What physically determines the coefficient of coupling, k?

The coefficient of coupling is primarily determined by the geometry and relative placement of the two inductors. If they are wound on a common, high-permeability core with all magnetic flux linking both coils, k approaches 1 (tight coupling). If they are far apart or oriented so their magnetic fields are perpendicular, very little flux links the secondary and k is close to 0 (loose coupling).

Why does the primary current waveform change when I connect the secondary circuit?

When the secondary circuit is closed, the induced current I₂ creates its own changing magnetic field. This field, in turn, induces an EMF back into the primary coil, an effect described by Lenz's law. This 'back EMF' alters the total voltage across the primary inductor, thereby changing the primary current. This demonstrates that mutual inductance is a two-way interaction.

Is this a perfect transformer model?

No, this is a simplified model of coupled air-core or linear-core inductors. A perfect, ideal transformer model assumes k=1, no winding resistance, and no leakage flux, allowing for simple voltage and current ratios. This simulator allows k to be less than 1 and includes the effects of the secondary load resistance, providing a more general analysis of mutual inductance.

What is the role of the load resistor R₂ in the secondary circuit?

The resistor R₂ completes the secondary circuit, allowing the induced EMF to drive a measurable current, I₂. It represents the useful load (e.g., a light bulb or device) powered by the coupled energy. Changing R₂ affects the amplitude of I₂ and, through mutual coupling, also influences the primary current's magnitude and phase.

Mutual Inductance

Mutual inductance describes the fundamental electromagnetic interaction between two separate electrical circuits. When the current in one circuit (the primary, L₁) changes, it generates a changing magnetic field. If this field links through the turns of a second, nearby circuit (the secondary, L₂), it induces an electromotive force (EMF) across it, as described by Faraday's law of induction. This simulator visualizes that process. The primary circuit contains an AC voltage source and its own inductance, L₁, driving a sinusoidal current, I₁(t). The secondary circuit consists of its inductance, L₂, and a load resistor, R₂. The core physics is governed by the coupled inductor equations: V₁ = L₁ (dI₁/dt) + M (dI₂/dt) and V₂ = M (dI₁/dt) + L₂ (dI₂/dt), where M is the mutual inductance. The strength of coupling is quantified by the dimensionless coefficient of coupling, k, where M = k√(L₁L₂) and 0 ≤ k ≤ 1. The simulator solves these equations to show the resulting primary and secondary currents in real-time. Key simplifications include ideal inductors (no parasitic resistance or capacitance) and a linear magnetic medium, ignoring core saturation and hysteresis. By adjusting k, L₁, L₂, R₂, and the source frequency, students can explore how these parameters affect the induced secondary current's amplitude and phase. They learn to distinguish between the self-induced EMF in the primary and the mutually induced EMF, observe the transformer action, and understand how loose coupling (low k) or a high secondary load (high R₂) diminishes the induced current.

Who it's for: Undergraduate physics and electrical engineering students studying electromagnetism, specifically electromagnetic induction and AC circuit theory.

Key terms

Mutual Inductance
Faraday's Law
Coefficient of Coupling
Induced Electromotive Force
Coupled Inductors
Transformer Action
Self Inductance
AC Circuit Analysis

Live graphs

How it works

Two magnetically linked inductors satisfy L₁ di₁/dt + M di₂/dt = V − R₁i₁ and M di₁/dt + L₂ di₂/dt = −R₂i₂ for a simple secondary loop. The coupling coefficient k sets M = k√(L₁L₂). Changing k or ω changes how much current is induced in the secondary — a compact alternative to the ideal-turns Transformer page.

Key equations

M = k √(L₁ L₂), 0 ≤ k < 1

L₁ i₁′ + M i₂′ = V − R₁ i₁ , M i₁′ + L₂ i₂′ = −R₂ i₂

Frequently asked questions

What physically determines the coefficient of coupling, k?: The coefficient of coupling is primarily determined by the geometry and relative placement of the two inductors. If they are wound on a common, high-permeability core with all magnetic flux linking both coils, k approaches 1 (tight coupling). If they are far apart or oriented so their magnetic fields are perpendicular, very little flux links the secondary and k is close to 0 (loose coupling).
Why does the primary current waveform change when I connect the secondary circuit?: When the secondary circuit is closed, the induced current I₂ creates its own changing magnetic field. This field, in turn, induces an EMF back into the primary coil, an effect described by Lenz's law. This 'back EMF' alters the total voltage across the primary inductor, thereby changing the primary current. This demonstrates that mutual inductance is a two-way interaction.
Is this a perfect transformer model?: No, this is a simplified model of coupled air-core or linear-core inductors. A perfect, ideal transformer model assumes k=1, no winding resistance, and no leakage flux, allowing for simple voltage and current ratios. This simulator allows k to be less than 1 and includes the effects of the secondary load resistance, providing a more general analysis of mutual inductance.
What is the role of the load resistor R₂ in the secondary circuit?: The resistor R₂ completes the secondary circuit, allowing the induced EMF to drive a measurable current, I₂. It represents the useful load (e.g., a light bulb or device) powered by the coupled energy. Changing R₂ affects the amplitude of I₂ and, through mutual coupling, also influences the primary current's magnitude and phase.