Transformers and coupled inductors

Intent

• Apply the ideal-transformer relations with the site’s 1:n turns-ratio convention.
• Add the magnetizing and leakage inductances to model a real transformer.
• Tell a transformer, which passes energy through, from a coupled inductor, which stores it.

Theory

The ideal transformer

Two windings on one core, with perfect coupling, no magnetizing current and no loss, form an ideal transformer. With the turns ratio written 1:n, where $n$ is the number of secondary turns divided by the number of primary turns, the voltage is multiplied by $n$ and the current divided by $n$ [Alexander & Sadiku 2017, Ch. 13], [Erickson & Maksimović 2020, Ch. 6]:

$$V_2 = n V_1$$

$V_2$

secondary voltage

$V_1$

primary voltage

$n$

turns ratio, secondary/primary

Ideal transformer: secondary voltage✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• ideal transformer: perfect coupling, no magnetizing current, no losses

Convention

turns ratio 1:n with $n$ the secondary turns divided by the primary turns (site convention)

Source

Alexander & Sadiku 2017, Ch. 13 (Magnetically Coupled Circuits); Erickson & Maksimović 2020, Ch. 6 (Converter Circuits)

Derivation

step-by-step sympy derivation

Equation id

xfmr.V2 · equations.yaml entry

$$I_2 = \frac{I_1}{n}$$

$I_2$

secondary current

$I_1$

primary current

$n$

turns ratio, secondary/primary

Ideal transformer: secondary current✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• ideal transformer: perfect coupling, no magnetizing current, no losses

Convention

turns ratio 1:n; the secondary current is taken as flowing out of the secondary's dotted terminal

Notes

the power is conserved: the product of the secondary voltage and current equals that of the primary

Source

Alexander & Sadiku 2017, Ch. 13 (Magnetically Coupled Circuits); Erickson & Maksimović 2020, Ch. 6 (Converter Circuits)

Derivation

step-by-step sympy derivation

Equation id

xfmr.I2 · equations.yaml entry

The power is conserved, and an impedance connected to the secondary appears at the primary divided by $n^2$ [Alexander & Sadiku 2017, Ch. 13].

Magnetizing inductance

A real core has a finite permeability, so it takes a current to set up the flux, even with the secondary open. This magnetizing current is modelled by an inductance $L_M$ in parallel with the primary of an ideal transformer, referred to the primary [Erickson & Maksimović 2020, Ch. 6]. It grows while a voltage is applied to the winding, which is why a transformer must be reset in every period (see the forward converter).

Leakage inductance

Not all of the flux links both windings. The flux that links only one of them appears as a leakage inductance in series with that winding [Erickson & Maksimović 2020, Ch. 10]. It slows the transfer of current from one winding to the other, and in a flyback its energy must be absorbed at every turn-off (see the flyback page).

Transformer or coupled inductor

A transformer transfers energy from the primary to the secondary while both windings conduct, and ideally stores none. A flyback’s coupled inductor works differently: its magnetizing inductance stores energy while the primary conducts and releases it through the secondary afterwards [Erickson & Maksimović 2020, Ch. 6]. The two are therefore designed for different goals: the transformer for a small magnetizing current, the coupled inductor for a specified magnetizing inductance and stored energy [Erickson & Maksimović 2020, Ch. 10].

Worked example

An example ideal transformer:

Ideal transformer example

Symbol	Value	How it is obtained
$n$ turns ratio, secondary/primary	0.25	given
$V_1$ primary voltage	48 V	given
$I_1$ primary current	500 mA	given
$V_2$ secondary voltage	12 V	Ideal transformer: secondary voltage
$I_2$ secondary current	2 A	Ideal transformer: secondary current

With a turns ratio of 0.25, the secondary has 12 V across it and carries 2 A: the voltage steps down by the same factor by which the current steps up.

Try it

▶ Open in the equation explorer: Ideal transformer: secondary voltage
Preset from the example transformer-ideal

Sweep $n$ over its logarithmic range: below one the transformer steps down, above one it steps up. Then compare with the flyback’s conversion ratio (flyback conversion ratio), in which the same factor multiplies the buck-boost ratio.

