In this article we will discuss about:- 1. Introduction to Process of Magnetism 2. Production of EMF 3. Faraday’s Laws of Electro-Magnetic Induction 4. Lenz’s Law 5. Induced EMF 6. AC Excitation 7. Magnetic Hysteresis 8. Energy Expended 9. Energy Stored.
Contents:
- Introduction to the Process of Magnetism
- Production of EMF
- Faraday’s Laws of Electro-Magnetic Induction
- Lenz’s Law
- Induced EMF in Magnetic Circuit
- AC Excitation in Magnetic Circuits
- Magnetic Hysteresis
- Energy Expended in a Magnetic Cycle
- Energy Stored in a Magnetic Field
1. Introduction to the Process of Magnetism:
The curve drawn giving relationship between induction density B and magnetising force H is known as B-H curve or magnetising curve. Fig. 9.24 shows the general shape of B-H curve of a magnetic material. In general it has four distinct regions oa, ab, be and the region beyond c.
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During the region oa the increase in flux density is very small, in region ab the flux density B increases almost linearly with the magnetising force H, in region be the increase in flux density is again small and in region beyond point c, the flux density B is almost constant.
The flat part of the B-H curve corresponds to magnetic saturation of the material. The non- linearity of the curve indicates that the relative permeability µr (B/µ0 H) of a magnetic material is not constant but depends very largely upon the flux density B.
The B-H curve depends on the material only and not upon the dimensions of a specific piece. Magnetising or B-H curves for silicon alloy (electrical sheet) steel, cast steel and cast iron etc., are shown in Fig. 9.25. In making magnetic calculations, values of H and µr corresponding to working flux density B are to be taken. The use of B-H curves permits the calculations of magnetic circuits with a fair degree of accuracy.
2. Production of EMF:
An electromotive force (emf) may be produced by:
(i) Chemical action
(ii) Heating of thermo-junction, or
ADVERTISEMENTS:
(iii) Electromagnetic action.
Here we will be discussing last method of producing emf (i.e., electro-magnetic induction) in detail. We know that when a current carrying conductor is suitably placed in a magnetic field a force is exerted on the conductor in a direction perpendicular to both the direction of flow of current and that of magnetic field. Faraday discovered that reverse of the above is also true i.e., an emf can be produced by moving a conductor in a magnetic field. This will be clear from the following experiment.
When two ends of a coil are connected to a galvanometer G and a bar magnet N S is brought nearer to the coil, deflection in the galvanometer is observed. The deflection of galvanometer indicates the flow of current through the coil and this deflection will persist so long as the magnet is moved. If the magnet movement is stopped, the deflection will reduce to zero. Now if the magnet is moved away from the coil, galvanometer will again show deflection but in opposite direction.
When the magnet is moved towards the coil, linking flux with the coil increases, therefore, an emf is induced in the coil, which causes flow of current through the coil and galvanometer shows deflection.
ADVERTISEMENTS:
Similarly when the magnet is moved away from the coil, the linking flux decreases, therefore, an emf is induced but in direction opposite to previous one. But when the movement of the magnet is stopped, though flux links with the coil, but there is no change in linking flux, therefore, no emf is induced in the coil.
The same results will be obtained if bar magnet is kept stationary and coil is moved towards or away from the bar magnet.
Hence, for the production of an emf, the change in the magnitude of the linking flux is essential.
ADVERTISEMENTS:
ADVERTISEMENTS:
3. Faraday’s Laws of Electro-Magnetic Induction:
Faraday’s First Law:
This law states that when the flux linking with the coil or circuit changes an emf is induced in it or whenever the magnetic flux is cut by the conductor an emf is induced in the conductor.
Faraday’s Second Law:
This law states that the magnitude of emf induced is directly proportional to the rate of change of flux linkage or to the product of number of turns and rate of change of flux linking the coil.
I.e. induced emf, e ∝ N. d ɸ/dt where N d ɸ/dt is the product of number of turns and rate of change of linking flux and is termed as rate of change of flux linkage.
4. Lenz’s Law:
This law states that the direction of induced emf is such that the current produced by it sets up a magnetic field opposing the motion or change producing it.
The above statement will be clarified from the following experiment:
When the bar magnet is moved towards the coil, as shown in Fig. 9.27, an emf is induced in the coil, which causes an electric current to flow through the coil in the direction shown in Fig. 9.27. Due to flow of this electric current through the coil magnetic field is produced, north pole adjacent to north pole of the bar magnet and south pole on the other end, which can be determined from helix rule.
