BJTs

BJT Symbol

Modes of Operation

Mode	EBJ	CBJ	$V_{CB}$	$V_{BE}$
Cutoff	Reverse	Reverse	Positive	Negative
Active	Forward	Reverse	Positive	Positive
Saturation	Forward	Forward	Negative	Positive

Thermal Voltage $V_T$

In order to avoid delving into device physics (you have to stop somewhere), the equation for thermal voltage won't be derived. Essentially, electrons that are hotter (moving faster) are at a higher potential energy. The equation for thermal voltage is:

$$V_T = \frac{kT}{q} $$ where:

$ k$ = the Boltzmann constant = 8.62×10^−5 eV/K
$ T$ = temperature in Kelvin
$ q$ = charge of an electron = 1.6022×10^-19 C

At room temperature ($T \approx$ 300 K), $V_T = 25.9$ mV.

Pragmatically, when you see $V_T$ in an equation, just know that it is dependant on temperature (times a constant).

Wiki: Boltzmann Constant

See Chapter 6 of Sedra/Smith

$$\beta = \frac{i_C}{i_B}$$ $$\beta \gg 1$$ $$\alpha_F = \frac{i_C}{i_E}$$ $$\alpha_F=\frac{\beta_F}{\beta_F+1} $$ $$\alpha_F \approx 1$$

Forward Active Mode (FAM)

Ebers-Moll Equation

$$i_C=I_S e^{v_{EB}/V_T}$$ where $I_S$ is the Saturation Current $$I_S=\frac{q A_E D_n n_i^2}{N_{AB} W_B}$$ Define the rest of the terms here

$$\beta_F=\frac{N_{DE} D_n W_E}{N_{AB} D_p W_B}$$ I think $\beta_F$ is the regular $\beta$ in FAM.

DC Analysis

Add stuff about beta, alpha

Small Signal Models: Hybrid Pi and T

Hybrid Pi Model

$$g_m = \frac{I_C}{V_T}$$ $I_C = $ DC collector current
$V_T = $ Thermal Voltage (add equation) $$r_\pi = \frac{\beta}{g_m}$$ $\beta = $ (refer to earlier section when it's written)
$g_m = $ ??

Hybrid Pi Model with Early Effect

$$r_o = ??$$

T Model

T Model with Early Effect

$$r_o = ??$$

Amplifier Configurations

Common Emitter

Common Collector (Emitter Follower)

Common Base

BJT Amplifier Formulas
p446 Sedra/Smith

Parameter Common Emitter Common Emitter with $ R_e $ Common Base Common Collector

$ R_{in} $ $ ( \beta + 1)r_e $ $ ( \beta + 1) (r_e + R_e)$ $ r_e $ $ (\beta +1)(r_e+R_L) $

$ A_{vo} $ $ -g_m R_C $ $ - \frac{g_m R_C}{1+g_m R_e} $ $ g_m R_C $ $ 1 $

$ R_o $ $ R_C $ $ R_C $ $ R_C $ $ r_e $

$ A_v $ $ -g_m (R_C || R_L) $
$ -\alpha \frac{R_C || R_L}{r_e} $ $ \frac{-g_m (R_C||R_L)}{1+g_m R_e} $
$ -\alpha \frac{R_C||R_L}{r_e+R_e} $ $ g_m(R_C||R_L) $
$ \alpha \frac{R_C||R_L}{r_e} $ $ \frac{R_L}{R_L+r_e} $

$ G_v $ $ -\beta \frac{R_C || R_L}{R_{sig} + (\beta + 1) r_e} $ $ -\beta \frac{R_C || R_L}{R_{sig} + (\beta + 1) (r_e+R_e)} $ $ \alpha \frac{R_C || R_L}{R_{sig}+r_e} $ $ \frac{R_L}{R_L+r_e+R_{sig}/(\beta +1)} $
$ G_{vo}=1 $
$ R_{out} = r_e + \frac{R_{sig}}{\beta + 1} $

Parameter	Common Emitter	Common Emitter with \( R_e \)	Common Base	Common Collector
\( R_{in} \)	\( ( \beta + 1)r_e \)	\( ( \beta + 1) (r_e + R_e)\)	\( r_e \)	\( (\beta +1)(r_e+R_L) \)
\( A_{vo} \)	\( -g_m R_C \)	\( - \frac{g_m R_C}{1+g_m R_e} \)	\( g_m R_C \)	\( 1 \)
\( R_o \)	\( R_C \)	\( R_C \)	\( R_C \)	\( r_e \)
\( A_v \)	\( -g_m (R_C \|\| R_L) \) \( -\alpha \frac{R_C \|\| R_L}{r_e} \)	\( \frac{-g_m (R_C\|\|R_L)}{1+g_m R_e} \) \( -\alpha \frac{R_C\|\|R_L}{r_e+R_e} \)	\( g_m(R_C\|\|R_L) \) \( \alpha \frac{R_C\|\|R_L}{r_e} \)	\( \frac{R_L}{R_L+r_e} \)
\( G_v \)	\( -\beta \frac{R_C \|\| R_L}{R_{sig} + (\beta + 1) r_e} \)	\( -\beta \frac{R_C \|\| R_L}{R_{sig} + (\beta + 1) (r_e+R_e)} \)	\( \alpha \frac{R_C \|\| R_L}{R_{sig}+r_e} \)	\( \frac{R_L}{R_L+r_e+R_{sig}/(\beta +1)} \) \( G_{vo}=1 \) \( R_{out} = r_e + \frac{R_{sig}}{\beta + 1} \)

Adam Gulyas

Embedded Hardware Engineer