Energy Conversion(2)

De La Salle University-Dasmariñas College of Engineering, Architecture, and Technology Engineering Department

MAGNETISM AND ELECROMAGNETISM Magnetic attract magnetic materials but not non-magnetic materials. Magnetism is a noncontact force (acts at a distance) MAGNETISM -

The ability to attract iron and steel. The knowledge of magnetism goes back to the Ancient Greeks who realized that a certain rock (Iodestone) attracted pieces of iron. When the hang a piece of this material, it rotates until it is pointing in a north-south direction of the earth.

-

Magnets are named after the town magnesia (a district in Thessaly) in Lydia, Asia Minor where the Iodestone was mined in ancient times. Natural permanents were called Lodestone (magnetic, 𝐹𝑒3 𝑂4) after Iodestar (or guiding star). Lodestone was first permanent magnetic material to be identified and studied. The regions near the ends of a magnet are called its poles.

Magnetic Materials:    

Iron Steel Nickel Cobalt

CLASSIFICATION OF MATTER ACCORDING TO THE MAGNETIC PROPERTY 1. Ferromagnetic - If materials such as cobalt, nickel or iron are put near a magnet they begin to act like another magnet. - Ferromagnetic materials are characterized by spontaneous magnetism that exists in the absence of a magnetic field. They can retain the ability to attract metals (particularly

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

those belonging to ferrous family) even after the magnetic field that induced magnetism to it has been removed. Iron is a soft ferromagnetic material. This means it will become -

magnetized very easily, but quickly loses its magnetic properties if the magnetized force is removed. Steel is more difficult to magnetize, but once it is magnetized, it retains its magnetic properties for a long time. Steel is called a “hard” ferromagnetic material. 2. Diamagnetic - Have the ability to slightly repel magnetic field. Faraday discovers these materials in 1845. He found that bismuth and glass are repelled from magnetic fields. 3. Paramagnetic - Also discovered by Faraday. He noted that some substances clearly not permanent magnets are nevertheless attracted by magnetic fields and these materials are named paramagnetic. MAGNET A magnet is any object that has a magnetic field. It attracts ferrous objects like pieces of iron, steel, nickel and cobalt. One of the most common magnets - the bar magnet - is a long, rectangular bar of uniform cross-section that attracts pieces of ferrous objects. The magnetic compass needle is also commonly used. The compass needle is a tiny magnet which is free to move horizontally on a pivot. One end of the compass needle points in the North direction and the other end points in the South direction. The end of a freely pivoted magnet will always point in the North-South direction. The end that points in the North is called the North Pole of the magnet and the end that points South is called the South Pole of the magnet. It has been proven by experiments that like magnetic poles repel each other whereas unlike poles attract each other.

The region around a magnet where a magnetic force can be felt is called the magnetic field.

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

The magnet field is strongest at the poles of a magnet.  Like poles repel

N N

S S

 Unlike poles attract N S

S N

MAGNETIC FIELD Magnetic field is the space surrounding a magnet, in which magnetic force is exerted. If a bar magnet is placed in such a field, it will experience magnetic force. However, the field will continue to exist even if the magnet is removed. The direction of magnetic field at a point is the direction of the resultant force acting on a hypothetical North Pole placed at that point. A magnetic field around a bar magnet has a shape and direction.

The magnetic field is represented using magnetic field lines (lines of force , flux lines) that show the shape, direction and strength of the field. HOW IS A MAGNETIC FIELD CREATED? When current flows in a wire, a magnetic field is created around the wire. From this it has been inferred that magnetic fields are produced by the motion of electrical charges. A magnetic field of a bar magnet thus results from the motion of negatively charged electrons in the magnet. Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The magnetic field (β) is defined in terms of force on moving charge in the Lorentz force law. The interaction of magnetic field with charge leads to many practical applications. Magnetic field sources are essentially dipolar in nature, having a north and south magnetic pole.

