Widlar Current Source - Analysis

Analysis

Figure 1 is an example Widlar current source using bipolar transistors, where the emitter resistor R₂ is connected to the output transistor Q₂, and has the effect of reducing the current in Q₂ relative to Q₁. The key to this circuit is that the voltage drop across the resistor R₂ subtracts from the base-emitter voltage of transistor Q₂, thereby turning this transistor off compared to transistor Q₁. This observation is expressed by equating the base voltage expressions found on either side of the circuit in Figure 1 as:

where β₂ is the beta-value of the output transistor, which is not the same as that of the input transistor, in part because the currents in the two transistors are very different. The variable I_B2 is the base current of the output transistor, V_BE refers to base-emitter voltage. This equation implies (using the Shockley diode law):

Eq. 1

where V_T is the thermal voltage.

This equation makes the approximation that the currents are both much larger than the scale currents I_S1, I_S2, an approximation valid except for current levels near cut off. In the following the distinction between the two scale currents is dropped, although the difference can be important, for example, if the two transistors are chosen with different areas.