Electrical Machines - Section 2
Magnetic Field Energy and Co-Energy
Magnetic-field energy is a fundamental concept in electromechanical energy conversion. When current flows through a coil, it establishes a magnetic field around it, and energy is stored in that field. For a linear magnetic system, the relationships among magnetomotive force (MMF), flux, permeance, inductance, field energy, and co-energy can be expressed using simple equations. These concepts are important because the stored magnetic energy and co-energy are later used to determine the force and torque produced in electromechanical devices.
Q1. What is Magnetic Linear System?
A1. A magnetic system is considered linear when the relationship between MMF and flux is linear.
The basic relationship is: F = N.i
where:
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F = magnetomotive force (MMF) in ampere-turns
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N = number of turns
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i = current in the coil
The flux can be expressed as: ϕ = F.P
where is the permeance of the magnetic circuit.
Q2. What is the Permeance of a Magnetic Circuit?
A2. Permeance is the measure of how easily a magnetic circuit permits magnetic flux to pass through it.
For a uniform magnetic path: P = µ0.µr.A/lm
where:
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µ0 = permeability of free space
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µr = relative permeability
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A = cross-sectional area
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lm = mean magnetic path length
A higher permeance allows a greater amount of flux to be produced for the same MMF.
Q3. What is Inductance of a Coil?
A3. The inductance of a coil is related to its number of turns and the permeance of the magnetic circuit.
The basic relationship is: L = N².P
It can also be written as: L = N.ϕ/i
Since, flux linkage is: λ = N.ϕ
we obtain: L = λ/i
where:
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L = inductance
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λ = flux linkage
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N = number of turns
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ϕ = magnetic flux
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i = current in the coil.
Q4. How to calculate Energy Stored in the Magnetic Field?
A4. When current is supplied to an inductive coil, energy is stored in its magnetic field.
For a linear magnetic system: Wfld = 0.5*L.i²
Using L = N².P and F = N.i,
this equation can be written as, Wfld = 0.5*F².P = 0.5*F.ϕ
These equations are useful for calculating the energy stored in a magnetic system.
Q5. How to calculate Energy from the F–ϕ Characteristic?
A5. For a linear magnetic system, the relationship between MMF and flux is a straight line.
The field energy is represented by the area under the characteristic:
For a linear system, this area is triangular. Therefore: Wfld = 0.5*F1.ϕ1
Thus, the graphical area under the F-ϕ curve represents the energy stored in the magnetic field.
Q6. What is Magnetic Field Co-Energy?
A6. Co-energy is another useful quantity associated with a magnetic system.
It is defined from the F-ϕ characteristic as:
For a linear magnetic system: W'fld = 0.5*F1.ϕ1
Therefore, for a linear system: Wfld = W'fld = 0.5*L.i²
Difference Between Field Energy and Co-Energy
Although field energy and co-energy have the same numerical value in a linear system, they are conceptually different quantities.
Field energy is associated with the area under the F-ϕ characteristic.
Co-energy is associated with the complementary area of the F-ϕ characteristic.
For nonlinear magnetic systems, field energy and co-energy generally do not have the same numerical value. The equality emphasized in this section applies to the linear case.
Q7. What are the Energy in Terms of Electrical Parameters?
A7. The most commonly used equation for a linear inductor is: Wfld = 0.5*L.i².
Therefore, if the inductance and current are known, the stored magnetic energy can be calculated directly.
For example, if: L = 2H, i = 3A.
then, Wfld = 0.5*(2)*(3)² = 9J.
Thus, the magnetic field stores 9 J of energy.
Practice MCQ
1. The MMF produced by a coil is given by:
A. F = N/i
B. F = N.i
C. F = i/N
D. F = N + i
2. The relationship between flux and MMF is:
A. ϕ = F/P
B. ϕ = F.P
C. ϕ = F + P
D. ϕ = P/F
3. The permeance of a uniform magnetic circuit is:
A. lm/(µ0.µr.A)
B. µ0.µr.A/lm
C. µ0.µr.A.lm
D. A/µ0.µr.lm
4. The inductance of a coil in terms of permeance is:
A. L = N.P
B. L = N/P
C. L = N².P
D. L = P/N²
5. Which equation gives the energy stored in a linear magnetic field?
A.W = L.i
B. W = L.i²
C. W = 0.5*L.i²
D. W = 0.5*L/i²
6. For a linear characteristic, the stored field energy is represented by:
A. The area of a rectangle
B. The area of a triangle under the characteristic
C. The slope of the characteristic
D. The intercept of the characteristic
Answer: 1. B, 2. B, 3. B, 4. C, 5. C, 6. B.
Practice Problems
Problem 1 — Calculate MMF
A coil has 800 turns and carries a current of 1.5 A. Calculate the magnetomotive force produced by the coil.
Use: F = N.i
Problem 2 — Calculate Permeance
A magnetic circuit has:
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Cross-sectional area, A = 4 x 10^-4 m²
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Magnetic path length, lm = 0.2m,
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Relative permeability, µr = 1000,
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µ0 = 4π x 10^-7 H/m
Calculate the permeance of the magnetic circuit.
Use: P = µ0.µr.A/lm
Problem 3 — Calculate Inductance
A coil has 500 turns and the magnetic circuit has a permeance of 2 x 10^-6 Wb/AT.
Calculate the inductance of the coil.
Use: L = N².P
Problem 4 — Calculate Magnetic Field Energy
An inductor has an inductance of 0.8 H and carries a current of 5 A. Calculate the energy stored in its magnetic field.
Use: Wfld = 0.5*L.i²
Problem 5 — Calculate Energy Using MMF and Flux
A magnetic circuit has an MMF of 1200 AT and produces a flux of 0.004 Wb.
Calculate the magnetic field energy for a linear magnetic system.
Use: Wfld = 0.5*F.ϕ