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: \(\Delta V_{BE}/\Delta T\)