# Widlar current source

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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, which means less area is needed in an integrated circuit.^{[1]}^{[2]}^{[3]}

The Widlar circuit may be used with bipolar transistors, MOS transistors, and even vacuum tubes. An example application is the 741 operational amplifier,^{[4]} and Widlar used the circuit as a part in many designs.^{[5]}

This circuit is named after its inventor, Bob Widlar, and was patented in 1967.^{[6]}^{[7]}

## Analysis

The figure is an example Widlar current source using bipolar transistors, where the emitter resistor R_{2} is connected to the output transistor Q_{2}, and has the effect of reducing the current in Q_{2} relative to Q_{1}. The key to this circuit is that the voltage drop across the resistor *R*_{2} subtracts from the base-emitter voltage of transistor *Q*_{2}, thereby turning this transistor off compared to transistor *Q*_{1}. This observation is expressed by equating the base voltage expressions found on either side of the circuit in the figure as:

where β_{2} 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.^{[8]} 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.

## Design procedure with specified currents

To design the mirror, the output current must be related to the two resistor values *R*_{1} and *R*_{2}. A basic observation is that the output transistor is in active mode only so long as its collector-base voltage is non-zero. Thus, the simplest bias condition for design of the mirror sets the applied voltage *V*_{A} to equal the base voltage *V*_{B}. This minimum useful value of *V*_{A} is called the *compliance voltage* of the current source. With that bias condition, the Early effect plays no role in the design.^{[9]}

These considerations suggest the following design procedure:

- Select the desired output current,
*I*_{O}=*I*_{C2}. - Select the reference current,
*I*_{R1}, assumed to be larger than the output current, probably considerably larger. (That is the purpose of the circuit.) - Determine the input collector current of
*Q*_{1},*I*_{C1}:

- Determine the base voltage
*V*_{BE1}using the Shockley diode law

- where
*I*_{S}is a device parameter sometimes called the*scale current*. - The value of base voltage also sets the compliance voltage
*V*_{A}=*V*_{BE1}. This voltage is the lowest voltage for which the mirror works properly.

- Determine
*R*_{1}:

* Determine the emitter leg resistance *R*_{2} using *Eq. 1* (to reduce clutter, the scale currents are chosen equal):

## Finding the current with given resistor values

The inverse of the design problem is finding the current when the resistor values are known. An iterative method is described next. Assume the current source is biased so the collector-base voltage of the output transistor *Q*_{2} is zero. The current through *R*_{1} is the input or reference current given as,

Rearranging, *I*_{C1} is found as:

*Eq. 2*

The diode equation provides:

*Eq. 3*

*Eq.1* provides:

These three relations are a nonlinear, implicit determination for the currents that can be solved by iteration.

- We guess starting values for
*I*_{C1}and*I*_{C2}. - We find a value for
*V*_{BE1}:

- We find a new value for
*I*_{C1}:

- We find a new value for
*I*_{C2}:

This procedure is repeated to convergence, and is set up conveniently in a spreadsheet. One simply uses a macro to copy the new values into the spreadsheet cells holding the initial values to obtain the solution in short order.

Note that with the circuit as shown, if V_{CC} changes, the output current will change. Hence, to keep the output current constant despite fluctuations in *V*_{CC}, the circuit should be driven by a constant current source rather than using the resistor *R*_{1}.

### Exact solution

The transcendental equations above can be solved exactly in terms of the Lambert W function.

## 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 *Q*_{2} increases the voltage drop across *R*_{2}, reducing the *V*_{BE} for *Q*_{2}, thereby countering the increase in current. This feedback means the output impedance of the circuit is increased, because the feedback involving *R*_{2} 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 the figure. A test current *I*_{x} is applied at the output, and the output resistance is then *R*_{O} = *V*_{x} / *I*_{x}. The transistor *Q*_{1} is replaced by its small-signal emitter resistance *r*_{E} because it is diode connected.^{[10]} The transistor *Q*_{2} 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 *R*_{2}:

Rearranging:

Using Kirchhoff's voltage law from the ground connection of *R*_{2} 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 *R*_{2} is large enough compared to the *r*_{π} of the output transistor. (Large resistances *R*_{2} 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 *R*_{2} tends to reduce this current further, causing a correlated increase in *r*_{π}. Therefore, a goal of *R*_{2} >> *r*_{π} can be unrealistic, and further discussion is provided below. The resistance *R*_{1}//*r*_{E} usually is small because the emitter resistance *r*_{E} usually is only a few ohms.

