Resistance and Resistors - Study Notes

Wattage and Power

• Watt is a measure of the amount of power used in a circuit.
• Formula: $P = E \times I$ where P is power (watts), E is voltage (volts), and I is current (amps).
• Other equivalent forms (derived from P = E I):
  • $P = I^2 \times R$
  • $P = \dfrac{E^2}{R}$
• Examples:
  • Ex 1: An electric iron connected to 120 V with a current draw of 8 A.
  • Power: $P = E \times I = 120 \text{ V} \times 8 \text{ A} = 960 \text{ W}$
  • Ex 2: An electric hair dryer with a power rating of 1000 W connected to 120 V.
  • Current drawn: $I = \dfrac{P}{E} = \dfrac{1000 \text{ W}}{120 \text{ V}} \approx 8.33 \text{ A}$

Ohm’s Law

• Ohm’s Law: in a DC circuit, current is directly proportional to voltage and inversely proportional to resistance.
• Key relationships:
  • $E = I \times R$
  • $I = \dfrac{E}{R}$
  • $R = \dfrac{E}{I}$
• Notes:
  • 1 volt is the pressure that pushes 1 ampere through 1 ohm.
  • The symbols used: I for current, E (or V) for voltage, R for resistance.
• Examples:
  • Ex 1: A circuit has R = 50 Ω, I = 2 A, E = ?
  • E = I × R = 2 \text{ A} \times 50 \text{ Ω} = 100 \text{ V}
  • $E = I \times R = 100 \text{ V}$
  • Ex 2: With V = 120 V and R = 30 Ω, current is I = V / R = 120 / 30 = 4 A.
  • $I = \dfrac{E}{R} = \dfrac{120}{30} = 4 \text{ A}$
  • Ex 3: With V = 240 V and I = 10 A, resistance is R = E / I = 240 / 10 = 24 Ω.
  • $R = \dfrac{E}{I} = \dfrac{240}{10} = 24 \text{ Ω}$

Summary of Key Electrical Quantities

• A coulomb is a measure of charge.
• An ampere (A) is 1 coulomb per second.
• The letter I stands for current (intensity) in Ohm’s law formulas.
• Voltage is electric pressure, potential difference, or electromotive force; E or V can represent voltage in Ohm’s law formulas.
• An ohm (Ω) is a unit of resistance (R).
• The watt (W) is a unit of power; represented by W or P in Ohm’s law formulas.
• Engineering notation is commonly used for electrical measurements.
• A complete circuit is required before current can flow.
• Short circuit: little or no resistance.
• Open circuit: infinite resistance.

Resistance in general

• Analogy: In a water system, a reducer controls flow. In an electric circuit, a resistor controls the flow of electrons.

Resistance: Basic Units and Symbols

• An ohm is the unit of resistance; symbol: Ω.
• The letter R represents resistance (in ohms).
• An ohm is the resistance that allows 1 A to flow when the applied voltage is 1 V.

Heat due to Resistance

• Whenever current flows through a resistance, heat is produced.
• This is why a wire heats up, electric ranges get hot, and lamp filaments get very hot due to resistance.
• Impedance is the AC counterpart of resistance (more commonly used in AC calculations).
• Impedance is the complex resistance that includes resistance and reactance.

Uses of Resistance

• Limit the flow of current in a circuit.
• Create a voltage divider arrangement to produce reference voltages.

Resistance of Conductors

• The resistance of a conductor depends on:
  • Type of material
  • Length of conductor
  • Cross-sectional area
  • Temperature
• Dependence:
  • Resistance is directly proportional to the length: $R \propto l$
  • Resistance is inversely proportional to cross-sectional area: $R \propto \dfrac{1}{A}$

Conductor Resistance: Formula

• Fundamental formula: $R = \dfrac{\rho \times l}{A}$
  • $\rho$ is resistivity (ohm-meters, Ω·m)
  • $l$ is length (meters)
  • $A$ is cross-sectional area (square meters)
• Resistivity captures how strongly a material opposes current flow.

