FAD1022 L34-L38 — Semiconductors & Op-Amps — Formula Sheet
Comprehensive formula sheet covering all equations from Lectures 34–38: Semiconductor Theory, Diodes, BJT Transistor Biasing (Fixed, Emitter-Stabilized, Voltage Divider), and Operational Amplifiers (Inverting & Non-inverting).
Lecture 34 — Semiconductor Theory and Diodes
Diode DC Series Configuration
Forward Bias (ON state) — Replace Si diode with $0.7,\text{V}$ source and apply KVL:
$$E - V_R - V_D = 0$$
- $E$ — Source voltage (V)
- $V_R$ — Voltage across series resistor (V)
- $V_D$ — Diode forward voltage / knee voltage (V); $0.7,\text{V}$ for Si, $0.3,\text{V}$ for Ge, $1.5,\text{V}$ for GaAs
Reverse Bias (OFF state) — Replace diode with open circuit:
$$I_D = I_R = 0,\text{A}$$
$$V_R = I_R R = 0,\text{V}$$
$$V_D = E - V_R = E$$
- $I_D$ — Diode current (A)
- $I_R$ — Current through resistor (A)
- $R$ — Series resistance ($\Omega$)
Half-Wave Rectifier
Positive cycle (diode forward biased):
$$V_m - V_D - V_O = 0 \quad \Rightarrow \quad V_O = V_m - V_D$$
- $V_m$ — Peak input voltage (V)
- $V_O$ — Output voltage (V)
- $V_D$ — Diode knee voltage (V)
Negative cycle (diode reverse biased):
$$V_O = 0$$
Average DC output voltage:
$$V_{DC} = 0.318,(V_m - V_D)$$
Lecture 35 — Transistor Biasing (Fixed & Emitter-Stabilized)
Fundamental BJT Current Relationships
Kirchhoff's Current Law (KCL) at the transistor node:
$$I_E = I_B + I_C$$
- $I_E$ — Emitter current (A)
- $I_B$ — Base current (A)
- $I_C$ — Collector current (A)
Current Gain (Beta, $\beta$):
$$\beta = \frac{I_C}{I_B} \quad \Rightarrow \quad I_C = \beta I_B$$
- $\beta$ — DC current gain (dimensionless); also denoted $h_{FE}$ (AC) or $h_{fe}$ (DC)
- $\beta$ is temperature-dependent
Alpha ($\alpha$):
$$\alpha = \frac{I_C}{I_E}$$
- $\alpha$ — Ratio of collector to emitter current (dimensionless)
Emitter Current in terms of Base Current:
$$I_E = I_B(\beta + 1)$$
Transistor Operating Regions
| Region | Emitter Junction | Collector Junction | Key Formula |
|---|---|---|---|
| Active (Linear) | Forward biased | Reverse biased | $I_C = \beta I_B$ |
| Saturation | Forward biased | Forward biased | $I_{C(\text{sat})}$ (max current) |
| Cutoff | Open / Reverse | Open / Reverse | $I_C = 0$, $I_B = 0$ |
1. Fixed-Bias Circuit
Base-Emitter Loop:
$$V_{CC} - I_B R_B - V_{BE} = 0$$
$$I_B = \frac{V_{CC} - V_{BE}}{R_B}$$
- $V_{CC}$ — Collector supply voltage (V)
- $R_B$ — Base resistor ($\Omega$)
- $V_{BE}$ — Base-emitter voltage; $\approx 0.7,\text{V}$ for Si transistors (V)
Collector-Emitter Loop:
$$V_{CC} - I_C R_C - V_{CE} = 0$$
$$V_{CE} = V_{CC} - I_C R_C$$
- $R_C$ — Collector resistor ($\Omega$)
- $V_{CE}$ — Collector-emitter voltage (V)
Saturation Current:
$$I_{C(\text{sat})} = \frac{V_{CC}}{R_C}$$
2. Emitter-Stabilized Bias Circuit
Base-Emitter Loop (with emitter resistor $R_E$):
$$V_{CC} - I_B R_B - V_{BE} - I_E R_E = 0$$
Substituting $I_E = I_B(\beta + 1)$:
$$I_B = \frac{V_{CC} - V_{BE}}{R_B + (\beta + 1)R_E}$$
- $R_E$ — Emitter resistor ($\Omega$)
Collector-Emitter Loop:
$$V_{CC} - I_C R_C - V_{CE} - I_E R_E = 0$$
Using $I_E \approx I_C$ (valid when $\beta \gg 1$):
$$V_{CE} = V_{CC} - I_C(R_C + R_E)$$
Saturation Current:
$$I_{C(\text{sat})} = \frac{V_{CC}}{R_C + R_E}$$
Voltage at Key Nodes:
$$V_E = I_E R_E$$
$$V_C = V_{CE} + V_E = V_{CC} - I_C R_C$$
$$V_B = V_{BE} + V_E = V_{CC} - I_B R_B$$
- $V_E$ — Emitter voltage relative to ground (V)
- $V_C$ — Collector voltage relative to ground (V)
- $V_B$ — Base voltage relative to ground (V)
Stability Design Approximation (for strong stability, when $(\beta + 1)R_E \gg R_B$):
$$(\beta + 1)R_E \geq 10 R_B$$
Under this condition:
$$I_C \approx \frac{V_{CC} - V_{BE}}{R_E}$$
$I_C$ becomes largely independent of $\beta$, providing excellent Q-point stability.
