Chapter 2 Magnetism and Electromagnetism Dayang Khadijah Hamzah ppkkp

Electromagnetism Magnetic Field - A permanent magnet on the table, cover it over with a sheet of smooth cardboard and sprinkle steel filings uniformly over the sheet. - Slight tapping of the latter causes the fillings to set themselves in curved chains between the poles as shown in figure below: A suspended permanent magnet Use of steel fillings for determining distribution of magnetic field

The shape and density of these chains enable one to form a picture of the magnetic condition of the space or ‘field’ around a bar magnet and lead to the idea of lines of magnetic flux. Noted that these lines of magnetic flux have no physical existence, they are purely imaginery.

Direction of Magnetic Field The direction of a magnetic field is taken as that in which the north-seeking pole of magnet points when the latter is suspended in the field. Thus, if bar magnet rests on a table and 4 compass needles are placed in positions indicated in figure below, it is found that the needles take up positions such that their axes coincide with the corresponding chain of fillings and their N poles are all pointing along the dotted line from the N pole of the magnet to its S pole.

The lines of magnetic flux are assumed to pass through the magnet, emerge from the N pole and return to the S pole. Use of compass needles for determining direction of magnetic field

Magnetic Field due to an electric current When a conductor carries an electric current, a magnetic field is produced around the conductor – phenomenon discovered by Oersted at Copenhagen in He found that when a wire carrying an electric current was placed above a magnetic field (figure below) and in line with the normal direction of the latter, the needle was deflected clockwise or anticlockwise, depending the direction of the current.

Thus, if the current is flowing away from us, as shown in figure below, the magnetic field has a clockwise direction and the lines of magnetic flux can be represented by concentric circles around the wire. Magnetic flux due to current in a straight conductor

In Figure below, we have a conductor in which drawn an arrow indicating direction of conventional current flow. If the current is flowing towards us, we indicate this by a dot equivalent to the approaching point of the arrow and if the current is flowing away then it is represented by a cross equivalent to the departing tail feathers of the arrow.

Current Conventions Right-hand screw rule

A convenient method of representing relationship between directions of current and its magnetic field is to placed a corkscrew or a woodscrew alongside the conductor carrying the current. In order that the screw is in the same direction as the current, towards right, it has to be turned clockwise when viewed from the left-hand side. Similarly, direction of the magnetic field, viewed from the same side, is clockwise around the conductor as indicated by the curved arrow F.

Force determination The force on the conductor can be measured for various currents and various densities of the magnetic field. It is found that Force on conductor proportional to current x (flux density) x (length of conductor) F(N)  flux density x l(m) x I(A) * The unit of flux density is taken as the density of a magnetic field such that conductor carrying I ampere at right angles to that field has a force of I N/m acting upon it.

Magnetic flux density Symbol : B Unit: Tesla (T) For a flux density of B teslas,force on conductor F = B l I Newton For a magnetic field having cross-sectional area of A square metres and a uniform flux-density of B teslas, the total flux in weber (Wb) is represented by Greek capital letter  (phi). Magnetic flux Symbol:  Unit: weber (Wb)  = BA T = 1 Wb/m 2

Flux Density 13

Permeability Permeability μ is a measure of the ease by which a magnetic flux can pass through a material (Wb/Am) Permeability of free space μ o = 4π x 10-7 (Wb/Am) Relative permeability: 14

Reluctance Reluctance: “resistance” to flow of magnetic flux Associated with “magnetic circuit” – flux equivalent to current What’s equivalent of voltage? 15

The relationship between current and magnetic field intensity can be obtained by using Ampere’s Law. Ampere’s Law Ampere’s Law states that the line integral of the magnetic field intensity, H around a closed path is equal to the total current linked by the contour. H: the magnetic field intensity at a point on the contour dl: the incremental length at that point If θ : the angle between vectors H and dl then 16

Ampere’s Law Consider a rectangular core with N winding Therefore 17

Relationship between B-H The magnetic field intensity, H produces a magnetic flux density, B everywhere it exists. - Permeability of the medium - Permeability of free space, - Relative permeability of the medium For free space or electrical conductor (Al or Cu) or insulators, is unity 18

Assumption: All fluxes are confined to the core The fluxes are uniformly distributed in the core Magnetic Equivalent Circuit The flux outside the toroid (called leakage flux), is so small (can be neglected) Use Ampere’s Law, F = Magnetomotive force (mmf) 19

