📚 DC FUNDAMENTALS • SET 1

Ohm's Law | Resistance | Resistivity | Conductance | NOTES ONLY

⚡ 1. OHM'S LAW

Ohm's Law states that the electric current through a conductor between two points is directly proportional to the voltage across the two points, provided the temperature remains constant.

V = I × R | I = V / R | R = V / I

V = Voltage (Volts) — Electrical pressure
I = Current (Amperes) — Flow of electrons
R = Resistance (Ohms, Ω) — Opposition to current

📌 Key Point: Current is directly proportional to voltage (I ∝ V) and inversely proportional to resistance (I ∝ 1/R). Graph of V vs I is a straight line through origin.

🔧 2. RESISTANCE (R)

Resistance is the property of a material that opposes the flow of electric current. It converts electrical energy into heat.

Factors Affecting Resistance

Length (l): R ∝ l (Longer wire = Higher resistance)
Cross-sectional Area (A): R ∝ 1/A (Thicker wire = Lower resistance)
Material: Different materials have different resistivities
Temperature: Resistance changes with temperature

R = ρ × (l / A)

where ρ (rho) = resistivity of the material (Ω-m)

📊 3. RESISTIVITY (ρ)

Resistivity is the resistance of a unit cube of a material (1m × 1m × 1m). It is a material property and does not depend on the shape or size of the conductor.

ρ = (R × A) / l (Unit: Ohm-metre, Ω-m)

Resistivity depends ONLY on:

Nature of the material
Temperature

Material	Resistivity at 20°C (Ω-m)	Classification
Silver (Ag)	1.59 × 10⁻⁸	Best conductor
Copper (Cu)	1.68 × 10⁻⁸	Excellent conductor
Gold (Au)	2.40 × 10⁻⁸	Good conductor
Aluminum (Al)	2.65 × 10⁻⁸	Good conductor
Tungsten (W)	5.60 × 10⁻⁸	Conductor (lamp filament)
Iron (Fe)	9.71 × 10⁻⁸	Fair conductor
Nichrome (alloy)	100 × 10⁻⁸	Heating element

⚠️ Important: Resistivity depends ONLY on material, NOT on length or area of cross section. Lower resistivity = Better conductor.

🔌 4. CONDUCTANCE (G) & CONDUCTIVITY (σ)

Conductance is the reciprocal of resistance — it measures how easily current flows through a material.

G = 1 / R (Unit: Siemens S or mho)

Conductivity (σ) is the reciprocal of resistivity.

σ = 1 / ρ (Unit: Siemens per metre, S/m)

Higher conductance = Lower resistance
Higher conductivity = Better conductor
For a conductor: G = σ × (A / l)

📋 5. QUICK REFERENCE TABLE

Quantity	Symbol	Unit	Formula
Voltage	V	Volt (V)	V = I × R
Current	I	Ampere (A)	I = V / R
Resistance	R	Ohm (Ω)	R = V / I = ρ × (l/A)
Resistivity	ρ	Ohm-metre (Ω-m)	ρ = (R × A) / l
Conductance	G	Siemens (S) / mho	G = 1 / R
Conductivity	σ	Siemens/metre (S/m)	σ = 1 / ρ

📝 6. IMPORTANT FORMULAS (REMEMBER)

V = I × R   |   I = V / R   |   R = V / I
R = ρ × (l / A)   |   ρ = (R × A) / l
G = 1 / R   |   σ = 1 / ρ

💡 7. KEY POINTS TO REMEMBER

Ohm's Law is valid only if temperature remains constant
Resistance is a property of the component (depends on dimensions)
Resistivity is a property of the material (independent of dimensions)
Silver is the best conductor (lowest resistivity)
Nichrome has high resistivity — used in heating elements
Conductance is the opposite of resistance (easier flow)

🔷 SET 2: TEMPERATURE COEFFICIENT & RTD 🔷

🌡️ 8. TEMPERATURE COEFFICIENT OF RESISTANCE (α)

The temperature coefficient of resistance (α) indicates how much the resistance of a material changes with temperature.

