:
     
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     QUESTION: I have a circuit that calls for a 1000 ohm Q81 temperature 
     compensating resistor. Is there some way I can use the 2000 ohm unit 
     you sell other than by using two in parallel to get 1000 ohms?
     
     ANSWER: Not only can you usually use our 2000 ohm units, but you can 
     also use just about any other value or other type of thermister if you 
     know what you are doing. This is probably a good place to present the 
     full ideas.
     
     First, and very briefly, the basics. The exponential stages of VCOs 
     and VCFs are formed using the relationship between the collector 
     current and the base-to-emitter voltage, the collector current being 
     an exponential function of the base-to-emitter voltage. This 
     relationship is very sensitive to temperature, and it is necessary to 
     temperature compensate. The first step in this compensation is to use 
     a matched pair of transistors against each other. This is a major 
     improvement and one we could not get along without. Even with this 
     done, there is another term which we can temperature compensate, and 
     it is worth going after this one. The exponential function is of the 
     form:
     
                e^(q * Vb / K(B) * T)
     
       (that's "e to the power of [q times Vb] divided by [K(B) times T]")
     
     where q is the charge of an electron, Vb is the voltage on the 
     transistor base of interest, K(B) is "Boltzmann's Constant", and T is 
     the "Absolute temperature". What we shall be concerned with here is 
     temperature compensating this term.
     
     It is our approach here not to try to control the temperature, but 
     rather to compensate for a change by making the Vb term temperature 
     sensitive as well. Thus we think of Vb(T), Vb as a function of T. 
     Since q and K(B) are constants, the exponential term will be constant 
     as long as the ratio Vb(T)/T is a constant. What is Vb(T)? It is first 
     of all the sum of control voltages scaled by some amount of about 
     0.018 so that a one volt change of any of the control voltages 
     provides about an 18mv change of Vb. The exact scale factor is 
     determined by trimming the volts/octave trimpot. Note that the trimpot 
     will take up only so much slack (we intentionally make the range small 
     so that we get good adjustment resolution), so if you change any 
     resistor in the control voltage input chain, you must adjust some 
     other resistor so that the result is still in the "target" range of 
     18mv per volt. Now to the question of actual temperature compensation.
     
     In order to know the significance of a change of temperature of 1 
     degree F for example, we must know how important it is to the 
     transistors as physical devices. Certainly a change of 20 degrees C 
     for a human being is rather drastic, and greatly alters our 
     performance. Yet, what of a purely physical device? These devices 
     "see" temperature with respect not to our thermometers, but with 
     respect to an "Absolute Zero" of temperature, the lowest temperature 
     possible anywhere. The absolute zero is a rather chilling -273 degrees 
     C (-460 degrees F). On the absolute scale (also called the Kelvin 
     scale), water freezes at +273 K and boils at +373 K. Human beings, 
     being composed in good part of water, prefer to live and play 
     synthesizers in a rather restricted range well within the limits of 
     the boiling and freezing points of the preferred liquid form. The 
     range is something like 50 degrees F to 100 degrees F which is 10 
     degrees C to 38 degrees C, or +283 K to +311 K. Thus, while we might 
     feel that a change in the range of 50 degrees F to 100 degrees F is 
     rather drastic, here we have a physical system with a linear response 
     to temperature (the K(B) * T term), and the change is, as a percentage 
     basis with reference to absolute zero, much less. If +300 K is a 
     "comfortable" temperature for the electrical system, then +283 K is 
     barely any change at all, and likewise for +311 K. Thus, when we 
     reduce the temperature dependence of the system to a linear one, and 
     try to keep the temperature limits to within tolerable human limits, 
     we are already pretty stable, with a remaining temperature response of 
     about 1 in 300 per degree Kelvin change.
     
     We see from the above that we basically expect our circuits to operate 
     around +300 K. A one degree change in temperature (K degrees are the 
     same size as C degrees) thus represents a change of one part in 300, 
     or about 0.33% per degree C, or about 3300ppm (parts per million) per 
     degree C. If we give the scale factor which determines Vb the same 
     temperature dependence, the ratio will remain constant. Thus, we need 
     a resistor that has a temperature coefficient of about 3300ppm per 
     degree C. The Tel Labs Q81 resistors have approximately this figure 
     (+3500ppm per degree C).Note that this figure is quite large compared 
     to ordinary resistors which generally run from about 600ppm per degree 
     C down to as low as 20ppm per degree C.
     
