Electrical Machines - Section 4
CATEGORIES OF ELECTROMECHANICAL MACHINES, AIR GAP AND INDUCED EMF
Section 4 focuses on the categories of electromechanical machines, the role of the air gap, and induced emf. It introduces different machine arrangements such as synchronous machines, permanent-magnet machines, reluctance motors, and switched-reluctance and stepper motors. It then explains why the air gap is an important region for magnetic energy storage. Finally, the section distinguishes between statically induced emf caused by changing current and dynamically induced emf associated with mechanical movement and position-dependent inductance.
1. Categories of Electromechanical Machines
Electromechanical machines can be classified according to how the magnetic field is produced and how the rotor or moving part interacts with it. The chapter discusses several important categories, including machines with stator and rotor windings, permanent-magnet machines, reluctance machines, and machines using switched stator currents.
1.1 One Stator Coil and One Rotor Coil
Consider a simple machine having one stator coil and one rotor coil. If the stator current is and the rotor current is , the electromagnetic torque is:
where:
T =-Lrairis sin sinθ
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Lra = mutual inductance
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ir= rotor current
-
is= stator current
-
θ= rotor position
At:
θ = 0°
the torque is zero even though the mutual inductance is maximum.
At:
θ = 90°
the mutual inductance becomes zero, while the torque reaches its maximum magnitude.
1.2 Synchronous Machine
A synchronous machine can be represented by two stator coils carrying suitable two-phase currents and a rotor carrying DC current.
The two-phase stator currents produce a rotating MMF. The rotor field interacts with this rotating magnetic field and produces torque.
When operating at synchronous speed, the torque can be unidirectional and constant.
1.3 Permanent-Magnet Machines
In a permanent-magnet machine, permanent magnets provide the required magnetic field.
The permanent magnet eliminates the need for a conventional rotor field winding in the arrangement discussed in the chapter.
1.4 Reluctance Motors
A reluctance motor does not require a rotor winding in the arrangement described.
Its operation is based on variation of magnetic permeance/reluctance with rotor position.
The rotor tends to move toward a position determined by the magnetic characteristics of the system.
1.5 Switched Reluctance and Stepper Motors
When stator currents are switched in a controlled sequence, the magnetic field changes from one position to another.
This principle is used in:
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Switched-reluctance motors
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Stepper motors
The switching of stator currents causes the rotor to move toward successive preferred magnetic positions.
2. Important Role of the Air Gap
The air gap is an essential part of an electrical machine because it separates the stator and rotor while allowing magnetic flux to pass between them.
The magnetic energy density is:
w = B²/2μ
where:
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w = magnetic energy density
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B = magnetic flux density
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μ = permeability
Since the permeability of air is much lower than that of the magnetic core, the air gap stores a significant amount of magnetic energy.
The chapter's example demonstrates that the air gap can store considerably more energy than the magnetic cores even though the core volume is much larger.
Key point
Maximum magnetic energy is stored in the air gap
The magnetic core primarily provides a low-reluctance path for the magnetic flux.
3. Statically Induced EMF
The general voltage equation given in the chapter is:
v₀(t)=ri + dλ/dt
Since:
λ=L(x)i
we can write:
v₀(t)=ri +L(x)di/dt + i dL(x)/dx dx/dt
The second term,
L(x)di/dt
represents the statically induced emf.
It occurs due to a change in current and therefore a change in flux linkage. It does not require mechanical movement.
4. Dynamically Induced EMF
The third term in the voltage equation is:
i dL(x)/dx dx/dt
This represents the dynamically induced emf, also called the speed emf.
For this component to exist:
dx/dt ≠ 0
and:
dl/dx ≠ 0
Therefore, mechanical movement and position-dependent inductance are required.
This component is particularly important in electromechanical energy conversion because it directly involves mechanical motion.
