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Principles of Electromechanical Systems
In this chapter, we lead you through a study of the mathematics and physics of electrical machines. After completing the chapter, you should be able to
Review the basic principles of electricity and magnetism. Understand the concepts of reluctance and magnetic circuits. Understand the properties of magnetic materials. Understand the principles of transformers. Calculate values of voltage, current, and turns for transformers using simple formulas Draw the equivalent circuit for a real transformer and find its parameters. Define reactive power and apparent power. Understand the basic concepts of electromagnetomechanical systems. Identify electrical devices in an everyday setting and be able to describe their basic operating characteristics.
REFERENCES
- Stephan J. Chapman, Electric Machinery Fundamentals, Third Edition, McGraw-Hill, 1999.
- Stephan J. Chapman, Electric Machinery and Power System Fundamentals, McGraw-Hill, 2002.
- J. R. Cogdell, Foundations of Electric Power, Prentice Hall, 1999.
- Mulukutla S. Sarma, Electric Machines, Second Edition, PWS Publishing Company, 1996.
- Riadh W. Y. Habash, Electromagnetic Fields and Radiation: Human Bioeffects and Safety, Marcel Dekker, 2002.
- Charles I. Hubert, Electric Machines: Theory, Operation, Applications, Adjustment, and Control, Second Edition, Prentice Hall, 2002.
- David K. Cheng, Field and Wave Electromagnetics, Second Edition, Addison Wesley, 1989.
- Stephen L. Herman, Electrical Studies for Trades, Second Edition, Delmar,
16.1 ELECTRICITY AND MAGNETISM
A connection between electricity and magnetism was discovered accidentally by H. C. Orsted, a Danish physicist, over 100 years ago, who noticed that a compass needle is deflected when brought into the vicinity of a current carrying wire. Thus, currents induce in their vicinity magnetic fields. As explained in Chapter 8, Faraday, who found that changing magnetic fields though loops of wire could cause currents to be induced, discovered a further connection between electricity and magnetism. Magnetism is one of the most phenomena in the electrical field. It is the force used to produce most of the electrical power in the world. The term magnetism is derived from Magnesia, the name of a region in Asia where lodestone, a naturally magnetic iron ore, was found in ancient times. The ancient Greeks and also the early Chinese knew about strange and rare stones with the power to attract iron. A steel needle stroked with such a “lodestone” became “magnetic” as well and later the Chinese found that such a needle, when freely suspended, pointed north south. The magnetic compass soon spread to Europe. Columbus used it when he crossed the Atlantic Ocean, noting not only that the needle deviated slightly from exact north (as indicated by the stars) but also that the deviation changed during the voyage. Around 1600 William Gilbert, physician to Queen Elizabeth I of England, proposed an explanation: the Earth itself was a giant magnet, with its magnetic poles some distance away from its geographic ones. It is evident from Figure 16-1 that in the upper (northern) half of the earth, the magnetic field is directed toward the earth; in the lower (southern) half, the field is directed away from the earth.
Figure 16-1 Earth may be thought of as a dipole magnet.
Imaginary dipole
16.2.2 Power
Power ( P ) is the rate of doing work. It may be defined as
dt
dW P =
For the special case of constant work, power is
t
W
P =
Power is measured in joules per second (watts). For the special case of a constant force, which is collinear with the direction of motion, power is
Fv dt
dr Fr F dt
d dt
dW P ( ) =
where v is one-dimensional linear velocity defined as
dt
dr v = (^) (16.8)
For rotational motion, assuming constant torque, we write
( θ ) T ω dT
dθ T T dt
d dt
dW P (^) =
where ω is the angular velocity
dt
dθ ω = (^) (16.10)
Example 16-
If an 80-kg person climb to the tope of a building (10 m) during a time of 30 seconds. Find the work and power.
