BJT formulas

This applies in active mode with no avalanche multiplication at the CBJ (multiplication factor $M=1$). When $I_{CBO}$ is negligible, $\alpha =I_C/I_E$. For a real device $\alpha <1$, and it is close to 1.

When $I_{CBO}$ is negligible, $\beta =I_C/I_B$. The second form holds when $\gamma$ and ${\alpha }_T$ are both close to 1, and it shows how the two losses add. Because $\alpha$ is close to 1, $\beta$ is very sensitive to small changes in $\alpha$. Carry enough significant figures in $1-\alpha$ before dividing.

The first relation is KCL for the whole device (currents taken into C and B and out of E for an npn). The others convert between $\alpha$ and $\beta$ and give any one of $I_E$, $I_B$, $I_C$ from another. A useful identity is $\beta +1=\frac{1}{1-\alpha }$.

$I_{CBO}$ is the collector current with the emitter open and the CBJ reverse-biased. $I_{CEO}$ is the collector current with the base open. $I_{CEO}$ is larger by a factor of $(\beta +1)$ because the leakage current, flowing into the base, is itself amplified. Both relations hold in active mode.

This follows from the reciprocity of the device. It reduces the four Ebers–Moll parameters to three independent ones, and it defines the transport saturation current $I_S$.

These hold when the EBJ is forward-biased with $V_{BE}$ several times $V_T$, so the $-1$ is negligible, and $V_{BC}\le 0$, so the CBJ diode term contributes at most a small leakage. They follow from the Ebers–Moll equations using $I_B=I_E-I_C=(1-{\alpha }_F)I_{ES}{e}^{V_{BE}/V_T}$ and $I_S={\alpha }_F{I}_{ES}$. This is the exponential law used for the active-mode $I_C$–$V_{BE}$ characteristic.

This first-order model holds in active mode at constant $V_{BE}$. $I_{C0}=I_S{e}^{V_{BE}/V_T}$ is the current extrapolated to $V_{CE}=0$. $V_A$ is the Early voltage. Extending the straight-line $I_C$–$V_{CE}$ characteristic back to the $V_{CE}$ axis gives a crossing at $V_{CE}=-V_A$.

It is evaluated at the DC operating point, in active mode. The middle expression follows exactly from the model above. The approximation $V_A/I_C$ is the one normally used, and it holds when $V_A\gg V_{CE}$.