5. General Voltage Equation
The complete voltage equation can therefore be separated into three parts:
v₀=ri +Ldi/dt + i dL/dx dx/dt
Resistance voltage
ri
Statically induced emf
estatic = L di/dt
Dynamically induced emf
edynamic = i dL/dx dx/dt
Thus:
v₀=ri + estatic + edynamic
6. Conditions for Electromechanical Energy Conversion
For the position-dependent inductance mechanism, the chapter identifies three important requirements:
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The inductance must vary with position
dL/dx ≠ 0
2. Mechanical movement must occur
dx/dt ≠ 0
3. The coil must carry current
i ≠ 0
When these conditions are satisfied, the dynamic-emf term becomes significant:
i dL/dx dx/dt
10 MCQ Problems — Section 4
1. Which machine operates based on variation of magnetic reluctance with rotor position?
A. Transformer
B. Reluctance motor
C. Induction heater
D. DC generator
2. Which machine uses permanent magnets to provide the magnetic field?
A. Permanent-magnet machine
B. Reluctance motor
C. Transformer
D. Step-down transformer
3. Switched-reluctance motors operate through:
A. Continuous DC voltage only
B. Controlled switching of stator currents
C. Permanent heating
D. Mechanical springs only
4. The magnetic energy density is:
A. B/2μ
B. B²/2μ
C. 2Bμ
D. B²μ
5. The major region of magnetic energy storage in an electrical machine is:
A. Shaft
B. Bearing
C. Air gap
D. Terminal
6. Which term represents statically induced emf?
A. ri
B. L di/dt
C. i dL/dx dx/dt
D. Li²
7. Dynamically induced emf depends on:
A. Mechanical movement and changing inductance with position
B. Resistance only
C. Current only
D. Temperature only
8. The dynamically induced emf is:
A. L di/dt
B. ri
C. i dl/dx dx/dt
D. Li
9. For electromechanical conversion through the dynamic-emf mechanism, which condition is necessary?
A. i = 0
B. dL/dx = 0
C. dx/dt = 0
D. i ≠ 0, dL/dx ≠ 0, dx/dt ≠ 0
10. The complete voltage equation is:
A. v = ri
B. v= Ldi/dt
C. v = ri + Ldi/dt + i(dL/dx)(dx/dt)
D. v = Li²
Answer: 1(A), 2(A), 3(B), 4(B), 5(C), 6(B), 7(A), 8(C), 9(D), 10(C).
Important Equations of Section 4
One stator–one rotor coil
T = - Lrairis sin sin θ
Magnetic energy density
w = B²/2π
General voltage equation
v₀ = ri = dλ/dt
Expanded voltage equation
v₀ = ri + L di/dt + dx/dt
Static emf
estatic = L di/dt
Dynamic emf
edynamic = i dL/dx dx/dt
Complete form
v₀ = ei + estatic + edynamic
Calculation Problems — Section 4
Problem 1 — What is Torque of a Simple Machine?
A machine has:
Lra = 0.5H, ir = 4A, is = 34
and:
θ = 30°.
Calculate the electromagnetic torque.
Use:
T =-Lrairis sin sin θ
Problem 2 — Wat is Air-Gap Energy Density?
An electrical machine has an air-gap flux density of:
B = 1.2T.
Assuming:
μ₀ = 4π × 10⁻⁷ H/m
calculate the magnetic energy density.
Use:
w = B²/2μ₀
Problem 3 — What is Statically Induced EMF?
A coil has an inductance of:
L = 0.5H
and the current changes at:
di/dt = 8A/s.
Calculate the statically induced emf.
Use:
estastic = L di/dt
Problem 4 — What is Dynamically Induced EMF?
A coil carries a current of:
i = 5A
Its inductance changes with position at:
dL/dx = 0.04H/m
and the mechanical part moves with:
dx/dt = 3m/s.
Calculate the dynamically induced emf.
Use:
edynamic = i dL/dx dx/dt
Problem 5 — What is Total Induced Voltage?
A coil has:
R = 2Ω, i = 3A, L = 0.4H di/dt = 5A/s, dL/dx = 0.02H
Calculate the total voltage using:
v₀ = ri + L di/dt + i dL/dx dx/dt.