Ohms Law for Magnetic Circuits

Introduction

Ohm’s law is widely known in electrical engineering for explaining the relationship between voltage, current, and resistance. A very similar concept exists in magnetic circuit analysis, where engineers study how magnetic flux is produced and controlled within magnetic materials. This concept is known as Ohm’s Law for Magnetic Circuits, or more formally Hopkinson’s Law. It provides a simple and powerful way to analyze magnetic systems such as transformers, electric motors, generators, relays, and inductors.

Concept of a Magnetic Circuit

A magnetic circuit is a closed path followed by magnetic flux. This path usually consists of ferromagnetic materials such as iron or steel, along with air gaps if present. When an electric current flows through a coil wound around a magnetic core, it produces a magnetic field that establishes magnetic flux within the core. The behavior of this flux depends on the geometry of the magnetic path and the magnetic properties of the materials used. Just as an electrical circuit requires a source to drive current, a magnetic circuit requires a magnetizing force to establish flux. This force is known as the magnetomotive force, and it plays the same role in magnetic circuits as voltage does in electrical circuits.

Ohm’s Law for Magnetic Circuits

Ohm’s law for magnetic circuits states that the magnetic flux produced in a magnetic circuit is directly proportional to the magnetomotive force applied and inversely proportional to the reluctance of the magnetic path. This relationship is expressed mathematically as: $$Φ = m.m.f / ℜ$$ Here, $Φ$ represents magnetic flux measured in webers, $m.m.f$ is the magnetomotive force measured in ampere-turns, and $ℜ$ is the reluctance of the magnetic circuit. This equation closely resembles the electrical Ohm’s law equation $I = V / R$, where current is equal to voltage divided by resistance. The analogy between electrical and magnetic quantities greatly simplifies the understanding of magnetic systems.

Magnetomotive Force (m.m.f)

The magnitude of MMF depends on the number of turns in the coil and the amount of current flowing through it. It is given by the expression: $$m.m.f = N × I$$ where $N$ is the number of turns of the coil and $I$ is the current in amperes. The unit of $m.m.f$ is ampere-turns. Increasing either the number of turns or the current will increase the magnetomotive force and hence increase the magnetic flux, provided the magnetic circuit does not reach saturation.

Reluctance of a Magnetic Circuit

Reluctance is the opposition offered by a magnetic circuit to the establishment of magnetic flux. It is similar in concept to electrical resistance but depends on different physical properties. The reluctance of a uniform magnetic path is given by: $$ℜ = l / (μA)$$ In this expression, $l$ is the length of the magnetic path, $A$ is the cross-sectional area, and $μ$ is the permeability of the material. From this relationship, it is clear that reluctance increases with longer magnetic paths and decreases with larger cross-sectional areas and higher permeability materials. Materials like iron and steel have high permeability and therefore low reluctance, while air has very low permeability and high reluctance. Recalling the equation introduced for Ohm's law for electric circuits. The same equation can be applied for magnetic circuits. For magnetic circuits, the effect desired is the flux $\Phi$. The cause is the magnetomotive force (mmf) , which is the external force (or "pressure") required to set up the magnetic flux lines within the magnetic material. The opposition to the setting up of the flux $\Phi$ is the reluctance $S$. Substituting, we have Although there is a great deal of similarity between electric and magnetic circuits, one must continue to realize that the flux $\Phi$ is not a "flow" variable such as current in an electric circuit. Magnetic flux is established in the core through the alteration of the atomic structure of the core due to external pressure and is not a measure of the flow of some charged particles through the core.