Magnetostatics Study Notes Part-2 for Electrical Engineering

From Maxwell’s equations we have in differential form:

where is the current density.

In integral form we have

Where ∂D_R denotes the boundary of the disk D_R.

In words, the line integral of the H field around the boundary of the disk is equal to the total current through the disk, NI. The positive sense of current is out of the page and the positive sense for the magnetic field is given by the right-hand rule and is counter-clockwise as we stated above.

Since H is uniform around the boundary of the disk, and since the length of the boundary is  we have

Solving for the magnetic field and flux, we have

We have defined the magneto-motive force (mmf) and the reluctance in the equation.

In magnetic circuits, magneto-motive force is analogous to voltage in electric circuits, reluctance is analogous to resistance, and flux is analogous to current. That is

The reluctance is proportional to the length of iron, inversely proportional to the cross sectional area of the iron, and inversely proportional to the permeability of the iron. A similar relationship holds for the resistance of a conductor as shown in the figure below.

Analogy between resistance and reluctance.

Computing Voltage and Inductance: The voltage around a single turn in the winding is given by another one of Maxwell’s equations. Integrating the electric field around a single turn gives us

For N turns we have, using our sign convention,

The figure below depicts a magnetic circuit having both iron and an air gap in the loop.

The cross sectional area, A, is uniform around the circuit, the length of the iron and air gap are as indicated, and a positive current produces flux φ in the direction shown.

The inductor with an air gap.

In this series connection, the reluctances of the iron and air add so that the circuit reluctance is

The flux depends on the current and the number of turns, N:

And the inductance at the coil terminals is:

Energy and Force Calculations: Since the power transferred into an inductor is, the total stored energy at time T in an inductor with zero initial currents is

Making the change of variable u = I(t) such that  we have

Referring again to the toroidal inductor reproduced below with inductance 

Toroidal inductor.

Self and Mutual Inductance of Simple Configurations

Self Inductance: The property of self-inductance is a particular form of electromagnetic induction.

Self-inductance is defined as the induction of a voltage in a current-carrying wire when the current in the wire itself is changing.

In the case of self-inductance, the magnetic field created by a changing current in the circuit itself induces a voltage in the same circuit. Therefore, the voltage is self-induced.

Self-inductance in terms of emf: A circuit can create changing magnetic flux through itself, which can induce an opposing voltage in itself. The size of that opposing voltage is:

V(opposing) = – L *change in I / change in time

where L is the self-inductance of the circuit, measured in henries.

Self-inductance in terms of Magnetic Flux: A coil carrying current has magnetic flux associated with it. The flux Ф is directly proportional to the current I.

Ф = LI.

Where L is the constant of proportionality, L is called as self Inductance. The ratio of magnetic flux to the current is called as Self Inductance (L).

Mutual Inductance:

• The changing magnetic field created by one circuit (the primary) can induce a changing voltage and/or current in a second circuit (the secondary).
• The mutual inductance, M, of two circuits, describes the size of the voltage in the secondary induced by changes in the current of the primary: V(secondary) = – M [change in primary current/change in time]
• The units of mutual inductance are Henry, abbreviated “H”.

The magnetic flux through a circuit can be related to the current in that circuit and the currents in other nearby circuits, assuming that there are no nearby permanent magnets.

The magnetic field produced by circuit 1 will intersect the wire in circuit 2 and create current flow.

The induced current flow in circuit 2 will have its own magnetic field which will interact with the magnetic field of circuit 1.

At some point P, the magnetic field consists of a part due to i₁ and a part due to i₂. These fields are proportional to the currents producing them.

The coils in the circuits are labelled L₁ and L₂and this term represents the self-inductance of each of the coils.

The values of L₁ and L₂ depend on the geometrical arrangement of the circuit (i.e. a number of turns in the coil) and the conductivity of the material. The constant M, called the mutual inductance of the two circuits, is dependent on the geometrical arrangement of both circuits.

In particular, if the circuits are far apart, the magnetic flux through circuit 2 due to the current i₁ will be small and the mutual inductance will be small. L₂ and M are constants.

We can write the flux, B through circuit 2 as the sum of two parts.

Φ_B2 = L₂i₂ + i₁M

The equation is similar to the one above can be written for the flux through circuit 1.

Φ_B1 = L₁i₁ + i₂M

Though it is certainly not obvious, it can be shown that the mutual inductance is the same for both circuits. Therefore, it can be written as follows:

M_1,2 = M_2,1

All the Best.

Sahi Prep Hai to Life Set Hai