3 Phase Resistance Calculation Formula Calculator
Calculate per-phase resistance for balanced three-phase circuits using line voltage, line current, and connection type. Ideal for quick engineering checks on resistive loads in star and delta systems.
Star: Rphase = VL / (√3 × IL)
Delta: Rphase = √3 × VL / IL
Also shown: Vphase, Iphase, and total three-phase power for a resistive load where power factor = 1.
Calculation Results
Enter your values and click Calculate Resistance to see per-phase resistance, phase voltage, phase current, and estimated three-phase power.
Expert Guide to the 3 Phase Resistance Calculation Formula
The 3 phase resistance calculation formula is one of the most practical relationships used in electrical design, troubleshooting, and training. Whether you are sizing a balanced heater bank, evaluating a motor test setup, checking a resistive load in an industrial panel, or simply trying to understand three-phase fundamentals, resistance calculation is a core skill. In a balanced three-phase system, each phase carries an equal share of the load, and that symmetry makes it possible to calculate resistance from a few easily measured quantities such as line voltage and line current.
At its simplest, resistance follows Ohm’s law: voltage divided by current. The challenge in three-phase work is that line quantities and phase quantities are not always the same. The exact formula depends on whether the load is connected in star (also called wye) or delta. That is why many calculation mistakes happen: the user measures line voltage and line current correctly, but applies single-phase logic without converting to phase values.
For a balanced and purely resistive star load, phase voltage is the line voltage divided by the square root of three, while phase current equals line current. That produces the formula:
Rphase = VL / (√3 × IL)
For a balanced and purely resistive delta load, phase voltage equals line voltage, while phase current is line current divided by the square root of three. That produces the formula:
Rphase = √3 × VL / IL
These expressions are valid when the load is balanced and effectively resistive. In many real systems, there is also reactance from inductance or capacitance, and that means impedance rather than pure resistance becomes the more accurate quantity. Still, resistance-based calculations remain extremely useful for resistive heating elements, lamp banks, test loads, and introductory power system analysis.
Why line and phase values matter
Three-phase circuits have two different “views” of voltage and current:
- Line voltage is measured between any two line conductors.
- Phase voltage is the voltage across one phase element.
- Line current is the current flowing in each supply conductor.
- Phase current is the current through one phase element.
In star systems, line current and phase current are equal, but line voltage is higher than phase voltage by a factor of √3. In delta systems, line voltage and phase voltage are equal, but line current is higher than phase current by a factor of √3. Those relationships are the bridge between measurements taken at the terminals and the actual resistance of each load branch.
| Connection Type | Voltage Relationship | Current Relationship | Resistance Formula from Line Values |
|---|---|---|---|
| Star (Wye) | Vphase = Vline / 1.732 | Iphase = Iline | Rphase = Vline / (1.732 × Iline) |
| Delta | Vphase = Vline | Iphase = Iline / 1.732 | Rphase = 1.732 × Vline / Iline |
Worked example using a common industrial supply
Suppose you have a balanced three-phase resistive load connected to a 415 V system drawing 20 A line current.
- Identify the connection type, star or delta.
- Convert line values to phase values if needed.
- Apply Ohm’s law at the phase level.
- Check total power using the three-phase power equation.
If the load is star connected:
- Vphase = 415 / 1.732 = 239.6 V
- Iphase = 20 A
- Rphase = 239.6 / 20 = 11.98 Ω
If the load is delta connected:
- Vphase = 415 V
- Iphase = 20 / 1.732 = 11.55 A
- Rphase = 415 / 11.55 = 35.94 Ω
Notice how the same line voltage and line current produce very different phase resistance values depending on the connection. That is not an error. It reflects the different current and voltage distribution patterns in star and delta networks.
Three-phase power and resistance
Resistance is often calculated together with power because many engineers want to confirm that the measured current is consistent with the expected load. For a balanced three-phase load, real power is:
P = √3 × VL × IL × power factor
For a purely resistive load, the power factor is approximately 1.00, so the equation becomes:
P = 1.732 × VL × IL
Using the 415 V and 20 A example, total power is:
P = 1.732 × 415 × 20 = 14,376 W, or about 14.38 kW.