Solution: In the MKS system of measure, each kg of mass exerts 9.8 newtons of force at the earth surface. Using Equation (16.2), we get
10 m 7.84 kNm kg
9.8N
= 80 kg× × =
W = Fr
A newton-meter is a joule
W = 7.84 kJ
To calculate the power, use Equation (16.6)
s
J
30s
7.84 kJ P = =
16.3 ELECTRICAL POWER
16.3.1 Sinusoidal Power Equation
We knew from the previous section that
t
W
P =
Since
t
Q
i =
then,
v i
t
Q
Q
W
t
W
P
= ×
= = × (16.11)
The voltage is used as the reference, assigning it a phase angle of 0o.
v = VP sin ( wt ) (16.12)
The current may be leading or lagging this voltage
i = IP sin( wt + θ) (16.13)
flux as shown in Figure 16-2. The MMF can be compared with EMF, the flux ( φ)
can be compared to current ( I ), and the reluctance (ℜ) of magnetic field is the counterpart of electrical resistance ( R )
Figure 16-2 Analogy between magnetic and electric circuits.
F
Where, φ = magnetic flux of circuit, measured in weber (Wb).
F = magnetomotive force of circuit, measured in ampere-turn. ℜ = reluctance of a circuit, measured in ampere-turn per weber (Wb).
The reluctance is represented by the following formula
μA
l ℜ = (16.20)
where l is the mean path length of the core measured in meters, A is the cross-
sectional area of the core, and μ is the magnetic permeability of material.
Reluctances in a magnetic circuit obey the same rules as resistances in an electric circuit. For example, the reluctance of a number of reluctances in series is given by
ℜeq =ℜ 1 +ℜ 2 +ℜ 3 +.. +ℜn (16.21)
and for reluctances in parallel
eq 1 2 3 n
F (^) m=NI^ φ^ ℜ
I
If there is an air gap in the flux path in a core, the effective cross-sectional area of the air gap will be larger than the cross-sectional area of the iron core on either side. The extra effective area is caused by the “fringing effect” of the magnetic field at the air gap as shown in Figure 16-3. Fringing occurs because the reluctances of different paths available near the air gap are quite comparable to each other, and the flux lines spread out. The effect of fringing increases with the length of the air gap. The effect of fringing is taken into consideration in magnetic circuit calculations by recognizing that the effective area of the air gap is greater than the actual area. However, the effect may be ignored when the air gap is small.
Figure 16-3 An air gap in the flux path in a core.
Table 16-1 provides an analogy between electric circuits and magnetic circuits.
Table 16-1 Analogy between Electric and Magnetic Circuits
Electrical Quantity Magnetic Quantity Electric field intensity E, V/m Voltage v , V Current i , I Current density J, A/m^2 Resistance R , Ω
Conductivity σ, 1/Ω-m
Magnetic field intensity H, A-turns/m Magnetomotive force Fm , A-turn
Magnetic flux φ, Wb
Magnetic flux density B, Wb/m Reluctance ℜ, A-turn/Wb
Permeability μ, Wb/A-m
Ferromagnetic core
Ferromagnetic core
Fringing Fringing
A
l
A
l
A
l
3 3
2 2
1 1
Now, we write the two loop equations
( ) 1 1 2 2 2 (^12 )^11
1 1 1 1 2 12
N I N I
N I
Solving these two simultaneous equations, we obtain
1 2 1 2 1 2
1 1 2 2 2 1 (^1) ℜℜ +ℜℜ +ℜℜ
N I N I
16.5 MAGNETIC MATERIALS
All materials show some magnetic effects. With the exception of ferromagnetic materials these effects are weak. Depending on their magnetic behavior, materials may be classified according to some of their basic magnetic properties, particularly whether or not they are magnetic and how they behave in the vicinity of an external magnetic field.
16.5.1 Non-magnetic Materials
Most materials we encounter have no obvious magnetic properties - they are said to be non-magnetic. In these materials, the magnetic fields of the individual atoms are randomly aligned and thus tend to cancel out, as shown in Figure 16-5. The relative permeability of such materials is relatively constant (around 1, more or less) and independent of the applied field. Examples of nonmagnetic materials are copper, brass, silver, water, air, aluminum, and biological tissues.
