Early effect

The Early effect (base-width modulation) is the slight rise in BJT collector current as the collector–emitter voltage increases, at fixed $v_{BE}$. It is the BJT analogue of the MOSFET’s Channel-length modulation, and it is the reason an active-mode BJT has a finite output resistance rather than behaving as a perfect current source.

Physical cause

In active mode the collector–base junction is reverse-biased. Raising $v_{CE}$ (with $v_{BE}$ held constant) increases that reverse bias, which widens the CBJ depletion region. The depletion region eats into the base, so the effective electrical base width $W$ — the neutral region the injected carriers must diffuse across — gets narrower. A narrower base means the carrier concentration gradient is steeper and the transit is shorter, so a larger fraction of injected carriers make it across before recombining. Result: $i_C$ rises slightly as $v_{CE}$ rises. Because the base width $W$ controls the saturation current, $I_S$ itself effectively depends on $v_{CE}$ through $W$.

The modified current law

Adding a first-order correction to [[BJT collector current|$i_C=I_S{e}^{v_{BE}/V_T}$]]:

$i_C=I_S{e}^{v_{BE}/V_T}\left(1+\frac{v_{CE}}{V_A}\right)$

where $V_A$ is the Early voltage, a device parameter typically 50–100 V for a discrete transistor. A larger $V_A$ means the curves are flatter (less base-width modulation) — good for high-gain amplifiers.

npn $i_C$ vs $v_{CE}$ family at constant $v_{BE}$: in the active region the slope is non-zero ($=1/r_o$); extrapolating the lines backward, they all intersect the $v_{CE}$ axis at the same point $-V_A$.

Output resistance

The non-zero slope of the $i_C$–$v_{CE}$ curve in the active region defines a small-signal output resistance. Differentiating the corrected law with respect to $v_{CE}$ at the bias point,

$\frac{1}{r_o}={\frac{\partial i_C}{\partial v_{CE}}|}_{v_{BE}}=\frac{I_S{e}^{V_{BE}/V_T}}{V_A}=\frac{I_C}{V_A}$

so

$r_o=\frac{V_A}{I_C}(\text{more exactly }\frac{V_A+V_{CE}}{I_C},\text{ but }{V}_A\gg V_{CE})$

This is identical in form to the MOSFET’s $r_o=V_A/I_D$. Geometric reading of the figure: every active-region line, extended backward, hits the $v_{CE}$ axis at $-V_A$ — exactly the construction used for the MOSFET in Channel-length modulation. At $I_C=1\text{mA}$ with $V_A=100\text{V}$, $r_o=100\text{k}\Omega$ — large, but not infinite, and it sets the ceiling on the gain of a Common-emitter amplifier ($A_v=-g_m(R_C\parallel r_o)$).

Active-mode large-signal model with the Early effect included: the collector current carries the $(1+v_{CE}/V_A)$ correction and an output resistance $r_o=V_A/I_C$ is added in parallel with the current source between collector and emitter.

For introductory DC hand analysis the Early effect is dropped (set $r_o\to \infty$) to give the simplified BJT large-signal model; it is restored only in small-signal gain and output-resistance calculations where it actually limits performance.