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

FLUX DENSITY (β) It is given by the flux passing per unit area through a plane at right angles to the flux. It is measured in Wb/𝑚2

β=

𝜱 𝑨

= µH = µ𝟎 µ𝒓 H

Direction of the magnetic field at any point is defined as the direction of motion of a change particle on which the magnetic field would not exert force. Magnitude of the magnetic field vector is proportional to the force acting on the moving charge, the magnitude of its velocity and the angle between velocity and magnetic field. Unit is the Tesla or Gauss SI

CGS

Wb/𝑚2 (Tesla)

Max/𝑐𝑚2 (Gauss)

ENG lines/𝑖𝑛2

FLUX PER UNIT POLE (Φ ) OR MAGNETIC LINES OF FORCE Just as an electric field is described by drawing the electric lines of force, in the same way, a magnetic field is described by drawing the magnetic lines of force. When a small north magnetic pole is placed in the magnetic field created by a magnet, it will experience a force. And if the North Pole is free, it will move under the influence of magnetic field. The path traced by a North magnetic pole free to move under the influence of a magnetic field is called a magnetic line of force. In other words, the magnetic lines of force are the lines drawn in a magnetic field along which a north magnetic pole would move. The direction of a magnetic line of force at any point gives the direction of the magnetic force on a north pole placed at that point. Since the direction of magnetic line of force is the direction of force on a North Pole, so the magnetic lines of force always begin on the N-pole of a magnet and end on the S-pole of the magnet. A small magnetic compass when moved along a line of force always sets itself along the line tangential to it. So, a line drawn from the South Pole of the compass to its North Pole indicates the direction of the magnetic field. Properties of the magnetic lines of force  The magnetic lines of force originate from the North Pole of a magnet and end at its South Pole.  The magnetic lines of force come closer to one another near the poles of a magnet but they are widely separated at other places.  The magnetic lines of force do not intersect (or cross) one another. When a magnetic compass is placed at different points on a magnetic line of force, it aligns itself along the tangent to the line of force at that point.

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

Magnetic Flux (φ) - It is the number of magnetic lines of forces in a magnetic field. -Maxwell-unit of magnetic flux equal to one line of force. - Weber- SI unit of magnetic flux equal to 108 lines or Maxwell. 1Wb = 1x108 Maxwell Conversion q = 1.602x10−19 C

1 𝑘𝑔𝑓 = 9.81 N

1 𝑙𝑏𝑓 = 4.4484 N

1 Tesla = 104 Gauss

1 N = 105 Dynes

Absolute and Relative Permeability of a medium Permeability - the ability of a material to conduct magnetic flux through it. Relative Permeability- ration of the permeability of material to the permeability of air or vacuum. The phenomena of magnetism and electromagnetism are dependent upon a certain property of the medium called its permeability. Every medium is supposed to possess two permeabilities:  Absolute permeability, µ𝑜  Relative permeability, µ𝑟 For measuring relative permeability, vacuum or free space is chose as the reference medium. It is allotted an absolute permeability of vacuum with reference to itself is unity. Hence, for free space, Absolute permeability, µ𝑜 = 4πx107 Henry/meter, constant 33 Relative permeability, µ𝑟 = 1 Now, take any medium other than vacuum. If its relative permeability, as compared to vacuum is µ𝑟 , then its abs. permeability is µ = µ𝑜 = µ𝑟 MAGNETISING FIELD STRENGTH/FORCE/MAGNETIC INTENSITY (H) -

Field strength at any point within a magnetic field is numerically equal to the force experienced by a N-pole of one Weber placed at that point. It should be noted that the field strength is a vector quantity having both magnitude and direction. mmf (magnetomotiveforce) per unit length of path of the magnetic flux. It is also called as the magnetizing force or the magnetic gradient OERSTED- cgs unit of magnetic field strength equal to gilbert per centimeter. AT/m – SI unit for H 1 oersted = 79.577 AT/m

a. Long Straight Wire

H=

𝑵𝑰 𝟐𝝅𝒓

where: r – distance N- Number of turns I – Current in Amperes (A) Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

b. Long Solenoid

H=

𝑵𝑰 𝒍

c. Circular Coil

H=

𝑵𝑰 𝟐𝒓

where: r – radius

d. Square Coil

H=

√𝟐𝑵𝑰 𝝅𝒂

where: a – distance from the corner

SAMPLE PROBLEMS 1. A solenoid 30cm long is wound with 300turns. What is the value of its field strength inside the solenoid, when the coil is carrying a current of 2 Amperes?

