Comprehensive Transistor Theory

Comprehensive Transistor Theory

1. Fundamental Current Relations

1.1 Base Current Relations

The transistor's fundamental operation relies on current amplification. In an NPN transistor, when a small current flows into the base terminal (\(I_B\)), it controls a much larger current (\(I_C\)) in the collector circuit.

This relationship is expressed as: \[ I_C = \beta I_B \] where \(\beta\) (beta) is the current gain or amplification factor.

Typical values for \(\beta\):

  • Power transistors: \(\beta \approx 10\)
  • General-purpose transistors: \(\beta \approx 100\)
  • Small-signal transistors: \(\beta\) can reach 1000

1.2 Kirchhoff's Current Law at Emitter

The emitter current is the sum of collector and base currents:

\[ I_E = I_C + I_B = I_B(1 + \beta) \]

2. Operating States

2.1 Cutoff State

The cutoff state represents the "OFF" condition of the transistor:

\[ V_{BE} < V_{BE(on)} \Rightarrow I_B = 0 \Rightarrow I_C = 0 \]

2.2 Saturation State

Saturation represents the "fully ON" state of the transistor:

\[ V_{BE} = V_{BE(on)} \] \[ 0 < I_C < \beta I_B \] \[ V_{CE} = R_{Sat} I_C \]

2.3 Load Line Equation

\[ I_C = \frac{V_{CC} - V_{CE}}{R_C} = -\frac{1}{R_C}V_{CE} + \frac{V_{CC}}{R_C} \]

3. Small Signal Analysis

3.1 Dynamic Emitter Resistance

\[ r_e' = \frac{V_T}{I_E} \] where thermal voltage is: \[ V_T = \frac{kT}{q} \approx 26\text{ mV at room temperature} \]

4. Temperature Effects

4.1 Base-Emitter Voltage

\[ \frac{\Delta V_{BE}}{\Delta T} \approx -2.1 \text{ mV/°C} \]

4.2 Collector Current Temperature Dependence

\[ I_C(T_2) = I_C(T_1) \cdot 2^{\frac{T_2-T_1}{10}} \]

5. Power Calculations

5.1 Total Power Dissipation

\[ P = V_{CE}I_C + V_{BE}I_B \]

5.2 Maximum Power Transfer Conditions

\[ V_{CE} = \frac{V_{CC}}{2} \] \[ I_C = \frac{V_{CC}}{2R_C} \]

6. Gain Calculations

6.1 Current Gain

\[ \beta = \frac{I_C}{I_B} \]

6.2 Small Signal Voltage Gain

\[ A_v = -\frac{R_C}{r_e'} \]

Important Design Considerations:

  • Maximum operating voltage: \(V_{CE(max)}\)
  • Maximum collector current: \(I_{C(max)}\)
  • Power dissipation: \(P_{max}\)
  • Temperature coefficient: \(\Delta V_{BE}/\Delta T\)