R_t = R₀ × (1 + α₀ × t)

R_t = Resistance at temperature t°C
R₀ = Resistance at 0°C
α₀ = Temperature coefficient at 0°C
t = Temperature difference (°C)

。

Material	Temperature Coefficient (α)	Behavior
Copper	+0.00393 /°C	Positive (R ↑ as T ↑)
Aluminum	+0.00403 /°C	Positive
Nichrome	+0.00017 /°C	Very small (almost constant)
Carbon	-0.0005 /°C	Negative (R ↓ as T ↑)
Germanium	Negative	Semiconductor

📈 9. POSITIVE vs NEGATIVE TEMPERATURE COEFFICIENT

Positive Temperature Coefficient (PTC)

Resistance increases as temperature increases
Metals: Copper, Aluminum, Iron, Silver, Gold, Tungsten
Reason: Increased atomic vibrations impede electron flow

Negative Temperature Coefficient (NTC)

Resistance decreases as temperature increases
Insulators & Semiconductors: Carbon, Germanium, Silicon
Reason: More electrons gain energy to jump to conduction band

📌 Key Point: Insulators have NTC (Negative Temperature Coefficient). As temperature increases, their resistance decreases.

🔬 10. RESISTANCE TEMPERATURE DETECTOR (RTD)

An RTD is a device whose resistance changes significantly with temperature. It is used for temperature measurement.

Typically made of Platinum (Pt100), Nickel, or Copper
Positive temperature coefficient (PTC)
Working principle: Pass a small current through it and measure voltage drop
Thermistor is a common type of RTD (usually NTC)

For Pt100: R₀ = 100Ω at 0°C
R₁₀₀ = 138.5Ω at 100°C

📌 Applications: Industrial temperature measurement, HVAC, automotive sensors, medical equipment

🧪 11. THERMISTORS

A thermistor is a type of RTD made from semiconductor materials. It is highly sensitive to temperature changes.

Type	Temperature Coefficient	Behavior	Application
NTC Thermistor	Negative	Resistance ↓ as T ↑	Temperature sensing, inrush current limiting
PTC Thermistor	Positive	Resistance ↑ as T ↑	Overcurrent protection, self-resetting fuses

🔌 12. INSULATORS

Insulators are materials that do not allow electric current to flow easily. They have very high resistivity.

Examples: Glass, Rubber, Plastic, Mica, Ceramic, Wood, Air
Resistivity: Very high (10¹⁰ to 10¹⁴ Ω-m)
Temperature Coefficient: Negative (NTC) — resistance decreases as temperature increases

⚠️ Important: Insulators have a negative temperature coefficient of resistance. As temperature increases, more electrons gain enough energy to move into the conduction band, which reduces resistance.

📊 13. OHMIC vs NON-OHMIC RESISTORS

Type	Definition	Examples
Ohmic Resistor	Follows Ohm's Law (V ∝ I) — constant resistance	Nichrome, Copper, Constantan, Manganin
Non-Ohmic Resistor	Does NOT follow Ohm's Law — resistance changes with voltage/current	Diode, Thermistor, LDR, Transistor

🔍 Example: Nichrome is an ohmic resistor (used in heating elements). Germanium, Diamond, Diode are non-ohmic.

📋 14. QUICK REFERENCE TABLE

Material Type	Temperature Coefficient	Resistance with ↑ Temperature
Metals (Cu, Al, Fe, Ag)	Positive (+)	Increases
Insulators (Glass, Rubber)	Negative (-)	Decreases
Semiconductors (C, Ge, Si)	Negative (-)	Decreases
Alloys (Nichrome, Constantan)	Very small (~0)}.\]
Almost constant

📝 15. KEY POINTS TO REMEMBER

Metals → Positive temperature coefficient (PTC)
Insulators & Semiconductors → Negative temperature coefficient (NTC)
Nichrome is an ohmic resistor with very small temperature coefficient
RTD (Pt100) is used for temperature measurement (PTC)
Thermistors are highly sensitive temperature sensors
At higher temperatures, insulators become less resistive (more conductive)

🔷 SET 3: ELECTRICAL ENERGY, POWER & BATTERY CAPACITY 🔷

⚡ 16. ELECTRICAL POWER (P)

Electrical Power is the rate at which electrical energy is consumed or converted into another form of energy (heat, light, mechanical, etc.).

P = V × I | P = I² × R | P = V² / R

P = Power (Watts, W)
V = Voltage (Volts, V)
I = Current (Amperes, A)
R = Resistance (Ohms, Ω)

📌 Key Point: 1 Watt = 1 Joule per second. Power is the rate of energy consumption.

🔋 17. ELECTRICAL ENERGY (E)

Electrical Energy is the total amount of electrical work done or power consumed over a period of time.

