Measuring Temperature with a Silicon Diode

Transcription

1 Measuring Teperature with a Silicon Diode Due to the high sensitivity, nearly linear response, and easy availability, we will use a 1N4148 diode for the teperature transducer in our easureents 10 Analysis of the theral dependence in the diode equation The current conducted through an ideal diode is described by the Shockley ideal diode equation: ( ) ] q VD I D I S [exp 1 n k B T where q is the electron charge, k B is Boltzann s constant, T is the teperature of the diode junction, and V D is the voltage across the diode (positive voltage indicates a forward-biased diode) n is a quality factor or eission coefficient that typically lies in the range of 1 2 and is usually assued to be approxiately 2 for a diode I S is the reverse bias saturation current given approxiately by ( I S A exp E ) g 2 k B T where A depends priarily on the geoetry and doping of the junction region and E g is the seiconductor band gap Tha band gap for silicon is usually given as 117 ev at 0 K and 111 ev at 300 K For the 1N4148 diode we usually easure a value of 120 ev Although the equation for I D, the diode current, appears to be very nonlinear (a product of exponentials appears to doinate the equation), it is possible to find operational paraeters that produce a very linear behavior, at least over a reasonable range of teperatures We can solve this for the voltage V D assuing that n 2: V D [ ( )] I ln DA + exp Eg 2 k B 2 k T B T + E g q 1

2 Since E g 120 ev and k B T 0025 ev at roo teperature, exp[ E g /(2 k B T )] and it can probably be neglected So we arrive at the expression V D ln ( I DA ) 2 kb T + E g q If the current through the diode is held constant, there is now a siple linear relationship between voltage and teperature If the diodes we will use were ideal, this expression would be all we need For real diodes, the functional for is still quite linear, eg, V T + b where and b are constants However, b ay be slightly different fro E g /q and the paraeter A in ln(i D /A) is unknown The constants and b will need to be deterined experientally for accurate teperature easureent 20 Checking the validity of the linear assuption It is advisable to verify that our assuption of linearity in the calibration equation is correct We have found that A 90 is a lower liit on the value for a typical 1N4148 diode Any nonlinearity in the equation will result priarily fro the logarithic ter in the nuerator: [ ( ID ln A + exp E )] g 2 k B T Figure 1 shows a graph of the teperature dependence of this ter for four values of I D assuing a value of A 90 A to help in deterining an appropriate diode current for linear behavior Fro this analysis, it is apparent that I D 10 µa would be adequate for easureents fro less than 70 K up to about 300 K (roo teperature) It becoes noticeably nonlinear by the tie we get to 370 K (boiling water) I D 100 µa will allow the use of the diode up to about 370 K but that is at the start of exponentially increasing error Since we are interested in teperatures fro 75 K (boiling liquid nitrogen) to 370 K, we can reasonably use I D 100 µa for our easureents 2

3 Figure 1: ln[i D /A + exp( E g /(2 k b T )]/ ln[i D /A] 1, the fractional error due to neglecting the exponential function in the logarithic ter of the voltage equation as a function of teperature for several values of the diode current I D These curves assue a value of A 90 A and E g 119 ev 30 Estiating the error in the easured teperature Any easureent is guaranteed to have soe error associated with it (review the Uncertainty, Errors, and Noise in Experiental Measureents handout) The obvious proble is that we can t easure a voltage with infinite precision But there are other concerns that ay crop up in a careful analysis of the errors There are two approaches to deterining the calibration constants for the diode 31 Calibrating with VT+b For this for of the equation, the constants are directly related to the theoretical equation derived earlier in this handout Rearranging to get the teperature fro the value of the diode voltage, T (V d b)/, the error analysis is slightly coplicated 3

4 ( ) 2 ( ) 2 ( ) 2 ( T ) 2 V D + V D b b + ( ) 2 ( ) 2 ( ) V D + b (VD b) ( ) 2 ( V 2 2 D + b 2 VD b ) + 2 Using the original equation for T and rearranging soewhat we get ( T ) 2 ( T T ) 2 V 2 D + b2 (V D b) ( ) 2 T ( VD 2 + b 2 ) + V D b ( ) 2 ( ) 2 T 2 This for of the equation is not useful if you are using teperatures in Celsius because of the proble of dividing by 0 It is useful with Kelvin teperatures The previous for without the explicit presence of T in the equation can be used with teperatures in Celsius As an exaple, data taken for a calibration in ice water resulted in the values T K and V D V For these values saples were acquired at saples/s and averaged to arrive at the final value For each saple, it was found that σ V 54 µv Since we averaged saples to arrive at V D, σ VD σ V / µv The Matheatica script NLSQFit240 will return estiated uncertainties in the fitting paraeters if we include uncertainties in the easureents in the third colun of the input file Doing so with the estiated uncertainties in the easureents resulted in V/K, σ V/K, b V, and σ b V Using these values in the above equation, we get ( T T ) 2 ( ) 2 + ( ) 2 ( ) 2 + ( ) 2 ( ) T T At T K we get T 0160 K If we use the easured V D in the fitting equation we get T K 4

5 This is not within the expected error, so the fit appears to be arginal (at least for ice water) 32 Calibrating with T V+b If you have fit your data using the equation T V D + b the error analysis is significantly easier: ( T ) 2 ( ) 2 ( ) 2 ( ) 2 V D + V D b b + ( V D ) 2 + ( b ) 2 + (V D ) 2 In this forat the fitting paraeters are 4206 K/V, σ 017 K/V, b 5069 K, and σ b 012 K Using the easured values at K with V D V and σ VD V we find T ( ) ( ) K and the teperature fro the easured voltage and fitting equation is T K The estiated error in the teperature is coparable for this ethod, but the difference between the easured teperature and the calculated teperature is larger than the first ethod 40 Estiating the effect of varying diode current on the easured teperature The diode current, I D will only affect the value of the constant Note that this is using the value of fro fitting the equation V D T + b Recall that 2 k B q ln ( ) ID A 5

6 (ignoring the sall exponential ter) and that A is a geoetry and coposition dependent constant of the diode (q and k B are still constants of the universe), so we only have I D dependence ( ) 2 ( ) 2 I D ( ) 2 2 kb q I D Using the original equation for and rearranging soe we get ( ) 2 ( ) 2 [ I D ln(i D /A) [ ] 2 I D ln(i D /A) ] 2 With the target value of I D 100 µa, the equation for and the fitting paraeters we can estiate the value of A Solving for A we get A I D exp[ q/(2 k B )] 983 for the value of given above Using these values, we get I D ln(i D /A) For a change in I D of 05 µa we get (approxiately) This value for is an order of agnitude saller than the value found fro the calibration of the diode It will not significantly affect the uncertainty in the teperature If we do include it, the uncertainty in the teperature at 273 K will increase by roughly 01 K This gives us a rough easureent of the accuracy with which we ust regulate the diode current [Modified: Deceber 20, 2017] 6