Widlar Current Source - Output Impedance

Output Impedance

An important property of a current source is its small signal incremental output impedance, which should ideally be infinite. The Widlar circuit introduces local current feedback for transistor . Any increase in the current in Q₂ increases the voltage drop across R₂, reducing the V_BE for Q₂, thereby countering the increase in current. This feedback means the output impedance of the circuit is increased, because the feedback involving R₂ forces use of a larger voltage to drive a given current.

Output resistance is found using a small-signal model for the circuit, shown in Figure 2. Transistor Q₁ is replaced by its small-signal emitter resistance r_E because it is diode connected. Transistor Q₂ is replaced with its hybrid-pi model. A test current I_x is attached at the output.

Using the figure, the output resistance is determined using Kirchhoff's laws. Using Kirchhoff's voltage law from the ground on the left to the ground connection of R₂:

Rearranging:

Using Kirchhoff's voltage law from the ground connection of R₂ to the ground of the test current:

or, substituting for I_b:

Eq. 4

According to Eq. 4, the output resistance of the Widlar current source is increased over that of the output transistor itself (which is r_O) so long as R₂ is large enough compared to the r_π of the output transistor (large resistances R₂ make the factor multiplying r_O approach the value (β +1)). The output transistor carries a low current, making r_π large, and increase in R₂ tends to reduce this current further, causing a correlated increase in r_π. Therefore, a goal of R₂ >> r_π can be unrealistic, and further discussion is provided below. The resistance R₁//r_E usually is small because the emitter resistance r_E usually is only a few ohms.