E = P × t | E = V × I × t

E = Energy (Joules, J)
P = Power (Watts, W)
t = Time (Seconds, s)

Practical unit: 1 kWh = 1000 Watts × 3600 seconds = 3.6 × 10⁶ Joules

🔍 Example: A 100W bulb running for 10 hours consumes:
Energy = 100W × 10h = 1000 Wh = 1 kWh (1 unit of electricity)

🔥 18. HEAT PRODUCED IN A RESISTOR (Joule's Law)

When current flows through a resistor, electrical energy is converted into heat energy. This is known as Joule Heating or I²R loss.

H = I² × R × t | H = V × I × t | H = (V² / R) × t

H = Heat produced (Joules, J)
I = Current (Amperes, A)
R = Resistance (Ohms, Ω)
t = Time (Seconds, s)

⚠️ Important: Heat produced is directly proportional to:

Square of current (I²)
Resistance (R)
Time (t)

🔋 19. BATTERY CAPACITY (Ah)

Battery capacity is the amount of charge a battery can store and deliver. It is measured in Ampere-hours (Ah).

Capacity (Ah) = Current (A) × Time (hours)

Time (hours) = Capacity (Ah) / Current (A)

1 Ah = 3600 Coulombs of charge
Higher Ah = Longer battery life
Example: A 200 Ah battery can supply 10A for 20 hours

🔍 Example: A wristwatch mercury cell has 200 mAh capacity and draws 15 μA.
Hours = 0.2 Ah / 0.000015 A = 13,334 hours ≈ 18.5 months

💡 20. RELATIONSHIP BETWEEN POWER, VOLTAGE, CURRENT, RESISTANCE

Given	Formula for Power (P)	Formula for Current (I)	Formula for Resistance (R)
V and I	P = V × I	I = P / V	R = V / I
V and R	P = V² / R	I = V / R	R = V² / P
I and R	P = I² × R	I = √(P / R)	R = P / I²

📊 21. POWER RATING OF RESISTORS

Resistors have a power rating (in Watts) indicating how much heat they can safely dissipate without damage.

Resistor TypeTypical Power RatingApplication Carbon Composition1/8W to 2WGeneral purpose electronics Metal Film1/8W to 5WPrecision circuits Wirewound5W to 500WHigh power applications, heating elements

⚠️ Important: If a resistor dissipates more power than its rating, it will overheat and burn out.

📋 22. QUICK REFERENCE TABLE

Quantity	Symbol	Unit	Formula
Power	P	Watt (W)	P = V × I = I²R = V²/R
Energy	E	Joule (J) / kWh	E = P × t = V × I × t
Heat	H	Joule (J)	H = I² × R × t
Battery Capacity	Ah	Ampere-hour (Ah)	Ah = I × t (hours)

📝 23. KEY POINTS TO REMEMBER

1 kWh = 3.6 × 10⁶ Joules (1 unit of electricity)
Heat produced (Joule's Law): H ∝ I² × R × t
If resistance is halved, heat produced is halved (at constant current)
Battery capacity (Ah) = Current (A) × Time (hours)
Bulb resistance from power rating: R = V² / P
Higher power rating = lower resistance (at same voltage)

🔷 SET 4: KIRCHHOFF'S LAWS & SERIES-PARALLEL CIRCUITS 🔷

🔀 24. KIRCHHOFF'S CURRENT LAW (KCL)

Kirchhoff's Current Law (KCL) states that the total current entering a node (junction) equals the total current leaving the node. It is based on the conservation of charge.

Σ I_in = Σ I_out

Also called the Junction Law
Charge cannot accumulate at a node
Sum of all currents at a node = 0 (with proper signs)

🔍 Example: If 5A enters a node, and two branches leave with 2A and 3A → 5A = 2A + 3A ✓

🔄 25. KIRCHHOFF'S VOLTAGE LAW (KVL)

Kirchhoff's Voltage Law (KVL) states that the sum of all voltages around any closed loop in a circuit equals zero. It is based on the conservation of energy.

Σ V_rise = Σ V_drop or Σ V = 0

Also called the Loop Law
Energy supplied by sources = Energy consumed by loads
V_rise = Voltage increase (batteries/sources)
V_drop = Voltage decrease (resistors/loads)

🔍 Example: In a series circuit with battery 12V and resistors 5V + 7V drop → 12V = 5V + 7V ✓

🔗 26. SERIES CIRCUIT CHARACTERISTICS

Components connected end-to-end so the same current flows through all of them.

