TheEarly effect, named after its discovererJames M. Early, is the variation in the effective width of the base in abipolar junction transistor (BJT) due to a variation in the applied base-to-collector voltage. A greaterreverse bias across the collector–base junction, for example, increases the collector–basedepletion width, thereby decreasing the width of thecharge carrier portion of the base.

In Figure 1, the neutral (i.e. active) base is green, and the depleted base regions are hashed light green. The neutral emitter and collector regions are dark blue and the depleted regions hashed light blue. Under increased collector–base reverse bias, the lower panel of Figure 1 shows a widening of the depletion region in the base and the associated narrowing of the neutral base region.

The collector depletion region also increases under reverse bias, more than does that of the base, because the collector is less heavily doped than the base. The principle governing these two widths ischarge neutrality. The narrowing of the collector does not have a significant effect as the collector is much longer than the base. The emitter–base junction is unchanged because the emitter–base voltage is the same.

Base-narrowing has two consequences that affect the current:

• 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 collector-base junction increases, which net current is called${\displaystyle I_{\text{CB0}}}$.

Both these factors increase the collector or "output" current of the transistor with an increase in the collector voltage, but only the second is called Early effect. This increased current is shown in Figure 2. Tangents to the characteristics at large voltages extrapolate backward to intercept the voltage axis at a voltage called theEarly voltage, often denoted by the symbolV_A.

In the forward active region the Early effect modifies the collector current (${\displaystyle I_{\mathrm{C}}}$) and the forwardcommon-emitter current gain (${\displaystyle {\beta }_{\mathrm{F}}}$), as typically described by the following equations:^[1][2]

where

• ${\displaystyle V_{\mathrm{C}\mathrm{E}}}$ is the collector–emitter voltage
• ${\displaystyle V_{\mathrm{B}\mathrm{E}}}$ is the base–emitter voltage
• ${\displaystyle I_{\mathrm{S}}}$ is the reverse saturation current
• ${\displaystyle V_{\mathrm{T}}}$ is the thermal voltage${\displaystyle \mathrm{k}\mathrm{T}/\mathrm{q}}$, which is approximately 26 mV; seethermal voltage: role in semiconductor physics
• ${\displaystyle V_{\mathrm{A}}}$ is theEarly voltage (typically15–150 V; less for smaller devices)
• ${\displaystyle {\beta }_{\mathrm{F}0}}$ is forward common-emitter current gain at zero bias.

Some models base the collector current correction factor on the collector–base voltageV_CB (as described inbase-width modulation) instead of the collector–emitter voltageV_CE.^[3] UsingV_CB may be more physically plausible, in agreement with the physical origin of the effect, which is a widening of the collector–base depletion layer that depends onV_CB. Computer models such as those used inSPICE use the collector–base voltageV_CB.^[4]

The Early effect can be accounted for insmall-signal circuit models (such as thehybrid-pi model) as a resistor defined as^[5]

${\displaystyle r_{\text{O}}=\frac{V_{\text{A}}+V_{\text{CE}}}{I_{\text{C}}}\approx \frac{V_{\text{A}}}{I_{\text{C}}}}$

in parallel with the collector–emitter junction of the transistor. This resistor can thus account for the finiteoutput resistance of a simplecurrent mirror or anactively loadedcommon-emitter amplifier.

In keeping with the model used inSPICE and as discussed above using${\displaystyle V_{CB}}$ the resistance becomes:

${\displaystyle r_{\text{O}}=\frac{V_{\text{A}}+V_{\text{CB}}}{I_{\text{C}}}}$

which almost agrees with the textbook result. In either formulation,${\displaystyle r_O}$ varies with DC reverse bias${\displaystyle V_{CB}}$, as is observed in practice.^{[citation needed]}

In theMOSFET the output resistance is given in Shichman–Hodges model^[6] (accurate for very old technology) as:

${\displaystyle r_{\text{O}}=\frac{1+\lambda V_{\text{DS}}}{\lambda I_{\text{D}}}=\frac{1}{I_{\text{D}}}\left(\frac{1}{\lambda }+V_{\text{DS}}\right)}$

where${\displaystyle V_{\text{DS}}}$ = drain-to-source voltage,${\displaystyle I_{\text{D}}}$ = drain current and${\displaystyle \lambda }$ =channel-length modulation parameter, usually taken as inversely proportional to channel lengthL.Because of the resemblance to the bipolar result, the terminology "Early effect" often is applied to the MOSFET as well.

The expressions are derived for a PNP transistor. For an NPN transistor, n has to be replaced by p, and p has to be replaced by n in all expressions below.The following assumptions are involved when deriving ideal current-voltage characteristics of the BJT^[7]

• Low level injection
• Uniform doping in each region with abrupt junctions
• One-dimensional current
• Negligible recombination-generation inspace 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 thediffusion equation.

${\displaystyle \frac{d^2\mathrm{\Delta }{p}_{\text{B}}(x)}{d{x}^2}=\frac{\mathrm{\Delta }{p}_{\text{B}}(x)}{L_{\text{B}}^2}}$

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

${\displaystyle \mathrm{\Delta }{p}_{\text{B}}(x)=C_1{e}^{\frac{x}{L_{\text{B}}}}+C_2{e}^{-\frac{x}{L_{\text{B}}}}}$

The following equations apply to the emitter and collector region, respectively, and the origins${\displaystyle 0}$,${\displaystyle 0^{\prime }}$, and${\displaystyle 0^{″}}$ apply to the base, collector, and emitter.

A boundary condition of the emitter is below:

${\displaystyle \mathrm{\Delta }{n}_{\text{E}}(0^{″})=n_{\text{E}O}\left(e^{\frac{1}{kT}q{V}_{\text{EB}}}-1\right)}$

The values of the constants${\displaystyle A_1}$ and${\displaystyle B_1}$ are zero due to the following conditions of the emitter and collector regions as${\displaystyle x^{″}\to 0}$ and${\displaystyle x^{\prime }\to 0}$.

Because${\displaystyle A_1=B_1=0}$, the values of${\displaystyle \mathrm{\Delta }{n}_{\text{E}}(0^{″})}$ and${\displaystyle \mathrm{\Delta }{n}_{\text{c}}(0^{\prime })}$ are${\displaystyle A_2}$ and${\displaystyle B_2}$, respectively.

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

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

${\displaystyle \mathrm{\Delta }{p}_{\text{B}}(x)=D_{1}x+D_2}$

The following are boundary conditions of${\displaystyle \mathrm{\Delta }{p}_{\text{B}}}$.

withW the base width. Substitute into the above linear relation.

${\displaystyle \mathrm{\Delta }{p}_{\text{B}}(x)=-\frac{1}{W}\left[\mathrm{\Delta }{p}_{\text{B}}(0)-\mathrm{\Delta }{p}_{\text{B}}(W)\right]x+\mathrm{\Delta }{p}_{\text{B}}(0)}$

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

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

Similarly, an expression of the collector current is derived.

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

• Small-signal model

• Transistor modeling

• Webarchive template wayback links
• All articles with dead external links
• Articles with dead external links from August 2019
• Articles with permanently dead external links
• Articles with short description
• Short description is different from Wikidata
• Use dmy dates from December 2023
• All articles with unsourced statements
• Articles with unsourced statements from November 2007