# Current mirror

A **current mirror** is an electric circuit designed to control a current through one active device by copying the current in another active device, thereby keeping the output current constant regardless of variations in loading. The current being 'copied' can be a varying signal current. Conceptually, an ideal current mirror is simply an ideal current amplifier. The current mirror is used to provide bias currents and active loads to circuits.

## Mirror characteristics

There are three main specifications that characterize a current mirror. The first is the current level it produces. The second is its AC output resistance, which determines how much the output current varies with the voltage applied to the mirror. The third specification is the minimum voltage drop across the mirror necessary to make it work properly. This minimum voltage is dictated by the need to keep the output transistor of the mirror in active mode. The range of voltages where the mirror works is called the **compliance range** and the voltage marking the boundary between good and bad behavior is called the **compliance voltage**. There are also a number of secondary performance issues with mirrors, for example, temperature and bias stability.

## Practical approximations

For small-signal analysis the current mirror can be approximated by its equivalent Norton impedance .

In large-signal hand analysis, a current mirror usually is approximated simply by an ideal current source. However, an ideal current source is unrealistic in several respects:

- it has infinite AC impedance, while a practical mirror has finite impedance
- it provides the same current regardless of voltage, that is, there are no compliance range requirements
- it has no frequency limitations, while a real mirror has limitations due to the parasitic capacitances of the transistors
- the ideal source has no sensitivity to real-world effects like noise, power-supply voltage variations and component tolerances.

## Circuit realizations of current mirrors

### Basic bipolar transistor mirror

The simplest bipolar current mirror consists of two transistors connected as shown in the figure. Transistor Q_{1} is connected so its collector-base voltage is zero. Consequently, the voltage drop across Q_{1} is *V*_{BE}, that is, this voltage is set by the diode law and Q_{1} is said to be **diode connected**. (See also Ebers-Moll model.) It is important to have Q_{1} in the circuit instead of a simple diode, because Q_{1} sets *V _{BE}* for the transistor Q

_{2}. If Q

_{1}and Q

_{2}are matched, that is, have substantially the same device properties, and if the mirror output voltage is chosen so the collector-base voltage of Q

_{2}also is zero, then the

*V*-value set by Q

_{BE}_{1}results in an emitter current in the matched Q

_{2}that is the same as the emitter current in Q

_{1}. Because Q

_{1}and Q

_{2}are matched, their β

_{0}-values also agree, making the mirror output current the same as the collector current of Q

_{1}. The current delivered by the mirror for arbitrary collector-base reverse bias

*V*

_{CB}of the output transistor is given by (see bipolar transistor):

- ,

where *V _{T}* =

**thermal voltage**(

*k*≈ 25 mV at 290K,

_{B}T/q*k*is the Boltzmann constant),

_{B}*I*= reverse saturation current, or scale current;

_{S}*V*= Early voltage. This current is related to the reference current

_{A}*I*when the output transistor

_{REF}*V*= 0 V by:

_{CB}as found using Kirchhoff's current law at the collector node of Q_{1}. The reference current supplies the collector current to Q_{1} and the base currents to both transistors — when both transistors have zero base-collector bias, the two base currents are equal. Parameter β_{0} is the transistor β-value for *V*_{CB} = 0 V.

#### Output resistance

If V_{CB} is greater than zero in output transistor Q_{2}, the collector current in Q_{2} will be somewhat larger than for Q_{1} due to the Early effect. In other words, the mirror has a finite output (or Norton) resistance given by the *r _{O}* of the output transistor, namely (see Early effect):

- ,

where *V _{A}* = Early voltage and

*V*= collector-to-base bias.

_{CB}#### Compliance voltage

To keep the output transistor active, *V _{CB}* ≥ 0 V. That means the lowest output voltage that results in correct mirror behavior, the compliance voltage, is

*V*=

_{OUT}*V*=

_{CV}*V*under bias conditions with the output transistor at the output current level

_{BE}*I*and with

_{C}*V*= 0 V or, inverting the

_{CB}*I-V*relation above:

where *V _{T}* = thermal voltage and

*I*= reverse saturation current (scale current).

_{S}#### Extensions and complications

When Q_{2} has *V _{CB}* > 0 V, the transistors no longer are matched. In particular, their β-values differ due to the Early effect, with

where V_{A} is the Early voltage and β_{0} = transistor β for V_{CB} = 0 V. Besides the difference due to the Early effect, the transistor β-values will differ because the β_{0}-values depend on current, and the two transistors now carry different currents (see Gummel-Poon model).

