Quick Answer
What is the electrical power formula?
P = V × I (Power = Voltage × Current)
| System | Formula | Example |
|---|---|---|
| DC | P = V × I | 12V × 5A = 60W |
| Single-Phase AC | P = V × I × PF | 120V × 10A × 0.9 = 1080W |
| Three-Phase AC | P = √3 × V × I × PF | 1.732 × 480V × 20A × 0.85 = 14,117W |
→ Use the Power Calculator for instant calculations.
After the Power Result
Use the formula answer to decide which workflow owns the next step:
| Result you calculated | Continue with |
|---|---|
| Watts, amps, or voltage for a basic circuit | Power Calculator for the exact input set |
| Motor kW, HP, or shaft output | Motor Power Calculator with efficiency and power factor |
| kWh or operating cost | Electricity Cost Calculator with runtime and utility rate assumptions |
| Low power factor or kVA-to-kW gap | Power Factor Calculator before sizing correction equipment |
| Long-run conductor loss | Voltage Drop Calculator before selecting wire size |
This keeps the formula page as the reference layer and sends project-specific sizing, cost, and conductor checks to the calculator built for that workflow.
For a chart record, use the Power Factor Triangle Chart when the result needs kW, kVAR, kVA, phase angle, and correction notes. Use the kVA to Amps Chart when an apparent-power result needs single-phase or three-phase line-current review before transformer, feeder, or equipment checks.
DC Power Formulas
Basic DC Power Equations
| Find | Formula | Units |
|---|---|---|
| Power (P) | P = V × I | Watts (W) |
| Power (P) | P = I² × R | Watts (W) |
| Power (P) | P = V² / R | Watts (W) |
| Voltage (V) | V = P / I | Volts (V) |
| Current (I) | I = P / V | Amperes (A) |
DC Power Wheel
All 12 formulas for DC calculations:
Power (P):
P = V × I
P = I² × R
P = V² / R
Voltage (V):
V = P / I
V = I × R
V = √(P × R)
Current (I):
I = P / V
I = V / R
I = √(P / R)
Resistance (R):
R = V / I
R = P / I²
R = V² / P
Single-Phase AC Power Formulas
Three Types of AC Power
| Type | Symbol | Unit | Formula | Description |
|---|---|---|---|---|
| Real Power | P | Watts (W) | P = V × I × PF | Actual work done |
| Reactive Power | Q | VAR | Q = V × I × sin(θ) | Energy stored in L/C |
| Apparent Power | S | VA | S = V × I | Total power in circuit |
Power Triangle Relationship
S² = P² + Q²
S (VA)
/|
/ |
/ | Q (VAR)
/θ |
------
P (W)
Where:
- P = Real Power (Watts)
- Q = Reactive Power (VAR)
- S = Apparent Power (VA)
- θ = Phase angle
- PF = cos(θ) = Power Factor
Single-Phase Formulas
| Calculate | Formula | Example |
|---|---|---|
| Real Power | P = V × I × PF | 120V × 15A × 0.9 = 1620W |
| Apparent Power | S = V × I | 120V × 15A = 1800VA |
| Reactive Power | Q = √(S² - P²) | √(1800² - 1620²) = 784.5VAR |
| Power Factor | PF = P / S | 1620W / 1800VA = 0.9 |
| Current | I = P / (V × PF) | 1620W / (120V × 0.9) = 15A |
Three-Phase Power Formulas
Balanced Three-Phase Systems
| Configuration | Real Power (P) | Apparent Power (S) |
|---|---|---|
| Line Values | P = √3 × V_L × I_L × PF | S = √3 × V_L × I_L |
| Phase Values | P = 3 × V_ph × I_ph × PF | S = 3 × V_ph × I_ph |
Where:
- √3 = 1.732 (square root of 3)
- V_L = Line voltage (line-to-line)
- I_L = Line current
- V_ph = Phase voltage (line-to-neutral)
- I_ph = Phase current
- PF = Power factor
Star (Wye) vs Delta Connection
| Relationship | Star (Y) | Delta (Δ) |
|---|---|---|
| V_line to V_phase | V_L = √3 × V_ph | V_L = V_ph |
| I_line to I_phase | I_L = I_ph | I_L = √3 × I_ph |
| Power | P = √3 × V_L × I_L × PF | P = √3 × V_L × I_L × PF |
Three-Phase Power Examples
Example 1: 480V Industrial Motor
Given: 480V, 3-phase, 50A, PF = 0.85
P = √3 × V × I × PF
P = 1.732 × 480V × 50A × 0.85
P = 35,353W = 35.4 kW
Example 2: 208V Commercial System
Given: 208V, 3-phase, 100A, PF = 0.9
P = √3 × V × I × PF
P = 1.732 × 208V × 100A × 0.9
P = 32,423W = 32.4 kW
Power Factor
Power Factor Formulas
| Calculate | Formula |
|---|---|
| Power Factor | PF = P / S = W / VA |
| Power Factor | PF = cos(θ) |
| True Power | P = S × PF |
| Apparent Power | S = P / PF |
| Reactive Power | Q = S × sin(θ) = S × √(1 - PF²) |
Power Factor Correction
To find capacitor kVAR needed:
kVAR = kW × (tan(θ₁) - tan(θ₂))
Where:
- θ₁ = arccos(current PF)
- θ₂ = arccos(target PF)
Example: Correct 100kW load from PF 0.7 to PF 0.95
θ₁ = arccos(0.7) = 45.57°
θ₂ = arccos(0.95) = 18.19°
kVAR = 100 × (tan(45.57°) - tan(18.19°))
kVAR = 100 × (1.02 - 0.33)
kVAR = 69 kVAR capacitor needed
→ Use Power Factor Calculator for correction sizing.
Power Conversion Formulas
Watts, kW, HP Conversions
| From | To | Formula |
|---|---|---|
| Watts | Kilowatts | kW = W / 1000 |
| Kilowatts | Watts | W = kW × 1000 |
| HP | Watts | W = HP × 746 |
| Watts | HP | HP = W / 746 |
| HP | kW | kW = HP × 0.746 |
| kW | HP | HP = kW / 0.746 = kW × 1.341 |
kW / kVA Conversions
| From | To | Formula |
|---|---|---|
| kW | kVA | kVA = kW / PF |
| kVA | kW | kW = kVA × PF |
Quick Reference: HP to kW
| HP | kW | HP | kW |
|---|---|---|---|
| 1 | 0.746 | 15 | 11.2 |
| 2 | 1.49 | 20 | 14.9 |
| 3 | 2.24 | 25 | 18.6 |
| 5 | 3.73 | 30 | 22.4 |
| 7.5 | 5.59 | 40 | 29.8 |
| 10 | 7.46 | 50 | 37.3 |
Common Voltage Systems
US Standard System Voltages
| System | Voltage | Typical Use |
|---|---|---|
| Single-Phase | 120V | Residential, lighting |
| Single-Phase | 240V | Residential appliances |
| Three-Phase | 208V | Commercial (Y from 120V) |
| Three-Phase | 240V | Commercial (Delta) |
| Three-Phase | 480V | Industrial, large motors |
| Three-Phase | 600V | Industrial (Canada/Heavy Industry) |
Voltage Relationships
| 3-Phase System | Line Voltage | Phase Voltage (Y) |
|---|---|---|
| 208V system | 208V | 120V |
| 240V system | 240V | 139V |
| 480V system | 480V | 277V |
| 600V system | 600V | 347V |
Worked Examples
Example 1: Calculate Heater Power
Given: 240V heater with 20Ω resistance
Solution:
P = V² / R
P = (240V)² / 20Ω
P = 57,600 / 20
P = 2,880W = 2.88 kW
Example 2: Industrial Motor Power
Given: 3-phase 480V motor, 30A per phase, PF = 0.87
Solution:
P = √3 × V × I × PF
P = 1.732 × 480V × 30A × 0.87
P = 21,706W = 21.7 kW
HP = 21.7 kW / 0.746 = 29.1 HP ≈ 30 HP motor
Example 3: Current from Power
Given: 5 kW load, 240V single-phase, PF = 0.95
Solution:
I = P / (V × PF)