Conductor Resistance (Example 1)

• Copper wire: length $l = 5\,\text{m}$, cross-section $A = 10\ \text{mm}^2 = 1\times 10^{-5}\ \text{m}^2$, resistivity \rho = 1.77\times 10^{-8}\ \Omega\text{·m}
• Resistance: $R = \dfrac{\rho \times l}{A} = \dfrac{(1.77\times 10^{-8})\times 5}{1\times 10^{-5}} \ = 8.85\times 10^{-3} \Omega \approx 0.0089\ \Omega$

Conductor Resistance (Example 1 - Aluminum)

• Aluminum wire: length $l = 1000\ \text{m}$, cross-section $A = 200\ \text{mm}^2 = 2\times 10^{-4}\ \text{m}^2$, resistivity $\rho = 2.83\times 10^{-8}\ \Omega\cdot\text{m}$
• Resistance: $R = \dfrac{\rho \times l}{A} = \dfrac{(2.83\times 10^{-8})\times 1000}{2\times 10^{-4}} = 0.1415\ \Omega$

Wire Resistance (Circular Mils and K)

• In some tables, length is in feet and cross-sectional area in circular mils (CM).
• General relation: $R = K \times \dfrac{L}{CM}$
  • $K$: resistivity in ohms-CM per foot
  • $L$: length in feet
  • $CM$: cross-sectional area in circular mils
• Note: CM is the cross-sectional area measure used for wires; one circular mil equals the area of a circle with diameter 1 mil.

Resistivity and Temperature Coefficients (Table Snapshot)

• Materials and typical parameters at 20°C:
  • Aluminum: $K = 17,\; \alpha = 0.004$ per °C
  • Carbon: $K \approx 22{,}000,\; \alpha \approx -0.0004$ per °C
  • Constantan: $K = 295,\; \alpha = 0.000002$ per °C
  • Copper: $K = 10.4,\; \alpha = 0.0039$ per °C
  • Gold: $K = 14,\; \alpha = 0.004$ per °C
  • Iron: $K = 60,\; \alpha = 0.0055$ per °C
  • Lead: $K = 126,\; \alpha = 0.0043$ per °C
  • Manganin: $K = 265,\; \alpha \approx 0.000000$ per °C
  • Mercury: $K = 590,\; \alpha = 0.00088$ per °C
  • Nichrome: $K = 675,\; \alpha = 0.0002$ per °C
  • Nickel: $K = 52,\; \alpha = 0.005$ per °C
  • Platinum: $K = 66,\; \alpha = 0.0036$ per °C
  • Silver: $K = 9.6,\; \alpha = 0.0038$ per °C
  • Tungsten: $K = 33.8,\; \alpha = 0.005$ per °C
• Note: Temperature coefficients indicate how resistivity changes with temperature.

Example 1 (Copper) and Example 2 (Aluminum) – Resistivity Tables

• Example 1 (from table): Find the resistance of a piece of copper wire with cross-section CM = 26{,}250, length = 550 ft, at 20°C.
• Example 2 (from table): An aluminum wire length 2{,}250 ft cannot have resistance greater than 0.2 Ω. What size aluminum wire is required?

Temperature Effects on Resistance

• For most conductors: increasing temperature increases resistance (positive temperature coefficient).
• Increase is approximately linear over a typical range.
• Semiconductors and insulators: resistance decreases with temperature (negative coefficient).

Resistor Types (Fixed Resistors)

• Fixed resistors have a single, unchangeable ohmic value.
• Common fixed resistor types:
  • Molded carbon composition
  • Carbon film
  • Metal film
  • Metal oxide
  • Wire-wound
  • Integrated circuit packages

Carbon Resistors (Composition Carbon Resistors)

• Composition carbon resistors are a common fixed type made from carbon graphite with a resin binder.