2. If a current of 5A flows through a long wire of radius 0.004 meter, what is the intensity of magnetic field produced 0.02 meter away from the surface of the wire?

3. A flat circular coil with 40 loops of wire has a diameter of 32 cm. What current must flow in its wires to produce a field of 3.0x 10−4 Wb/𝑚2 ?

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

MAGNETIC FORCES FORCE ON A CHARGE  The amount of attraction or repulsion between charged objects can be put in quantitative terms by the introduction of the electric force. The simplest case to consider is the force between two points charges (charges with a negligible size)

F = qvβsinƟ (N)

where: q – charge in Coulomb Ɵ – angle between wire and magnetic field v – velocity in m/s β – flux density in Tesla

FORCE ON A CURRENT CARRYING CONDUCTOR LYING IN A MAGNETIC FIELD    



The magnetic force on a charged particle depends on the relative orientation of the particle's velocity and the magnetic field. A magnetic force cannot change the speed of a charged particle, only its direction. When a charged particle enters a uniform magnetic field in a direction perpendicular to that field, its motion is continuously changed by the magnetic force A current consists of many small charged particles running through a wire. If immersed in a magnetic field, the particles will be experience a force; they can transmit this force to the wire through which they travel. The force on a section of wire of length L carrying a current I through a magnetic field B is

F = βILsinƟ (N)

where: β – Tesla I – Current in Ampere (A) L – length in meter (m)

F=

𝛃𝐈𝐋𝐬𝐢𝐧Ɵ 𝟏𝟎

(Dynes)

where: β – Gauss I – Current in Ampere (A) L – length in centimeter (cm)

F=

𝛃𝐈𝐋𝐬𝐢𝐧Ɵ 𝟏𝟏.𝟑𝒙𝟏𝟎𝟔

(𝒍𝒃𝒇 )

where: β – lines I – Current in Ampere (A) L – length in in/ft

Because forces are easy to measure, it is the force exerted on a current-carrying wire which is used to define the SI unit of current, the ampere.

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

FORCE BETWEEN TWO PARALLEL CONDUCTORS  Current in the same direction. The field strength in the space between the conductors is decreased due to the two fields there being in the opposition to each other. Hence, the two conductors are attached towards each other.  Current in the opposite direction. The field strength is increased in the space between the two conductors due to the two fields being in the same direction there. Because of the lateral repulsion of the lines of force, the two conductors expensive a mutual force of repulsion.

F=

µ𝟎 µ𝒓 𝑰𝟏 𝑰𝟐 𝒍 𝟐𝝅𝒅

where:

µ𝟎 - constant permeability, const 33 µ𝒓 - relative permeability 𝑙 - length in meter (m)

F=

𝟐𝒙𝟏𝟎−𝟕 µ𝒓 𝑰𝟏 𝑰𝟐 𝒍 𝒅

I – current in amperes (A) d – distance between two conductors

SAMPLE PROBLEMS 1. An armature conductor 12cm long moves right angle to the magnetic flux of 1.20 Tesla and carrying 5A. What is the force experienced by the conductor?

2. Two straight parallel wires 2m long and 3mm apart carries a current of 8A in opposite direction. Calculate the force between these conductors?

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

3. A coil of moving instrument is wound with 250 turns of wire. The flux density in the gap is 0.085 Tesla and the effective length of the coil side in the air gap is 5cm. Find the force doing acting on each coil side when carrying current of 60mA? In Dynes.

LORENTZ RIGHT HAND RULE The Lorentz Force Law can be used to describe the effects of a charged particle moving in a constant magnetic field. In an open right hand, the direction of the four fingers points to the direction of the magnetic field, the thumb pointing perpendicular to the four fingers points to the direction of the magnetic force in a positive charge is in the direction in which your open palm would push.