Now since similar poles repel each other, the movement of the bar magnet towards the coil is opposed by the field produced. Mechanical energy is expended in moving the bar magnet towards the coil against this force of repulsion and this expended energy is converted into electrical energy.
The same results will be observed when the bar magnet is moved away from the coil.
When the bar magnet is moved away from the coil, an emf is induced in the coil, current flows through it in the direction opposite to that in which it was flowing when the magnet was moved towards the coil, so magnetic field will be produced of opposite polarity i.e., south pole adjacent to north pole of the bar magnet.
Therefore, force of attraction will be exerted between the coil and bar magnet, which will oppose the movement of the bar magnet away from the coil. Mechanical energy is expended in this case also and is converted into electrical energy.
5. Induced EMF in Magnetic Circuit:
Let the flux linking with the coil of turns N be changed by an amount in short time dt.
EMF induced, e = Rate of change of flux linkage = Number of turns x rate of change of flux
= N dɸ/dt
A minus sign is required to be placed before the right hand side quantity of above expression just to indicate the phenomenon explained by Lenz’s law; therefore, expression for induced emf may be written as:
E = – N d ɸ/d t volts … (9.7)
Example 1:
A coil has 400 turns. Find the induced voltage in it if the flux changes from 0.2 mWb to 1 mWb in 0.2 second.
Solution:
Change of flux in 0.2 s, d ɸ = 0.001 – 0.0002 = 0.0008 Wb
Rate of change of flux, d ɸ/d t = 0.0008/0.2 = 0.004 Wb/s
Induced voltage, e = N d ɸ/d t = 400 × 0.004 = 1.6 V Ans.
Dynamically Induced EMF:
We have learnt that when the flux linking with the coil or a circuit change, an emf is induced in the coil or circuit.
EMF can be induced by changing the flux linking in two ways:
(i) By increasing or decreasing the magnitude of the current producing the linking flux. In this case there is no motion of the conductor or of coil relative to the field and, therefore, emf induced in this way is known as statically induced emf.
(ii) By moving a conductor in a uniform magnetic field and emf produced in this way is known as dynamically induced emf.
Consider a conductor of length I metres placed in a uniform magnetic field of density B Wb/m2, as shown in Fig. 9.29 (a).
Let this conductor be moved with velocity v m/s in the direction of the field, as shown in Fig. 9.29 (b). In this case no flux is cut by the conductor; therefore, no emf is induced in it.
Now if this conductor is moved with velocity v m/s in a direction perpendicular to its own length and perpendicular to the direction of the magnetic field, as shown in Fig. 9.29 (c) flux is cut by the conductor, therefore, an emf is induced in the conductor.
Area swept per second by the conductor;
= I × v m2/s
Flux cut per second
= Flux density × area swept per second = Blv
Rate of change of flux,
dɸ/dt = Flux cut per second = BIv Wb/s
Induced emf, e = dɸ/dt = Blv volts
If the conductor is moved with velocity v metres per second in a direction perpendicular to its own length and at an angle 0 to the direction of magnetic field, as shown in Fig. 9.29 (d).
The magnitude of emf induced is proportional to the component of the velocity in a direction perpendicular to the direction of the magnetic field and induced emf is given by:
e = B/v sin 0 volts
The direction of this induced emf is given by Fleming’s right hand rule.
If the thumb, forefinger and middle finger of right hand are held mutually perpendicular to each other, forefinger pointing into the direction of the field and thumb in the direction of motion of conductor then the middle finger will point in the direction of the induced emf as shown in Fig. 9.31 (a).
Fig. 9.31 (b) illustrates another way of determination of induced emf, known as right hand flat palm rule. This law states that if right hand is so placed in a magnetic field along the conductor that the magnetic lines of force emerging from the north pole enter the palm and the thumb points in the direction of motion of conductor, the other four fingers will give the direction of induced emf or current.
Example 2:
Find the induced emf in a conductor of length 150 cm moving at an angle of 30″ to the direction of uniform magnetic field of flux density 1.2 Wb/m2 with a velocity of 60 m/s.
Solution:
Flux density B = 1.2 T
Length of conductor, I = 150 cm = 1.5 m
Velocity of conductor, v = 60 m/s
Angle of movement of conductor from the direction of magnetic field,
θ = 30°
EMF induced, e = B/v sin θ = 1.2 × 1.5 × 60 × sin 30° = 108 × 0.5 = 54 V Ans.
