Formula Sheet: DC Circuits
Comprehensive formula sheet extracted from FAD1022 Tutorial 4 — DC Circuits
1. Ohm's Law
1.1 Basic Form
$$V = IR$$
1.2 Rearranged Forms
$$I = \frac{V}{R}, \quad R = \frac{V}{I}$$
| Variable | Description | SI Unit |
|---|---|---|
| $V$ | Potential difference (voltage) | $\text{V}$ (Volts) |
| $I$ | Electric current | $\text{A}$ (Amperes) |
| $R$ | Resistance | $\Omega$ (Ohms) |
2. Electromotive Force (EMF) and Terminal Voltage
2.1 Terminal Voltage of a Real Cell/Battery
$$V_{terminal} = \varepsilon - Ir$$
2.2 Current from a Real Source
$$I = \frac{\varepsilon}{R_{external} + r}$$
2.3 Terminal Voltage (Alternative Forms)
$$V_{terminal} = IR_{external}$$
$$V_{terminal} = \varepsilon \frac{R_{external}}{R_{external} + r}$$
| Variable | Description | SI Unit |
|---|---|---|
| $\varepsilon$ | Electromotive force | $\text{V}$ |
| $V_{terminal}$ | Terminal voltage | $\text{V}$ |
| $r$ | Internal resistance | $\Omega$ |
| $R_{external}$ | External load resistance | $\Omega$ |
| $I$ | Current drawn from source | $\text{A}$ |
Note: For an ideal source (no internal resistance), $V_{terminal} = \varepsilon$.
3. Power in DC Circuits
3.1 General Power Formula
$$P = IV$$
3.2 Power Dissipated in a Resistor
$$P = I^2 R = \frac{V^2}{R}$$
3.3 Power Supplied by EMF Source
$$P_{source} = I\varepsilon$$
3.4 Power Lost to Internal Resistance
$$P_{lost} = I^2 r$$
3.5 Useful Power Delivered to Load
$$P_{delivered} = I^2 R_{external} = IV_{terminal}$$
| Variable | Description | SI Unit |
|---|---|---|
| $P$ | Power | $\text{W}$ (Watts) |
4. Resistor Combinations
4.1 Resistors in Series
$$R_{eq} = R_1 + R_2 + R_3 + \dots = \sum_{i} R_i$$
Key property: Same current $I$ through each resistor. Total voltage adds: $V_{total} = V_1 + V_2 + V_3 + \dots$
4.2 Resistors in Parallel
$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots = \sum_{i} \frac{1}{R_i}$$
For two resistors:
$$R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$$
Key property: Same voltage $V$ across each resistor. Total current adds: $I_{total} = I_1 + I_2 + I_3 + \dots$
4.3 Current Divider (Two Parallel Resistors)
$$I_1 = I_{total} \frac{R_2}{R_1 + R_2}$$
$$I_2 = I_{total} \frac{R_1}{R_1 + R_2}$$
5. Kirchhoff's Laws
5.1 Kirchhoff's Current Law (KCL) — Junction Rule
$$\sum I_{in} = \sum I_{out}$$
Or equivalently:
$$\sum I = 0 \quad \text{(algebraic sum of currents at a junction)}$$
Physical basis: Conservation of charge. No charge accumulates at a junction.
5.2 Kirchhoff's Voltage Law (KVL) — Loop Rule
$$\sum_{loop} \Delta V = 0$$
Physical basis: Conservation of energy. The net change in potential around any closed loop is zero.
5.3 Sign Conventions for KVL
| Component | Traveling WITH current | Traveling AGAINST current |
|---|---|---|
| Resistor ($R$) | $-IR$ (potential drop) | $+IR$ (potential rise) |
| EMF source ($\varepsilon$) | $+\varepsilon$ (negative to positive) | $-\varepsilon$ (positive to negative) |
6. Circuit Analysis Methods
6.1 Single Loop Circuit
$$I = \frac{\sum \varepsilon}{\sum R}$$
6.2 Potential Difference Between Two Points
$$V_A - V_B = \sum \text{(potential changes from A to B)}$$
7. Summary of Key Relationships
| Concept | Formula |
|---|---|
| Ohm's Law | $V = IR$ |
| Terminal voltage | $V_{terminal} = \varepsilon - Ir$ |
| Power (general) | $P = IV$ |
| Power (resistor) | $P = I^2 R = \dfrac{V^2}{R}$ |
| Resistors in series | $R_{eq} = \sum R_i$ |
| Resistors in parallel | $\dfrac{1}{R_{eq}} = \sum \dfrac{1}{R_i}$ |
| KCL (Junction Rule) | $\sum I_{in} = \sum I_{out}$ |
| KVL (Loop Rule) | $\sum \Delta V = 0$ |
Related Concepts
- DC Circuits
- Ohm's Law
- Kirchhoff's Current Law
- Kirchhoff's Voltage Law
- EMF
- Internal Resistance
- Terminal Voltage
- Resistor Network
- Series Circuit
- Parallel Circuit
Related Lectures
- FAD1022 L10-L13 — DC Circuits