Common-emitter current gain

The common-emitter current gain is represented by β_F or the h-parameter h_FE; it is approximately the ratio of the collector's direct current to the base's direct current in forward-active region. (The F subscript is used to indicate the forward-active mode of operation.) It is typically greater than 50 for small-signal transistors, but can be smaller in transistors designed for high-power applications. Both injection efficiency and recombination in the base reduce the BJT gain.

Common-base current gain

Another useful characteristic is the common-base current gain, α_F. The common-base current gain is approximately the gain of current from emitter to collector in the forward-active region. This ratio usually has a value close to unity; between 0.980 and 0.998. It is less than unity due to recombination of charge carriers as they cross the base region.

Alpha and beta are related by the following identities:

${\displaystyle {\begin{aligned}\alpha _{\text{F}}&={\frac {I_{\text{C}}}{I_{\text{E}}}},&\beta _{\text{F}}&={\frac {I_{\text{C}}}{I_{\text{B}}}},\\\alpha _{\text{F}}&={\frac {\beta _{\text{F}}}{1+\beta _{\text{F}}}}&\iff \beta _{\text{F}}&={\frac {\alpha _{\text{F}}}{1-\alpha _{\text{F}}}}.\end{aligned}}}$

Beta is a convenient figure of merit to describe the performance of a bipolar transistor, but is not a fundamental physical property of the device. Bipolar transistors can be considered voltage-controlled devices (fundamentally the collector current is controlled by the base–emitter voltage; the base current could be considered a defect and is controlled by the characteristics of the base–emitter junction and recombination in the base). In many designs beta is assumed high enough so that base current has a negligible effect on the circuit. In some circuits (generally switching circuits), sufficient base current is supplied so that even the lowest beta value a particular device may have will still allow the required collector current to flow.

Ebers–Moll model

Ebers–Moll model for an NPN transistor.^[28] I_B, I_C and I_E are the base, collector and emitter currents; I_CD and I_ED are the collector and emitter diode currents; α_F and α_R are the forward and reverse common-base current gains.

Ebers–Moll model for a PNP transistor

Approximated Ebers–Moll model for an NPN transistor in the forward active mode. The collector diode is reverse-biased so I_CD is virtually zero. Most of the emitter diode current (α_F 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:

${\displaystyle {\begin{aligned}I_{\text{E}}&=I_{\text{ES}}\left(e^{\frac {V_{\text{BE}}}{V_{\text{T}}}}-1\right)\\I_{\text{C}}&=\alpha _{\text{F}}I_{\text{E}}\\I_{\text{B}}&=\left(1-\alpha _{\text{F}}\right)I_{\text{E}}\end{aligned}}}$

The base internal current is mainly by diffusion (see Fick's law) and

${\displaystyle J_{n\,({\text{base}})}={\frac {1}{W}}qD_{n}n_{bo}e^{\frac {V_{\text{EB}}}{V_{\text{T}}}}}$

where

• ${\displaystyle V_{\text{T}}}$ is the thermal voltage ${\displaystyle kT/q}$ (approximately 26 mV at 300 K ≈ room temperature).
• ${\displaystyle I_{\text{E}}}$ is the emitter current
• ${\displaystyle I_{\text{C}}}$ is the collector current
• ${\displaystyle \alpha _{\text{F}}}$ is the common base forward short-circuit current gain (0.98 to 0.998)
• ${\displaystyle I_{\text{ES}}}$ is the reverse saturation current of the base–emitter diode (on the order of 10⁻¹⁵ to 10⁻¹² amperes)
• ${\displaystyle V_{\text{BE}}}$ is the base–emitter voltage
• ${\displaystyle D_{n}}$ is the diffusion constant for electrons in the p-type base
• W is the base width

The ${\displaystyle \alpha }$ and forward ${\displaystyle \beta }$ parameters are as described previously. A reverse ${\displaystyle \beta }$ is sometimes included in the model.

