# * Self-inductance * Mutual inductance * Transformers. PPT No. 32

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1 * Self-inductance * Mutual inductance * Transformers PPT No. 32

2 Inductance According to Faraday s Electromagnetic Induction law, induction of an electromotive force occurs in a circuit by varying the magnetic flux linked with the circuit. From this law, the concept of Inductance may be derived as the property of an electric circuit by which an electromotive force is induced in it as the result of a changing magnetic flux.

3 Inductance Inductance L may be defined in terms of the electromotive force generated to oppose a change in current ΔI in the given time duration Δt according to Faraday s law as follows: Inductance is measured in S.I. unit called henry (H):

4 Inductance 1 henry is 1 volt-second / ampere. If the rate of change of current in a circuit ΔI / Δt is one ampere per second and the resulting electromotive force is one volt, then the inductance of the circuit is one henry.

5 Inductance Since emf results due to the rate of change of magnetic flux, inductance L may also be defined as a measure of the amount of magnetic flux φ produced for a given electric current I as L= φ /I where the inductance L is one henry, if current I of one ampere, produces magnetic flux φ of one weber. There are two types of inductances: 1 Self inductance 2. Mutual Inductance

6 Self-inductance 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 L is the constant of proportionality. L is called the self inductance. Thus the ratio of the magnetic flux to the current is the self-inductance

7 Self-inductance

8 Self-inductance Self-inductance in terms of emf According to Faraday s laws of electromagnetic induction when the electric current flowing through a wire changes, magnetic flux associated with it changes, electromotive force (emf) is induced that opposes the change in flux and current The opposing induced emf is called as the back emf. EMF is due to a change in flux in that circuit (wire) itself, hence it is called as Self inductance.

9 Self-inductance Self-inductance in terms of emf Suppose the current flowing in the circuit increases from 0 to I. It changes by an amount di in a time duration dt then the magnetic flux linking the circuit changes by an amount If the current is increasing then the induced emf always acts to reduce the current, and vice versa. the self inductance L of a circuit is necessarily a positive number

10 Self-inductance of A solenoid Consider a helically wound solenoid of length l and cross-sectional area A. Length is much greater than its diameter. n is the number of turns per unit length. Current I is flowing through the solenoid. hence the field within the solenoid is approximately constant

11 Self-inductance of A solenoid The magnetic flux across each turn of area A is = A μ 0 n I The total number of turns = n l turns, The total magnetic flux is φ= A μ 0 n 2 I l (Using the definition of Self inductance: L= φ / I = > L= Aμ 0 n 2 l

12 Mutual Inductance Mutual Inductance in Terms of Magnetic Flux Consider two coils in proximity of each other. When a steady current flows in the I coil, its magnetic field is linked to the II coil also and vise a versa. They are called as inductively coupled coils. Though there is no direct physical coupling between the two coils, coupling is provided entirely due to the magnetic field generated by the current flowing in the I coil.

13 Mutual Inductance The flux φ 2 through the second coil is directly proportional to the current I1 flowing in the I coil. Hence, φ 2 = M 21 I 1 (M 21 = constant of proportionality) M 21 is called the mutual inductance of coil 2 with respect to coil 1. Similarly the flux φ 1 through the I coil is directly proportional to the current I 2 flowing in the II coil. Hence, φ 1 = M 12 I 2 (M 12 = constant of proportionality). M 12 is called the mutual inductance of coil 1 with respect to coil 2. It can be proved that M 12 = M 21

14 Mutual Inductance Mutual Inductance in Terms of Magnetic Flux The flux linking coil 2 when a current flows in coil 1 is exactly the same as the flux linking coil 1 when the same current flows in coil 2. M 12 = M 21 = M M is called as the Mutual Inductance between two coils. It implies that the mutual inductance of two coupled coils is a purely geometric quantity, dependent on the sizes, shapes, and relative orientations of the coils

15 Mutual Inductance Mutual Inductance in Terms of Magnetic Flux If the magnetic field is steady and is not changing, there will be no induced voltage in the second coil. However, if the current in the first coil is changing by opening/ closing a switch to stop/start the current, there will be a change in the magnetic flux associated with the second coil and a voltage will be induced in it according to Faraday's law.

16 Mutual Inductance emf induced in Coil C 2 due to the Current-flow in Coil C 1

17 Mutual Inductance Mutual Inductance in Terms of Magnetic Flux If current I 1 flowing in coil 1 changes by an amount di 1 in time duration dt, the flux linking coil 2 changes by dφ 2 = M di 1 during same duration, then an emf is produced in second coil because of the change in flux linking it It can be written as (φ 2 = M 21 I 1 = M I 1 )

18 Mutual Inductance Thus, the emf generated in the II coil due to the current I 1 flowing in the I coil is directly proportional to the rate at which that current changes. Similarly for the emf generated in the I coil due to the current I 2 flowing in the II coil is directly proportional to the rate at which that current changes.