Lecture 36 — Voltage Divider Bias
Approximation Condition
For the approximate analysis to be valid:
$$\beta R_E \geq 10 R_{B2}$$
Equivalently:
$$R_{TH} \leq 0.1,\beta R_E$$
where $R_{TH}$ is the Thevenin resistance of the base divider:
$$R_{TH} = R_{B1} \parallel R_{B2} = \frac{R_{B1} \cdot R_{B2}}{R_{B1} + R_{B2}}$$
- $R_{B1}$, $R_{B2}$ — Voltage divider resistors ($\Omega$)
- $R_{TH}$ — Thevenin equivalent resistance ($\Omega$)
Approximate Analysis Procedure
| Step | Formula | Description |
|---|---|---|
| 1 | $V_B = \dfrac{R_{B2},V_{CC}}{R_{B1} + R_{B2}}$ | Base voltage from voltage divider rule |
| 2 | $V_E = V_B - V_{BE}$ | Emitter voltage |
| 3 | $I_E = \dfrac{V_E}{R_E}$ | Emitter current |
| 4 | $I_C \cong I_E$ | Collector current (approximation) |
| 5 | $V_{CE} = V_{CC} - I_C(R_C + R_E)$ | Collector-emitter voltage |
Bias Configuration Comparison
| Quantity | Fixed Bias | Emitter-Stabilized | Voltage Divider (Approx.) |
|---|---|---|---|
| $I_B$ | $\dfrac{V_{CC} - V_{BE}}{R_B}$ | $\dfrac{V_{CC} - V_{BE}}{R_B + (\beta+1)R_E}$ | Not directly calculated |
| $I_C$ | $\beta I_B$ | $\beta I_B$ | $\cong I_E = \dfrac{V_B - V_{BE}}{R_E}$ |
| $V_{CE}$ | $V_{CC} - I_C R_C$ | $V_{CC} - I_C(R_C + R_E)$ | $V_{CC} - I_C(R_C + R_E)$ |
Lecture 37 — Op-Amp: Inverting Amplifier
Inverting Amplifier Gain
Current through input resistor equals current through feedback resistor (no current enters op-amp input):
$$I = \frac{V_{in}}{R_1} = -\frac{V_{out}}{R_f}$$
Output voltage:
$$V_{out} = -\frac{R_f}{R_1},V_{in}$$
- $V_{in}$ — Input voltage (V)
- $V_{out}$ — Output voltage (V)
- $R_1$ — Input resistor ($\Omega$)
- $R_f$ — Feedback resistor ($\Omega$)
- Negative sign indicates 180° phase inversion
Lecture 38 — Op-Amp: Non-Inverting Amplifier
Non-Inverting Amplifier Gain
Current through $R_1$:
$$I = \frac{V_{in}}{R_1}$$
Voltage across feedback resistor $R_f$:
$$V_{R_f} = I \cdot R_f = V_{in} \cdot \frac{R_f}{R_1}$$
Output voltage:
$$V_{out} = V_{in} + V_{R_f} = V_{in}\left(1 + \frac{R_f}{R_1}\right)$$
$$V_{out} = \left(\frac{R_1 + R_f}{R_1}\right)V_{in}$$
- Output is in phase with input (0° phase shift)
- Gain is always $\geq 1$
Inverting vs Non-Inverting Comparison
| Configuration | Gain Formula | Phase |
|---|---|---|
| Inverting | $V_{out} = -\dfrac{R_f}{R_1},V_{in}$ | $180°$ (inverted) |
| Non-inverting | $V_{out} = \left(1 + \dfrac{R_f}{R_1}\right)V_{in}$ | $0°$ (non-inverted) |
Quick Reference: Diode Knee Voltages
| Semiconductor | Knee Voltage $V_D$ |
|---|---|
| Germanium (Ge) | $0.3,\text{V}$ |
| Silicon (Si) | $0.7,\text{V}$ |
| Gallium Arsenide (GaAs) | $1.5,\text{V}$ |
Quick Reference: BJT Operating Regions Summary
| Region | $V_{BE}$ | $V_{CE}$ | $I_C$ | Application |
|---|---|---|---|---|
| Cutoff | $< 0.7,\text{V}$ | $= V_{CC}$ | $\approx 0$ | Open switch |
| Active | $\approx 0.7,\text{V}$ | $> V_{CE(sat)}$ | $\beta I_B$ | Amplifier |
| Saturation | $\approx 0.7,\text{V}$ | $\approx 0.2,\text{V}$ | $I_{C(sat)}$ | Closed switch |
Source: FAD1022 L34-L38 — Semiconductors & Op-Amps Formula sheet compiled for finals revision. All equations extracted from lecture notes.