#### Current dependence of output resistance

In the figure:

- Top Panel: Circuit output resistance
*R*_{O}*vs.*DC output current*I*_{C2}using the design formula of*Eq. 5*for*R*_{2}; - Center Panel: Resistance
*R*_{O2}in output transistor emitter leg; - Bottom Panel: Feedback factor contributing to output resistance. Current in the reference transistor
*Q*_{1}is held constant, thereby fixing the compliance voltage. Plots assume*I*_{C1}= 10 mA,*V*_{A}= 50 V,*V*_{CC}= 5 V,*I*_{S}= 10 fA, β_{1, 2}= 100 independent of current.

The current dependence of the resistances *r*_{π} and *r*_{O} is discussed in the article hybrid-pi model. The current dependence of the resistor values is:

- in ohms, and
- is the output resistance due to the Early effect when
*V*_{CB}= 0 V (device parameter*V*_{A}is the Early voltage).

From earlier in this article (setting the scale currents equal for convenience):

*Eq. 5*

Consequently, for the usual case of small *r*_{E}, and neglecting the second term in *R*_{O} with the expectation that the leading term involving *r*_{O} is much larger:

*Eq. 6*

where the last form is found by substituting *Eq. 5* for *R*_{2}. *Eq. 6* shows that a value of output resistance much larger than *r*_{O} of the output transistor results only for designs with *I*_{C1} >> *I*_{C2}. The trade-off figure shows that the circuit output resistance *R*_{O} is not determined so much by feedback as by the current dependence of the resistance *r*_{O} of the output transistor. (The output resistance in the figure varies four orders of magnitude, while the feedback factor varies only by one order of magnitude.)

Increase of *I*_{C1} to increase the feedback factor also results in increased compliance voltage, not a good thing as that means the current source operates over a more restricted voltage range. So, for example, with a goal for compliance voltage set, placing an upper limit upon *I*_{C1}, and with a goal for output resistance to be met, the maximum value of output current *I*_{C2} is limited.

The center panel in the trade-off figure shows the design trade-off between emitter leg resistance and the output current: a lower output current requires a larger leg resistor, and hence a larger area for the design. An upper bound on area therefore sets a lower bound on the output current and an upper bound on the circuit output resistance.

*Eq. 6* for *R*_{O} depends upon selecting a value of *R*_{2} according to *Eq. 5*. That means *Eq. 6* is not a *circuit behavior* formula, but a *design value* equation. Once *R*_{2} is selected for a particular design objective using *Eq. 5*, thereafter its value is fixed. If circuit operation causes currents, voltages or temperatures to deviate from the designed-for values; then to predict changes in *R*_{O} caused by such deviations, *Eq. 4* should be used, not *Eq. 6*.

## Notes

- ↑ PR Gray, PJ Hurst, SH Lewis & RG Meyer (2001).
*Analysis and design of analog integrated circuits*, 4rth Edition. John Wiley and Sons, §4.4.1.1 pp. 299–303. ISBN 0-471-32168-0. - ↑ AS Sedra and KC Smith (2004).
*Microelectronic circuits*, 5th Edition. Oxford University Press, Example 6.14, pp. 654–655. ISBN 0-19-514251-9. - ↑ MH Rashid (1999).
*Microelectronic circuits: analysis and design*. PWS Publishing Co., 661–665. ISBN 0-534-95174-0. - ↑ AS Sedra and KC Smith (2004).
*§9.4.2, p. 899*, 5th Edition. ISBN 0-19-514251-9. - ↑ See, for example, Figure 2 in
*IC voltage regulators*. - ↑ RJ Widlar: US Patent Number 03320439; Filed May 26, 1965; Granted May 16, 1967:
*Low-value current source for integrated circuits* - ↑ See Widlar:
*Some circuit design techniques for linear integrated circuits*and*Design techniques for monolithic operational amplifiers* - ↑ PR Gray, PJ Hurst, SH Lewis & RG Meyer (2001).
*Figure 2.38, p. 115*. ISBN 0-471-32168-0. - ↑ Of course, one might imagine a design where the output resistance of the mirror is a major consideration. Then a different approach is necessary.
- ↑ In a
*diode-connected transistor*the collector is short-circuited to the base, so the transistor collector-base junction has no time-varying voltage across it. As a result, the transistor behaves like the base-emitter diode, which at low frequencies has a small-signal circuit that is simply the resistor*r*_{E}=*V*_{T}/*I*_{E}, with*I*_{E}the DC Q-point emitter current. See diode small-signal circuit.