Editing Bipolar junction transistor (section)

== Theory and modeling ==
[[File:NPN Band Diagram Equilibrium.svg|thumb|[[Band diagram]] for NPN transistor at equilibrium]]
[[File:NPN Band Diagram Active.svg|thumb|Band diagram for NPN transistor in active mode, showing injection of electrons from emitter to base, and their overshoot into the collector]]

BJTs can be thought of as two diodes (p–n junctions) sharing a common region that minority carriers can move through. A PNP BJT will function like two diodes that share an N-type cathode region, and the NPN like two diodes sharing a P-type anode region.  Connecting two diodes with wires will not make a BJT, since minority carriers will not be able to get from one p–n junction to the other through the wire.

Both types of BJT function by letting a small current input to the base control an amplified output from the collector. The result is that the BJT makes a good switch that is controlled by its base input. The BJT also makes a good amplifier, since it can multiply a weak input signal to about 100 times its original strength. Networks of BJTs are used to make powerful amplifiers with many different applications.

In the discussion below, focus is on the NPN BJT. In what is called active mode, the base–emitter voltage <math>V_{\text{BE}}</math> and collector–base voltage <math>V_{\text{CB}}</math> are positive, forward biasing the emitter–base junction and reverse-biasing the collector–base junction. In this mode, electrons are injected from the forward biased n-type emitter region into the p-type base where they diffuse as minority carriers to the reverse-biased n-type collector and are swept away by the electric field in the reverse-biased collector–base junction.

For an illustration of forward and reverse bias, see [[Diode#Current–voltage characteristic|semiconductor diodes]].

=== Large-signal models ===
In 1954, [[Jewell James Ebers]] and [[John L. Moll]] introduced their [[mathematical model]] of transistor currents:<ref>{{cite journal|doi=10.1109/jrproc.1954.274797|title=Large-Signal Behavior of Junction Transistors|journal=Proceedings of the IRE|volume=42|issue=12|pages=1761–1772|date=1954|last1=Ebers|first1=J.|last2=Moll|first2=J.|s2cid=51672011}}</ref>

==== Ebers–Moll model <span class="anchor" id="Ebers-Moll model"></span><span class="anchor" id="Ebers-Moll"></span><span class="anchor" id="Ebers–Moll"></span> ====
[[File:Ebers-Moll model schematic (NPN).svg|frame|Ebers&ndash;Moll model for an NPN transistor.<ref>
{{cite book |title=Microelectronic Circuits |edition=2nd |first1=Adel S. |last1=Sedra |first2=Kenneth C. |last2=Smith |isbn=978-0-03-007328-1 |date=1987 |page=[https://archive.org/details/microelectronicc0000sedr/page/903 903] |publisher=Holt, Rinehart, and Winston |url=https://archive.org/details/microelectronicc0000sedr/page/903 }}</ref> ''I''<sub>B</sub>, ''I''<sub>C</sub> and ''I''<sub>E</sub> are the base, collector and emitter currents; ''I''<sub>CD</sub> and ''I''<sub>ED</sub> are the collector and emitter diode currents; ''α''<sub>F</sub> and ''α''<sub>R</sub> are the forward and reverse common-base current gains.]]
[[File:Ebers-Moll model schematic (PNP).svg|frame|Ebers&ndash;Moll model for a PNP transistor]]
[[File:Approximated Ebers Moll.svg|frame|Approximated Ebers&ndash;Moll model for an NPN transistor in the forward active mode. The collector diode is reverse-biased so ''I''<sub>CD</sub> is virtually zero. Most of the emitter diode current (''α''<sub>F</sub> is nearly 1) is drawn from the collector, providing the amplification of the base current.]]

The DC emitter and collector currents in active mode are well modeled by an approximation to the Ebers–Moll model:
: <math>\begin{align}
  I_\text{E} &= I_\text{ES} \left(e^\frac{V_\text{BE}}{V_\text{T}} - 1\right) \\
  I_\text{C} &= \alpha_\text{F} I_\text{E} \\
  I_\text{B} &= \left(1 - \alpha_\text{F}\right) I_\text{E}
\end{align}</math>

