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Theory and modeling

In the discussion below, focus is on the npn bipolar transistor. In the npn transistor in what is called active mode the base-emitter voltage $V_{\text{BE}}$ and collector-base voltage $V_{\text{CB}}$ are positive, forward biasing the emitter-base junction and reverse-biasing the collector-base junction. In active mode of operation, electrons are injected from the forward biased n-type emitter region into the p-type base where they diffuse to the reverse biased n-type collector and are swept away by the electric field in the reverse biased collector-base junction. For a figure describing forward and reverse bias, see the end of the article semiconductor diodes.

Large-signal models

Ebers–Moll model

The DC emitter and collector currents in active mode are well modeled by an approximation to the Ebers–Moll model:

Ebers–Moll Model for NPN Transistor

Ebers–Moll Model for PNP Transistor

$I_{\text{E}} = I_{\text{ES}} \left(e^{\frac{V_{\text{BE}}}{V_{\text{T}}}} - 1\right)$

$I_{\text{C}} = \alpha_T I_{\text{ES}} \left(e^{\frac{V_{\text{BE}}}{V_{\text{T}}}} - 1\right)$

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

$J_n(\text{base}) = \frac{q D_n n_{bo}}{W} e^{\frac{V_{\text{EB}}}{V_{\text{T}}}}$

where

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

The collector current is slightly less than the emitter current, because the value of $\alpha_T$ is very close to 1.0. In the BJT a small amount of base–emitter current causes a larger amount of collector–emitter current. The ratio of the allowed collector–emitter current to the base–emitter current is called current gain, β or $h_{\text{FE}}$. A β value of 100 is typical for small bipolar transistors. In a typical configuration, a very small signal current flows through the base–emitter junction to control the emitter–collector current. β is related to α through the following relations:

$\alpha_T = \frac{I_{\text{C}}}{I_{\text{E}}}$

$\beta_F = \frac{I_{\text{C}}}{I_{\text{B}}}$

$\beta_F = \frac{\alpha_T}{1 - \alpha_T}\iff \alpha_T = \frac{\beta_F}{\beta_F+1}$

Emitter Efficiency : $\eta = \frac{J_n(\text{base})}{J_{\text{E}}}$; that is, the ratio of current injected into the base to the current in the emitter; the two differ due to backward injection from the base into the emitter and to recombination. See carrier generation and recombination.

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.

$i_{\text{C}} = I_{\text{S}}\left(e^{\frac{V_{\text{BE}}}{V_{\text{T}}}} - e^{\frac{V_{\text{BC}}}{V_{\text{T}}}}\right) - \frac{I_{\text{S}}}{\beta_R}\left(e^{\frac{V_{\text{BC}}}{V_{\text{T}}}} - 1\right)$

$i_{\text{B}} = \frac{I_{\text{S}}}{\beta_F}\left(e^{\frac{V_{\text{BE}}}{V_{\text{T}}}} - 1\right) + \frac{I_{\text{S}}}{\beta_R}\left(e^{\frac{V_{\text{BC}}}{V_{\text{T}}}} - 1\right)$

$i_{\text{E}} = I_{\text{S}}\left(e^{\frac{V_{\text{BE}}}{V_{\text{T}}}} - e^{\frac{V_{\text{BC}}}{V_{\text{T}}}}\right) + \frac{I_{\text{S}}}{\beta_F}\left(e^{\frac{V_{\text{BE}}}{V_{\text{T}}}} - 1\right)$

where

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

Base-width modulation

Top: pnp base width for low collector-base reverse bias; Bottom: narrower pnp base width for large collector-base reverse bias. Light colors are depleted regions.

As the applied collector–base voltage ($V_{\text{BC}}$) 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.

In the forward-active region, the Early effect modifies the collector current ($i_{\text{C}}$) and the forward common emitter current gain ($\beta_F$) as given by:Bản mẫu:Fact

$i_{\text{C}} = I_{\text{S}} \, e^{\frac{v_{\text{BE}}}{V_{\text{T}}}} \left(1 + \frac{V_{\text{CB}}}{V_{\text{A}}}\right)$

$\beta_F = \beta_{F0}\left(1 + \frac{V_{\text{CB}}}{V_{\text{A}}}\right)$

$r_{\text{o}} = \frac{V_{\text{A}}}{I_{\text{C}}}$

where:

• $V_{\text{CB}}$ is the collector–base voltage
• $V_{\text{A}}$ is the Early voltage (15 V to 150 V)
• $\beta_{F0}$ is forward common-emitter current gain when $V_{\text{CB}}$ = 0 V
• $r_{\text{o}}$ is the output impedance
• $I_{\text{C}}$ is the collector current

Current–voltage characteristics

The following assumptions are involved when deriving ideal current-voltage characteristics of the BJT

• Low level injection
• Uniform doping in each region with abrupt junctions
• One-dimensional current flow
• Negligible recombination-generation in space charge regions
• Negligible electric fields outside of space charge regions.