Magnetic Equivalent Circuit Where; N – no of turns of coil i – current in the coil H – magnetic field intensity l – mean length of the core 20

Magnetomotive Force, F Coil generates magnetic field in ferrous torroid Driving force F needed to overcome torroid reluctance Magnetic equivalent of ohms law Circuit Analogy 21

Magnetomotive Force The MMF is generated by the coil Strength related to number of turns and current, measured in Ampere turns (At) 22

Field Intensity The longer the magnetic path the greater the MMF required to drive the flux Magnetomotive force per unit length is known as the “magnetizing force” H Magnetizing force and flux density related by: 23

Magnetomotive Force Electric circuit: Emf = V = I x R Magnetic circuit: mmf = F = Φ x = (B x A) x l μ A = (B x A) x l μ = B x= H x l 24

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= Φ x l μ A = x x 2 x = =

= Φ x = 4 x x = 17.6 I = F N = = 44 mA 28

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Circuit Analogy 31

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Electromagnetic Induction In 1831, Michael Faraday discover electromagnetic induction namely method of obtaining an electric current with the aid of magnetic flux. He wound two coils A & C on a steel ring R as figure below, and found that when switch S was closed, deflection was obtained on galvanometer G and when S was opened, G was deflected in reverse direction. Then, he found that when permanent magnet NS was moved relatively to coil C, galvanometer G was deflected in one direction when the magnet moved towards the coil and in reverse direction when the magnet was withdrawn. And this experiment convinced Faraday that an electric current could be produced by the movement of magnetic flux relative to a coil.

Alternatively, we can say that when a conductor cuts or is cut by magnetic flux, an e.m.f. is generated in the conductor and the magnitude of e.m.f. is proportional to the rate at which the conductor cuts or is cut by the magnetic flux. Electromagnetic Induction

Direction of induced e.m.f… available for deducing the direction or generated e.m.f. namely, a) Fleming’s right-hand rule b) Lenz’s Law Flemming’s right-hand rule

a) Fleming’s right-hand rule If the first finger of the right is pointed in the direction of the magnetic flux, as figure below and if the thumb is pointed in the direction of the motion of the conductor relative to the magnetic field Two methods, then the second finger held at right angles to both the thumb and the first finger represents the direction of the e.m.f.

Field or Flux with First finger, Motion of the conductor relative to the field with the M in thuMb and e.m.f. with the E in sEcond finger.

A moving electric field creates a magnetic field that rotates around it A moving magnetic field creates an electric field that rotates around it The Right Hand Rule helps describe this 40

First define positive electric current as flowing from the positive (+) end of a battery, through an electric circuit, and back into the negative (-) end. Next define a magnetic field as always pointing away from a North pole and towards a South pole. Curl your fingers in the direction of the rotating field. Current Field Lines 41

Extend your thumb. It now points in the direction of the other field. If your fingers are curling along with a rotating electric field, your thumb will point in the direction of the magnetic field and vice versa. 42

Wire carrying current out of page Wire carrying current into page Representing Currents X 43

Wire carrying current out of page Wire carrying current into page The magnetic field of two wires X 44

Faraday’s Law First Law Whenever the magnetic flux linked with a coil changes, an emf (voltage) is always induced in it. Or Whenever a conductor cuts magnetic flux, an emf (voltage) is induced in that conductor. 45

Faraday’s Law Second Law The magnitude of the induced emf (voltage) is equal to the rate of change of flux-linkages. where  a b If a magnetic flux, , in a coil is changing in time (n turns), hence a voltage, e ab is induced e = induced voltage N = no of turns in coil d  = change of flux in coil dt = time interval 46

Voltage Induced from a time changing magnetic field 47

The Right Hand Rule Wire carrying current out of page Wire carrying current into page The magnetic field of two wires X 48

The Right Hand Rule Both of the wire carrying current out of page The magnetic field of two wires 49

The Right Hand Rule The overall field around a coil is the sum of the fields around each individual wire The magnetic field of a coil 50

The Right Hand Rule The magnetic field around a solenoid resembles that of a bar magnet. Inside the solenoid the field lines are parallel to one another. We say it is a uniform field. The magnetic field of a solenoid 51

b) Lenz’s Law In 1834, Heinrich Lenz, a German physicist enunciated a simple rule, known as Lenz’s Law: The direction of an induced e.m.f. is always such that it tends to set up a current opposing the motion or the change of flux responsible for inducing that e.m.f.

