Faraday's law and inductors

Intent

• Relate a winding’s current to the magnetic field intensity in its core (Ampère’s law).
• Compute the flux-density swing from the voltage applied to a winding and its duration (Faraday’s law).
• Explain why the flux swing in a converter’s magnetic components is set by volt-seconds, not by the load current.

Theory

Ampère’s law

The magnetic field intensity integrated around a closed path equals the current that the path encloses. For $N$ turns on a core without a gap, with the field uniform along the effective path length, this gives [Erickson & Maksimović 2020, Ch. 10]:

$$H = \frac{N I_w}{l_e}$$

$H$

magnetic field intensity

$I_w$

winding current

$N$

number of turns

$l_e$

effective magnetic path length

Field intensity in an ungapped core from Ampère's law✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• ungapped core with the field intensity uniform along the path l_e

Source

Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory)

Derivation

step-by-step sympy derivation

Equation id

mag.H_ampere · equations.yaml entry

Faraday’s law

A changing flux through a winding induces a voltage equal to the number of turns times the rate of change of the flux. With the flux density uniform over the core’s effective area, a constant voltage applied to the winding for the on-time changes the flux density linearly [Erickson & Maksimović 2020, Ch. 10], [Dixon 2001]:

$$\Delta B = \frac{V_w t_\mathrm{on}}{N A_e}$$

$\Delta B$

flux-density swing, peak-to-peak

$A_e$

effective core area

$N$

number of turns

$V_w$

winding voltage during t_on

$t_\mathrm{on}$

switch on-time

Flux-density swing from Faraday's law✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• flux density uniform over A_e

Convention

$\Delta B$ is the peak-to-peak swing; the AC amplitude used in core-loss formulas is $\Delta B/2$

Source

Dixon 2001; Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory)

Derivation

step-by-step sympy derivation

Equation id

mag.dB_faraday · equations.yaml entry

This is the magnetic counterpart of the inductor law on the circuit-laws page: a constant voltage makes the current ramp there, and the flux density here.

Volt-seconds set the flux swing

The flux swing depends only on the volt-seconds applied, the turns and the core area, not on the current drawn by the load. In a transformer, which carries no dc magnetizing current, the volt-seconds alone set the flux. In a filter inductor, the dc current sets the average flux density and the volt-seconds set the swing around it (peak flux density). In steady state, volt-second balance brings the flux back to its starting value in every period (volt-second balance).

Worked example

An example winding on a core without a gap:

Wound-core example

Symbol	Value	How it is obtained
$N$ number of turns	20	given
$I_w$ winding current	2 A	given
$l_e$ effective magnetic path length	50 mm	given
$V_w$ winding voltage during ton	12 V	given
$t_\mathrm{on}$ switch on-time	5 µs	given
$A_e$ effective core area	100 mm²	given
$H$ magnetic field intensity	800 A/m	Field intensity in an ungapped core from Ampère's law
$\Delta B$ flux-density swing, peak-to-peak	30 mT	Flux-density swing from Faraday's law

The winding current produces a field intensity of 800 A/m in the core, and the applied volt-seconds swing the flux density by 30 mT peak-to-peak.

Try it

▶ Open in the equation explorer: Flux-density swing from Faraday's law
Preset from the example inductor-core

Sweep $N$: the flux swing falls in inverse proportion to the turns. More turns in the same winding area, however, raise the winding resistance [Erickson & Maksimović 2020, Ch. 11].

Bench exercise

Drive a winding on a ferrite core with a square wave from a function generator through a small series resistor, and wind a second winding of a few turns on the same core as a sense winding. The oscilloscope’s integral of the sense-winding voltage, divided by its turns and the core area, is the flux density. Check that its peak-to-peak swing follows the equation above, then double the drive frequency and check that the swing halves.

Gotchas

• Turns cut both ways. At a given voltage, more turns lower the flux swing; at a given current, more turns raise the field intensity and the flux density (Ampère's law).
• Net volt-seconds walk the flux into saturation. A winding with a nonzero average voltage over a period has a flux that grows every period until the core saturates [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.
• 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.
• Lecture notes 6.334 Lecture Notes — D. Perreault, MIT OpenCourseWare · retrieved 2026-09-23
  Chapter-style notes on dc/dc converters, magnetics and isolated converters.

Quiz

1. A 20-turn winding on a core without a gap, with a $0.05\,\mathrm{m}$ effective path length, carries $1\,\mathrm{A}$. What is the field intensity in the core?

   1. $20\,\mathrm{A/m}$
   2. $400\,\mathrm{A/m}$
   3. $1000\,\mathrm{A/m}$
   4. $4000\,\mathrm{A/m}$
   Answer and explanation

   Correct answer: B The enclosed current is $20\,\mathrm{A}$ turns, spread over $0.05\,\mathrm{m}$ of path.

2. A constant $12\,\mathrm{V}$ is applied for $10\,\mu\mathrm{s}$ to a 24-turn winding on a core of $1\,\mathrm{cm^2}$ effective area. What is the flux-density swing?

   1. $0.005\,\mathrm{T}$
   2. $0.05\,\mathrm{T}$
   3. $0.5\,\mathrm{T}$
   4. $5\,\mathrm{T}$
   Answer and explanation

   Correct answer: B The volt-seconds, $120\,\mu\mathrm{Vs}$, divided by the turns times the area, $24 \times 10^{-4}\,\mathrm{m^2}$.

3. The switching frequency of a converter is doubled at the same duty ratio and voltages. What happens to the flux swing in its transformer?

   1. It doubles
   2. It halves
   3. It is unchanged
   4. It falls to a quarter
   Answer and explanation

   Correct answer: B The on-time halves, so the volt-seconds per period, and with them the flux swing, halve.

4. What sets the flux swing in a transformer that carries no dc magnetizing current?

   1. The load current
   2. The volt-seconds applied to a winding, its turns and the core area
   3. The core's permeability alone
   4. The secondary current
   Answer and explanation

   Correct answer: B Faraday's law ties the rate of change of the flux to the winding voltage, whatever the load.

5. The turns on a given core are doubled while the winding current stays the same. What happens to the field intensity?

   1. It halves
   2. It is unchanged
   3. It doubles
   4. It quadruples
   Answer and explanation

   Correct answer: C The enclosed current, the turns times the winding current, doubles, and the path length is the same.

6. Why must the average voltage across a transformer winding be zero in steady state?

   1. To keep the output voltage constant
   2. Any net volt-seconds would make the flux grow every period until the core saturates
   3. To avoid winding loss
   4. Because the secondary is isolated
   Answer and explanation

   Correct answer: B The flux change per period is proportional to the net volt-seconds; only zero net volt-seconds brings it back to its starting value.