This is especially useful when checking heater arrays, immersion elements, or test resistors. If the calculated resistance implies a certain current draw, and the power meter shows a very different result, then one of the following may be true:
- The load is not balanced.
- The load is not purely resistive.
- Measurements were taken under transient conditions.
- Supply voltage is sagging under load.
- One or more branches are damaged or drifting in value.
Typical real-world supply values and what they imply
Engineers often encounter standard nominal system voltages that vary by region and application. In the United States, 208 V line-to-line systems are common in commercial buildings, while 480 V is common in industrial facilities. In many other countries, 400 V or 415 V three-phase systems are standard low-voltage industrial supplies. The formulas do not change, but the resulting resistance values obviously do.
| Nominal 3-Phase System | Common Use | Star Resistance at 10 A | Delta Resistance at 10 A |
|---|---|---|---|
| 208 V | US commercial buildings | 12.01 Ω per phase | 36.03 Ω per phase |
| 400 V | IEC low-voltage networks | 23.09 Ω per phase | 69.28 Ω per phase |
| 415 V | Industrial distribution in many regions | 23.96 Ω per phase | 71.88 Ω per phase |
| 480 V | US industrial facilities | 27.71 Ω per phase | 83.14 Ω per phase |
These values assume a balanced resistive load drawing 10 A line current. The table is useful because it shows how quickly resistance changes with system voltage and with connection type. If you mistakenly assume star when the load is actually delta, your resistance estimate can be off by a factor of about three.
How to calculate 3 phase resistance step by step
- Measure or obtain the line-to-line voltage.
- Measure the line current in one conductor.
- Verify the load is balanced or close to balanced.
- Determine whether the load is star or delta connected.
- Convert line values to phase values using the proper relationship.
- Apply Ohm’s law, R = V / I, using phase voltage and phase current.
- Optionally confirm by calculating total power.
Common mistakes to avoid
- Using line voltage directly for a star phase resistor: star phase voltage is lower than line voltage.
- Using line current directly for a delta phase resistor: delta phase current is lower than line current.
- Ignoring power factor: a motor is not a pure resistor, so resistance formulas alone can mislead.
- Mixing units: kV, V, A, mA, and kA must be converted consistently.
- Assuming balance when there is unbalance: if phase currents differ significantly, use per-phase analysis instead.
When resistance is not enough
In AC systems, many loads have both resistive and reactive components. That means the total opposition to current is impedance, represented by Z, not only resistance R. For example, induction motors, long cable runs, transformers, and power factor correction equipment all add reactance. If your measured voltage, current, and power do not align with a pure resistance model, you may need impedance calculations, phasor analysis, or power quality measurements.
Still, resistance calculations remain highly valuable in specific scenarios:
- Three-phase heater banks
- Load banks for testing generators and UPS systems
- Resistive oven and furnace elements
- Short practical exercises in electrical training
- Quick field checks for balanced resistive branches
Authoritative references for further study
If you want deeper technical context, review these authoritative resources:
- NIST.gov: Unit conversions and SI guidance
- Energy.gov: Industrial energy and electrical system resources
- Michigan State University hosted educational references and related academic power system materials
Final takeaway
The 3 phase resistance calculation formula is straightforward once you separate line quantities from phase quantities. For star, divide line voltage by √3 before using Ohm’s law. For delta, divide line current by √3 before using Ohm’s law. In compact form, that gives you:
- Star: Rphase = VL / (1.732 × IL)
- Delta: Rphase = 1.732 × VL / IL
When used correctly, these formulas provide a fast and reliable method for estimating per-phase resistance in balanced three-phase resistive circuits. Use them with accurate measurements, clear knowledge of the connection type, and a healthy awareness that AC systems with reactance require impedance-based analysis instead.