The base internal current is mainly by diffusion (see [[Fick's law]]) and
: <math>J_{n\,(\text{base})} = \frac{1}{W} q D_n n_{bo} e^{\frac{V_\text{EB}}{V_\text{T}}}</math>
where
* <math>V_{\text{T}}</math> is the [[Boltzmann constant#Thermal voltage|thermal voltage]] <math>kT/q</math> (approximately 26&nbsp;mV at 300&nbsp;K ≈ room temperature).
* <math>I_{\text{E}}</math> is the emitter current
* <math>I_{\text{C}}</math> is the collector current
* <math>\alpha_\text{F}</math> is the common base forward short-circuit current gain (0.98 to 0.998)
* <math>I_{\text{ES}}</math> is the reverse saturation current of the base–emitter diode (on the order of 10<sup>−15</sup> to 10<sup>−12</sup> amperes)
* <math>V_{\text{BE}}</math> is the base–emitter voltage
* <math>D_n</math> is the diffusion constant for electrons in the p-type base
* ''W'' is the base width

The <math>\alpha</math> and forward <math>\beta</math> parameters are as described previously. A reverse <math>\beta</math> is sometimes included in the model.

The unapproximated Ebers&ndash;Moll equations used to describe the three currents in any operating region are given below. These equations are based on the transport model for a bipolar junction transistor.<ref name=Sedra>{{cite book |first1=A.S. |last1=Sedra |first2=K.C. |last2=Smith |title=Microelectronic Circuits |date=2004 |edition=5th |publisher=Oxford |location=New York |isbn=978-0-19-514251-8 |pages=Eqs.&nbsp;4.103&ndash;4.110, p.&nbsp;305 |no-pp=true }}</ref>
: <math>\begin{align}
  i_{\text{C}} &= I_\text{S} \left[                   \left(e^\frac{V_\text{BE}}{V_\text{T}} - e^\frac{V_\text{BC}}{V_\text{T}}\right) - \frac{1}{\beta_\text{R}} \left(e^\frac{V_\text{BC}}{V_\text{T}} - 1\right) \right]\\
  i_{\text{B}} &= I_\text{S} \left[ \frac{1}{\beta_\text{F}} \left(e^\frac{V_\text{BE}}{V_\text{T}} - 1                               \right) + \frac{1}{\beta_\text{R}} \left(e^\frac{V_\text{BC}}{V_\text{T}} - 1\right) \right]\\
  i_{\text{E}} &= I_\text{S} \left[                   \left(e^\frac{V_\text{BE}}{V_\text{T}} - e^\frac{V_\text{BC}}{V_\text{T}}\right) + \frac{1}{\beta_\text{F}} \left(e^\frac{V_\text{BE}}{V_\text{T}} - 1\right) \right]
\end{align}</math>
where
* <math>i_\text{C}</math> is the collector current
* <math>i_\text{B}</math> is the base current
* <math>i_\text{E}</math> is the emitter current
* <math>\beta_\text{F}</math> is the forward common emitter current gain (20 to 500)
* <math>\beta_\text{R}</math> is the reverse common emitter current gain (0 to 20)
* <math>I_\text{S}</math> is the reverse saturation current (on the order of 10<sup>−15</sup> to 10<sup>−12</sup> amperes)
* <math>V_\text{T}</math> is the thermal voltage (approximately 26&nbsp;mV at 300 K ≈ room temperature).
* <math>V_\text{BE}</math> is the base–emitter voltage
* <math>V_\text{BC}</math> is the base–collector voltage

===== Base-width modulation =====
{{Main|Early effect}}
[[File:Early effect (NPN).svg|frame|Top: NPN base width for low collector–base reverse bias; Bottom: narrower NPN base width for large collector–base reverse bias. Hashed regions are [[Depletion width|depleted regions]].]]

As the collector–base voltage (<math>V_\text{CB} = V_\text{CE} - V_\text{BE}</math>) varies, the collector–base depletion region varies in size. An increase in the collector–base voltage, for example, causes a greater reverse bias across the collector–base junction, increasing the collector–base depletion region width, and decreasing the width of the base. This variation in base width often is called the ''[[Early effect]]'' after its discoverer [[James M. Early]].

Narrowing of the base width has two consequences:
* There is a lesser chance for recombination within the "smaller" base region.
* The charge gradient is increased across the base, and consequently, the current of minority carriers injected across the emitter junction increases.

Both factors increase the collector or "output" current of the transistor in response to an increase in the collector–base voltage.

===== Punchthrough =====
When the base–collector voltage reaches a certain (device-specific) value, the base–collector depletion region boundary meets the base–emitter depletion region boundary. When in this state the transistor effectively has no base. The device thus loses all gain when in this state.