Further, Q_{2} may get substantially hotter than Q_{1} due to the associated higher power dissipation. To maintain matching, the temperature of the transistors must be nearly the same. In integrated circuits and transistor arrays where both transistors are on the same die, this is easy to achieve. But if the two transistors are widely separated, the precision of the current mirror is compromised.

Additional matched transistors can be connected to the same base and will supply the same collector current. In other words, the right half of the circuit can be duplicated several times with various resistor values replacing R_{2} on each. Note, however, that each additional right-half transistor "steals" a bit of collector current from Q_{1} due to the non-zero base currents of the right-half transistors. This will result in a small reduction in the programmed current.

An example of a mirror with emitter degeneration to increase mirror resistance is found in two-port networks.

For the simple mirror shown in the diagram, typical values of will yield a current match of 1% or better.

### Basic MOSFET current mirror

The basic current mirror can also be implemented using MOSFET transistors, as shown in the adjacent figure. Transistor *M*_{1} is operating in the saturation or active mode, and so is *M*_{2}. In this setup, the output current *I*_{OUT} is directly related to *I*_{REF}, as discussed next.

The drain current of a MOSFET *I*_{D} is a function of both the gate-source voltage and the drain-to-gate voltage of the MOSFET given by *I*_{D} = *f* (*V*_{GS}, *V*_{DG}), a relationship derived from the functionality of the MOSFET device. In the case of transistor *M*_{1} of the mirror, *I*_{D} = *I*_{REF}. Reference current *I*_{REF} is a known current, and can be provided by a resistor as shown, or by a "threshold-referenced" or "self-biased" current source to insure that it is constant, independent of voltage supply variations.^{[1]}

Using *V*_{DG}=0 for transistor *M*_{1}, the drain current in *M*_{1} is *I*_{D} = *f* (*V*_{GS},*V*_{DG}=0), so we find: *f* (*V*_{GS}, 0) = *I*_{REF}, implicitly determining the value of *V*_{GS}. Thus *I*_{REF} sets the value of *V*_{GS}. The circuit in the diagram forces the same *V*_{GS} to apply to transistor *M*_{2}. If *M*_{2} also is biased with zero *V*_{DG} and provided both transistors *M*_{1} and *M*_{2} have good matching of their properties, such as channel length, width, threshold voltage *etc.*, the relationship *I*_{OUT} = *f* (*V*_{GS},*V*_{DG}=0 ) applies, thus setting *I*_{OUT} = *I*_{REF}; that is, the output current is the same as the reference current when *V*_{DG}=0 for the output transistor, and both transistors are matched.

The drain-to-source voltage can be expressed as *V*_{DS}=*V*_{DG} +*V*_{GS}. With this substitution, the Shichman-Hodges model provides an approximate form for function *f* (*V*_{GS},*V*_{DG}):^{[2]}

where, is a technology related constant associated with the transistor, *W/L* is the width to length ratio of the transistor, *V*_{GS} is the gate-source voltage, *V*_{th} is the threshold voltage, λ is the channel length modulation constant, and *V*_{DS} is the drain source voltage.

#### Output resistance

Because of channel-length modulation, the mirror has a finite output (or Norton) resistance given by the *r _{O}* of the output transistor, namely (see channel length modulation):

- ,

where *λ* = channel-length modulation parameter and *V _{DS}* = drain-to-source bias.

#### Compliance voltage

To keep the output transistor resistance high, *V _{DG}* ≥ 0 V.

^{[3]}See Baker.

^{[4]}That means the lowest output voltage that results in correct mirror behavior, the compliance voltage, is

*V*=

_{OUT}*V*=

_{CV}*V*for the output transistor at the output current level with

_{GS}*V*= 0 V, or using the inverse of the

_{DG}*f*-function,

*f*:

^{ −1}- .

For Shichman-Hodges model, *f ^{ -1}* is approximately a square-root function.

#### Extensions and reservations

A useful feature of this mirror is the linear dependence of *f* upon device width *W*, a proportionality approximately satisfied even for models more accurate than the Shichman-Hodges model. Thus, by adjusting the ratio of widths of the two transistors, multiples of the reference current can be generated.

