Early effect

The Early effect, named after its discoverer James M. Early, is the variation in the effective width of the base in a bipolar junction transistor (BJT) due to a variation in the applied base-to-collector voltage. A greater reverse bias across the collector–base junction, for example, increases the collector–base depletion width, thereby decreasing the width of the charge carrier portion of the base.
Explanation.
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 base is more heavily doped than the collector. The principle governing these two widths is charge 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:
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 the Early voltage, often denoted by the symbol VA.
Large-signal model.
In the forward active region the Early effect modifies the collector current (formula_2) and the forward common-emitter current gain (formula_3), as typically described by the following equations:
Where
Some models base the collector current correction factor on the collector–base voltage "V"CB (as described in base-width modulation) instead of the collector–emitter voltage "V"CE. Using "V"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 on "V"CB. Computer models such as those used in SPICE use the collector–base voltage "V"CB.
Small-signal model.
The Early effect can be accounted for in small-signal circuit models (such as the hybrid-pi model) as a resistor defined as
in parallel with the collector–emitter junction of the transistor. This resistor can thus account for the finite output resistance of a simple current mirror or an actively loaded common-emitter amplifier.
In keeping with the model used in SPICE and as discussed above using formula_11 the resistance becomes:
which almost agrees with the textbook result. In either formulation, formula_13 varies with DC reverse bias formula_11, as is observed in practice.
In the MOSFET the output resistance is given in Shichman–Hodges model (accurate for very old technology) as:
where formula_16 = drain-to-source voltage, formula_17 = drain current and formula_18 = channel-length modulation parameter, usually taken as inversely proportional to channel length "L".
Because of the resemblance to the bipolar result, the terminology "Early effect" often is applied to the MOSFET as well.
Current–voltage characteristics.
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
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.
A solution of this equation is below, and two boundary conditions are used to solve and find formula_20 and formula_21.
The following equations apply to the emitter and collector region, respectively, and the origins formula_23, formula_24, and formula_25 apply to the base, collector, and emitter.
A boundary condition of the emitter is below:
The values of the constants formula_28 and formula_29 are zero due to the following conditions of the emitter and collector regions as formula_30 and formula_31.
Because formula_33, the values of formula_34 and formula_35 are formula_36 and formula_37, respectively.
Expressions of formula_39 and formula_40 can be evaluated.
Because insignificant recombination occurs, the second derivative of formula_42 is zero. There is therefore a linear relationship between excess hole density and formula_43.
The following are boundary conditions of formula_45.
with W the base width. Substitute into the above linear relation.
With this result, derive value of formula_48.
Use the expressions of formula_48, formula_39, formula_52, and formula_53 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.