A Widlar current source is a modification of the basic two-transistor current mirror that incorporates an emitter degeneration resistor for only the output transistor, enabling the current source to generate low currents using only moderate resistor values. The Widlar circuit may be used with bipolar transistors, MOS transistors, and even vacuum tubes. An example application is the 741 operational amplifier, and Widlar used the circuit as a part in many designs. This circuit is named after its inventor, Bob Widlar, and was patented in 1967.
DC analysis
Figure 1 is an example Widlar current source using bipolar transistors, where the emitter resistance 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 resistance 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:
V B = V BE1 = V BE2 + ( β 2 + 1 ) I B2 R 2 ⇒
1 R 2 ( V BE1 − V BE2 ) = ( β 2 + 1 ) I B2 , {\displaystyle {\begin{aligned}&V_{\text{B}}=V_{\text{BE1}}=V_{\text{BE2}}+(\beta _{2}+1)I_{\text{B2}}R_{2}\\\Rightarrow {}&{\frac {1}{R_{2}}}\left(V_{\text{BE1}}-V_{\text{BE2}}\right)=(\beta _{2}+1)I_{\text{B2}}\ ,\end{aligned}}}
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 IB2 is the base current of the output transistor, VBE refers to base-emitter voltage. This equation implies (using the Shockley diode equation): Eq. 1
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