Statically Induced EMF:
Statically induced emf may be:
(a) Self-induced emf or
(b) Mutually induced emf
(a) Self-Induced EMF:
When the current flowing through the coil is changed, the flux linking with its own winding changes and due to the change in linking flux with the coil, an emf, known as self-induced emf is induced.
Since according to Lenz’s law, an induced emf acts to oppose the change that produces it, a self-induced emf is always in such a direction as to oppose the change of current in the coil or circuit in which it is induced. This property of the coil or circuit due to which it opposes any change of the current in the coil or circuit, is known as self-inductance.
Consider a solenoid of N turns, length l metres, area of x-section a square metres and of relative permeability µr. When the solenoid carries a current of i amperes, a magnetic field of flux Ni/l/µ0µr a webers is set up around the solenoid and links with it.
If the current flowing through the solenoid is changed, the flux produced by it will change and, therefore, an emf will be induced.
The quantity N2µrµ0a/l is a constant for any given coil or circuit and is called coefficient of self-inductance. It is represented by symbol L and is measured in henries.
Coefficient of Self Induction:
The coefficient of self-induction (L) can be determined from any one of the following three relations:
First Method:
In case the dimensions of the solenoid are given, the coefficient of self-induction may be determined from the relation:
Second Method:
In case the magnitude of induced emf in a coil for a given rate of change of current in the coil is known, self-inductance of the coil may be determined from the following relation:
Third Method:
In case the number of turns of the coil and flux produced per ampere of current in the coil is known, the self-inductance of the coil may be determined from the following relation:
The above relation can be derived as follows:
Magnetic flux produced in a coil of N turns, length l metres, area of x-section a metres2 and relative permeability µr when carrying a current of i amperes is given by:
From the above relation it is obvious that the self-inductance of a coil or circuit is equal to weber-turns per ampere in the coil or circuit.
In the above relation if N ɸ = 1 Wb-turn and i = 1 A then L = 1 H.
Hence a coil is said to have a self-inductance of one henry if a current of 1 A, when flowing through it, produces flux linkage of 1 wb-turn in it.
(b) Mutually Induced EMF:
Consider two coils A and B placed closed together so that the flux created by one coil completely links with the other coil. Let coil A have a battery and switch S and coil B be connected to the galvanometer G.
When switch S is opened, no current flows through coil A, so no flux is created in coil A, i.e., no flux links with coil B, therefore, no emf is induced across coil B, the fact is indicated by galvanometer zero deflection.
Now when the switch S is closed current in coil A starts rising from zero value to a finite value, the flux is produced during this period and increases with the increase in current of coil A, therefore, flux linking with the coil B increases and an emf, known as mutually induced emf, is produced in coil B, the fact is indicated by galvanometer deflection.
As soon as the current in coil A reaches its finite value, the flux produced or flux linking with coil B becomes constant, so no emf is induced in coil B, and galvanometer pointer returns back to zero position. Now if the switch S is opened, current will start decreasing, resulting in decrease in flux linking with coil B, an emf will be again induced but in direction opposite to previous one, this fact will be shown by the galvanometer deflection in opposite direction.
Hence whenever the current in coil A changes, the flux linking with coil B changes and an emf, known as mutually induced -emf is induced in coil B.
Consider coil A of turns N1 wound on a core of length l metres, area of cross-section a square metres and relative permeability µr. When the current of i, amperes flows through it, a flux of N1 i1/l/µ0µra is set up around the coil. Let whole of the flux produced due to flow of current in coil A be linked with the coil B having N2 turns and placed near by coil A.
Mutually induced emf, em = – Rate of change of flux linkage of coil B
= – N2 × rate of change of flux in coil A
The quantity N1N2 a µ0 µr/l is called the coefficient of mutual induction of coil B with respect to coil A. It is represented by symbol M and is measured in henrys.
Coefficient of Mutual Induction:
Mutual inductance may be defined as the ability of one coil of circuit to induce an emf in a nearby coil by induction when the current flowing in the first coil is changed. The action is also reciprocal i.e., the change in current flowing through second coil will also induce an emf in the first coil. The ability of reciprocal induction is measured in terms of the coefficient of mutual induction M.