Color Code for Resistors

• Value often determined by color bands; colors represent numbers.
• Two main coding schemes: 4-band and 5-band resistors.

Color Code Mapping

• Digit colors (0–9): Black=0, Brown=1, Red=2, Orange=3, Yellow=4, Green=5, Blue=6, Violet=7, Gray=8, White=9
• Multiplier and tolerance bands are interpreted based on the band position and color.

5-Band Resistors (±1%) vs 4-Band Resistors (±10%)

• 5-band scheme: Band 1 = 1st digit, Band 2 = 2nd digit, Band 3 = 3rd digit, Band 4 = multiplier, Band 5 = tolerance.
• 4-band scheme: Band 1 = 1st digit, Band 2 = 2nd digit, Band 3 = multiplier, Band 4 = tolerance.
• Wide spacing appears in some diagrams between bands.

Resistance Value Calculation (4-band example)

• Example bands: brown, green, red, silver.
  • 1st digit = brown = 1
  • 2nd digit = green = 5
  • Multiplier = red = 10^2 = 100
  • Value: $(15) \times 100 = 1500$ Ω
  • Tolerance band = silver = ±10%
  • Result: $1500 \text{ Ω} \;\pm 10\%$

Gold and Silver as Multipliers

• Gold as multiplier means divide the combined first two digits by 10 (× 0.1).
• Silver as multiplier means divide the combined first two digits by 100 (× 0.01).
• Example: bands orange, white, gold, gold.
  • Digits: orange=3, white=9 → 39
  • Multiplier: gold → × 0.1 → 39 × 0.1 = 3.9 Ω
  • Tolerance (4th band): gold → ±5%
  • Note: The example text also mentions a tolerance of 65% in one place, but the standard 4-band code uses ±5% for gold; here it is given as 5% tolerance.

Example 3: Interpreting a 1% Resistor

• Five-band example: first three bands are digits, fourth is multiplier, fifth is tolerance.
• Bands: brown, black, black, brown, 1%
  • 1st digit = brown = 1
  • 2nd digit = black = 0
  • 3rd digit = black = 0
  • Multiplier = brown = ×10
  • Value: 1000 Ω
  • Tolerance = 1%

Example 4: Five-Band Resistor with Tolerance and Reliability

• Five-band resistor: red, orange, violet, red, brown
  • First three bands: red=2, orange=3, violet=7 → digits 237
  • Fourth band: red = multiplier ×100
  • Value: 237 × 100 = 23{,}700 Ω
  • Fifth band: brown = tolerance of 1%
• Notes on the fifth band:
  • Brown: 1% tolerance
  • Orange or yellow as the fifth band indicate reliability levels (orange = good reliability, yellow = space-flight reliability) in some contexts.

Variable Resistors

• Resistance can be varied (e.g., knobs for volume, lighting).
• Construction:
  • Two fixed terminals at ends of resistive element
  • Center terminal connects to a wiper that moves along the resistive element when the shaft rotates
• Used for potentiometers and rheostats

Conductance

• Conductance: measure of a material’s ability to allow charge flow.
• Conductance is the reciprocal of resistance:
  • $G = \dfrac{1}{R}$
• Unit: Siemens (S)

Measuring Resistance — The Ohmmeter

• Usually part of a multimeter.
• Steps:
  • Remove all power sources from the circuit.
  • Isolate the component under test.
  • Connect the probes across the component.
  • Ensure the ohmmeter is set to the correct range.

Measuring Resistance — Short Circuit and Open Circuit

• Short circuit: a low-resistance path exists between two points; the ohmmeter will indicate very low or zero resistance when measuring a short.
• Open circuit: the conductor is broken between the test points; the ohmmeter will indicate infinite resistance.

Note on Sources

• Images in slides are from Delmar’s Standard Textbook of Electricity, 6th edition.
• These notes summarize content from the ELT103-Fall 2017 presentation on Resistance and Resistors.