Resistance

Learning Objectives

By the end of this module, you will be able to:

• Define resistance and explain its effect in a circuit.
• Calculate the resistance of a conductor based on its physical properties.
• Identify common resistor types: carbon composition, wirewound, film, and surface mount.
• Determine the tolerance range of a resistor.
• Describe how a variable resistor (potentiometer and rheostat) operates.
• Decode a resistor's value using the color code or alphanumeric SMD code.
• Identify the three types of resistor circuits.
• Calculate total resistance in series, parallel, and series-parallel circuits.

1. What is Resistance?

Resistance is the opposition to the flow of electric current (electron flow) in a circuit. It is a fundamental property of materials.

Think of it like friction in a water pipe. A narrow, rough pipe has high resistance and restricts water flow, while a wide, smooth pipe has low resistance and allows water to flow easily.

• Symbol: $R$
• Unit: Ohm
• Abbreviation: The Greek letter Omega ($\Omega$)

Factors Affecting Resistance

The resistance of a conductor is determined by four main factors:

1. Material (Resistivity $\rho$): Different materials have different inherent abilities to resist electron flow. This property is called resistivity ($\rho$). Materials with low resistivity are conductors (e.g., copper, silver), while those with high resistivity are insulators (e.g., rubber, glass).
2. Length (L): Resistance is directly proportional to the length of the conductor. A longer wire has more resistance.
3. Cross-Sectional Area (A): Resistance is inversely proportional to the cross-sectional area. A thicker wire (larger area) has less resistance.
4. Temperature (T): For most conductors, resistance increases as temperature increases.

The relationship is described by the formula:

$$R = \rho \frac{L}{A}$$

Where:

• $R$ is resistance in Ohms ($\Omega$)
• $\rho$ (rho) is resistivity in Ohm-meters ($\Omega \cdot m$)
• $L$ is the length in meters ($m$)
• $A$ is the cross-sectional area in square meters ($m^2$)

Resistivity of Common Materials

The table below shows the resistivity of various materials. A lower number indicates a better conductor.

Material	Resistivity ($\rho$) at 20°C ($\Omega \cdot m$)	Relative to Silver
Silver	$1.59 \times 10^{-8}$	1.00
Copper	$1.68 \times 10^{-8}$	1.06
Gold	$2.44 \times 10^{-8}$	1.53
Aluminum	$2.82 \times 10^{-8}$	1.77
Tungsten	$5.60 \times 10^{-8}$	3.52
Iron	$9.71 \times 10^{-8}$	6.11

Conductance

Conductance is the opposite of resistance. It is the measure of how easily a material allows electrons to flow.

• Symbol: $G$
• Unit: Siemens ($S$)
• An older, informal unit is the Mho ($\mho$), which is "ohm" spelled backward.

Conductance is simply the reciprocal of resistance:

$$G = \frac{1}{R} \quad \text{or} \quad R = \frac{1}{G}$$

2. Resistors (The Component)

A resistor is an electronic component specifically designed to have a precise value of resistance. Its primary purpose is to control current, divide voltage, and manage signal levels within a circuit.

Resistors are classified into two main categories: fixed and variable.

Power Rating & Tolerance

Two critical parameters for any resistor are its power rating and tolerance.

• Power Rating: Measured in Watts (W), this indicates how much heat the resistor can safely dissipate before being damaged. A higher wattage rating means a physically larger resistor.
• Tolerance: This is the percentage of error in the resistor's manufactured resistance value. Common tolerances are $\pm1\%$, $\pm5\%$, and $\pm10\%$. For example, a $100\Omega$ resistor with a $\pm5\%$ tolerance can have an actual value between $95\Omega$ and $105\Omega$.

Fixed Resistor Types

Type	Key Characteristics	Common Use Cases
Carbon Composition	Inexpensive, durable, but low precision.	General-purpose, non-critical applications.
Wirewound	High precision and high power ratings.	High-current circuits, power supplies.
Film Resistors	Good precision and stability. Metal film is more precise than carbon film.	Audio circuits, measurement devices.
Surface Mount (SMD)	Very small, designed for automated circuit board assembly.	Modern electronics (phones, computers).

Variable Resistors

A variable resistor allows its resistance value to be adjusted, typically by turning a knob or a screw.

• They work by moving a conductive wiper across a resistive track.
• The resistance changes based on the wiper's position along the track.
• Their resistance can change linearly (a steady change) or logarithmically (slow change at first, then rapid). Logarithmic types are often used for audio volume controls.

When used in a circuit, they are commonly called a potentiometer or a rheostat.