The coefficient of mutual induction (M) can be determined from any one of the following three relations:
First Method:
In case the dimensions of the coils are given, the coefficient of mutual induction may be determined from the relation:
Second Method:
In case the magnitude of induced emf in the second coil for a given rate of change of current in the first coil is known, mutual inductance between the coils may be determined from the following relation:
Third Method:
In case the number of the turns of the coil and flux linking with this coil per ampere of the current in another coil is known, the mutual inductance of the coil may be determined from the following relation:
6. AC Excitation in Magnetic Circuits:
The magnetic circuits of transformers, ac machines and many other electromagnetic devices are excited from ac rather than dc sources. With dc excitation, the steady-state current is determined by the impressed voltage and the resistance of the circuit, the inductance entering only into transient processes (building- up and decaying processes of current at the switching on/off instants).
The magnetic flux in the magnetic circuit then adjusts itself in accordance with this value of current so that the relationship imposed by B- H or magnetisation curve is satisfied.
With ac excitation, however, inductance enters into the steady-state performance as well; the result for most magnetic circuits, although not for all, is that, to a close approximation, the flux is determined by the impressed voltage and frequency, and the magnetizing current must adjust itself in accordance with this flux so that the relationship imposed by the magnetisation curve is satisfied.
Except where preservation of linear relationship is of great importance, the normal working flux density in a magnetic circuit is kept beyond the linear portion of the magnetisation curve for the overall circuit (i.e., partially saturating the circuit).
This is done so as to affect the economic utilisation of magnetic material. Thus accurate analysis cannot be predicted on constant self-inductance. Equivalent circuits containing parameters that do remain substantially constant are used instead. The reactive effect of the time-varying flux on the exciting circuit can readily be shown from Faraday’s law,
e = N dɸ/dt
Consider an iron core excited by a winding having N turns and carrying a current of i amperes (Fig. 9.34). A magnetic flux ɸ is produced by the exciting current i.
Let the magnetic flux ɸ vary sinusoidally with time t as in:
ɸ = ɸmax sin 2 π f t … (9.17)
ɸmax being the maximum value of flux in the cycle and f is the supply frequency.
The induced emf in accordance with Faraday’s law is:
e = N dɸ/dt = 2 π f N ɸmax cos 2 π f i … (9.18)
And its effective or rms value is:
Erms = 2 π/√2 f N ɸmax = 4.44 f N ɸmax volts … (9.19)
The polarity of the emf must, in accordance with Lenz’s law, oppose the change in flux and therefore is as shown in Fig. 9.34 when the flux is increasing. Since the current produces the flux, the two may be considered in phase. From Eq. (9.18), the induced voltage leads the flux, and hence the exciting current by π/2 radians or 90°. The induced emf and the coil resistance drop oppose the applied voltage.
The resistance drop does not exceed a few per cent of the applied voltage in ac machines, most transformers, and many other electromagnetic devices. To a close approximation, resistance drop may be neglected and the induced emf E and applied voltage V may be considered equal in magnitude. The flux ɸmax is then determined by the applied voltage V in accordance with Eq. (9.19), even if maintenance of this flux requires a magnetising current far in excess of rated current for the device.
7. Magnetic Hysteresis:
If a magnetic substance is magnetised in a strong magnetic field, it retains a considerable portion of magnetism after the magnetic force has been withdrawn. The phenomenon of lagging of magnetisation or induction flux density behind the magnetising force is known as magnetic hysteresis.
Let a core of specimen of iron be wound with a number of turns of a wire and current be passed through the solenoid. A magnetic field of intensity H proportional to the current flowing through the solenoid is produced. Let magnetising force H is increased from zero to a certain maximum value and then gradually reduced to zero.
If the values of flux density B in the core corresponding to various values of magnetising force H are determined and B-H curves are drawn for increasing and decreasing values of magnetising force H then it will be observed that B-H curve obtained for decreasing values of H lies above that obtained for increasing values of H.
While decreasing the magnetising force H, when H is brought to zero the induction density B, is represented by OC and is called as residual magnetism. The power of retaining the residual magnetism is called the retentivity of the material.
Now if the direction of flow of current is reversed, the magnetising force H is reversed. Let the current be increased in the negative direction until the induction density B becomes zero. At this instant i.e. when B = 0, the demagnetising force H = OD, which is required to neutralize the residual magnetism, and is known as coercive force.
If the demagnetising force H is further increased to the previous maximum value and again gradually decreased to zero, reversed and further increased in original or positive direction to the maximum value, a closed loop ACDEFGA is obtained which is usually known as hysteresis loop or magnetic cycle.