Figure 16-5 Non-magnetic material.
16.5.2 Permanent Magnetic Materials
Permanent magnets are magnets that do not require any power or force to maintain their field. The basic purpose of any magnet is to store energy or to convert energy from one form to another. In a permanent magnet, however, the magnetic fields of the individual atom are aligned in one preferred direction, giving rise to a net magnetic field, as shown in Figure 16-6.
Figure 16-6 Permanent magnet.
16.5.3 Ferromagnetic Materials
In this material, there are domains in which the magnetic fields of the individual atoms align, but the orientation of the magnetic fields of the domains is random, giving rise to no net magnetic field. This is illustrated in Figure 16-7 (a). A significant property of ferromagnetic materials is that when an external magnetic field is applied to them, the magnetic fields of the individual domains tend to line up in the direction of this external field. This is because of the nature of magnetic forces, which causes the external magnetic field to be enhanced. This is illustrated in Figure 16-7 (b). The relative permeability of ferromagnetic materials varies over a wide range for various applied fields. Examples of these materials are iron, nickel, cobalt, and mumetal. The advantage of using a ferromagnetic material for cores in electric machines and transformers is that of getting more flux for a given MMF with iron than with air. Another area where ferromagnetic materials are employed is in magnetic recording devices, such as for cassette tapes, floppy discs for computers, and the
16.6 FARADAY’S LAW
In 1831, Michael Faraday in London found that a magnetic field could produce current in a closed circuit when the magnetic flux linking the circuit is changing. This phenomenon is known as electromagnetic induction. Faraday concluded from his experiment that the induced current was proportional, not to the magnetic flux itself, but to its rate of change. Consider the closed wire loop shown in Figure 16-8. A magnetic field with magnetic flux density B is normal to the plane of the loop. If the direction of B is upward and decreasing in value, a current I will be generated in the upward direction. If B^ is directed upward but its value is increasing in magnitude, the direction of the current will be opposite.
Loop
B B (decreasing) (increasing) (a) (b)
Figure 16-8 Induced currents due to magnetic flux density B.
When B is decreasing, the current induced in the loop is in such a direction as to produce a field, which tends to increase B [Figure 16-8 (a)]. However, when B is increasing, the current induced in the loop is in such a direction as to produce a field opposing B [Figure 16-8 (b)]. Therefore, the induced current in the loop is always in such a direction as to produce flux opposing the change in B. This phenomenon is called Lenz’s law. As the magnetic field changes, it produces an E field. Integrating E field around a loop yields an electromotive force , or e , measured in volts as follows
e = (^) ∫ E. d I (16.23)
e appears between the two terminals, if the loop is open circuit. This is the basic
for the operation of an electric generator. A quantitative relation between the EM force induced in a closed loop and the magnetic field producing e can be developed. This is represented by
e = - dt
d φ (16.24)
where φ = (^) ∫∫ B. d s is the total flux in webers (Wb). If a coil has N turns and
if the same flux passes through all of them, then we write Equation (16.24) as
dt
dφ e = - N
Equation (16.25) may be written as
e = dt
d − (^) ∫∫ B. ds (^) (16.26)
where d s is surface element measured in square meter (m^2 ) and t is time measured in seconds (s). Although Joseph Henry in Albany, New York also discovered the result shown in Equation (16.25), the credit is still attributed to Faraday. Both Faraday and Henry discovered the above finding independently at about the same time, however, it is known as Faraday’s law of induction. Faraday’s law is well known through its importance in motors, generators, transformers, induction heaters, and other similar devices. Also, Faraday’s law provides the foundation for the electromagnetic theory. The total time derivative in Equation (16.26) operates on B , as well as the differential surface area d s. Therefore, e can be generated under three conditions: a time-varying magnetic field linking a stationary loop; a moving loop with a time-varying area; and a moving loop in a time-varying magnetic field.
