Quick Answer
What are the key three-phase formulas?
| Calculate | Formula |
|---|---|
| Power (kW) | P = √3 × V_L × I_L × PF ÷ 1000 |
| Current (A) | I = P × 1000 ÷ (√3 × V × PF) |
| Apparent Power (kVA) | S = √3 × V_L × I_L ÷ 1000 |
Where √3 = 1.732
→ Use the 3-Phase Calculator for instant calculations.
After the Formula Result
Use the formula result as the first calculation, then close the workflow with the right calculator handoff:
| If the result is... | Next check |
|---|---|
| A motor running-current estimate | Compare against the Motor Current Calculator and nameplate data |
| A service or feeder load in kVA | Review current and transformer relationships with the Transformer Calculator |
| A long circuit run | Check the same current in the Voltage Drop Calculator |
| A low power factor load | Continue with the Power Factor Calculator before sizing correction equipment |
For code-sensitive conductor, breaker, or equipment decisions, verify the adopted NEC edition, equipment listings, manufacturer instructions, and AHJ requirements before treating the formula result as final.
For a chart record, use the Single-Phase vs Three-Phase Chart when the next question is which voltage and phase model applies. Use the kVA to Amps Chart when the formula result needs line-current documentation before transformer, feeder, load-bank, or equipment review.
Three-Phase Power Formulas
Real Power (Watts/kW)
| Method | Formula | When to Use |
|---|---|---|
| Line Values | P = √3 × V_L × I_L × PF | Most common (line-to-line voltage) |
| Phase Values | P = 3 × V_ph × I_ph × PF | When phase values known |
Example: 480V, 100A, PF = 0.85
P = √3 × 480V × 100A × 0.85
P = 1.732 × 480 × 100 × 0.85
P = 70,666W = 70.7 kW
Apparent Power (VA/kVA)
| Method | Formula |
|---|---|
| Line Values | S = √3 × V_L × I_L |
| Phase Values | S = 3 × V_ph × I_ph |
Reactive Power (VAR/kVAR)
Q = √3 × V_L × I_L × sin(θ)
Q = √(S² - P²)
Line vs Phase Values
Understanding the Difference
| Term | Description | Symbol |
|---|---|---|
| Line Voltage | Voltage between any two lines (L-L) | V_L or V_LL |
| Phase Voltage | Voltage from line to neutral (L-N) | V_ph or V_LN |
| Line Current | Current in the supply line | I_L |
| Phase Current | Current through each winding | I_ph |
Star (Wye) Connection Formulas
V_L = √3 × V_ph
V_ph = V_L / √3
I_L = I_ph
Common Wye Voltages:
| Line Voltage | Phase Voltage |
|---|---|
| 208V | 120V |
| 400V | 230V |
| 480V | 277V |
| 600V | 347V |
Delta Connection Formulas
V_L = V_ph
I_L = √3 × I_ph
I_ph = I_L / √3
Star vs Delta Comparison
Voltage and Current Relationships
| Parameter | Star (Y) | Delta (Δ) |
|---|---|---|
| Line Voltage | V_L = √3 × V_ph | V_L = V_ph |
| Phase Voltage | V_ph = V_L ÷ √3 | V_ph = V_L |
| Line Current | I_L = I_ph | I_L = √3 × I_ph |
| Phase Current | I_ph = I_L | I_ph = I_L ÷ √3 |
| Power | P = √3 × V_L × I_L × PF | P = √3 × V_L × I_L × PF |
| Neutral | Available | Not available |
When to Use Each
| Configuration | Best For |
|---|---|
| Star (Wye) | Distribution systems, single-phase loads from 3-phase, motors that need reduced starting current |
| Delta | Motor running, higher starting torque, transmission (no neutral needed) |
Three-Phase Current Calculations
Current from Power (Most Common)
I = P / (√3 × V × PF)
Example: 50 kW motor, 480V, PF = 0.87
I = 50,000W / (1.732 × 480V × 0.87)
I = 50,000 / 723.5
I = 69.1A per phase
Current from kVA
I = kVA × 1000 / (√3 × V)
Example: 75 kVA transformer, 480V
I = 75,000 / (1.732 × 480)
I = 75,000 / 831.4
I = 90.2A
Current from HP (Motors)
I = HP × 746 / (√3 × V × Eff × PF)
Example: 50 HP motor, 480V, Eff = 0.92, PF = 0.87
I = 50 × 746 / (1.732 × 480 × 0.92 × 0.87)
I = 37,300 / 665.6
I = 56.0A
Three-Phase Voltage Calculations
Standard Three-Phase Voltages
| Nominal | Line-to-Line | Line-to-Neutral | Use |
|---|---|---|---|
| 208/120V | 208V | 120V | Commercial |
| 240V | 240V | 139V | Commercial Delta |
| 480/277V | 480V | 277V | Industrial |
| 600/347V | 600V | 347V | Industrial (Canada) |
Voltage Drop in Three-Phase
V_drop = √3 × I × (R × cos(θ) + X × sin(θ)) × L
For approximate calculation (copper wire):
V_drop (%) = (√3 × I × L × ρ) / (A × V) × 100
Where:
- I = Line current (A)
- L = One-way length (feet)
- ρ = Resistivity constant
- A = Wire cross-section area
→ Use Voltage Drop Calculator for accurate sizing.