==== Gummel–Poon charge-control model ====
The Gummel–Poon model<ref>{{cite journal|doi=10.1002/j.1538-7305.1970.tb01803.x|title=An Integral Charge Control Model of Bipolar Transistors|journal=Bell System Technical Journal|volume=49|issue=5|pages=827–852|date=1970|last1=Gummel|first1=H. K.|last2=Poon|first2=H. C.}}</ref> is a detailed charge-controlled model of BJT dynamics, which has been adopted and elaborated by others to explain transistor dynamics in greater detail than the terminal-based models typically do.<ref>{{cite web|url=http://ece-www.colorado.edu/~bart/book/book/chapter5/ch5_6.htm#5_6_2|title=Bipolar Junction Transistors|access-date=August 10, 2016|archive-date=February 7, 2009|archive-url=https://web.archive.org/web/20090207001659/http://ece-www.colorado.edu/~bart/book/book/chapter5/ch5_6.htm#5_6_2|url-status=dead}}</ref> This model also includes the dependence of transistor <math>\beta</math>-values upon the direct current levels in the transistor, which are assumed current-independent in the Ebers–Moll model.<ref name=Sedra2>{{cite book |author1=A.S. Sedra |author2=K.C. Smith |title=Microelectronic Circuits |url=https://archive.org/details/microelectronicc00sedr_571 |url-access=limited |date=2004 |edition=5th |publisher=Oxford |location=New York |isbn=978-0-19-514251-8 |page=[https://archive.org/details/microelectronicc00sedr_571/page/n538 509] }}</ref>

=== Small-signal models ===

==== Hybrid-pi model ====
{{Main|hybrid-pi model}}
[[File:Hybrid-pi detailed model.svg|thumb|Hybrid-pi model]]

The hybrid-pi model is a popular [[Electronic circuit|circuit]] model used for analyzing the [[small signal]] and AC behavior of bipolar junction and field effect [[transistors]]. Sometimes it is also called ''Giacoletto model'' because it was introduced by [[Lawrence J. Giacoletto|L.J. Giacoletto]] in 1969. The model can be quite accurate for low-frequency circuits and can easily be adapted for higher-frequency circuits with the addition of appropriate inter-electrode [[capacitance]]s and other parasitic elements.

{{Clear}}

==== h-parameter model ====
[[File:BJT h-parameters (generalised).svg|frame|Generalized h-parameter model of an NPN BJT.<br />''Replace '''x''' with '''e''', '''b''' or '''c''' for CE, CB and CC topologies respectively.'']]

Another model commonly used to analyze BJT circuits is the ''[[two-port network#Hybrid parameters (h-parameters)|h-parameter]]'' model, also known as the hybrid equivalent model, closely related to the [[hybrid-pi model]] and the [[admittance parameters|y-parameter]] [[two-port network#Y-parameters (admittance parameters)|two-port]], but using input current and output voltage as independent variables, rather than input and output voltages. This two-port network is particularly suited to BJTs as it lends itself easily to the analysis of circuit behavior, and may be used to develop further accurate models. As shown, the term ''x'' in the model represents a different BJT lead depending on the topology used. For common-emitter mode the various symbols take on the specific values as:
* Terminal 1, base
* Terminal 2, collector
* Terminal 3 (common), emitter; giving ''x'' to be ''e''
* ''i''<sub>i</sub>, base current (''i''<sub>b</sub>)
* ''i''<sub>o</sub>, collector current (''i''<sub>c</sub>)
* ''V''<sub>in</sub>, base-to-emitter voltage (''V''<sub>BE</sub>)
* ''V''<sub>o</sub>, collector-to-emitter voltage (''V''<sub>CE</sub>)
and the h-parameters are given by:
* ''h''<sub>ix</sub> = ''h''<sub>ie</sub> for the common-emitter configuration, the '''i'''nput impedance of the transistor (corresponding to the base resistance ''r''<sub>pi</sub>).
* ''h''<sub>rx</sub> = ''h''<sub>re</sub>, a '''r'''everse transfer relationship, it represents the dependence of the transistor's (input) ''I''<sub>B</sub>&ndash;''V''<sub>BE</sub> curve on the value of (output) ''V''<sub>CE</sub>. It is usually very small and is often neglected (assumed to be zero) at DC.
* ''h''<sub>fx</sub> = ''h''<sub>fe</sub>, the "forward" current-gain of the transistor, sometimes written ''h<sub>21</sub>''. This parameter, with lower case "fe" to imply small signal (AC) gain, or more often with capital letters for "FE" (specified as ''h''<sub>FE</sub>) to mean the "large signal" or DC current-gain (''β''<sub>DC</sub> or often simply ''β''), is one of the main parameters in datasheets, and may be given for a typical collector current and voltage or plotted as a function of collector current. See below.
* ''h''<sub>ox</sub> = 1/''h''<sub>oe</sub>, the output impedance of transistor. The parameter ''h''<sub>oe</sub> usually corresponds to the output admittance of the bipolar transistor and has to be inverted to convert it to an impedance.

