 Current mirror

A current mirror is a circuit designed to copy a current through one active device by controlling the current in another active device of a circuit, keeping the output current constant regardless of loading. The current being 'copied' can be, and sometimes is, a varying signal current. Conceptually, an ideal current mirror is simply an ideal inverting current amplifier that reverses the current direction as well or it is a currentcontrolled current source (CCCS). The current mirror is used to provide bias currents and active loads to circuits.
Contents
Mirror characteristics
There are three main specifications that characterize a current mirror. The first is the transfer ratio (in the case of a current amplifier) or the output current magnitude (in the case of a constant current source CCS). 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 output part of 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 stability.
Practical approximations
For smallsignal analysis the current mirror can be approximated by its equivalent Norton impedance .
In largesignal hand analysis, a current mirror is usually and simply approximated 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 realworld effects like noise, powersupply voltage variations and component tolerances.
Circuit realizations of current mirrors
Basic idea
A bipolar transistor can be used as the simplest currenttocurrent converter but its transfer ratio would highly depend on temperature variations, β tolerances, etc. To eliminate these undesired disturbances, a current mirror is composed of two cascaded currenttovoltage and voltagetocurrent converters placed at the same conditions and having reverse characteristics. They have not to be obligatory linear; the only requirement is their characteristics to be mirrorlike (for example, in the BJT current mirror below, they are logarithmic and exponential). Usually, two identical converters are used but the characteristic of the first one is reversed by applying a negative feedback. Thus a current mirror consists of two cascaded equal converters (the first  reversed and the second  direct).
Basic BJT current mirror
If a voltage is applied to the BJT baseemitter junction as an input quantity and the collector current is taken as an output quantity, the transistor will act as an exponential voltagetocurrent converter. By applying a negative feedback (simply joining the base and collector) the transistor can be "reversed" and it will begin acting as the opposite logarithmic currenttovoltage converter; now it will adjust the "output" baseemitter voltage so as to pass the applied "input" collector current.
The simplest bipolar current mirror (shown in Figure 1) implements this idea. It consists of two cascaded transistor stages acting accordingly as a reversed and direct voltagetocurrent converters. Transistor Q_{1} is connected to ground. Its collectorbase voltage is zero as shown. 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 EbersMoll model.) It is important to have Q_{1} in the circuit instead of a simple diode, because Q_{1} sets V_{BE} for 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 collectorbase voltage of Q_{2} is also zero, then the V_{BE}value set by Q_{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 collectorbase reverse bias V_{CB} of the output transistor is given by (see bipolar transistor):

 ,
where I_{S} = reverse saturation current or scale current, V_{T} = thermal voltage and V_{A} = Early voltage. This current is related to the reference current I_{REF} when the output transistor V_{CB} = 0 V by:
as found using Kirchhoff's current law at the collector node of Q_{1}:

 I_{REF} = I_{C} + I_{B1} + I_{B2}.
The reference current supplies the collector current to Q_{1} and the base currents to both transistors — when both transistors have zero basecollector bias, the two base currents are equal, I_{B1}=I_{B2}=I_{B}.
Parameter β_{0} is the transistor βvalue 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_{CB} = collectortobase bias.
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_{BE} under bias conditions with the output transistor at the output current level I_{C} and with V_{CB} = 0 V or, inverting the IV relation above:
where V_{T} = thermal voltage and I_{S} = reverse saturation current or scale current.
Extensions and complications
When Q_{2} has V_{CB} > 0 V, the transistors no longer are matched. In particular, their βvalues 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 βvalues will differ because the β_{0}values depend on current, and the two transistors now carry different currents (see GummelPoon 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 righthalf transistor "steals" a bit of collector current from Q_{1} due to the nonzero base currents of the righthalf 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 twoport networks.
For the simple mirror shown in the diagram, typical values of β 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 Figure 2. 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 gatesource voltage and the draintogate 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 "thresholdreferenced" or "selfbiased" current source to ensure 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} is also biased with zero V_{DG} and provided 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 draintosource voltage can be expressed as V_{DS}=V_{DG} +V_{GS}. With this substitution, the ShichmanHodges model provides an approximate form for function f (V_{GS},V_{DG}):^{[2]}
where, K_{p} is a technology related constant associated with the transistor, W/L is the width to length ratio of the transistor, V_{GS} is the gatesource 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 channellength 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 = channellength modulation parameter and V_{DS} = draintosource bias.
Compliance voltage
To keep the output transistor resistance high, V_{DG} ≥ 0 V.^{[nb 1]} (see Baker).^{[3]} That means the lowest output voltage that results in correct mirror behavior, the compliance voltage, is V_{OUT} = V_{CV} = V_{GS} for the output transistor at the output current level with V_{DG} = 0 V, or using the inverse of the ffunction, f^{ −1}:

 .
For ShichmanHodges model, f ^{1} is approximately a squareroot 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 ShichmanHodges 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 ShichmanHodges model^{[4]} 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 currentvoltage characteristics. 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 and the very poor modeling of V_{ds} drain voltage dependence provided by λV_{ds}. Another failure of the equations that proves very significant is the inaccurate dependence upon the channel length L. A significant source of Ldependence stems from λ, as noted by Gray and Meyer, who also note that λ usually must be taken from experimental data.^{[5]}
Feedback assisted current mirror
Figure 3 shows a mirror using negative feedback to increase output resistance. Because of the op amp, these circuits are sometimes called gainboosted current mirrors. Because they have relatively low compliance voltages, they also are called wideswing current mirrors. A variety of circuits based upon this idea are in use,^{[6]}^{[7]}^{[8]} particularly for MOSFET mirrors because MOSFETs have rather low intrinsic output resistance values. A MOSFET version of Figure 3 is shown in Figure 4 where MOSFETs M_{3} and M_{4} operate in Ohmic mode to play the same role as emitter resistors R_{E} in Figure 3, and MOSFETs M_{1} and M_{2} operate in active mode in the same roles as mirror transistors Q_{1} and Q_{2} in Figure 3. An explanation follows of how the circuit in Figure 3 works.
The operational amplifier is fed the difference in voltages V_{1}  V_{2} at the top of the two emitterleg resistors of value R_{E}. This difference is amplified by the op amp and fed to the base of output transistor Q_{2}. If the collector base reverse bias on Q_{2} is increased by increasing the applied voltage V_{A}, the current in Q_{2} increases, increasing V_{2} and decreasing the difference V_{1}  V_{2} entering the op amp. Consequently, the base voltage of Q_{2} is decreased, and V_{BE} of Q_{2} decreases, counteracting the increase in output current.
If the op amp gain A_{v} is large, only a very small difference V_{1}  V_{2} is sufficient to generate the needed base voltage V_{B} for Q_{2}, namely
Consequently, 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 I_{C1} in Q_{1}, which in turn is set by the reference current as
where β_{1} for transistor Q_{1} and β_{2} for Q_{2} differ due to the Early effect if the reverse bias across the collectorbase of Q_{2} is nonzero.
Output resistance
An idealized treatment of output resistance is given in the footnote.^{[nb 2]} A smallsignal analysis for an op amp with finite gain A_{v} but otherwise ideal is based upon Figure 5 (β, r_{O} and r_{π} refer to Q_{2}). To arrive at Figure 5, notice that the positive input of the op amp in Figure 3 is at AC ground, so the voltage input to the op amp is simply the AC emitter voltage 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 smallsignal base current I_{b} as:
Combining this result with Ohm's law for R_{E}, V_{e} can be eliminated, to find:^{[nb 3]}
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 smallsignal analysis of the MOSFET circuit of Figure 4 is obtained from the bipolar analysis by setting β = g_{m} r_{π} in the formula for R_{out} and then letting r_{π} → ∞. The result is
This time, R_{E} is the resistance of the sourceleg MOSFETs M_{3}, M_{4}. Unlike Figure 3, however, as A_{v} is increased (holding R_{E} fixed in value), R_{out} continues to increase, and does not approach a limiting value at large A_{v}.
Compliance voltage
For Figure 3, a large op amp gain achieves the maximum R_{out} with only a small R_{E}. A low value for R_{E} means V_{2} also is small, allowing a low compliance voltage for this mirror, only a voltage V_{2} larger than the compliance voltage of the simple bipolar mirror. For this reason this type of mirror also is called a wideswing current mirror, because it allows the output voltage to swing low compared to other types of mirror that achieve a large R_{out} only at the expense of large compliance voltages.
With the MOSFET circuit of Figure 4, like the circuit in Figure 3, the larger the op amp gain A_{v}, the smaller R_{E} can be made at a given R_{out}, and the lower the compliance voltage of the mirror.
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 multitransistor mirror circuits are used both with bipolar and MOS transistors. These circuits include:
 the Widlar current source
 the Wilson current source
 the Cascoded current sources
Notes
 ^ 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_{DG} ≥ 0 V.
 ^ 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,

 ^ Notice that as A_{v} → ∞, V_{e} → 0 and I_{b} → I_{X}.
See also
 Current source
 Widlar current source
 Wilson current source
 Bipolar junction transistor
 MOSFET
 Channel length modulation
 Early effect
 {{nmos current mirror{}
References
 ^ Paul R. Gray, Paul J. Hurst, Stephen H. Lewis, Robert G. Meyer (2001). Analysis and Design of Analog Integrated Circuits (Fourth Edition ed.). New York: Wiley. p. 308–309. ISBN 0471321680.
 ^ Gray et al.. Eq. 1.165, p. 44. ISBN 0471321680.
 ^ R. Jacob Baker (2010). CMOS Circuit Design, Layout and Simulation (Third ed.). New York: WileyIEEE. pp. 297, §9.2.1 and Figure 20.28, p. 636. ISBN 9780470881323.
 ^ NanoDotTek Report NDT14082007, 12 August 2007
 ^ Gray et al.. p. 44. ISBN 0471321680.
 ^ R. Jacob Baker. § 20.2.4 pp. 645–646. ISBN 9780470881323.
 ^ 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 ed.). Boston, Mass.: Kluwer Academic. p. §6.1, p. 105–108. ISBN 1402077726. http://books.google.com/books?id=IuLsny9wKIIC&pg=PA110&dq=gain+boost+wide++%22current+mirror%22#PPA107,M1.
 ^ W. M. C. Sansen (2006). Analog design essentials. New York ; Berlin: Springer. p. §0310, p. 93. ISBN 0387257462.
External links
 Patent (1968) by RJ Widlar: Biasing scheme especially suited for integrated circuits
 Current mirrors
 4QD tec  Current sources and mirrors Compendium of circuits and descriptions
Categories: Analog circuits
 Electronic design
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