It is important to characterize the minority diffusion currents induced by injection of carriers.

With regard to pn-junction diode, a key relation is the diffusion equation.

$\frac{d^2 \Delta p_{\text{B}} (x)}{dx^2} = \frac{\Delta p_{\text{B}} (x)}{L_{\text{B}}^2}$

A solution of this equation is below, and two boundary conditions are used to solve and find $C_1$ and $C_2$.

$\Delta p_{\text{B}} (x) = C_1 e^{x/L_{\text{B}}} + C_2 e^{-x/L_{\text{B}}}$

The following equations apply to the emitter and collector region, respectively, and the origins $0$, $0'$, and $0''$ apply to the base, collector, and emitter.

$\Delta n_{\text{B}} (x'') = A_1 e^{x''/L_{\text{B}}} + A_2 e^{-x''/L_{\text{B}}}$

$\Delta n_{\text{c}} (x') = B_1 e^{x'/L_{\text{B}}} + B_2 e^{-x'/L_{\text{B}}}$

A boundary condition of the emitter is below:

$\Delta n_{\text{E}} (0'') = n_{\text{E}O} ( \exp (q V_{\text{EB}} / kT) - 1)\;$

The values of the constants $A_1$ and $B_1$ are zero due to the following conditions of the emitter and collector regions as $x'' \rightarrow 0$ and $x' \rightarrow 0$.

$\Delta n_{\text{E}} (x'') \rightarrow 0$

$\Delta n_{\text{c}} (x') \rightarrow 0$

Because $A_1 = B_1 = 0$, the values of $\Delta n_{\text{E}} (0'')$ and $\Delta n_{\text{c}} (0')$ are $A_2$ and $B_2$, respectively.

$\Delta n_{\text{E}} (x'') = n_{\text{E}0} (\exp (q V_{\text{EB}} / kT) - 1) \exp(-x''/L_{\text{E}})\;$

$\Delta n_{\text{C}} (x') = n_{\text{C}0} (\exp (q V_{\text{CB}} / kT) - 1) \exp(-x'/L_{\text{C}})\;$

Expressions of $I_{\text{E}n}$ and $I_{\text{C}n}$ can be evaluated.

$I_{\text{E}n} = - q A D_{\text{E}} \frac{d \Delta_{\text{E}} (x'')}{dx} |_{x''=0''}$

$I_{\text{C}n} = -q A \frac{D_{\text{C}}}{L_{\text{C}}} n_{\text{C}0} ( \exp ( q V_{\text{CB}} / kT) - 1)$

Because insignificant recombination occurs, the second derivative of $\Delta p_{\text{B}} (x)$ is zero. There is therefore a linear relationship between excess hole density and $x$.

$\Delta p_{\text{B}} (x) = D_1 x + D_2\;$

The following are boundary conditions of $\Delta p_{\text{B}}$.

$\Delta p_{\text{B}} (0) = D_2\;$

$\Delta p_{\text{B}} (W) = D_1 W + \Delta p_{\text{B}} (0)\;$

Substitute into the above linear relation.

$\Delta p_{\text{B}} (x) = - \left ( \frac{ \Delta p_{\text{B}} (0) - \Delta p_{\text{B}} (W) }{W} \right ) x + \Delta p_{\text{B}} (0)$.

With this result, derive value of $I_{\text{E}p}$.

$I_{\text{E}p} (0) = - q A D_{\text{B}} \frac{d \Delta p_{\text{B}}}{dx}|_{x=0}$

$I_{\text{E}p} (0) = \frac{q A D_{\text{B}}}{W} \left ( \Delta p_{\text{B}} (0) - \Delta p_{\text{B}} (W) \right )$

Use the expressions of $I_{\text{E}p}$, $I_{\text{E}n}$, $\Delta p_{\text{B}}(0)$, and $\Delta p_{\text{B}}(W)$ to develop an expression of the emitter current.

