• Reference voltages or current that exhibit little dependence on temperature prove essential in many analogue circuits. It is interesting to note that, since most process parameters vary with temperature, if a reference is independent, and then it is usually process independent.
• To generate a quantity that remains constant with temperature, postulate that if two quantities having opposite temperature coefficients (TCs) I added with proper weighting, the result display a zero TC.

• For example, for two voltages V1 and V2 that vary in opposite directions with temperature α1 and α2 such that

  

• Obtaining a reference voltage VREF = α1V1 + α2V2 , with zero TC.

• Among various device parameters in semiconductor technologies, the characteristics of bipolar transistor have proven the most reproducible and well defined quantities that can provide positive and negative TCs. Even though many parameters of MOS devices have been considered for the task of reference generation, bipolar devices still forms the core of such circuits.

1. Negative TC voltage:

• The base-emitter voltage of bipolar transistor exhibits a negative TC. The expression for TC is as follows-

• For a bipolar device we can write,

  

• The saturation current, Is α μ KT ni^2 where μ denotes mobility of minority carriers and ni is the intrinsic minority carrier concentration of silicon. The temperature dependence of these quantities is represented as,

  

• Where, b is proportionality factor. Rearranging equation (1), we can write,

  

• To find temperature coefficient, partially differentiate VBE with respect to T. Assume that Ic is constant with respect to T.

  

• From equation (3) we have,

  

• From equation (5) and (6) we can write,

  

• This is the expression for TC of VBE voltage.

2. Positive TC voltage:

• It is observed that if two bipolar transistors operate at unequal current densities, then the difference between their base-emitter voltages is directly proportional to the absolute temperature. Consider an example, as shown in figure below-

  

• Here, two identical transistors (with reverse saturation currents, IS1 = IS2) are biased at collector currents of nI0 and I0 and their base currents are negligible then,

  

• Therefore, temperature coefficient of ΔVBE will be,

  

• Therefore, temperature coefficient of ΔVBE is positive.