---
title: "Ideal transformer no-load, load, ratios"
subject: "Electrical Machine"
chapter: 2
key: "semester_4/electrical_machine:ideal-transformer-ratios-derivation"
source: https://www.bctengineer.com/q/semester_4/electrical_machine/ideal-transformer-ratios-derivation
site: BCT Engineering
---

# Ideal transformer no-load, load, ratios

Electrical Machine · Chapter 2: Transformer

- Asked in: 2071 Magh, 4+4 marks; 2070 Magh, 8 marks (asked 2 times)
- Video: [Ideal Transformer On No Load | Basic Concepts | Electrical Machines](https://www.youtube.com/watch?v=-31ANzeRim0) — ENGINEERING TUTORIAL
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/electrical_machine/ideal-transformer-ratios-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machine/chapter-2

## Question

Explain the working of an ideal transformer under (i) no-load and (ii) loaded conditions and derive expressions for voltage and current ratios relating to transformer turns ratio.

## Short answer

No-load: both windings link same flux, so V₂/V₁ = N₂/N₁ = K. On load, ampere-turn balance N₁I₁ = N₂I₂ gives I₂/I₁ = N₁/N₂ = 1/K. Thus voltage ratio equals turns ratio, current ratio is its inverse.

## Answer

An **ideal transformer** has no winding resistance, no leakage flux, no core loss and infinite core permeability (so the magnetising current is negligible). Its two windings have N1 (primary) and N2 (secondary) turns.

#### (i) No-load operation
```text
  I0 ≈ 0    +-------+  
 o--+ ( ) Φ ( )+--o  open circuit
 V1 |N1  core  N2|  V2 = E2
 o--+ ( )   ( )+--o
```
- The primary V1 is applied; the secondary is open.
- A tiny magnetising current Iμ flows, lagging V1 by 90°. It sets up the alternating flux Φ in the core.
- The flux links both windings and induces emfs in them by Faraday's law: E1 = −N1 dΦ/dt, E2 = −N2 dΦ/dt.
- With no resistance, E1 = V1 (it opposes V1 as a counter-emf) and V2 = E2.
- Since both windings link the same flux:

**E2/E1 = V2/V1 = N2/N1 = K** (voltage ratio, K = transformation ratio)

#### (ii) Loaded operation
- When a load is connected, the secondary current I2 flows, and produces an mmf N2 I2 that opposes the core flux.
- The primary draws an additional current I1 to cancel it. As the flux (and hence the magnetising mmf) stays unchanged, the net mmf is zero:  N1 I1 − N2 I2 = 0.

**I2/I1 = N1/N2 = 1/K** (current ratio)

#### Power relation
V1 I1 = V2 I2: input VA equals output VA, because the transformer is ideal.

| Quantity | Relation |
|---|---|
| Voltage ratio | V2/V1 = N2/N1 = K |
| Current ratio | I2/I1 = N1/N2 = 1/K |
| Impedance seen from primary | Z1 = Z_L/K² |

Step-up: K > 1 (voltage rises, current falls). Step-down: K < 1.

## Check yourself

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

1. Which is NOT an assumption of an ideal transformer?
   A. No leakage flux
   B. Infinite core permeability
   C. Finite core loss that depends on load
   D. No winding resistance
   Answer: C. An ideal transformer has no core loss either. (Key point: Ideal assumptions)

2. Why is E2/E1 = N2/N1?
   A. Currents are equal
   B. Both windings link the same flux
   C. Resistances equal
   D. Frequencies differ
   Answer: B. E = -N dPhi/dt for each with the same Phi, so the ratio is the turns ratio. (Key point: Voltage ratio)

3. Why is I2/I1 = N1/N2?
   A. Impedance is equal
   B. Voltage is constant
   C. Net mmf is zero: N1 I1 = N2 I2
   D. Power is zero
   Answer: C. The flux stays unchanged, so primary load mmf balances secondary mmf. (Key point: Current ratio)

4. What impedance does the primary see for load ZL?
   A. ZL K
   B. ZL / K^2
   C. ZL K^2
   D. ZL / K
   Answer: B. Z1 = V1/I1 = (V2/K)/(K I2) = ZL/K^2. (Key point: Impedance)

## Related in Electrical Machine

- [Ideal transformer no-load and loaded, ratios](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aideal-transformer-voltage-current-ratio) — Electrical Machines, chapter 2, asked 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
- [No-load and loaded operation, constant flux](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aideal-transformer-no-load-loaded-flux) — Electrical Machines, chapter 2, asked 3×