$\Delta p_{\text{B}}(W) = p_{\text{B}0} \exp ( q V_{\text{CB}}/kT)\;$

$\Delta p_{\text{B}}(0) = p_{\text{B}0} \exp ( q V_{\text{EB}}/kT))\;$

$I_{\text{E}} = qA \left ( \left ( \frac{D_{\text{E}} n_{\text{E}0}}{L_{\text{E}}} + \frac{D_{\text{B}} p_{\text{B}0}}{W} \right ) \left ( \exp \left ( \frac{q V_{\text{EB}}}{kT} \right ) - 1 \right ) - \left ( \frac{D_{\text{B}}}{W} p_{\text{B}0} \right ) \left ( \exp \left ( \frac {q V_{\text{CB}}}{k T} \right ) - 1 \right ) \right )$

Similarly, an expression of the collector current is derived.

$I_{\text{C}p} (W) = I_{\text{E}p} (0)\;$

$I_{\text{C}} = I_{\text{C}p} (W) + I_{\text{C}n} (0')\;$

$I_{\text{C}} = q A \left ( \left ( \frac{D_{\text{B}}}{W} p_{\text{B}0} \right ) ( \exp (q V_{\text{EB}} / kT ) -1 ) - \left ( \frac{ D_{\text{C}} n_{\text{C}0} }{ L_{\text{C}} } + \frac{ D_{\text{B}} p_{\text{B}0} }{ W } \right ) ( \exp (q V_{\text{CB}} / kT ) -1 ) \right )$

An expression of the base current is found with the previous results.

$I_{\text{B}} = I_{\text{E}} - I_{\text{C}}\;$

$I_{\text{B}} = q A \left ( \frac{D_{\text{E}}}{L_{\text{E}}} n_{\text{E}0} ( \exp ( q V_{\text{EB}} / kT ) - 1) + \frac{D_{\text{C}}}{L_{\text{C}}} n_{\text{C}0} ( \exp ( q V_{\text{CB}} / k T ) - 1 ) \right )$

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.

Also, it is almost always theoretically intervened that the transistor will gain voltage when supplied to the base, however it is exponentially effective to the collector and the emitter for the voltage increase.

Gummel–Poon charge-control model

The Gummel–Poon model 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 [1]. This model also includes the dependence of transistor $\beta$-values upon the dc current levels in the transistor, which are assumed current-independent in the Ebers–Moll model.

Small-signal models

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, 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 the BJT lead depending on the topology used. For common-emitter mode the various symbols take on the specific values as:

• x = 'e' because it is a common-emitter topology
• Terminal 1 = Base
• Terminal 2 = Collector
• Terminal 3 = Emitter
• i_in = 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 – The input impedance of the transistor (corresponding to the emitter resistance r_e).
• h_rx = h_re – Represents the dependence of the transistor's I_B–V_BE curve on the value of V_CE. It is usually very small and is often neglected (assumed to be zero).
• h_fx = h_fe – The current-gain of the transistor. This parameter is often specified as h_FE or the DC current-gain (β_DC) in datasheets.
• h_ox = h_oe – The output impedance of transistor. This term is usually specified as an admittance 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 analyzes. 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). It should also be noted that the h-parameter model as shown is suited to low-frequency, small-signal analysis. For high-frequency analyzes the inter-electrode capacitances that are important at high frequencies must be added.

• Forward-active (or simply, active): The emitter–base junction is forward biased and the base–collector junction is reverse biased. Most bipolar transistors are designed to afford the greatest common-emitter current gain, $\beta_F$, in forward-active mode. If this is the case, the collector–emitter current is approximately proportional to the base current, but many times larger, for small base current variations.
• Reverse-active (or inverse-active or inverted): By reversing the biasing conditions of the forward-active region, a bipolar transistor goes into reverse-active mode. In this mode, the emitter and collector regions switch roles. Because most BJTs are designed to maximize current gain in forward-active mode, the $\beta_F$ in inverted mode is several (2–3 for the ordinary germanium transistor) times smaller. This transistor mode is seldom used, usually being considered only for failsafe conditions and some types of bipolar logic. The reverse bias breakdown voltage to the base may be an order of magnitude lower in this region.
• Saturation: With both junctions forward-biased, a BJT is in saturation mode and facilitates high current conduction from the emitter to the collector. This mode corresponds to a logical "on", or a closed switch.
• Cutoff: In cutoff, biasing conditions opposite of saturation (both junctions reverse biased) are present. There is very little current flow, which corresponds to a logical "off", or an open switch.
• Avalanche breakdown region

Although these regions are well defined for sufficiently large applied voltage, they overlap somewhat for small (less than a few hundred millivolts) biases. For example, in the typical grounded-emitter configuration of an NPN BJT used as a pulldown switch in digital logic, the "off" state never involves a reverse-biased junction because the base voltage never goes below ground; nevertheless the forward bias is close enough to zero that essentially no current flows, so this end of the forward active region can be regarded as the cutoff region.