Consider the application of Lenz’s Law. We find that when S is closed and the battery has the polarity shown, the direction of the magnetic flux in the ring is clockwise. The current in C must try to produce a flux in an anticlockwise direction tending to oppose the growth of the flux due to A namely the flux which is responsible for the e.m.f. induced in C. But an anticlockwise flux in the ring would require the current in C to be passing through the coil from X to Y. Hence, this must also be the direction of the e.m.f induced in C.

Lenz’s Law Lenz’s law states that the polarity of the induced voltage is such that the voltage would produce a current that opposes the change in flux linkages responsible for inducing that emf. If the loop is closed, a connected to b, the current would flow in the direction to produce the flux inside the coil opposing the original flux change. The direction (polarity) of induced emf (voltage) can be determined by applying Lenz’s Law. Lenz’s law is equivalent to Newton’s law. 54

N S I Lenz’s Law 55

MAGNETIC CIRCUITS Introduction Mutual Inductance Energy in a coupled circuit

Introduction When two loops with or without contacts between them affect each other through the magnetic field generated by one of them, are said to be magnetically coupled. TRANSFORMER is an electrical device designed on the basis of the concept of magnetic coupling.

Mutual Inductance Mutual inductance happens when two inductors or coils are in close proximity to each other, the magnetic flux caused by current in one coil links with other coil, thereby inducing voltage in the latter. Consider a single inductor, a coil with N turns. When current i flows through the coil, a magnetic flux  is produced around it.

According to Faraday’s Law, the voltage v induced in the coil is proportional to the number of turns N and the time rate of change of the magnetic flux  : v = N d  dt

The flux  is produced by the current i so that any change in  is caused by a change in current. Hence it can be written as V = L di dt which is the voltage-current relationship for the inductor. The inductance L of the inductor is thus given by L = N d  dt This inductance commonly known as self- inductance

Consider two coils with self-inductances L 1 & L 2 that are in close proximity with each other. Coil 1 has N 1 turns, while coil 2 has N 2 turns. Assume that second inductor carries no current.

Magnetic flux  1 emanating from coil 1 has 2 components: one component  11 links only coil 1 and other component  12 links both coils. Hence,  1 =  11 +  12 Although the two coils are separated, they are said to be magnetically coupled.

Since flux  1 links coil 1, the voltage induced in coil 1 is: V 1 = N 1 d  dt Only flux  12 links coil 2, the voltage induced in coil 2 is: V 2 = N 2 d  12 dt

Continued… As the fluxes are caused by the current i 1 flowing in coil 1, hence V 1 = N 1 d  1 di 1 = L 1 di 1 di 1 dt dt Similarly v2 can be written as; V 2 = N 2 d  12 di 1 = M 21 di 1 di 1 dt dt Where M 21 = N 2 d  12 di 1 where M 21 = N 2 d  12 dt

M 21 is known as the mutual inductance of coil 2 with respect to coil 1. Subscript 21 indicates that the inductance M 21 relates the voltage induced in coil 2 to the current in coil1. Thus the open- circuit mutual voltage (induced voltage) across coil 2 is V 2 = M 21 di 1 dt

Thus the open-circuit mutual voltage (induced voltage) across coil 2 is V 1 = M 12 di 2 dt

M is the mutual inductance between two coils. Both self-inductance L and mutual inductance M is measured in henrys (H). We can conclude that mutual inductance is the ability of one inductor to induce a voltage across a neighbouring inductor. M 12 = M 21 = M

Since it is convenient to show the construction of coils on a circuit, we apply the dot convention in circuit analysis. A dot is placed in the circuit at one end of each two magnetically coupled coils to indicate the direction of magnetic flux if current enters that dotted terminal of the coil. Dot convention stated that if the current enters the dotted terminal of one coil, reference polarity of the mutual voltage in the second coil is +ve at the dotted terminal of the second coil.