It is to be noted from the hysteresis loop that B lags behind H. The two never attain zero value simultaneously.
Hysteresis is especially pronounced in materials of high residual magnetism, such as hard steel. In most cases, hysteresis is a liability as it causes dissipation of heat, waste of energy, and humming due to change in polarity and rotation of element magnets in the material.
Hysteresis loops for hard steel, wrought iron and cast steel and for alloyed sheet steel are shown in Fig. 9.37.
Loop (a) is for hard steel. Due to its high retentivity power and large coercive force, this material is well suited for permanent magnets. Since the area of hysteresis loop for hard steel is large, hard steel is not suitable for rapid reversals of magnetisation. Certain alloys of steel, aluminium and nickel known as alnico alloys are extremely suitable for permanent magnets.
Loop (b) is for wrought iron and cast steel which rises steeply. Hence these materials have high magnetic permeability and good retentivity, therefore, these materials are suitable for cores of electromagnets.
Loop (c) is for iron, low carbon steel, silicon alloys, permalloy or Mumetal sheets. Since the permeability of these materials is very high and hysteresis losses are very low, these materials are most suitable for transformer cores and armatures, which are subjected to rapid reversals of magnetisation. Silicon alloys and permalloy (78.5% Ni; 21% iron with small quantities of copper, molybdenum, chromium, cobalt and manganese etc.) are better for use as compared to iron and low carbon, steel.
8. Energy Expended in a Magnetic Cycle:
During each magnetic cycle, the energy expended in the specimen is proportional to the area of the closed loop.
Consider a ring of specimen of circumference I metres, cross-sectional area a metres2 and having N turns of an insulated wire. Let the current flowing through the wire be of I amperes.
Magnetising force, H = NI/l
Or I = Hl/N
Let the flux density at this instant be B.
Total flux through the ring ɸ
= B × a Webers
If the current is increased to increase the magenetising force H and induction density B, Induced emf in the coil, e’ = – Number of turns on coil x rate of change of flux-
According to Lenz’s law this induced emf will oppose the flow of current, therefore, in order to maintain the current I in the solenoid, the source of supply must have an equal and opposite emf-
Now al is the volume of ring and HdB is the area of elementary strip of B-H curve shown in Fig. 9.39 and ʃ HdB is the total area enclosed by the hysteresis loop.
Energy consumed/cycle = Volume of the ring × area of the loop … (9.20)
Energy consumed per cycle per cubic metre of volume = Area of the hysteresis loop
This energy expended in taking a specimen through a magnetic cycle is wasted and since it appears as heat, it is termed as hysteresis loss.
9. Energy Stored in a Magnetic Field:
When a coil is connected to an electric source, the current flowing in the circuit gradually increases from zero to its final value, and a magnetic field is established. Consequently, a portion of the electrical energy supplied by the electric source is stored as magnetic field, while remainder, supplied during the rise of current, is dissipated from the magnetising coil as heat.
After the magnetic field has been established, and the current has attained its maximum or steady value, any more energy given to it will be dissipated as heat. In other words, no additional energy is required to maintain the magnetic field, once the steady- state has reached. If, however, the circuit of a steady-state established magnetic field is broken, the energy stored in it will be spent in generating an induced emf or current.
Figure 9.40 depicts an iron-cored coil when the resistance r of the coil is lumped outside so that the exciting coil is devoid of any resistance (pure, lossless). Let the inductance of the coil be L Henrys and a current of i amperes be flowing through it at any instant t. At this instant the current is rising at the rate of di/dt amperes per second. This induces an emf e in the coil.
The induced emf in the coil is given by the expression:
Multiplying both sides of above equation by i, we have the power input to the coil:
Which is positive when both I and di/dt have the same sign, else it is negative? If the coil current were zero at t = 0 and has attained the value of I amperes at t = T, the energy input to the coil during this interval of T seconds is:
It should be noted that the total stored energy in the magnetic field depends upon the final or steady- state value of the current and is independent of the manner in which the current has increased or time it has taken to grow.
Equation (9.21) can be written as:
Correspondingly the stored energy is:
Equation (9.21) can also be written as:
Where S is the reluctance of the magnetic circuit and ɸ is the flux established in that magnetic circuit.
In case of an air-gap in the core, air-gap reluctance being far larger than that of the core, major portion of the field energy would reside in the air-gap.