The implications of this expression include: 1. The force is perpendicular to both the velocity (v) of the charge (q) and the magnetic field (B) 2. The magnitude of the force F=qvBsinθ where θ is the angle<180 degrees between the velocity and the magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero. 3. The direction of the force is given by the right hand rule. The force relationship above is in the form of a vector product.

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

B – Magnetic field

(Wb/𝑚2 (Tesla)

Max/𝑐𝑚2 (Gauss)

lines/𝑖𝑛2

F – Force

N

Dynes

𝑙𝑏𝑓

v- velocity/speed

m/s

cm/s

in/s,ft/s

FLEMING LEFT AND RIGHT HAND RULE Whenever, a current carrying conductor comes under a magnetic field, there will be force acting on the conductor and on the other hand, if a conductor is forcefully brought under a magnetic field, there will be an induced current in that conductor. In both of the phenomenons, there is a relation between magnetic field, current and force. This relation is directionally determined by Fleming Left Hand rule and Fleming Right Hand rule respectively. 'Directionally' means these rules do not show the magnitude but show the direction of any of the three parameters (magnetic field, current, force) if the direction of other two are known. Fleming Left Hand rule is mainly applicable for electric motor and Fleming Right Hand rule is mainly applicable for electric generator. In late 19th century, John Ambrose Fleming introduced both these rules and as per his name, the rules are well known as Fleming left and right hand rule

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

Fleming Left Hand rule

It is found that whenever a current carrying conductor is placed inside a magnetic field, a force acts on the conductor, in a direction perpendicular to both the directions of the current and the magnetic field. In the figure it is shown that, a portion of a conductor of length L placed vertically in a uniform horizontal magnetic field strength H, produced by two magnetic poles N and S. If i is the current flowing through this conductor, the magnitude of the force acts on the conductor is, F = BIL

Fleming Right Hand rule

As per Faraday's law of electromagnetic induction, whenever a conductor moves inside a magnetic field, there will be an induced current in it. If this conductor gets forcefully moved inside the magnetic field, there will be a relation between the direction of applied force, magnetic field and the current. This relation among these three directions is determined by Fleming Right Hand Rule This rule states "Hold out the right hand with the first finger, second finger and thumb at right angle to each other. If forefinger represents the direction of the line of force, the thumb points in the direction of motion or applied force, then second finger points in the direction of the induced current.

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

SAMPLE PROBLEMS 1. Using the right hand rule, find the direction of the missing information in the diagram

2. Each of the lettered dots shown in the figure represents an electric charge of 5µC moving at speed 3x10^6 m/s in the direction shown. Determine the magnetic force (magnitude and direction) acting one each charges due to the 0.15 Tesla magnetic fields that points in the positive y-direction?

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

TORQUE ON A FLAT COIL IN A UNIFORM MAGNETIC FIELD When a current carrying loop is placed across a magnetic field, it has the tendency to be rotated either clockwise or counter-clockwise dependent on the direction of the magnetic field and the current. Its direction of rotation is determined using the right hand rule. Before considering the mathematical nature of the forces on currents in magnetic fields it is worth just looking at the simple magnetic field diagrams that give rise to these effects. These are shown in Figure 1. (a) is the field between two magnets, (b) the field due to a current in a straight wire and (c) the resulting field if they are put together. This last field is known as the "catapult" field because it tends to catapult the wire out of the field in the direction shown by the arrow.

T = INAβsinƟ (N-m)

where: N- turns I – current in amperes (A) A – area of the coil Ɵ – angle between magnetic field and a perpendicular to the

plane of the coil SAMPLE PROBLEM 1. A rectangular loop 10cm high and 5cm wide is placed in magnetic field of 0.01 Tesla. If the loop contains turns and carries a current of 50mA. What is the torque on it? Assume that the face of the loop is parallel to the field? Given 250 turns.

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

MAGNETIC CIRCUIT -

A closed path in which magnetic induction or flux flows. A system of magnetic conductors which magnetism maybe established upon the application of magnetomotive force (mmf)

MAGNETOMOTIVE FORCE (mmf) -

A force that sets up or tends to set up magnetic flux in a magnetic circuit. By an electric current through a number of turns of a wire produces it

mmf = NI (if the unit is in AT) = 0.4πNI (if the unit is in Gilberts)

RELUCTANCE (R) -

Property of material that opposes flux flow. It is equal to the ration of the mmf in a magnetic circuit to the magnetic flux through any cross section of the magnetic circuit.

