A Wilson current mirror is a three-terminal circuit (Fig. 1) that accepts an input current at the input terminal and provides a "mirrored" current source or sink output at the output terminal. The mirrored current is a precise copy of the input current.
It may be used as a Wilson current source by applying a constant bias current to the input branch as in Fig. 2. The circuit is named after George R. Wilson, an integrated circuit design engineer who worked for Tektronix.Sedra, A.S. & Smith, K.C.: "Microelectronic Circuits, 6th Ed.", OUP (2010), pp. 539 - 541. Wilson devised this configuration in 1967 when he and Barrie Gilbert challenged each other to find an improved current mirror overnight that would use only three transistors. Wilson won the challenge.Gilbert, B., "Bipolar Current Mirrors," in "Analogue IC Design: the Current-Mode Approach," Eds. Toumazou, C., Lidgey, F. J. & Haigh, D. G., Peter Peregrinus Ltd. (1990), , pp. 268-275.
Circuit operation
There are three principal metrics of how well a current mirror will perform as part of a larger circuit. The first measure is the static error, i.e., the difference between the input and output currents expressed as a fraction of the input current. Minimizing this difference is critical in such applications of a current mirror as the differential to single-ended output signal conversion in a differential amplifier stage because this difference controls the common mode and power supply rejection ratios. The second measure is the output impedance of the current source or equivalently its inverse, the output conductance. This impedance affects stage gain when a current source is used as an active load and affects common mode gain when the source provides the tail current of a differential pair The last metric is the pair of minimum voltages from the common terminal, usually a power rail connection, to the input and output terminals that are required for proper operation of the circuit. These voltages affect the headroom to the power supply rails that are available for the circuitry in which the current mirror is embedded.
An approximate analysis due to Gilbert shows how the Wilson current mirror works and why its static error should be very low. Transistors Q1 and Q2 in Fig. 1 are a matched pair sharing the same emitter and base potentials and therefore have and . This is a simple two-transistor current mirror with as its input and as its output. When a current is applied to the input node (the connection between the base of Q3 and collector of Q1), the voltage from that node to ground begins to increase. As it exceeds the voltage required to bias the emitter-base junction of Q3, Q3 acts as an emitter follower or common collector amplifier and the base voltage of Q1 and Q2 begins to rise. As this base voltage increases, current begins to flow in the collector of Q1. All increases in voltage and current stop when the sum of the collector current of Q1 and base current of Q3 exactly balance . Under this condition all three transistors have nearly equal collector currents and therefore approximately equal base currents. Let . Then the collector current of Q1 is ; the collector current of Q2 is exactly equal to that of Q1 so the emitter current of Q3 is . The collector current of Q3 is its emitter current minus the base current so . In this approximation, the static error is zero.
Difference of input and output currents A more exact formal analysis shows the expected static error. We assume: All transistors have the same current gain β. Q1 and Q2 are matched, and they share the same base-emitter voltage, so their collector currents are equal.
Therefore, and . The base current of Q3 is given by, and the emitter current by,
... (1)
From the sum of currents at the node shared by the emitter of Q3, the collector of Q2 and the bases of Q1 and Q2, the emitter current of Q3 must be:
... (2)
Equating the expressions for in (1) and (2) gives:
... (3)
The sum of currents at the input node implies that . Substituting for from (3) leads to: or .
Because is the output current, the static error, the difference between the input and output currents, is:
... (4)
With NPN transistors, the current gain, , is of the order of 100, and, in principle, the mismatch is about 1:5000.
For the Wilson current source of Fig. 2, the input current of the mirror is . The base-emitter voltages, , are typically between 0.5 and 0.75 volts so some authors approximate this result as . The output current is thus substantially dependent only on VCC and R1 and the circuit acts as a constant current source, that is, the current remains constant with variations in the impedance of the load. However, variations in VCC or changes in the value of R1 due to temperature will be reflected in variations in the output current. This method of direct generation of a reference current from the power supply using a resistor rarely has adequate stability for practical applications and more complex circuits are used to provide reference currents independent of temperature and supply voltages.
Equation (4) substantially underestimates the differences between the input and output currents that are generally found in this circuit for three reasons. First, the emitter-collector voltages of the inner current mirror formed by Q1 and Q2 are not the same. Transistor Q2 is diode-connected and has , which is typically on the order of 0.6 to 0.7 volts. The collector emitter voltage of Q1 is higher by the base-emitter voltage of Q3 and therefore is about twice the value across Q2. The Early effect (base-width modulation) in Q1 will force its collector current to be slightly higher than that of Q2. This problem can be essentially eliminated by the addition of a fourth transistor, shown as Q4 in the improved Wilson current mirror of Fig. 4a. Q4 is diode-connected in series with the collector of Q1, lowering its collector voltage until it is approximately equal to for Q2.
