Basics

Charge

Quantized & conserved. Measured in Coulomb ($\text{C}$). $$

$$1\,\text{C} = \text{Charge of}\;6.25 \times 10^{18} \;\text{electrons}$$

Time invariant charge is denoted as $Q$. And time varying charge is denoted as $q$.

Current

Amount of charge flowing through a point in unit time. Conventional current (opposite to electron flow) flows from positive to negative potentials.

$$I = \frac{\text{d}Q}{\text{d}t}$$

Time invariant current is denoted as $I$. And time varying current is denoted as $i$.

Voltage

Voltage at a point is the work that must be done against the electric field to move a unit positive charge from infinity to that point.

$$V = \frac{E}{Q}$$

Time invariant voltage is denoted as $V$. And time varying voltage is denoted as $v$.

Voltage difference between 2 points is the work that must be done against the electric field to move a unit positive charge from one point to another.

$$V_{AB} = V_A - V_B$$

Double subscript notation

Voltage is heigher at node $a$. Current is flowing from $a$ to $b$.

$$V_{ab} = V_a - V_b$$

Electric Circuit

Types of circuits

• Closed circuit - the electricity flows
• Open circuit - the electricity doesn’t flow. current = $0$. $\infty$ resistance.
• Short circuit - very large current. $0$ resistance.

Power

$$p= \frac{\text{d}w}{\text{d}t}= \frac{\text{d}w}{\text{d}q} \frac{\text{d}q}{\text{d}t} = vi$$

Total Work

$$w = \int_{t_0}^{t} {p\,\text{d}t} = \int_{t_0}^{t} {vi\,\text{d}t}$$

When v and i are constant

$$w = vi \int_{t_0}^{t} {\text{d}t} = vi(t - t_0)$$

Electrical Load

Something that consumes electrical energy.

Linear loads

Loads that have a linear relationship between the applied voltage and the current. Can be expressed using a combination of resistors, capacitors and inductors only.

Note

If a AC sinusoidal voltage is applied across a load, current through the load will also be sinusoidal iff the load is linear.

Non-linear loads

• Diodes
• Superconductors
• Varistors (voltage-dependent resistors)
• Non-linear inductors