R=

𝒍 µ𝒐 µ𝒓 𝑨

=v

𝒍 𝑨

units: AT/Wb; Gilbert/Max

Where: 𝑙 − Mean length of the magnetic path (m) µ𝑜 - Free space of permeability µ𝑜 − Relative permeability A – Cross sectional area of the magnetic path (sq. m) 1

v – reluctivity ; the reciprocal of permeability = µ 𝑙 - mean length / circumference 𝑙 − πd ; where d = mean diameter 𝑙 − 2πr; where r = radius 𝑅𝑐𝑔𝑠 = 79.577𝑥 106 𝑅𝑚𝑘𝑠 Permeance (P) – reciprocal of reluctance 𝟏

P=𝑹=

Implies the ease or readiness with which magnetic flux is developed.

µ𝟎 µ𝒓 𝑨 𝒍

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

FLUX ( 𝜱) -

Used to indicate the apparent stress in the space surrounding an energized coil or magnet.

𝜱=

𝒎𝒎𝒇 𝑹

=

𝑵𝑰 𝑹

=

𝟎.𝟒𝝅𝑵𝑰 𝑹

SAMPLE PROBLEMS 1. A certain laminated steel core has a permeability of 3000. The length is 5cm and the cross sectional area is 2sq.m. What is the reluctance?

2. A magnetic circuit consists of silicon steel of 3000 permeability and air gap. The length of the steel core is 10cm and the air gap is 2cm. Both have the same cross section of 1.5 sq.cm. A current of ½ Ampere flows through the windings to produce 2351 Maxwell flux. How many turns are there in the coil?

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

3. A solenoid has 250 turns. What is the magnetomotive force (mmf) in Gilbert when the current is 0.12 Amp?

4. A coil with 900 turns is wound over a magnetic core with a reluctance of 10000 AT/Wb. If a current of 2A is passed through the coil, determine the flux density inside the coil?

UNITS SYMBOL

MKS

CGS

β

Wb/𝑚2 (Tesla)

Max/𝑐𝑚2 ( Gauss)

𝛷

Wb ( Weber)

Max (Maxwell)

A

𝑚2

𝑐𝑚2

µ0

Const 33

1

µ𝑟

-

-

H

AT/m

Oersted

R

AT/Wb

Gilbert/Max

mmf

AT

Gilberts Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

ENERGY STORED IN A MAGNETIC FIELD The energy stored in a magnetic field is equal to the work needed to produce a current through the inductor. When a conductor carries a current, a magnetic field surrounding the conductor is produced. The resulting magnetic flux is proportional to the current. If the current changes, the change in magnetic flux is proportional to the time-rate of change in current by a factor called inductance (L). Since nature abhors rapid change, a voltage (electromotive force, EMF) produced in the conductor opposes the change in current, which is also proportional to the change in magnetic flux. Thus, inductors oppose change in current by producing a voltage that,in turn, creates a current to oppose the change in magnetic flux; the voltage is proportional to the change in current. Due to energy conservation, the energy needed to drive the original current must have an outlet. For an inductor, that outlet is the magnetic field—the energy stored by an inductor is equal to the work needed to produce a current through the inductor.

𝟏

𝟏

W = 𝟐 R𝜱𝟐

𝜷𝟐

W = 𝟐 (LA) [ µ ]

Where: w – energy stored in Joules (J) µ - permeability of core

𝟏

W = 𝟐 𝑳𝑰𝟐 𝛷 - Flux 𝛽 – Magnetic flux density

R – Reluctance L – Inductance I – Current

FARADAY’S LAW A law that states an electrical field is induced in any system in which a magnetic field is changing with time.  FARADAY’S FIRST LAW OF ELECTROMAGNETIC INDUCTION. Whenever the flux linking a coil or current changes, an emf is induced in it.  FARADAY’S SECOND LAW OF ELECTROMAGNETIC INDUCTION. The magnitude of the induced emf is proportional to the rate of change of flux linkages.