As shown, the h-parameters have lower-case subscripts and hence signify AC conditions or analyses. For DC conditions they are specified in upper-case. For the CE topology, an approximate h-parameter model is commonly used which further simplifies the circuit analysis. For this the ''h''<sub>oe</sub> and ''h''<sub>re</sub> parameters are neglected (that is, they are set to infinity and zero, respectively). The h-parameter model as shown is suited to low-frequency, small-signal analysis. For high-frequency analyses the inter-electrode capacitances that are important at high frequencies must be added.

===== Etymology of ''h''<sub>FE</sub> =====
The ''h'' refers to its being an h-parameter, a set of parameters named for their origin in a '''''h'''ybrid equivalent circuit'' model (see above). As with all h parameters, the choice of lower case or capitals for the letters that follow the "h" is significant; lower-case signifies "small signal" parameters, that is, the slope the particular relationship; upper-case letters imply "large signal" or [[Direct Current|DC]] values, the ratio of the voltages or currents. In the case of the very often used ''h''<sub>FE</sub>:
* ''F'' is from '''''F'''orward current amplification'' also called the current gain. 
* ''E'' refers to the transistor operating in a ''common '''E'''mitter'' (CE) configuration. 
So h<sub>FE</sub> (or hFE) refers to the (total; DC) collector current divided by the base current, and is dimensionless. It is a parameter that varies somewhat with collector current, but is often approximated as a constant; it is normally specified at a typical collector current and voltage, or graphed as a function of collector current.

Had capital letters not been used for used in the subscript, i.e. if it were written ''h<sub>fe</sub>'' the parameter indicate small signal ([[Alternating Current|AC]]) current gain, i.e. the slope of the Collector current versus Base current graph at a given point, which is often close to the hFE value unless the test frequency is high.

=== Industry models <span class="anchor" id="Mextram"></span><span class="anchor" id="HICUM"></span><span class="anchor" id="HiCuM"></span><span class="anchor" id="Modella"></span><span class="anchor" id="VBIC"></span> ===
{{Expand section|date=January 2015}}

The Gummel–Poon SPICE model is often used, but it suffers from several limitations. For instance, reverse breakdown of the base–emitter diode is not captured by the SGP (SPICE Gummel–Poon) model, neither are thermal effects (self-heating) or quasi-saturation.<ref>{{cite web |title=VBIC Description and Derivation Details<!--title from here: https://designers-guide.org/vbic/references.html--> |url=https://designers-guide.org/vbic/documents/VbicText.pdf |website=Designer's Guide}}</ref> These have been addressed in various more advanced models which either focus on specific cases of application (Mextram, HICUM, Modella) or are designed for universal usage (VBIC).<ref>{{Cite web |url=http://www.silvaco.com/content/kbase/smartspice_device_models.pdf |title=SmartSpice Analog Circuit Simulator|publisher=Silvaco |access-date=January 15, 2015 |archive-date=March 5, 2016 |archive-url=https://web.archive.org/web/20160305020435/http://www.silvaco.com/content/kbase/smartspice_device_models.pdf }}</ref><ref name="Gildenblat2010" >{{cite book |editor=Gennady Gildenblat |title=Compact Modeling: Principles, Techniques and Applications |date=2010 |publisher=Springer Science & Business Media |isbn=978-90-481-8614-3 |at=Part&nbsp;II: Compact Models of Bipolar Junction Transistors, pp. 167–267 cover Mextram and HiCuM in-depth }}</ref><ref name="Schröter2010">{{cite book |first=Michael |last=Schröter |title=Compact Hierarchical Bipolar Transistor Modeling with Hicum |date=2010 |publisher=World Scientific |isbn=978-981-4273-21-3 }}</ref><ref>{{Cite web |url=http://joerg-berkner.de/Fachartikel/pdf/2002_ICCAP_UM_Berkner_Compact_Models_4_BJTs.pdf |title=Compact Models for Bipolar Transistors, Berkner |access-date=January 16, 2015 |archive-url=https://web.archive.org/web/20150116091931/http://joerg-berkner.de/Fachartikel/pdf/2002_ICCAP_UM_Berkner_Compact_Models_4_BJTs.pdf |archive-date=January 16, 2015 }}</ref>