The unapproximated Ebers–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.^[29]

${\displaystyle {\begin{aligned}i_{\text{C}}&=I_{\text{S}}\left[\left(e^{\frac {V_{\text{BE}}}{V_{\text{T}}}}-e^{\frac {V_{\text{BC}}}{V_{\text{T}}}}\right)-{\frac {1}{\beta _{\text{R}}}}\left(e^{\frac {V_{\text{BC}}}{V_{\text{T}}}}-1\right)\right]\\i_{\text{B}}&=I_{\text{S}}\left[{\frac {1}{\beta _{\text{F}}}}\left(e^{\frac {V_{\text{BE}}}{V_{\text{T}}}}-1\right)+{\frac {1}{\beta _{\text{R}}}}\left(e^{\frac {V_{\text{BC}}}{V_{\text{T}}}}-1\right)\right]\\i_{\text{E}}&=I_{\text{S}}\left[\left(e^{\frac {V_{\text{BE}}}{V_{\text{T}}}}-e^{\frac {V_{\text{BC}}}{V_{\text{T}}}}\right)+{\frac {1}{\beta _{\text{F}}}}\left(e^{\frac {V_{\text{BE}}}{V_{\text{T}}}}-1\right)\right]\end{aligned}}}$

where

• ${\displaystyle i_{\text{C}}}$ is the collector current
• ${\displaystyle i_{\text{B}}}$ is the base current
• ${\displaystyle i_{\text{E}}}$ is the emitter current
• ${\displaystyle \beta _{\text{F}}}$ is the forward common emitter current gain (20 to 500)
• ${\displaystyle \beta _{\text{R}}}$ is the reverse common emitter current gain (0 to 20)
• ${\displaystyle I_{\text{S}}}$ is the reverse saturation current (on the order of 10⁻¹⁵ to 10⁻¹² amperes)
• ${\displaystyle V_{\text{T}}}$ is the thermal voltage (approximately 26 mV at 300 K ≈ room temperature).
• ${\displaystyle V_{\text{BE}}}$ is the base–emitter voltage
• ${\displaystyle V_{\text{BC}}}$ is the base–collector voltage

Base-width modulation

Main article: Early effect

Top: NPN base width for low collector–base reverse bias; Bottom: narrower NPN base width for large collector–base reverse bias. Hashed regions are depleted regions.

As the collector–base voltage (${\displaystyle V_{\text{CB}}=V_{\text{CE}}-V_{\text{BE}}}$) 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.

Gummel–Poon charge-control model

The Gummel–Poon model^[30] 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.^[31] This model also includes the dependence of transistor ${\displaystyle \beta }$-values upon the direct current levels in the transistor, which are assumed current-independent in the Ebers–Moll model.^[32]

Hybrid-pi model

Hybrid-pi model

The hybrid-pi model is a popular 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 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 capacitances and other parasitic elements.

h-parameter model

Generalized h-parameter model of an NPN BJT.
Replace x with e, b or c for CE, CB and CC topologies respectively.

Another model commonly used to analyze BJT circuits is the h-parameter model, also known as the hybrid equivalent model, closely related to the hybrid-pi model and the y-parameter 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 behaviour, 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_i, base current (i_b)
• i_o, collector current (i_c)
• V_in, base-to-emitter voltage (V_BE)
• V_o, collector-to-emitter voltage (V_CE)

and the h-parameters are given by:

• h_ix = h_ie for the common-emitter configuration, the input impedance of the transistor (corresponding to the base resistance r_pi).
• h_rx = h_re, a reverse transfer relationship, it represents the dependence of the transistor's (input) I_B–V_BE curve on the value of (output) V_CE. It is usually very small and is often neglected (assumed to be zero) at DC.
• h_fx = h_fe, the "forward" current-gain of the transistor, sometimes written h₂₁. This parameter, with lower case "fe" to imply small signal (AC) gain, or more often with capital letters for "FE" (specified as h_FE) to mean the "large signal" or DC current-gain (β_DC 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_ox = 1/h_oe, the output impedance of transistor. The parameter h_oe 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_oe and h_re 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.