Example 16-
Consider a coil of wire wrapped around an iron core. If the flux in the core is given by the equation
φ =0.5 sin ω t Wb
Figure 16-9 A simple transformer.
One of the coils in the conventional transformer is called primary winding , which is connected to the input AC power supply. A second coil is called secondary winding , which supplies electric power to the driven load. In Figure 16-9, one winding of the transformer has been connected to an AC supply, and the other winding has been connected to a load. As current increases from zero to its peak positive point, a magnetic field expands outward around the coil. When the current decreases from its peak point toward zero, the magnetic field collapses. When the current increases toward its negative peak, the magnetic field again expands, but with an opposite polarity of the previous one. The field once again collapses when the current decreases from its negative peak toward zero. This regular expanding and collapsing of the magnetic field cuts the windings of the primary and induces a voltage into it. The induced voltage opposes the applied voltage and limits the current flow of the primary. When a coil induces a voltage into itself, it is known as self- induction. Another term that we should be familiars with that plays an important role in understanding transformers is called mutual inductance. Mutual inductance is when you take two coils, apply current to one coil only, do not matter which one as long as it is AC current, and then place close together both coils (as long as they do not electrically touch each other) and mutual inductance will take place. This is where the expanding and collapsing flux magnetic fields of the first coil with the current will cut across the winding of the second coil without the current, and voltage will be induced in the second coil. In order to restrict most of the flux to a defined path linking the windings, the core is usually made of ferromagnetic material. In order to reduce the losses caused by eddy currents in the core, the core is usually comprised of a stack of thin laminations.
Secondary leakage flux
Load ZS
Primary leakage flux
VP e (^) P
NP turns NS turns
eS vS
16.7.3 Ideal Transformer
If a time-varying voltage source is connected to the primary winding, then by
virtue of Faraday’s law, a corresponding time-varying flux d φ /dt is established
and a counter (induced) voltage eP is developed in the primary coil L (^) P
dt
d v (^) P eP NP
Where NP is the primary winding turns. Remember here that the resistance drop in the winding is neglected, so the counter voltage eP equals the applied voltage at the primary winding vP. As seen from Figure 16-8, the flux links also coil L (^) S and a counter voltage eS is induced across the output coil
dt
d v (^) S eS NS
Where NS is the secondary winding turns. A basic law concerning transformers is that all values of a transformer are proportional to its turns ratio. Turns ratio is the ratio of the voltages across the primary and secondary windings of an ideal transformer. By combining Equation (16.27) and Equation (16.28), we will get the turns ratio
a N
N
v
v S
P S
P = =
Where a is the turns ratio. Let us now consider the case in which the waveforms of the applied voltage and flux are sinusoidal. In this case, the flux is given by
φ = φmaxsin ω t (16.30)
Where φmax is the maximum value of the flux, and ω is 2 π f. Accordingly,
Equation (16.30) is written as
ω N t dt
d
e P NP P φ ω
= = maxcos (16.31)
The RMS value of the induced voltage is given by
Figure 16-10 Equivalent circuit viewed from the primary winding.
VP
Load Z (^) L
VP
L S
P P L Z N
N
Z
2
L S
P P L (^) N Z
N
Z
2
Example 16-
A step-down transformer has 300 turns of wire on the primary and 60 turns of wire on secondary. This is a ratio of 5:1 (300/60 = 5). Assume that 120 V is connected to the primary winding. What is the voltage of the secondary winding?
Solution: Use Equation (16.29) to find the voltage of the secondary winding
24 V
S
S
S
P S
P
v
v
N
N
v
v
Example 16-
Assume that a load of 5 Ω is connected to secondary winding of the transformer of Example 16-6. Calculate the current flow in the secondary and primary windings.
Solution: The current flow of the secondary can be computed using Ohm’s law since the voltage and impedance
4 A
R
V
I =
Combine Equation (16.29) and Equation (16.34), which gives
P
S S
P i
i v
v
Apply Equation (16.31) to find the current in the primary winding