Alternatively, if a current leaves the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is -ve at the dotted terminal of the second coil

The current i 1 is enters the dotted terminal of coil 1 and v 2 is -ve at the dotted terminal of coil 2. Hence, the mutual voltage is – Mdi 1 /dt The sign of mutual voltage v 2 is determined by the reference polarity for v 2 and the direction of i 1. Since i 1 enters the dotted terminal of coil 1 and v 2 is +ve at the dotted terminal of coil 2, the mutual voltage is +Mdi 1 /dt

The current i 2 is leaves the dotted terminal of coil 2 and v 1 is +ve at the dotted terminal of coil 1. Hence, the mutual voltage is – Mdi 2 /dt The sign of mutual voltage v 1 is determined by the reference polarity for v 1 and the direction of i 2. Since i 2 leaves the dotted terminal of coil 2 and v 1 is - ve at the dotted terminal of coil 1, the mutual voltage is +Mdi 2 /dt

Dot convention for coils in series, the sign indicates the polarity of the mutual voltage (series-aiding connection) Dot convention for coils in series, the sign indicates the polarity of the mutual voltage (series-opposing connection) L = L 1 + L 2 + 2M L = L 1 + L 2 - 2M

Time-domain analysis of a circuit containing coupled coils As in figure above, applying KVL to coil 1 gives V 1 = i 1 R 1 + L 1 di 1 /dt + M di 2 /dt Coil 2, KVL gives V 2 = i 2 R 2 + L 2 di 2 /dt + M di 1 /dt

Continued… Frequency-domain analysis of a circuit containing coupled coils As in figure above, applying KVL to coil 1 gives V = (Z1 + jωL 1 )I 1 – jωMI 2 Coil 2, KVL gives 0 = – jωMI 1 + (ZL + jωL 2 )I 2

Example 1 Solution: For coil 1, KVL gives (-j4 + j5)I 1 – j3I 2 = > jI 1 -3jI 2 = (1) For coil 2, KVL gives -j3I 1 + (12 + j6)I 2 = > I 1 = (2 – j4)I (2) Substitute eqn (2) into (1) (j2 + 4 – j3)I 2 = (4 - j)I 2 = 12 Or I 2 = 12/(4 - j) = 2.91   (3) From eqn (2) & (3) I 1 = (2 – j4)I 2 =   A

Energy in a coupled Circuit Consider the circuit above, assume current i 1 and i 2 are zero initially, so that energy stored in coils is zero. If we let i 1 increase from 0 to I 1 while maintain i 2 = 0, the power in coil 1 is P 1 (t) = v 1 i 1 = i 1 L 1 di 1 /dt The energy stored in an inductor is; W = ½ Li 2

The energy stored in the circuit is: W 1 = ½ L 1 I 1 2 Now, we maintain i 1 = I 1, i 2 increase from 0 to I 2, the mutual voltage induced in coil 1 is M 12 di 2 /dt while the mutual voltage induced in coil 2 is zero since i 1 does not change. The power in the coil is; P 2 (t) = I 1 M 12 di 2 /dt + i 2 L 2 di 2 /dt The energy stored in the circuit is W 2 = M 12 I 1 I 2 + ½ L 2 I 2 2 So, the total energy stored in the coils is W = w 1 + w 2 = ½ L 1 I ½ L 2 I M 12 I 1 I 2

Since the total energy stored should be the same, we can conclude that M 12 = M 21 = M and W = 1/2 L 1 I ½ L 2 I MI 1 I 2 (This equation based on assumption that the coil currents both entered the dotted terminals) If one current enters one dotted terminal while the other current leaves the dotted terminal, the mutual voltage is negative, so that the mutual energy MI 1 I 2 is also negative. So, w = ½ L 1 I ½ L 2 I 2 2 – MI 1 I 2 Hence, instantaneous energy stored in the circuit as below; W = ½ L 1 i ½ L 2 i 2 2 ± Mi 1 i 2

To establish an upper limit for Mutual Inductance, M. The energy stored cannot be negative because the circuit is passive. The extent to which the mutual inductance M approaches the upper limit is specified by the coefficient of coupling k, given by M = k√L 1 L 2 where 0 ≤ k ≤ 1 The coupling coefficient k is a measure of the magnetic coupling between two coils.

If entire flux produced by one coil links another coil, then k = 1 & we have 100% coupling or the coils are said to be perfectly coupled. For k 0.5, they are said to be tightly coupled. a) Windings loosely compound b) Windings tightly coupled