Property	Rule	Formula
Current	Same through all components	I_T = I₁ = I₂ = I₃
Voltage	Divided among components	V_T = V₁ + V₂ + V₃
Resistance	Additive	R_T = R₁ + R₂ + R₃
Power	Additive	P_T = P₁ + P₂ + P₃

Voltage Division Rule: V_x = V_T × (R_x / R_T)

📌 Key Point: In series circuit, total resistance is always greater than the largest individual resistance.

🔀 27. PARALLEL CIRCUIT CHARACTERISTICS

Components connected across the same voltage source, so same voltage appears across each.

Property	Rule	Formula
Voltage	Same across all components	V_T = V₁ = V₂ = V₃
Current	Divided among branches	I_T = I₁ + I₂ + I₃
Resistance	Reciprocal sum	1/R_T = 1/R₁ + 1/R₂ + 1/R₃
Conductance	Additive	G_T = G₁ + G₂ + G₃
Power	Additive	P_T = P₁ + P₂ + P₃

Current Division Rule: I_x = I_T × (R_T / R_x) or I_x = I_T × (G_x / G_T)

📌 Key Point: In parallel circuit, total resistance is always less than the smallest individual resistance.

🔗 28. SERIES-PARALLEL COMBINATIONS

Most practical circuits are combinations of series and parallel connections.

Steps to Solve Series-Parallel Circuits:

Identify which resistors are in series and which are in parallel
Simplify parallel combinations first (using 1/R_T = 1/R₁ + 1/R₂)
Simplify series combinations (using R_T = R₁ + R₂)
Repeat until only one equivalent resistance remains
Use voltage and current division rules as needed

🔍 Example: Three resistors 2Ω, 3Ω, 6Ω.
• All in series: R_T = 2+3+6 = 11Ω
• All in parallel: 1/R_T = 1/2+1/3+1/6 = 1 → R_T = 1Ω
• 2Ω in series with parallel combination of 3Ω & 6Ω: R_p = (3×6)/(3+6)=2Ω, R_T = 2+2=4Ω

📊 29. SPECIAL FORMULAS

ConfigurationEqual Resistors (n, each = R)Two Resistors (R₁, R₂) SeriesR_T = n × RR_T = R₁ + R₂ ParallelR_T = R / nR_T = (R₁ × R₂) / (R₁ + R₂)

📝 30. KEY POINTS TO REMEMBER

KCL: Sum of currents entering = Sum of currents leaving
KVL: Sum of voltages around a closed loop = 0
In series: Current same, voltage divides, R_T > largest R
In parallel: Voltage same, current divides, R_T < smallest R
For n equal resistors in series: R_T = nR
For n equal resistors in parallel: R_T = R/n
Parallel combination formula: R_p = (R₁×R₂)/(R₁+R₂)

🔷 SET 5: THEVENIN, NORTON & SUPERPOSITION THEOREMS 🔷

🔧 31. THEVENIN'S THEOREM

Thevenin's Theorem states that any linear circuit containing voltage sources, current sources, and resistances can be replaced by a single voltage source (V_TH) in series with a single resistance (R_TH) connected across the load.

V_TH = Open-circuit voltage across the load terminals
R_TH = Equivalent resistance seen from load terminals (all sources deactivated)

Deactivating sources: Voltage sources → short circuit, Current sources → open circuit
Thevenin equivalent simplifies complex circuits for load analysis

🔍 Example: A 5V battery with 2Ω internal resistance → V_TH = 5V, R_TH = 2Ω

🔧 32. NORTON'S THEOREM

Norton's Theorem states that any linear circuit can be replaced by a single current source (I_N) in parallel with a single resistance (R_N) connected across the load.

I_N = Short-circuit current through the load terminals
R_N = Equivalent resistance seen from load terminals (same as R_TH)

Norton equivalent is the dual of Thevenin equivalent
R_TH = R_N (always equal)

🔄 33. SOURCE TRANSFORMATION

Source transformation is the process of converting a voltage source in series with a resistor into a current source in parallel with the same resistor, and vice versa.

