Resistors

Definition

A resistor is a passive linear dipole such that the voltage across its terminals is proportional to the current passing through it.

Symbol

IEC symbol

or

IEEE symbol

Resistance value

The resistance unit is the Ohm ($\Omega$). It can also be defined with Siemens ($S$) as the reciprocal of the resistance.

$$\Omega = \frac{1}{S}$$

Formulas

$$U = R \times I$$

where:

• $U$: voltage in Volts (V)
• $I$: current in Amperes (A)
• $R$: resistance in Ohms ($\Omega$)

Or using the conductance:

$$I = G \times U$$

where:

• $G = \frac{1}{R}$ is the conductance in Siemens (S)

The resistance of a wire-like conductor with a uniform cross-section is given by:

$$R = \rho \times \frac{l}{A}$$

where:

• $\rho$: resistivity of the material ($\Omega.m$)
• $l$: length of the wire (m)
• $A$: cross-sectional area of the wire (m²)

Series Connection

A circuit composed of multiple resistors connected in series can be reduced to a single equivalent resistance $R_e$, given by:

$$R_e = R_1 + R_2 + \dots + R_n$$

Parallel Connection

A circuit composed of multiple resistors connected in parallel can be reduced to a single equivalent resistance $R_e$, given by:

$$\frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$$

Formula for two resistors in parallel

For two resistors:

$$\frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2}$$

The formula can be simplified to:

$$R_e = \frac{R_1 * R_2}{R_1 + R_2}$$

Details

$$\frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2}$$$$\frac{1}{R_e} * (R_1 * R_2) = (\frac{1}{R_1} + \frac{1}{R_2}) * (R_1 * R_2)$$$$\frac{R_1 * R_2}{R_e} = \frac{R_1 * R_2}{R_1} + \frac{R_1 * R_2}{R_2}$$$$\frac{R_1 * R_2}{R_e} = R_2 + R_1$$$$\frac{1}{R_e} = \frac{R_2 + R_1}{R_1 * R_2}$$$$R_e = \frac{R_1 * R_2}{R_2 + R_1} = \frac{R_1 * R_2}{R_1 + R_2}$$