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Circuits, electronics and noise
Delta-to-star resistor conversion
Star arm Rₐ that replaces a delta network of three resistors R_AB, R_BC and R_CA, as seen from terminal A.
- Ref
- ELN-05
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- Ra [Ω]
A delta of three resistors can be replaced by a star (wye) with arms Rₐ = R_AB R_CA/ΣR, R_b = R_AB R_BC/ΣR and R_c = R_BC R_CA/ΣR, where ΣR is the sum of the three delta resistors. Each arm’s product is of the two delta resistors that meet at its terminal. The calculator gives the arm at terminal A; the other two are in the derived rows.
Star arm Rₐ that replaces a delta network of three resistors R_AB, R_BC and R_CA, as seen from terminal A. The other two arms, R_b and R_c, are in the derived rows and use the same rule.
Calculator
- Star arm at terminal B10 Ω
- Star arm at terminal C30 Ω
- Sum of the delta resistors180 Ω
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A delta of 30, 60 and 90 Ω is equivalent to a star of 15, 10 and 30 Ω.
- A delta of three equal resistors R becomes a star of three equal resistors R/3.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Ω) | Local step | Change in result |
|---|---|---|---|---|---|---|
| RAB Delta resistor between A and B | 1 to 100,000 Ω | 0.596 | 89.87 | 89.27 | +10% | +8.2 % |
| RCA Delta resistor between C and A | 1 to 100,000 Ω | 0.3297 | 29.97 | 29.64 | +10% | +4.76 % |
| RBC Delta resistor between B and C | 1 to 100,000 Ω | 0.02697 | 22.31 | 22.29 | +10% | −3.23 % |
Sweep of Delta resistor between C and A
| Delta resistor between C and A (Ω) | Star arm at terminal A (Ω) |
|---|---|
| 1 | 0.3297 |
| 4.217 | 1.343 |
| 17.78 | 4.95 |
| 74.99 | 13.64 |
| 316.2 | 23.35 |
| 1,334 | 28.1 |
| 5,623 | 29.53 |
| 23,710 | 29.89 |
| 100,000 | 29.97 |
Sweep one variable
Variables and how each one contributes
Star arm at terminal A
What the equation gives you. Everything else feeds it.
Delta resistor between A and B
The delta resistor between terminals A and B, in Ω. Raises both arms at A and B. Raising it by 10 % (to 33) moves the result by +8.2 %, from 15 to 16.23 Ω.
Delta resistor between B and C
The delta resistor between terminals B and C, in Ω. Raises the arms at B and C and, through the sum, lowers the arm at A. Raising it by 10 % (to 66) moves the result by −3.23 %, from 15 to 14.52 Ω.
Delta resistor between C and A
The delta resistor between terminals C and A, in Ω. Raises both arms at C and A. Raising it by 10 % (to 99) moves the result by +4.76 %, from 15 to 15.71 Ω.
Worked example with the values above
- Sum of delta resistors: 30 + 60 + 90 = 180 Ω.
- Product at terminal A: R_AB R_CA = 30 × 90 = 2,700.
- Rₐ = 2,700 / 180 = 15 Ω; the others are 10 Ω and 30 Ω.
Assumptions and limits
- Ideal sources and linear resistors with positive values, all in base SI units.
- The circuit is exactly as described: no extra loads, and instruments that draw no current.
Common mistakes
- Using the wrong pair of delta resistors in the product: each arm uses the two resistors that meet at its terminal.
- Dividing by a single resistor instead of the sum of all three.
- Forgetting that the reverse (star to delta) conversion has a different form.
Statistics and uncertainty
Resistor values are only known to their tolerance, typically 1 % to 5 %, so a result built from them carries that scatter. Ratios such as dividers and bridges are less sensitive to a common drift than to differences between the resistors.
Where it is used
- Sensor interfaces, bias networks, bridge circuits and mixers.
- Reducing a network to simpler series and parallel parts.
Maths you need
- Ohm’s law: V = I R
- Kirchhoff’s current law: currents into a node add to zero
- Series and parallel resistors: R₁ + R₂ and R₁R₂/(R₁ + R₂)