I = 5000W / (240V × 0.95)
I = 5000 / 228
I = 21.9A
Example 4: kVA Sizing for Transformer
Given: Load 150 kW, PF = 0.8
Solution:
kVA = kW / PF
kVA = 150 kW / 0.8
kVA = 187.5 kVA
Select next standard size: 200 kVA transformer or 225 kVA transformer
Power Loss in Conductors
Line Loss Formula
P_loss = I² × R
Where:
- P_loss = Power lost in wire (Watts)
- I = Current (Amperes)
- R = Wire resistance (Ohms - check AWG charts)
Percentage Loss
% Loss = (P_loss / P_total) × 100
Example: 50A circuit, 0.5Ω total round-trip wire resistance
P_loss = (50A)² × 0.5Ω = 1,250W
If load is 10kW:
% Loss = (1250 / 10000) × 100 = 12.5% (This is too high! NEC requires <3% to 5% drop max)
→ Use Voltage Drop Calculator to properly size conductors.
Common Mistakes to Avoid
| Mistake | Why It's Wrong | Correct Approach |
|---|---|---|
| Ignoring power factor | Undersized equipment | Always include PF for AC circuits containing motors/transformers |
| Using P=VI for 3-phase | Missing √3 factor | Use P = √3 × V × I × PF for three-phase power |
| Confusing kW and kVA | These differ by PF | kW = kVA × PF |
| Using 1 HP = 1 kW | Inaccurate (1 HP = 746W) | 1 HP = 0.746 kW exactly |
Related Calculators
| Calculator | Use When... |
|---|---|
| Power Calculator | Basic V, I, P calculations |
| 3-Phase Power Calculator | Three-phase systems |
| Power Factor Calculator | PF correction sizing |
| Motor Power Calculator | Motor load calculations |
Summary
Key Formulas:
| System | Power Formula |
|---|---|
| DC | P = V × I = I²R = V²/R |
| Single-Phase AC | P = V × I × PF |
| Three-Phase AC | P = √3 × V × I × PF |
Key Conversions:
- 1 HP = 746W = 0.746 kW
- kVA = kW / PF
- √3 = 1.732
FAQ
What is the difference between kW and kVA?
kW (kilowatts) is real power that does actual work. kVA (kilovolt-amperes) is apparent power, the total power in the circuit. They differ by the power factor: kW = kVA × PF. Equipment like transformers and utility generators are rated in kVA.
Why use √3 in three-phase calculations?
The √3 (1.732) factor mathematically accounts for the 120° phase shift between the three phases in a balanced three-phase system. It correctly converts between line and phase values ensuring an accurate power calculation.
How do I calculate motor running power?
Use P = √3 × V × I × PF for three-phase motors, where I is the actual running current under load. Or use the nameplate HP rating × 0.746 for rated maximum output power in kW. Remember that actual electrical power input is higher due to motor efficiency losses.
What is a good power factor?
Residential loads typically run at PF 0.85-0.95. Industrial targets are usually strictly maintained above 0.9 or 0.95 to avoid costly utility penalties. Unity (1.0) is mathematically ideal but practically rare in factories. Anything below 0.85 usually dictates a firm need for correction capacitors.
How do I convert between HP and kW?
Multiply HP by 0.746 to get kW. Divide kW by 0.746 (or multiply by 1.341) to calculate HP. These conversions are for mechanical power equivalents; electrical input may be slightly higher due to the motor's internal efficiency.