---
title: "Parallel transformers and per-unit impedance load sharing"
subject: "Electrical Machines I"
chapter: 2
key: "semester_4/electrical_machines_i:parallel-operation-per-unit-impedance"
source: https://www.bctengineer.com/q/semester_4/electrical_machines_i/parallel-operation-per-unit-impedance
site: BCT Engineering
---

# Parallel transformers and per-unit impedance load sharing

Electrical Machines I · Chapter 2: Transformer

- Asked in: 2075 Bhadra, 2+6 marks
- Video: [Load Sharing - Parallel Transformers](https://www.youtube.com/watch?v=evPttWAzB7w) — Moving Electrons
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/electrical_machines_i/parallel-operation-per-unit-impedance
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines_i/chapter-2

## Question

Mention the conditions for parallel operation of transformer. "In parallel operation of transformer if we want to share load according to their capacity we have to match the per unit impedance of transformer according to their base." Justify this statement with mathematical expression.

## Short answer

Parallel operation requires same voltage ratio, polarity, per-unit impedance, and X/R ratio. Equal per-unit impedance on each transformer's own kVA base makes load share proportional to rating; otherwise the unit with smaller per-unit impedance overloads.

## Answer

#### Conditions for parallel operation
1. **Same voltage ratio** (same primary and secondary voltage ratings), otherwise a circulating current flows even at no load.
2. **Same polarity**, otherwise a dead short circuit occurs through the secondaries.
3. **Same per-unit (percentage) impedance**, so that load is shared in proportion to kVA ratings.
4. **Same X/R ratio**, so that both operate at the same power factor as the load.
5. For three-phase units: **same phase sequence** and **same phase displacement** (same vector group).

Conditions 1, 2 and 5 are essential; 3 and 4 are desirable for proper load sharing.

#### Justification: load sharing depends on per-unit impedance
Consider transformers A and B with equal voltage ratios in parallel, with equivalent impedances $Z_A$ and $Z_B$ (referred to secondary), supplying load current $I$ at voltage $V$.

```text
        I_A    Z_A
   +---->----[====]---+
   |                  |      I
 E +                  +---->---- load (V)
   |    I_B    Z_B    |
   +---->----[====]---+
```

Both have the same EMF $E$ and same terminal voltage $V$, so the impedance drops are equal:
$$
I_A Z_A = I_B Z_B \quad\Rightarrow\quad \frac{I_A}{I_B} = \frac{Z_B}{Z_A}
$$
With $I = I_A + I_B$:
$$
I_A = I\,\frac{Z_B}{Z_A + Z_B}, \qquad I_B = I\,\frac{Z_A}{Z_A + Z_B}
$$
Multiplying by $V$ gives kVA shares:
$$
S_A = S\,\frac{Z_B}{Z_A + Z_B}, \qquad S_B = S\,\frac{Z_A}{Z_A + Z_B}
$$
So the current shared by each is **inversely proportional to its ohmic impedance**.

Now write each impedance in per unit on its **own** rating. With rated currents $I_{A,r}$, $I_{B,r}$:
$$
z_A = \frac{I_{A,r}Z_A}{V}, \qquad z_B = \frac{I_{B,r}Z_B}{V}
$$
Substitute $Z_A = \frac{z_A V}{I_{A,r}}$ and $Z_B = \frac{z_B V}{I_{B,r}}$ into $I_A Z_A = I_B Z_B$:
$$
\begin{aligned}
I_A\frac{z_A V}{I_{A,r}} &= I_B\frac{z_B V}{I_{B,r}} \\
\frac{I_A/I_{A,r}}{I_B/I_{B,r}} &= \frac{z_B}{z_A}
\end{aligned}
$$
$I_A/I_{A,r}$ and $I_B/I_{B,r}$ are the fractional loadings of A and B. For **both to carry the same fraction of their rating** (load shared in proportion to kVA capacity):
$$
\frac{I_A}{I_{A,r}} = \frac{I_B}{I_{B,r}} \iff z_A = z_B
$$
Hence the per-unit impedances, each on its own base, must be equal. If they are not, the transformer with the **smaller** per-unit impedance takes more than its share and may be overloaded while the other is under-used. If also $X/R$ is equal, the impedance angles match and both currents are in phase with the load current, so no transformer works at a worse pf.

**Example:** a 500 kVA unit with $z = 0.05$ pu and a 250 kVA unit with $z = 0.05$ pu share 600 kVA as 400 kVA and 200 kVA (both 80% loaded).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which conditions are essential for parallel operation?
   A. Same per-unit impedance only
   B. Same kVA rating and same size
   C. Same weight and cooling
   D. Same voltage ratio, polarity, and (for three-phase) phase sequence and displacement
   Answer: D. Voltage ratio, polarity and phase relationship must match; impedance matching is desirable for proper sharing. (Key point: Essential conditions)

2. With equal voltage ratios, how do currents divide between A and B?
   A. I_A/I_B = Z_A/Z_B
   B. In proportion to resistance only
   C. Equally
   D. I_A/I_B = Z_B/Z_A
   Answer: D. Equal impedance drops give I_A Z_A = I_B Z_B. (Key point: Current sharing)

3. What must hold for load shared in proportion to kVA ratings?
   A. Ohmic impedances equal
   B. Per-unit impedances on own bases equal
   C. Iron losses equal
   D. Rated currents equal
   Answer: B. Fractional loadings are equal iff z_A = z_B. (Key point: Per-unit condition)

4. If the per-unit impedances differ, what happens?
   A. The one with larger per-unit impedance takes more
   B. Both are equally loaded
   C. The one with smaller per-unit impedance is overloaded relatively
   D. Circulating current at no load
   Answer: C. Smaller per-unit impedance takes a greater share of its rating. (Key point: Unequal per-unit impedance)

## Related in Electrical Machines I

- [Load sharing of 50 kVA and 25 kVA transformers in parallel](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Aparallel-50kva-25kva-60kva-load) — Electrical Machines I, chapter 2
- [Need and conditions for parallel operation of transformers](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Atransformer-parallel-operation-conditions) — Electrical Machines I, chapter 2
- [Conditions for parallel operation of transformers](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Atransformer-parallel-conditions-reasons) — Electrical Machines I, chapter 2
- [Transformer no-load operation and ideal transformer properties](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Atransformer-no-load-ideal-properties) — Electrical Machines I, chapter 2
- [Transformer no-load and load operation, constant core flux](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Aideal-transformer-noload-load-constant-flux) — Electrical Machines I, chapter 2, asked 5×
- [Real transformer no-load operation and amp-turn balance](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Areal-transformer-noload-amp-turn-balance) — Electrical Machines I, chapter 2

## Related in other subjects

- [Load sharing of two transformers in parallel](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aparallel-transformers-load-sharing) — Electrical Machines, chapter 2
- [Conditions for parallel operation of transformers](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aparallel-operation-conditions) — Electrical Machines, chapter 2
- [Per-unit transformer impedance same on both sides](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Apu-transformer-impedance-same-sides) — Power System Analysis I, chapter 3, asked 4×
- [Load sharing between two stations in parallel](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Aload-sharing-two-stations) — Electric Machines II, chapter 1