It must be recognized that the Shichman-Hodges model^{[5]} is accurate only for rather dated technology, although it often is used simply for convenience even today. Any quantitative design based upon new technology uses computer models for the devices that account for the changed current-voltage characteristics.^{[6]} Among the differences that must be accounted for in an accurate design is the failure of the square law in *V*_{GS} for voltage dependence of the current, and the very poor modeling of *V*_{DS} drain voltage dependence provided by λ*V*_{DS}, leading to abysmal output resistance estimates. Another failure of the equations that proves very significant is the inaccurate dependence upon the channel length *L*. A significant source of *L*-dependence stems from λ, as noted by Gray and Meyer, who also note that λ usually must be taken from experimental data.^{[7]}

### Feedback assisted current mirror

The adjacent figure shows a bipolar mirror using negative feedback to increase output resistance. Because of the op amp, these circuits are sometimes called **gain-boosted current mirrors**. Because they have relatively low compliance voltages, they also are called **wide-swing current mirrors**. A variety of circuits based upon this idea are in use,^{[8]}^{[9]}^{[10]} particularly for MOSFET mirrors because MOSFETs have rather low intrinsic output resistance values. A MOSFET version of the bipolar circuit is shown in the figure underneath where MOSFETs *M _{3}* and

*M*operate in Ohmic mode to play the same role as emitter resistors

_{4}*R*in the bipolar circuit, and MOSFETs

_{E}*M*and

_{1}*M*operate in active mode in the same roles as mirror transistors

_{2}*Q*and

_{1}*Q*in the bipolar mirror. An explanation follows of how the bipolar circuit works.

_{2}The operational amplifier is fed the difference in voltages *V _{1} - V_{2}* at the top of the two emitter-leg resistors of value

*R*. This difference is amplified by the op amp and fed to the base of output transistor

_{E}*Q*. If the collector base reverse bias on

_{2}*Q*is increased by increasing the applied voltage

_{2}*V*, the current in

_{A}*Q*increases, increasing

_{2}*V*and decreasing the difference

_{2}*V*entering the op amp. Consequently, the base voltage of

_{1}- V_{2}*Q*is decreased, and

_{2}*V*of

_{BE}*Q*decreases, counteracting the increase in output current.

_{2}If the op amp gain *A _{v}* is large, only a very small difference

*V*is sufficient to generate the needed change in base voltage

_{1}- V_{2}*ΔV*for

_{B}*Q*, namely

_{2}Consequently, *V _{1}* is held close to

*V*and the currents in the two leg resistors are held nearly the same, and the output current of the mirror is very nearly the same as the collector current

_{2}*I*in

_{C1}*Q*, which in turn is set by the reference current as

_{1}where β_{1} for transistor *Q _{1}* and β

_{2}for

*Q*differ due to the Early effect if the reverse bias across the collector-base of

_{2}*Q*is non-zero.

_{2}#### Output resistance

An idealized treatment of output resistance is given in the footnote.^{[11]} A small-signal analysis for an op amp with finite gain *A*_{v} but otherwise ideal is based upon the adjacent small-signal circuit (β, r_{O} and *r _{π}* refer to

*Q*). To arrive at this circuit, notice that the positive input of the op amp in the bipolar circuit is at AC ground, so the voltage input to the op amp is simply the AC emitter voltage

_{2}*V*

_{e}applied to its negative input, resulting in a voltage output of −

*A*

_{v}

*V*

_{e}. Using Ohm's law across the input resistance r

_{π}determines the small-signal base current

*I*

_{b}as:

Combining this result with Ohm's law for *R*_{E}, *V*_{e} can be eliminated, to find:^{[12]}

Kirchhoff's voltage law from the test source *I*_{X} to the ground of *R*_{E} provides:

Substituting for *I*_{b} and collecting terms the output resistance *R*_{out} is found to be:

For a large gain *A _{v} >> r_{π} / R_{E}* the maximum output resistance obtained with this circuit is

a substantial improvement over the basic mirror where *R _{out} = *r

_{O}

*.*

The small-signal analysis of the MOSFET version of this wide-swing mirror is obtained from the bipolar analysis by setting β = *g _{m} r_{π}* in the formula for

*R*and then letting

_{out}*r*→ ∞. The result is

_{π}This time, *R _{E}* is the resistance of the source-leg MOSFETs M

_{3}, M

_{4}. Unlike the bipolar version, however, as

*A*is increased (holding

_{v}*R*fixed in value),

_{E}*R*continues to increase, and does not approach a limiting value at large

_{out}*A*.

