The Early effect is the variation in the width of the base in a BJT due to a variation in the applied base-to-collector voltage, named after its discoverer James M. Early. A greater reverse bias across the collector–base junction, for example, increases the collector–base depletion width, decreasing the width of the charge neutral portion of the base.
In Figure 1 the neutral base width is dark blue, and the depleted base regions are light blue. The neutral emitter and collector regions are rust and the depleted regions pink. Under increased collector–base reverse bias, the lower panel of Figure 1 shows 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. The principle governing these two widths is charge neutrality. 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 emitter junction increases.
Both these factors increase the collector or "output" current of the transistor with an increase in the collector voltage. 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.
Contents |
Large-signal model
In the forward active region the Early effect modifies the collector current (<math>I_\mathrm{C}</math>) and the forward common-emitter current gain (<math>\beta_\mathrm{F}</math>), as typically described by the following equations:[1][2]
- <math> I_\mathrm{C} = I_\mathrm{S} e^{\frac{V_\mathrm{BE}}{V_\mathrm{T}}} \left(1 + \frac{V_\mathrm{CE}}{V_\mathrm{A}}\right)</math>
- <math> \beta_\mathrm{F} = \beta_\mathrm{F0}\left(1 + \frac{V_\mathrm{CE}}{V_\mathrm{A}}\right)</math>
Where
- <math>V_\mathrm{CE}</math> is the collector–emitter voltage
- <math>V_\mathrm{T}</math> is the thermal voltage <math>\mathrm{kT/q}</math>; see thermal voltage: role in semiconductor physics
- <math>V_\mathrm{A}</math> is the Early voltage (typically 15 V to 150 V; smaller for smaller devices)
- <math>\beta_\mathrm{F0}</math> is forward common-emitter current gain at zero bias.
Some models base the collector current correction factor on the collector–base voltage VCB (as described in base-width modulation) instead of the collector–emitter voltage VCE.[3] Using VCB 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 VCB. Computer models such as those used in PSPICE use the collector–base voltage VCB.[4].
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 (see [5]
- <math>r_O=\frac{V_A+V_{CE}}{I_C} \ \approx \frac{V_A}{I_C} \ </math>
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 <math>V_{CB}</math> the resistance becomes:
- <math>r_O=\frac{V_A +V_{CB}}{I_C} \ </math>,
which almost agrees with the textbook result. In either formulation, <math>r_O</math> varies with DC reverse bias <math>V_{CB}</math>, as is observed in practice. In the MOSFET the output resistance is given in Shichman–Hodges model[6] (accurate for very old technology) as:
- <math>r_O = \begin{matrix} \frac {1+\lambda V_{DS}}{\lambda I_D} \end{matrix} =\begin{matrix} \frac {1/\lambda +V_{DS}} {I_D} \end{matrix}</math>,
where <math>V_{DS}</math> = drain-to-source voltage, <math>I_D</math> = drain current and <math>\lambda</math> = 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.
References and notes
- ^ R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design. McGraw-Hill Professional, p. 317. ISBN 0072505036.
- ^ Massimo Alioto and Gaetano Palumbo (2005). Model and Design of Bipolar and Mos Current-Mode Logic: CML, ECL and SCL Digital Circuits. Springer. ISBN 1402028784.
- ^ Paolo Antognetti and Giuseppe Massobrio (1993). Semiconductor Device Modeling with Spice. McGraw-Hill Professional. ISBN 0071349553.
- ^ Orcad PSpice Reference Manual named PSpcRef.pdf, p. 209. This manual is included with the free version of Orcad PSpice, but they do not maintain a copy on line. If the link given here expires, try Googling PSpcRef.pdf.
- ^ R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design, Second Edition, McGraw-Hill Professional, Eq. 13.31, p. 891. ISBN 0-07-232099-0.
- ^ NanoDotTek™ Report NDT14-08-2007, 12 August 2007 [1]