19 Mutual Inductance M is a measure of the mutual induction between two magnetically coupled coils. It is the ratio of the induced electromotive force to the rate of change of current producing it. It is also called as the coefficient of mutual induction. The SI unit of mutual inductance is called henry (H). One henry is equivalent to a volt-second per ampere. M is a purely geometric quantity. It depends only on the size, number of turns, relative position, and relative orientation of the two coils.

20 Mutual Inductance Mutual Inductance between Two Wires Suppose two insulated wires wound on the same core have the parameters A: cross-sectional area A l: the length l of the core, N 1 and N 2 : the number of turns of two wires I 1 and I 2 : the currents flowing in the wires, then the flux Φ 2 linking the II wire is given by

21 Mutual Inductance Mutual Inductance between Two Wires The mutual inductance of the II wire with respect to the I The mutual inductance of the I wire with respect to the II The mutual inductance between the two wires is given by

22 Mutual Inductance Mutual Inductance between Two Wires M is a geometric quantity which depends on the dimensions material of the core and the manner of winding around the core. However it does not depend on the value of currents flowing through the wires.

23 Applications of Mutual Inductance One of the important application of Mutual Inductance is in a machine like Transformer. Transformer is a device that transforms alternating electrical voltage from one level to another level and transfers electrical energy from one circuit to another through inductively coupled conductors.

24 Transformer Principle of Transformer Transformer operates on the principle of electromagnetic induction -Mutual Induction When the electric current through a circuit changes, the magnetic flux associated with the circuit in proximity (i.e. magnetically coupled circuit) changes and an electromotive force (EMF) is induced in coupled circuit.

25 Transformer Construction of Transformer The principle parts of a hypothetical ideal transformer are as follows: The Primary: A coil or winding connected to an AC source serves as an Input. The Secondary: A coil or winding in close proximity to the primary winding, but electrically isolated from it. It is connected to a load to which electrical power is to be delivered. Ideally, primary and secondary windings have zero resistance

26 Transformer

27 Transformer

28 Transformer The Core, On it both the primary and secondary coils are wound It is made up of a material of very high magnetic permeability, such as iron. It provides the best path of low/ negligible reluctance for the magnetic flux and minimizes loss in magnetic and electrical energy. In Iron-core transformers, core can be made by stacking layers of thin steel/ iron laminations insulated from each other by a coat of non-conducting paint/ varnish to reduce eddy current losses.

29 Transformer The Core (contd) Generally, with high frequency (above 20 khz) voltage sources, air-core transformers are used while with low frequency (below 20 khz) voltage sources, Iron-core transformers are used. The iron-core transformer provides better power transfer than the air-core transformer. The Enclosure It protects the above components from dirt, moisture, and mechanical damage

30 Transformer Transformer Working The alternating current (A.C.) that flows through the primary winding establishes a time-varying magnetic flux in the primary and voltage V p is induced in the primary due to self-induction. The flux in the primary is linked to the secondary winding also, through the transformer core. The varying magnetic flux in the primary induces a varying electromotive force (emf) or "voltage" V s in the secondary winding due to the effect called as mutual induction.

31 Transformer Transformer Working In accordance with Faraday's law of induction, Induced voltages are proportional to the rate of change of flux in the windings: V p and V s are the induced emfs in Primary & Secondary, N p and N s are their numbers of turns respectively, and are the time derivatives of the magnetic flux φ p and φ s linking the primary and secondary windings respectively.

32 Transformer Transformer Working As the core of an ideal transformer has zero reluctance all the magnetic flux produced by the primary winding also links the secondary, The emfs are equal in magnitude to the measured terminal voltages. so This is the well-known basic equation for stepping up or stepping down transformer voltage.

33 Transformer Working Transformer The ratio of transformation of primary to secondary voltage is therefore the same as the ratio of the number of turns in the windings; alternatively, the volts-per-turn ratio is the same in both windings. If the secondary coil is connected to a load, current flows through it. Thus electrical power is transferred from the primary circuit to the secondary circuit. Ideally, the transformer is perfectly efficient; all the incoming energy is transformed from the primary circuit to the magnetic field and into the secondary circuit.

34 Transformer

35 Transformer Transformer Working In the ideal condition, the incoming /input electric power must equal the outgoing / output power i.e. P input = P output P input = I P V P and P output = I S V S P input = P output I P V P = I S V S Varying the ratio of the number of turns between the primary and the secondary windings, the ratio of the input and output voltages and currents can be controlled

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