Editing Current mirror (section)

=== Basic BJT current mirror ===
If a voltage is applied to the BJT base-emitter junction as an input quantity and the collector current is taken as an output quantity, the transistor will act as an ''exponential voltage-to-current 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 current-to-voltage converter''; now it will adjust the "output" base-emitter 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'' voltage-to-current converters. The emitter of transistor Q<sub>1</sub> is connected to ground. Its collector and base are tied together, so its collector-base voltage is zero. Consequently, the voltage drop across Q<sub>1</sub> is ''V''<sub>BE</sub>, that is, this voltage is set by the [[diode modelling#Shockley diode model|diode law]] and Q<sub>1</sub> is said to be [[diode-connected transistor|diode connected]]. (See also [[bipolar transistor#Ebers.E2.80.93Moll model|Ebers-Moll model]].) It is important to have Q<sub>1</sub> in the circuit instead of a simple diode, because Q<sub>1</sub> sets ''V''<sub>BE</sub> for transistor Q<sub>2</sub>. If Q<sub>1</sub> and Q<sub>2</sub> 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<sub>2</sub> is also zero, then the ''V''<sub>BE</sub>-value set by Q<sub>1</sub> results in an emitter current in the matched Q<sub>2</sub> that is the same as the emitter current in Q<sub>1</sub>{{citation needed|date=January 2016}}. Because Q<sub>1</sub> and Q<sub>2</sub> are matched, their ''β''<sub>0</sub>-values also agree, making the mirror output current the same as the collector current of Q<sub>1</sub>.

The current delivered by the mirror for arbitrary collector-base reverse bias, ''V''<sub>CB</sub>, of [[bipolar junction transistor|the output transistor]] is given by:
: <math> I_\text{C} = I_\text{S} \left( e^{\frac{V_\text{BE}}{V_\text{T}}} - 1 \right) \left(1 + \frac{V_\text{CE}}{V_\text{A}}\right),</math>
where ''I<sub>S</sub>'' is the reverse saturation current or scale current; ''V''<sub>T</sub>, the [[Boltzmann constant#Thermal voltage|thermal voltage]]; and ''V''<sub>A</sub>, the [[Early effect|Early voltage]]. This current is related to the reference current ''I''<sub>ref</sub> when the output transistor ''V''<sub>CB</sub> = 0&nbsp;V by:
: <math> I_\text{ref} = I_C \left( 1 + \frac{2}{\beta_0} \right),</math>
as found using [[Kirchhoff's current law]] at the collector node of Q<sub>1</sub>:
: <math> I_\text{ref} = I_C + I_{B1} + I_{B2} \ .</math>

The reference current supplies the collector current to Q<sub>1</sub> and the base currents to both transistors – when both transistors have zero base-collector bias, the two base currents are equal, I<sub>B1</sub> = I<sub>B2</sub> = I<sub>B</sub>.
: <math> I_\text{ref} = I_C + I_B + I_B = I_C + 2 I_B = I_C \left(1 + \frac {2} {\beta_0} \right),</math>

Parameter ''β''<sub>0</sub> is the transistor ''β''-value for ''V''<sub>CB</sub> = 0&nbsp;V.

==== Output resistance ====
If ''V''<sub>BC</sub> is greater than zero in output transistor Q<sub>2</sub>, the collector current in Q<sub>2</sub> will be somewhat larger than for Q<sub>1</sub> due to the [[Early effect]]. In other words, the mirror has a finite output (or Norton) resistance given by the ''r''<sub>o</sub> of the output transistor, namely:
: <math> R_N = r_o = \frac{V_A + V_{CE}}{I_C} \ ,</math>

where ''V''<sub>A</sub> is the Early voltage; and ''V''<sub>CE</sub>, the collector-to-emitter voltage of output transistor.

==== Compliance voltage ====
To keep the output transistor active, ''V''<sub>CB</sub> ≥ 0&nbsp;V. That means the lowest output voltage that results in correct mirror behavior, the compliance voltage, is ''V<sub>OUT</sub>'' = ''V''<sub>CV</sub> = ''V''<sub>BE</sub> under bias conditions with the output transistor at the output current level ''I''<sub>C</sub> and with ''V''<sub>CB</sub> = 0&nbsp;V or, inverting the ''I''–''V'' relation above:
: <math>V_{CV} = V_T \ln\left(\frac{I_C}{I_S} + 1\right),</math>
where ''V''<sub>T</sub> is the [[Boltzmann constant#Thermal voltage|thermal voltage]]; and ''I''<sub>S</sub>, the reverse saturation current or scale current.

==== Extensions and complications ====
When Q<sub>2</sub> has ''V''<sub>CB</sub> > 0&nbsp;V, the transistors no longer are matched. In particular, their ''β''-values differ due to the Early effect, with
: <math>\begin{align}
 \beta_1 &= \beta_0 \\
 \beta_2 &= \beta_0 \left(1 + \frac{V_{CB}}{V_A}\right),
\end{align}</math>
where ''V''<sub>A</sub> is the [[Early effect|Early voltage]] and ''β''<sub>0</sub> is the transistor ''β'' for ''V''<sub>CB</sub> = 0&nbsp;V. Besides the difference due to the Early effect, the transistor ''β''-values will differ because the ''β''<sub>0</sub>-values depend on current, and the two transistors now carry different currents (see ''[[Gummel–Poon model]]'').

Further, Q<sub>2</sub> may get substantially hotter than Q<sub>1</sub> due to the associated higher power dissipation. To maintain matching, the temperature of the transistors must be nearly the same. In [[integrated circuit]]s 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. Note, however, that each additional right-half transistor "steals" a bit of collector current from Q<sub>1</sub> due to the non-zero base currents of the right-half transistors. This will result in a small reduction in the programmed current.

See also an [[two-port network#Example: bipolar current mirror with emitter degeneration|example of a mirror with emitter degeneration to increase mirror resistance]].

For the simple mirror shown in the diagram, typical values of <math>\beta</math> will yield a current match of 1% or better.
[[File:Simple MOSFET mirror.PNG|thumbnail|200px|Figure 2: An n-channel MOSFET current mirror with a resistor to set the reference current I<sub>REF</sub>; V<sub>DD</sub> is positive voltage.]]