Formula & Calculator
Star-Delta Line-Phase Relationship
Relates line and phase voltages/currents for star (wye) and delta three-phase connections.
Interpretation
Star‑delta relationships: In star (Y), V_L = √3·V_ph and I_L = I_ph. In delta (Δ), V_L = V_ph and I_L = √3·I_ph.
These relationships are used to convert between phase and line quantities.
Example: Star with V_ph=230V gives V_L = √3×230 ≈ 398V; Delta with I_ph=10A gives I_L = √3×10 ≈ 17.32A.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| V_L | Line-to-line RMS voltage | V |
| V_ph | Phase (winding) RMS voltage | V |
| I_L | Line RMS current | A |
| I_ph | Phase (winding) RMS current | A |
| √3 | Square root of 3 (1.732) — conversion factor | dimensionless |
What it means
In a three‑phase system, the relationships between line and phase quantities differ for star and delta connections. In star (Y): V_L = √3 · V_ph and I_L = I_ph. In delta (Δ): V_L = V_ph and I_L = √3 · I_ph. These relationships are used to convert between line and phase values when analysing circuits or connecting loads. For example, a motor with a star‑connected winding rated at 230V phase can be connected to a 400V line (since 230×√3 ≈ 400). In delta, the same motor would require 230V line voltage. Example: A star‑connected load with phase voltage 230V has line voltage 398V. A delta‑connected load with phase current 10A has line current 17.32A. Understanding these relationships is essential for proper motor connection and transformer selection.
Worked example
Star‑Delta Connection – Practical Example
Real‑World| Parameter | Value |
|---|---|
| Connection | Delta (Δ) |
| VL | 400 V |
| Iph | 10 A |
| Relations | VL = Vph, IL = √3·Iph |
Common mistakes
Watch for unit mismatches (W vs kW, single- vs three-phase) and remember to include power factor or efficiency where the formula requires it.Applications
Star‑delta relationships relate phase and line quantities in three‑phase systems. In star, V_L = √3·V_ph and I_L = I_ph; in delta, V_L = V_ph and I_L = √3·I_ph. These conversions are essential for analysing three‑phase circuits. Engineers use them to connect loads and generators properly, to calculate equivalent circuit parameters, and to design transformers.
- Three‑phase circuit analysis and design
- Transformer connection and conversion
- Motor and generator connection verification
- Load balancing and phase current calculations
- Educational understanding of three‑phase connections
Frequently Asked Questions
For a star connection: V_L = √3 V_ph and I_L = I_ph. For a delta connection: V_L = V_ph and I_L = √3 I_ph.
In star, the line voltage is the vector sum of two phase voltages, giving √3. In delta, the line current is the vector sum of two phase currents, giving √3.
For both star and delta, the total power is P = √3 V_L I_L cos θ. This is consistent if the relationships are used.
Common errors: 1) mixing up star and delta, 2) using the wrong factor (√3 vs 3), 3) applying the relationships to unbalanced systems, 4) forgetting that the relationships are for balanced systems.
Use the formulas: for star, V_ph = V_L/√3, I_ph = I_L; for delta, V_ph = V_L, I_ph = I_L/√3.
Designing three‑phase circuits, selecting transformers, and understanding motor connections.
For star, Z_ph = V_ph / I_ph. For delta, Z_ph = V_ph / I_ph. The line quantities differ.
Star‑delta starting reduces the starting current and torque, often used for large motors.