Handy Equations for BJT Amplifiers

Saturday, 2014 February 08

It all starts with Shockley's diode equation

\[ \begin{align} I_C = I_S e^{\frac{V_{BE}}{V_T}} \end{align} \]

The basic small-signal parameters \(g_m\), \(r_\pi\), and \(r_o\) are simply found using partial derivatives.

\[ \begin{align} g_m &= \frac{\partial I_C}{\partial V_{BE}} = \frac{I_C}{V_T} \\ r_\pi &= \frac{\partial V_{BE}}{\partial I_B} = \frac{\beta}{g_m} = \frac{\beta V_T}{I_C} = \frac{V_T}{I_B} \end{align} \]

For \(r_o\), however, Shockley's diode equation will need to be adjusted to account for the effect of Early voltage, \(V_A\), and the collector-emitter voltage, \(V_{CE}\), at the forward-active region.

\[ \begin{align} I_C = I_S e^{\frac{V_{BE}}{V_T}} \left( 1 + \frac{V_{CE}}{V_A} \right) \end{align} \]

The small-signal output resistance can now be computed.

\[ \begin{align} \frac{1}{r_o} &= \frac{\partial V_{CE}}{\partial I_C} = \frac{I_C}{V_A} \\ r_o &= \frac{V_A}{I_C} \end{align} \]

Common emitter

\[ \begin{align} G_m &= g_m \\ R_i &= r_\pi \\ R_o &= r_o || R_C \\ a_v &= -g_m \left( r_o || R_C \right) \\ a_i &= g_m r_\pi = \beta \end{align} \]

Common base

\[ \begin{align} \alpha &= \frac{\beta}{\beta + 1} \\ r_e &= \frac{\alpha}{g_m} \end{align} \]

For the ideal case where \(r_o \rightarrow \infty\) and \(r_b \rightarrow 0\),

\[ \begin{align} G_m &= g_m \\ R_i &= r_e \\ R_o &= R_C \\ a_v &= g_m R_C \\ a_i &= g_m r_e = \alpha \end{align} \]

When \(r_b \gt 0\),

\[ \begin{align} G_m &= \frac{g_m}{1 + \frac{r_b}{r_\pi}} \\ R_i &= r_e \left( 1 + \frac{r_b}{r_\pi} \right) \end{align} \]

When \(r_o\) is finite,

\[ \begin{align} R_i &= r_e + \frac{\alpha \left( R_C || R_L \right)}{g_m r_o} \\ R_o &= R_C || \left[ \frac{r_o + R_S \left( 1 + \frac{g_m r_o }{\alpha} \right)}{1 + \frac{R_S}{r_\pi} } \right] \\ R_o &\simeq R_C || \left( \frac{g_m r_o}{\alpha} R_S \right) \end{align} \]

References

  1. Gray, et al., Analysis and Design of Analog Integrated Circuits, 5th ed. (2010)