Bench exercise

Drive the primary of a small signal transformer from a function generator with a sine wave within its rated frequency range, with the secondary open. Measure both winding voltages to find the turns ratio. Then measure the primary current with the secondary open, which is the magnetizing current, and with the secondary shorted, which is limited by the leakage inductance, and estimate both inductances from the voltage, current and frequency.

Gotchas

• Check the turns-ratio convention. Data sheets and textbooks write turns ratios in different directions; this site always uses 1:n with $n$ the secondary turns divided by the primary turns.
• The magnetizing current is not load current. It flows with no load at all, and the net volt-seconds on the winding must be zero in every period, or the magnetizing current grows until the core saturates [Erickson & Maksimović 2020, Ch. 6].
• Mind the dots. Reversing one winding inverts the secondary voltage; in a flyback the winding polarity is chosen so that the diode blocks while the switch is on [Erickson & Maksimović 2020, Ch. 6].

Go deeper

• Book chapter Basic Magnetics Theory (Ch. 10) — R. W. Erickson, D. Maksimović · retrieved 2026-09-23
  Magnetic circuits, the transformer model with magnetizing and leakage inductance, and core and winding loss mechanisms.
• Book chapter Converter Circuits (Ch. 6) — R. W. Erickson, D. Maksimović · retrieved 2026-09-23
  How the basic topologies relate, and transformer-isolated versions including the flyback and forward converters.
• Course Magnetics for Power Electronic Converters — University of Colorado Boulder (Coursera) · retrieved 2026-09-23
  Magnetic circuits, core and winding losses, and inductor and transformer design.

Quiz

1. An ideal transformer with turns ratio 1:n, $n = 0.2$, has $50\,\mathrm{V}$ across its primary. What is the secondary voltage?

   1. $2.5\,\mathrm{V}$
   2. $10\,\mathrm{V}$
   3. $50\,\mathrm{V}$
   4. $250\,\mathrm{V}$
   Answer and explanation

   Correct answer: B The secondary voltage is $n$ times the primary voltage.

2. The secondary of the same transformer delivers $5\,\mathrm{A}$. What is the primary current?

   1. $0.2\,\mathrm{A}$
   2. $1\,\mathrm{A}$
   3. $5\,\mathrm{A}$
   4. $25\,\mathrm{A}$
   Answer and explanation

   Correct answer: B The secondary current is the primary current divided by $n$, so the primary current is $n$ times $5\,\mathrm{A}$. The power is the same on both sides.

3. A $10\,\Omega$ resistor is connected to the secondary of an ideal transformer with $n = 0.5$. What resistance does the primary see?

   1. $2.5\,\Omega$
   2. $5\,\Omega$
   3. $20\,\Omega$
   4. $40\,\Omega$
   Answer and explanation

   Correct answer: D An impedance on the secondary appears at the primary divided by $n^2$, here $0.25$.

4. What element models the current that a real transformer draws with its secondary open?

   1. The leakage inductance
   2. The magnetizing inductance
   3. The winding capacitance
   4. The load resistance
   Answer and explanation

   Correct answer: B Setting up the flux in a core of finite permeability takes a current, modelled by the magnetizing inductance in parallel with the primary.

5. How does a flyback's coupled inductor differ from a forward converter's transformer?

   1. It has no secondary winding
   2. It stores energy in its magnetizing inductance and releases it later, instead of passing energy through while both windings conduct
   3. It needs no core
   4. It has no leakage inductance
   Answer and explanation

   Correct answer: B In the flyback the primary and the secondary conduct in turn; the energy waits in the magnetizing inductance in between.

6. In the 1:n convention used on this site, what is $n$?

   1. The primary turns divided by the secondary turns
   2. The secondary turns divided by the primary turns
   3. The total number of turns
   4. The ratio of the winding resistances
   Answer and explanation

   Correct answer: B The site follows Erickson's convention: 1:n with $n$ the secondary turns over the primary turns.