Second, the Wilson current mirror is susceptible to mismatches in the current gain, , of its transistors, particularly the match between and the current gains of the matched pair Q1 and Q2. Accounting for differences among all three transistors, one can show that where is the harmonic mean of the current gains of Q1 and Q2 or . Beta mismatches of five percent or more are reported to be common, causing an order of magnitude increase in the static error.
Finally, the collector current in a bipolar transistor for low and moderate emitter currents conforms closely to the relation where is the thermal voltage and is a constant dependent on temperature, doping concentrations, and collector-emitter voltage. Matched currents in transistors Q1 and Q2 depend on conformity to the same equation but observed mismatches in are geometry dependent and range from percent. Such differences between Q1 and Q2 lead directly to static errors of the same percentage for the entire mirror. Careful layout and transistor design must be used to minimize this source of error. For example, Q1 and Q2 may each be implemented as a pair of paralleled transistors arranged as a cross-coupled quad in a common-centric layout to reduce effects of local gradients in current gain. If the mirror is to be used at a fixed bias level, matching resistors in the emitters of this pair can transfer some of the matching problem from the transistors to those resistors.
Input and output impedances and frequency response
A circuit is a current source only to the extent that its output current is independent of its output voltage. In the circuits of Fig. 1 and Fig. 2, the output voltage of importance is the potential from the collector of Q3 to ground. The measure of that independence is the output impedance of the circuit, the ratio of a change in output voltage to the change in current it causes. Fig. 3 shows a small signal model of a Wilson current mirror drawn with a test voltage source, , attached to the output. The output impedance is the ratio: . At low frequency this ratio is real and represents an output resistance.
In Fig. 3, transistors Q1 and Q2 are shown as forming a standard two-transistor current mirror. It is sufficient for calculating the output impedance to assume that the output current of this current mirror sub-circuit, , is equal to the input current, , or . Transistor Q3 is represented by its low-frequency hybrid-pi model with a current controlled dependent current source for the collector current.
The sum of currents at the emitter node of Q3 implies that:
... (5)
Because the dynamic resistance of the diode-connected transistor Q2, the input resistance of the two-transistor current mirror, is much smaller than , the test voltage, , effectively appears across the collector-emitter terminals of Q3. The base current of Q3 is . Using equation (5) for , the sum of currents at the collector node of Q3 becomes . Solving for the output impedance gives:
... (6)
In a standard two-transistor current mirror, the output impedance would be the dynamic early resistance of the output transistor, the equivalent of which in this case is . The Wilson current mirror has an output impedance that is higher by the factor , on the order of 50 times.
The input impedance of a current mirror is the ratio of the change in input voltage (the potential from the input terminal to ground in Fig. 1 and Fig. 2) to the change in input current that causes it. Since the change in output current is very nearly equal to any change in input current, the change in the base-emitter voltage of Q3 is . Eq. (3) shows that the collector of Q2 changes by nearly the same amount, so . The input voltage is the sum of the base-emitter voltages of Q2 and Q3; the collector currents of Q2 and Q3 are nearly equal implying that . The input impedance is . Using the standard formula for leads to:
... (7)
where is the usual thermal voltage, the product of the Boltzmann constant and absolute temperature divided by the charge of an electron. This impedance is twice the value of for the standard two-transistor current mirror.
Current mirrors are frequently used in the signal path of an integrated circuit, for example, for differential to single-ended signal conversion within an operational amplifier. At low bias currents, the impedances in the circuit are high enough that the effect of frequency may be dominated by device and parasitic capacitances shunting the input and output nodes to ground, lowering the input and output impedances. The collector-base capacitance, , of Q3 is one component of that capacitive load. The collector of Q3 is the output node of the mirror and its base is the input node. When any current flows in , that current becomes an input to the mirror and the current is doubled at the output. Effectively the contribution from Q3 to the total output capacitance is . If the output of the Wilson mirror is connected to a relatively high impedance node, the voltage gain of the mirror may be high. In that case the input impedance of the mirror may be affected by the Miller effect because of , although the low input impedance of the mirror mitigates this effect.