Voltage Source → Current Source: I_s = V_s / R, R same in parallel
Current Source → Voltage Source: V_s = I_s × R, R same in series

🔍 Example: 5V source with 2Ω series resistance → 2.5A source with 2Ω parallel resistance

⚡ 34. SUPERPOSITION THEOREM

Superposition Theorem states that in a linear circuit with multiple independent sources, the response (voltage or current) in any element is the algebraic sum of the responses caused by each source acting alone.

To use superposition:
Consider one source at a time
Deactivate all other sources: Voltage sources → short circuit, Current sources → open circuit
Calculate the contribution of that source
Repeat for all sources
Add all contributions (with proper signs)

📌 Key Point: Superposition applies only to linear circuits (resistors, capacitors, inductors). It does NOT apply to power calculations (P = I²R).

📊 35. COMPARISON: THEVENIN vs NORTON

Feature	Thevenin	Norton
Equivalent	Voltage source in series with R_TH	Current source in parallel with R_N
R_TH vs R_N	R_TH = R_N	R_N = R_TH
Value	V_TH = Open-circuit voltage	I_N = Short-circuit current
Conversion	I_N = V_TH / R_TH	V_TH = I_N × R_N

📝 36. KEY POINTS TO REMEMBER

Thevenin: V_TH (open circuit voltage), R_TH (resistance with sources deactivated)
Norton: I_N (short circuit current), R_N = R_TH
Source transformation: V_S in series with R ↔ I_S in parallel with R
Superposition: Each source considered alone, others deactivated
Deactivate voltage sources → short circuit
Deactivate current sources → open circuit
Superposition does NOT apply to power

🔷 SET 6: ADVANCED NUMERICALS & EQUIVALENT RESISTANCE 🔷

🧮 37. EQUIVALENT RESISTANCE TRICKS

For Parallel Resistors:

Equivalent resistance is always less than the smallest resistor in parallel
For two resistors: R_eq = (R₁ × R₂) / (R₁ + R₂)
For equal resistors (n, each = R): R_eq = R / n

For Series Resistors:

Equivalent resistance is always greater than the largest resistor in series
For equal resistors (n, each = R): R_eq = n × R

💡 38. MAXIMUM POWER TRANSFER THEOREM

Maximum Power Transfer Theorem states that maximum power is delivered to the load when the load resistance equals the source resistance (R_L = R_TH).

P_max = V_TH² / (4 × R_TH)

Used in communication circuits and matching networks
Efficiency at maximum power transfer = 50%

🔍 Example: If V_TH = 10V, R_TH = 5Ω, then R_L = 5Ω for max power.
P_max = 10² / (4×5) = 100/20 = 5W

⚠️ 39. FUSE RATING & OVERCURRENT PROTECTION

A fuse is a protective device that melts and breaks the circuit when current exceeds its rated value.

Fuse rating = Maximum current it can carry without melting
If current exceeds rating → Fuse melts (blows) → Circuit opens
Protects equipment from overcurrent and short circuits

📌 Key Point: Fuse rating means the fuse will melt if current exceeds that value for a specified time.

🔍 40. OPEN vs SHORT CIRCUIT

Condition	Resistance	Current	Voltage	Ohmmeter Reading
Open Circuit	Infinite (∞)	Zero	Full source voltage	Infinite (∞)
Short Circuit	Zero (0)	Very high	Zero	Zero (0)

📋 41. QUICK REFERENCE: SERIES vs PARALLEL

Parameter	Series Circuit	Parallel Circuit
Current	Same in all components	Divided among branches
Voltage	Divided among components	Same across all branches
Resistance	R_T = R₁ + R₂ + ...	1/R_T = 1/R₁ + 1/R₂ + ...
Conductance	1/G_T = 1/G₁ + 1/G₂ + ...	G_T = G₁ + G₂ + ...
Power	P_T = P₁ + P₂ + ...	P_T = P₁ + P₂ + ...
Voltage Divider	V_x = V_T × (R_x/R_T)	Not applicable
Current Divider	Not applicable	I_x = I_T × (R_T/R_x)

📝 42. KEY FORMULAS SUMMARY

Ohm's Law: V = I × R
Power: P = V × I = I²R = V²/R
Energy: E = P × t
Series R: R_T = R₁ + R₂ + ...
Parallel R: 1/R_T = 1/R₁ + 1/R₂ + ...
Two Resistors Parallel: R_T = (R₁ × R₂)/(R₁ + R₂)
Voltage Divider: V_x = V_T × (R_x/R_T)
Current Divider: I_x = I_T × (R_T/R_x)

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