INDUCED EMF – it is the voltage generated by a conductor or coil moving in magnetic field. e=N

𝒅𝜱 𝒅𝒕

where: e – induced emf (Volt) N – number of turns 𝑑𝛷 𝑑𝑡

- rate of change of flux (Weber per second) Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

INDUCED EMF A voltage can be developed in a wire by moving the wire across a magnetic field so that flux cutting results (Faraday’s Principle) e = βLv sin Ɵ where: e – induced emf (volt) β – flux density at the location of the conductor (Tesla) L – length of the conductor (meter) v – relative velocity (meter per second) 𝒅𝒊

e = L 𝒅𝒕

where: e – self- induced emf (volt) L – self inductance (Henry) di/dt – rate of change of current ( Ampere per second)

SAMPLE PROBLEMS 1. Find the electromotive force in a conductor of length 50cm moving perpendicular at a velocity 590m/s to a region of flux density 1 Tesla?

2. A magnetic coil produces 100,000 Maxwells with 2000 turns and with a current of 2 Amp. The current is cut-off and the flux collapses in 0.01sec. What is the average voltage that will appear across the coil?

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

3. The flux density emanating from a pole of a generator is 20,000 Gauss. A conductor one meter long cuts the flux perpendicularly at a speed of 40m per second. What voltage is developed?

INDUCTANCE A property of an electric circuit by which emf is induced in it as the result of changing magnetic flux. it is also a circuit element, typically a conducting coil, in which emf is generated by electromagnetic induction. Self Inductance – the ratio of emf produced in a circuit by self induction to the rate of change of current producing it, expressed in Henries (H)

L=

𝑵𝜱 𝑰

L=

µ𝟎 µ𝒓 𝑨𝑵𝟐

L=

𝒍

𝑵𝟐 𝑹

Where: L – inductance (Henry) µ0 - permeability of free space (const 33, 4𝜋 𝑥 10−7 Henry per meter µ𝑟 - relative permeability of the core used A – cross sectional are of the magnetic path( square meter) N – number of turns φ – flux (Weber) I – Current (Ampere) 𝑙 - mean length of the magnetic path (meter) R – reluctance of the magnetic path (AT/Weber) Mutual Inductance – the ratio of emf in a circuit to the corresponding change of current in a neighboring circuit. Measures the mutual induction between two magnetically linked circuits, given as the ratio of the induced emf to the rate of charge of current producing it, measured in Henries (H) M=

𝑵𝜱 𝑰

M=

µ𝟎 µ𝒓 𝑨𝑵𝟏 𝑵𝟐 𝒍

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

Series Coil with mutual inductance Series aiding – sources of electromotive force (emf) which give the ability to the current to flow in the same direction. 𝐿𝑇𝐴 = 𝐿1 + 𝐿2 + 2M Series opposing – sources of electromotive force (emf) which give the ability to the current to flow in opposite direction. 𝐿𝑇𝑂 = 𝐿1 + 𝐿2 - 2M Parallel Coil with mutual inductance Parallel aiding – this is where connected coils increasing the total equivalent inductance. 𝐿 𝐿 − 𝑀2

𝐿𝑇𝐴 = 𝐿 1+ 2𝐿 1

2 − 2𝑀

Parallel opposing – this is where connected coils decreasing the total equivalent inductance compared to coils that have zero mutual inductance. 𝐿 𝐿 − 𝑀2

𝐿𝑇𝐴 = 𝐿 1+ 2𝐿 1

2 + 2𝑀

Where: 𝐿1 , 𝐿2 - self inductance in H (Henry) M – Mutual inductance M=

𝑳𝑻𝑨 − 𝑳𝑻𝟎 𝟒

Coupling Factor/ Coefficient of coupling k=

𝑴 √𝑳𝟏 𝑳𝟐

SAMPLE PROBLEMS 1. Two coils in a network are positioned such that there is 80% coupling between them. If the inductance of one coil is 20mH and the inductance of the other coil is 16mH. Find the mutual inductance?

