UNIT 2 Magnetic Circuits

Production of Magnetic Field We know that any current carrying conductor produces a magnetic field. The basic law governing the production of a magnetic field by a current is Ampere's law: H = magnetic field l = core length Inet = net current F = Magnetomotive force = Ni A magnetic field is characterized either by (H) the magnetic field intensity or by (B) the magnetic flux density vector. These two vectors are connected by a rather simple relation: μ0 =4π ×10 H/ m is called the absolute permeability of free space and μr, a dimensionless quantity called the relative permeability of a medium (or a material).

Electric Circuit Analogy Magnetic Circuit Analogy flux () in a given area is given by: Hence The flow of magnetic flux induced in the ferromagnetic core can be made analogous to an electrical circuit hence the name magnetic circuit. The analogy is as follows: Electric Circuit Analogy Magnetic Circuit Analogy

Referring to the magnetic circuit analogy, F is denoted as magnetomotive force (mmf) which is similar to Electromotive force in an electrical circuit (emf). Therefore, we can safely say that F is the prime mover or force which pushes magnetic flux around a ferromagnetic core at a value of Ni (refer to ampere’s law). Hence F is measured in ampere turns. Hence the magnetic circuit equivalent equation is as shown: (similar to V=IR) The element of R in the magnetic circuit analogy is similar in concept to the electrical resistance. It is basically the measure of material resistance to the flow of magnetic flux. Reluctance in this analogy obeys the rule of electrical resistance (Series and Parallel Rules).

Magnetically Coupled Circuits Magnetic circuits are fundamental component of Electrical Machines and transformers. Magnetically coupled circuits are two or more electric circuits that are not connected directly but through a coupling established by magnetic field.

Magnetically Coupled Circuits Conductively coupled circuit means that one loop affects the neighboring loop through current conduction. Magnetically coupled circuit means that two loops, with or without contacts between them, affect each other through the magnetic field generated by one of them. Based on the concept of magnetic coupling, the transformer is designed for stepping up or down ac voltages or currents.

Faraday’s Law: ‘If a flux passes through a turn of a coil of wire, voltage will be induced in the turn of the wire that is directly proportional to the rate of change in the flux with respect of time’ If there is N number of turns in the coil with the same amount of flux flowing through it, hence: Note the negative sign at the equation above which is in accordance to Lenz’ Law which states: ‘The direction of the build-up voltage in the coil is as such that if the coils were short circuited, it would produce current that would cause a flux opposing the original flux change.’

Self Inductance

Mutual Inductance (1/4)

Mutual Inductance (2/4)

Mutual Inductance (3/4) We will see that M12=M21=M. Mutual coupling only exists when the inductors or coils are in close proximity, and the circuits are driven by time-varying sources. Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H). The dot convention states that a current entering the dotted terminal induces a positive polarity of the mutual voltage at the dotted terminal of the second coil.

Mutual Inductance (4/4)

DOT CONVENTION i1 + _ i1 + _ i2 + _ i2 + _

Series-Aiding Connection + _ v1 v2

Series-Opposing Connection + _ v1 v2

Example 1

Circuit Model for Coupled Inductors

Example 2

Example 3 For the circuit below, find the ratio of the output voltage across the 400 ohm resistor to the source voltage expressed using phasor notation.

Example 3

Example 4

Example 4

Example 5

Example 6 For the circuit below, find i1(t), i2(t), and i3(t) if f = 50 Hz