# 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

$$g_m = \frac{I_C}{V_T}$$
\(I_C = \) DC collector current

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

\(g_m = \) ??

#### 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 |
---|---|---|---|---|

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

\( A_{vo} \) | \( -g_m R_C \) | \( - \frac{g_m R_C}{1+g_m R_e} \) | \( g_m R_C \) | \( 1 \) |

\( R_o \) | \( R_C \) | \( R_C \) | \( R_C \) | \( r_e \) |

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

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