     There are several positions into which the temperature compensating 
     resistor can be placed so that the scale is properly compensated. Two 
     examples are shown below:
     
     (diagram A shows op-amp summer with 100K input resistors and 2K Q81 
     feedback resistor, driving a 100 ohm pot to a 400 ohm resistor (Rx) to 
     ground, Vb taken from the pot wiper)
     
     (diagram B shows op-amp summer with 100K input resistors and feedback 
     consists of a 91K plus 25K trimmer in series, driving a 110K resistor 
     (Ry) into a 2K Q81 (Rz) resistor to ground, with Vb taken from the 
     junction of the 110K and 2K resistors)
     
     In circuit A, our standard circuit, the temperature compensating 
     resistor (Q81) is in the feedback loop of the op-amp summer, and 
     directly controls the scale factor. Vb is then determined by setting 
     the v/Oct trimmer, Rx being a stable resistor. In circuit B, we have a 
     setup similar to that used with the SSM2030 VCO IC. In the original 
     SSM application note, Ry is a precision 54.9K resistor while Rz is a 
     1000 ohm Q81. Since Ry and Rz form a voltage divider, it is only the 
     ratio that is important. Thus, in circuit B, we have doubled both 
     resistors, and Ry needs not be precision because we can trim the 
     volts/octave response with the 25K trimmer in the summer. The only 
     difference in changing from 1K to 2K for the Q81 is that the 
     "equivalent source resistance" going into the base changes from about 
     1K to about 2K, a negligable difference for the high-Beta transistors 
     generally used. Rz should not get too large however. Note that if Rz 
     is a 1,87K resistor, Ry should be about 100K.
     
     We will give another example to illustrate two points at once. Suppose 
     you do not have any Q81 resistors, but do have some other sort of 
     thermisters. Can you use them? First of all, the temperature 
     coefficient must be 3300ppm per degree C or larger, either positive or 
     negative. Note that negative temperature coefficients are just fine if 
     we use them in the correct place. In circuit B, if Rz is a stable 
     resistor and Ry has a -3300ppm per degree C coefficient, things work 
     just the same. For our example, let's suppose we have a thermister of 
     80K resistance and a temperature coefficient of -7800ppm per degree C. 
     Our first step is to adjust the temperature coefficient downward with 
     a series stable resistor. Note that if a 109K resistor (stable) is put 
     in series with the 80K thermister, the total resistance is 189K. A 1 
     degree C change in temperature will decrease the total resistance by 
     0.78% * 80K or 624 ohms, which is 0.33% of the 189K total. This makes 
     the total resistance have a temperature coefficient of -3300ppm per 
     degree C, the desired magnitude. The calculation of the value of the 
     stable resistor Rb is given below:
     
     (diagram shows Ra in series with Rb, where rB is stable)
     
     Rb = Ra * ((|x| / 3.3 x 10E3) - 1)
     
     (that's Rb equals Ra times (a fraction minus one), where the fraction 
     is the absolute value of x divided by 3300)
     
     where |x| is the magnitude of the temperature coefficient (forget the 
     + or - sign). The next and final step is to place the series 
     combination in an appropriate place in the circuit. If the overall 
     temperature coefficient is positive, we use it like a Q81 resistor. If 
     it is negative, we use it in a different position, typically the upper 
     leg of a voltage divider. For our example, we arrived at a 189K 
     resistor with a temperature coefficient of -3300ppm per degree C. We 
     can use this as an Ry type of resistor in circuit B above. We must 
     maintain the proper scale factor of about 0.0182, so this means that:
     
     Rz = 0.0182 (Ra + Rb)
     
     which comes out to 3.6K in this case. The final circuit is shown in 
     figure C below:
     
     (diagram C shows an op-amp summer with 100K input resistors and a 100K 
     feedback resistor, driving the 109K/80K tempco chain, driving a 3.6K 
     resistor to ground, with Vb taken from the junction of the 3.6K 
     resistor and the tempco)
     
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     (end of article)
     
     I opted not to include ASCIImatics since the circuits are pretty easy. 
     Hope this is informative.
     
     - Gene
     gstopp at fibermux.com