Motor Full Load Current (FLC) Tables
Three-Phase Motor FLC at 460V
Based on NEC Table 430.250:
| HP | FLC (A) | HP | FLC (A) |
|---|---|---|---|
| 1 | 2.1 | 30 | 40 |
| 2 | 3.0 | 40 | 52 |
| 3 | 4.8 | 50 | 65 |
| 5 | 7.6 | 60 | 77 |
| 7.5 | 11 | 75 | 96 |
| 10 | 14 | 100 | 124 |
| 15 | 21 | 125 | 156 |
| 20 | 27 | 150 | 180 |
| 25 | 34 | 200 | 240 |
Motor FLC at Different Voltages
| HP | 208V | 230V | 460V | 575V |
|---|---|---|---|---|
| 10 | 30.8 | 28 | 14 | 11 |
| 25 | 74.8 | 68 | 34 | 27 |
| 50 | 143 | 130 | 65 | 52 |
| 100 | 273 | 248 | 124 | 99 |
→ Use Full Load Current Calculator for NEC table FLC values, or Motor Current Calculator for formula-current comparison.
Transformer Sizing
Three-Phase Transformer kVA
kVA = √3 × V × I / 1000
Or from load:
kVA = kW / PF
Standard 3-Phase Transformer Sizes
15, 30, 45, 75, 112.5, 150, 225, 300, 500, 750, 1000, 1500, 2000, 2500 kVA
Sizing Example:
Load: 200 kW, PF = 0.85
kVA = 200 / 0.85 = 235.3 kVA
Select next size: 300 kVA transformer
Worked Examples
Example 1: Calculate 3-Phase Power
Given: 480V system, 50A per line, PF = 0.9
Solution:
P = √3 × V × I × PF
P = 1.732 × 480 × 50 × 0.9
P = 37,413W = 37.4 kW
Example 2: Current for Known Load
Given: 100 kW load, 208V, 3-phase, PF = 0.85
Solution:
I = P / (√3 × V × PF)
I = 100,000 / (1.732 × 208 × 0.85)
I = 100,000 / 306.2
I = 326.6A
Example 3: Star to Delta Conversion
Given: Star-connected motor, V_ph = 230V, I_ph = 10A
For Star:
V_L = √3 × V_ph = 1.732 × 230 = 398.4V
I_L = I_ph = 10A
If reconnected in Delta at same line voltage:
V_ph = V_L = 398.4V
I_ph = V_ph / Z = 398.4 / 23 = 17.3A (assuming same impedance)
I_L = √3 × I_ph = 1.732 × 17.3 = 30A
Power increases 3× in Delta!
Example 4: Generator Sizing
Given: Facility load 150 kW, PF = 0.8, want 25% reserve
Solution:
Required kVA = kW / PF = 150 / 0.8 = 187.5 kVA
With 25% reserve: 187.5 × 1.25 = 234.4 kVA
Select: 250 kVA generator
Unbalanced Load Calculations
Single-Phase Load on Three-Phase
For single-phase load connected line-to-neutral:
I_line = kW × 1000 / (V_ph × PF)
Balancing Three-Phase Loads
To balance loads across phases:
- Calculate total load per phase
- Move loads to equalize current
- Target within 10% imbalance
Common Mistakes to Avoid
| Mistake | Why It's Wrong | Correct Approach |
|---|---|---|
| Forgetting √3 factor | Power calculation off by 1.732 | Always use √3 for 3-phase |
| Using wrong voltage | Line vs phase confusion | Clarify V_LL or V_LN before calculating |
| Ignoring power factor | Undersized conductors | Include PF for AC loads |
| Same current Star/Delta | Currents differ by √3 | Apply correct conversion |
Related Calculators
| Calculator | Use When... |
|---|---|
| 3-Phase Power Calculator | Power and current calculations |
| Full Load Current Calculator | Motor FLC lookup |
| Motor Current Calculator | Formula-current comparison |
| Transformer Sizing | Transformer kVA selection |
| Voltage Drop Calculator | Wire sizing for distance |
Summary
Essential Three-Phase Formulas:
| Calculate | Formula |
|---|---|
| Power | P = √3 × V_L × I_L × PF |
| Current | I = P / (√3 × V × PF) |
| kVA | S = √3 × V_L × I_L |
Star vs Delta:
- Star: V_L = √3 × V_ph, I_L = I_ph
- Delta: V_L = V_ph, I_L = √3 × I_ph
Remember: √3 = 1.732
FAQ
What does √3 represent in three-phase?
√3 (1.732) is the mathematical factor that relates line and phase values due to the 120° phase shift between phases. It appears in all balanced three-phase power calculations.
How do I convert single-phase to three-phase current?
Three-phase current is lower than single-phase for the same power. Divide single-phase current by √3 (1.732) for equivalent three-phase current at the same voltage.
What is the difference between Star and Delta?
Star (Wye) has a neutral connection and different line/phase voltages (V_L = √3 × V_ph). Delta has no neutral, and line voltage equals phase voltage, but line current is √3 times phase current.
Why is 480V common for industrial?
480V three-phase offers efficient power transmission with lower currents than 208V or 240V systems. Lower current means smaller conductors and lower losses for heavy industrial loads.
How do I calculate amps from HP for a 3-phase motor?
Use I = (HP × 746) / (√3 × V × Eff × PF) or refer to NEC Table 430.250 for standard FLC values. Always use nameplate data when available.