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magnetic-circuit

Magnetic circuit

A magnetic circuit is a closed path containing a magnetic flux. It generally contains magnetic elements such as permanent magnets, ferromagnetic materials, and electromagnets, but may also contain air gaps and other materials.

Some examples of magnetic circuits are:

Magnetic circuit laws

If Phi is the magnetic flux in the circuit, Theta is the magnetomotive force F applied to the circuit, and R_m is the reluctance of the circuit, then it follows from Ampère's law that:

Phi = frac F R

This is analogous to Ohm's law in electrical circuits, where the current is equal to the voltage (sometimes called electromotive force) divided by the resistance of the circuit. Here, magnetic flux, magnetomotive force and reluctance are analogous to current, voltage and resistance respectively.

If A is the area, μ is the permeability of the material, and l is the length

R_m = frac{l}{mu A}

This is similar to the equation for electrical resistance in materials, with permeability being analogous to conductivity. Longer, thinner geometries with low permeabilities lead to higher reluctance. Low reluctance, like low resistance in electric circuits, is generally preferred.

Magnetic circuits obey other laws that are similar to electrical circuit laws. For example, the total reluctance R_T of reluctances R_1, R_2, dots in series is:

R_T = R_1 + R_2 + dots

(this also follows from Ampère's law and is analogous to Kirchhoff's voltage law for adding resistances in series). Also, the sum of magnetic fluxes Phi_1, Phi_2, dots into any node is always zero:

Phi_1 + Phi_2 + dots = 0.
This follows from Gauss's law and is analogous to Kirchhoff's current law for analysing electrical circuits.

Together, the three laws above form a complete system for analysing magnetic circuits, in a manner similar to electric circuits. Comparing the two types of circuits shows that:

  • The equivalent to resistance R is the reluctance Rm
  • The equivalent to current I is the magnetic flux Φ
  • The equivalent to voltage V is the magnetomotive Force F

Magnetic circuits can be solved for the flux in each branch by application of the magnetic equivalent of Kirchhoff's Voltage Law (KVL) for pure source/resistance circuits. Specifically, whereas KVL states that the voltage excitation applied to a loop is equal to the sum of the voltage drops (resistance times current) around the loop, the magnetic analogue states that the magnetomotive force (achieved from ampere-turn excitation) is equal to the sum of MMF drops (product of flux and reluctance) across the rest of the loop. (If there are multiple loops, the current in each branch can be solved through a matrix equation--much as a matrix solution for mesh circuit branch currents is obtained in loop analysis--after which the individual branch currents are obtained by adding and/or subtracting the constituent loop currents as indicated by the adopted sign convention and loop orientations.) Per Ampère's law, the excitation is the product of the current and the number of complete loops made and is measured in ampere-turns. Stated more generally:

M = N,I = oint vec{H} cdot dvec{l}

(Note that, per Stokes's theorem, the closed line integral of H dot dl around a contour is equal to the open surface integral of curl H dot dA across the surface bounded by the closed contour. Since, from Maxwell's equations, curl H = J, the closed line integral of H dot dA evaluates to the total current passing through the surface. This is equal to the excitation, NI, which also measures current passing through the surface, thereby verifying that the net current flow through a surface is zero ampere-turns in a closed system that conserves energy.)

More complex magnetic systems, where the flux is not confined to a simple loop, must be analysed from first principles by using Maxwell's equations.

References

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