How do they control voltage and current?

Potentiometer (Voltage Controller)

A potentiometer (or "pot") is used to create an adjustable voltage divider. It has three terminals: two ends of the resistive track and one for the wiper.

• How it works: A source voltage is applied across the two outer terminals. The output voltage is taken between the wiper and one of the outer terminals. As the wiper moves, it "taps off" a different voltage level, from 0V up to the full source voltage.

Rheostat (Current Controller)

A rheostat is used to control the current in a circuit. It only uses two terminals: one end of the resistive track and the wiper.

• How it works: The rheostat is connected in series with the load (e.g., an LED or motor). By changing the resistance, you change the total resistance of the circuit path. According to Ohm's Law ($I = V/R$), increasing the resistance decreases the current flow, and vice-versa.

3. Decoding Resistor Values

Axial Resistor Color Code (4-Band)

Color	Digit	Multiplier	Tolerance
Black	0	$10^0$	-
Brown	1	$10^1$	$\pm1\%$
Red	2	$10^2$	$\pm2\%$
Orange	3	$10^3$	-
Yellow	4	$10^4$	-
Green	5	$10^5$	$\pm0.5\%$
Blue	6	$10^6$	$\pm0.25\%$
Violet	7	$10^7$	$\pm0.1\%$
Grey	8	$10^8$	-
White	9	$10^9$	-
Gold	-	$10^{-1}$	$\pm5\%$
Silver	-	$10^{-2}$	$\pm10\%$

Example: A resistor has the bands: Orange, White, Red, Gold.

• Band 1 (Orange): 3
• Band 2 (White): 9
• Band 3 (Red): Multiplier of $10^2$ (or add 2 zeros)
• Band 4 (Gold): Tolerance of $\pm5\%$

The value is $39 \times 10^2 = 3900 \Omega$ or $3.9 k\Omega$, with a tolerance of $\pm5\%$.

Surface Mount (SMD) Alphanumeric Code

SMD resistors use a printed code because they are too small for color bands.

• 3-Digit Code: The first two digits are the significant figures, and the third is the multiplier (the power of 10).
  • 472 = $47 \times 10^2 = 4700\Omega = 4.7k\Omega$
• 4-Digit Code: Used for higher precision resistors. The first three digits are significant figures, and the fourth is the multiplier.
  • 1003 = $100 \times 10^3 = 100,000\Omega = 100k\Omega$
• EIA-96 Code (1% Tolerance): A more complex system with a two-digit number code followed by a letter multiplier. (e.g., 01A = $100\Omega$)

4. Resistors in Circuits

Resistors can be connected in three primary configurations: series, parallel, and series-parallel (also called compound or combination).

Series Circuits

In a series circuit, components are connected end-to-end, providing only one path for the current.

The total resistance ($R_T$) is the sum of all individual resistances.

$$R_T = R_1 + R_2 + R_3 + \dots + R_n$$

Example:

• $R_1 = 10\Omega$
• $R_2 = 20\Omega$
• $R_3 = 30\Omega$

$R_T = 10\Omega + 20\Omega + 30\Omega = 60\Omega$

Parallel Circuits

In a parallel circuit, components are connected across the same two points, providing multiple paths for the current.

The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances.

$$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}$$

Shortcut for Two Resistors: A common and much faster formula for only two resistors in parallel is "product over sum":

$$R_T = \frac{R_1 \times R_2}{R_1 + R_2}$$

Key Takeaway

The total resistance in a parallel circuit is always less than the value of the smallest individual resistor.

Series-Parallel (Compound) Circuits

These circuits are a combination of series and parallel configurations. To find the total resistance, you simplify the circuit in steps.

Strategy:

1. Identify any groups of resistors that are in parallel.
2. Calculate the equivalent resistance ($R_{eq}$) for each parallel group.
3. Redraw the circuit, replacing each parallel group with its single equivalent resistor.
4. The circuit is now a simple series circuit. Add up all the resistances to find the total.

Example: $R_1 = 50\Omega$ is in series with a parallel group of $R_2 = 100\Omega$ and $R_3 = 100\Omega$.

1. Solve the parallel group ($R_2$ and $R_3$): $R_{eq} = \frac{R_2 \times R_3}{R_2 + R_3} = \frac{100 \times 100}{100 + 100} = \frac{10000}{200} = 50\Omega$
2. Add the series components: The circuit is now just $R_1$ in series with the $50\Omega$ equivalent resistor. $R_T = R_1 + R_{eq} = 50\Omega + 50\Omega = 100\Omega$