_{v}#### Compliance voltage

For the bipolar circuit, a large op amp gain achieves the maximum *R _{out}* with only a small

*R*. A low value for

_{E}*R*means

_{E}*V*also is small, allowing a low compliance voltage for this mirror, only a voltage

_{2}*V*larger than the compliance voltage of the simple bipolar mirror. For this reason this type of mirror also is called a

_{2}*wide-swing current mirror*, because it allows the output voltage to swing low compared to other types of mirror that achieve a large

*R*only at the expense of large compliance voltages.

_{out}With the MOSFET circuit, like the bipolar circuit, the larger the op amp gain *A _{v}*, the smaller

*R*can be made at a given

_{E}*R*, and the lower the compliance voltage of the mirror.

_{out}### Other current mirrors

There are many sophisticated current mirrors that have higher output resistances than the basic mirror (more closely approach an ideal mirror with current output independent of output voltage) and produce currents less sensitive to temperature and device parameter variations and to circuit voltage fluctuations. These multi-transistor mirror circuits are used both with bipolar and MOS transistors. These circuits include:

## References

- ↑
Paul R. Gray, Paul J. Hurst, Stephen H. Lewis, Robert G. Meyer (2001).
*Analysis and Design of Analog Integrated Circuits*, Fourth Edition. New York: Wiley. ISBN 0471321680. - ↑
Gray
*et al.*.*Eq. 1.165, p. 44*. ISBN 0471321680. - ↑ Keeping the output resistance high means more than keeping the MOSFET in active mode, because the output resistance of real MOSFETs only begins to increase on entry into the active region, then rising to become close to maximum value only when
*V*≥ 0 V._{DG} - ↑
R. Jacob Baker (2008).
*CMOS Circuit Design, Layout and Simulation*, Revised Second Edition. New York: Wiley-IEEE, p. 297, §9.2.1 and Figure 20.28, p. 636. ISBN 978-0-470-22941-5. - ↑ NanoDotTek Report NDT14-08-2007, 12 August 2007 [1]
- ↑
A popular MOSFET computer model is BSIM4, see: BSIM4.6.4 MOSFET Model: User's manual.
*BSIM 4 web site*. Electrical Engineering and Computer Sciences Department, UC Berkeley (2009). Retrieved on 2011-05-20. This 170-page manual describes a model much more complex than a simple algebraic formula. - ↑
Gray
*et al.*.*p. 44*. ISBN 0471321680. - ↑
R. Jacob Baker.
*§ 20.2.4 pp. 645–646*. ISBN 978-0-470-22941-5. - ↑
Ivanov VI and Filanovksy IM (2004).
*Operational amplifier speed and accuracy improvement: analog circuit design with structural methodology*, The Kluwer international series in engineering and computer science, v. 763. Boston, Mass.: Kluwer Academic. ISBN 1-4020-7772-6. - ↑
W. M. C. Sansen (2006).
*Analog design essentials*. New York ; Berlin: Springer. ISBN 0-387-25746-2. - ↑ An idealized version of the argument in the text, valid for infinite op amp gain, is as follows. If the op amp is replaced by a nullor, voltage
*V*_{2}=*V*_{1}, so the currents in the leg resistors are held at the same value. That means the emitter currents of the transistors are the same. If the*V*_{CB}of Q_{2}increases, so does the output transistor β because of the Early effect: β = β_{0}( 1 +*V*_{CB}/*V*_{A}). Consequently the base current to Q_{2}given by*I*_{B}=*I*_{E}/ (β + 1) decreases and the output current*I*_{out}=*I*_{E}/ (1 + 1 / β) increases slightly because β increases slightly. Doing the math,_{O}= (*V*_{A}+*V*_{CB}) /*I*_{out}. That is, the ideal mirror resistance for the circuit using an ideal op amp nullor is*R*_{out}= ( β + 1 ) r_{O}, in agreement with the value given later in the text when the gain → ∞. - ↑ Notice that as
*A*_{v}→ ∞,*V*_{e}→ 0 and*I*_{b}→*I*_{X}.