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

2. A current of 2 Amp through a coil sets up flux linkages of 4Wb-turns. What is the inductance of the coil?

3. Two coils of inductance 𝐿1 = 1.16 mH, 𝐿2 = 2 mH are connected in series. Find the total energy stored when the steady current is 2 Amp?

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

ENERGY CONVERTION GRADING RUBRIC FOR PROBLEM SET Name: _________________________________ InstructorName:_____________________

Course/Section:_______________ Date:_______________________

Problem Set No: __________ Component

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At least one component contains a minor error.

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Minor issues of tone, voice, spelling, punctuation, or formatting.

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Exceedingly confusing writing.

Major precision errors that cause meaningful ambiguity in the interpretation of the solution, which can only be resolved with difficulty (if at all)

Severely underspecified instructions, definitions, claims, or arguments

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Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

DIRECTION: On a long plain bond paper copy and solve the following problems completely. Use engineering lettering and use straight edge for figures. Failure to follow direction will get your work rejected.

1.

The force acting on a pole of 3Wb is 12N. The magnetic intensity of the magnetic field is __.

2.

A wire 12cm long and carrying a current of 30A is placed in between the pole face of a magnet whose magnetic flux density is 0.9 Tesla. If the wire is inclined at an angle 60degrees from the plane of the magnetic field, what is the force exerted on the wire?

3.

The reluctance of a non-magnetic circuit is 12 units. How much flux will be set up if surrounded by a coil 600 turns carrying a current of 3A.

4.

The relative permeability of a certain silicon steel is 4500. A certain magnetic loop consists of a silicon steel of 10cm square, 20cm long and an air gap of ¼ cm. What is the reluctance of the magnetic circuit?

5.

A coil with 900 turns is wound over a magnetic core with a reluctance of 10,000 AT/Wb. If a current of 0.5A is pass through the coil, how much is the magnetic flux that the coil generates?

6.

A given magnetic circuit has a magnetic field intensity of 400AT/m. If the length of the magnetic path is doubled maintaining the same magnetomotive force, how much is the new magnetic field intensity?

7.

A magnetomotive force is supplied by a current of one ampere through 100 turns. The magnetic circuit consists of a steel core of 1000 permeability, 10cm long and 4 sq. cm. area and an air gap one cm long. What is the field intensity at the air gap?

8.

A non magnetic ring having a cross sectional area of 10 cm2 is uniformly wound with 300 turns of a given wire. If a current of 1A is passed through the coil, 2.4µWb of flux is generated inside the ring. Determine the average diameter of the ring.

9.

A coil with 250 turns is wound over a 200cm a cylindrical iron core whose relative permittivity is 250. If a current of 2A is pass through the coil, determine the flux density in the core.

10.

A toroidal core with a mean circumference of 100cm and a cross sectional area of 10cm2 is wound with 500 turns of wire. What current would be required to generate a flux of 1 mWb in the core. Assume the core has a relative permeability of 800. Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra

11.

A magnetic ring (relative permitivitty=800) has a mean radius of 10cm and a cross sectional area of 5cm2. An air gap measuring 1.5mm is cut in the ring. Determine the required mmf in order to produce a flux of 0.25mWb in the air gap.

12.

A magnetic ring with a mean diameter of 25cm and a cross sectional area of 5cm 2 is wound with a coil of 600 turns. An air gap 4mm is made by cutting a section of the ring. A current of 10A is passed through the coil. Determine the energy stored in the air gap. Assume relative permittivity of the ring to be 1000.

13.

How much is the inductance of a coil that induces 500V when the current changes at the rate of 5mA in 2µs?

14.

The energy (Wo) stored in a coil is dependent in the inductance (L) of the coil and the current flowing. If the inductance were doubled with the same current flowing, what would be the resulting stored energy?

15.

A 6.0 H coil whose resistance is 12 ohms is connected in series with a 24 ohms resistor and to a 144 V battery and a switch. The switch is closed at t=0. Determine the energy stored in the magnetic field at steady state.

Second Semester, S. Y. 2016 – 2017 Engr. Noemi Q. Guerra