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Introduction to Magnetic Systems
Magnetic systems are important in electrical machines, because the electromechanical energy conversion process involves the conversion of electrical energy to magnetic energy and then to mechanical energy i.e.:
Figure 1: Electromechanical Energy Conversion Process
Electromagnetic energy conversion depends on the production of two magnetic fields, which interact to develop torque in the machine. The two magnetic fields can be produced by: a. Electric currents in coils b. Electric currents in coils and permanent magnets Hence an analysis of the energy conversion process involves: a. Materials used to produce magnetic fields b. Development and analysis of magnetic circuits associated with the system.
Before going further, it is essential to review some fundamental relationships.
1. CHARGE AND CURRENT q = Quantity of electricity or charge i = Current = Rate of flow of charge
∴ i = dq / dt A
and q =∫ idt
2. FORCE CAUSED BY MAGNETIC FIELD
A particle carrying a charge ‘ q ’ and moving at a velocity ‘ v ’ in a magnetic field of
strength ‘B’ is subjected to a force F, (Fig. 2) where F r^ = q ( v rΛ B r)
Electrical Energy
Magnetic Energy
Mechanical Energy
Figure 2
Figure 3
The magnitude of the force F is given by:
F = qvB sin θ ,
where θ = angle between B and v,
and F^ r^ is perpendicular to the plane in which v^ r^ and B^ r^ lies. The direction of F^ r^ is given by the right hand screw rule. The moving charge ‘ q ’ may form part of a current flowing in a conductor of length ‘ l ’. For an elementary length of conductor dl , measured in the direction of the
current, Fig. 3, and where vectors dl
r and B^ r^ are orthogonal, dF = dq .( dl / dt ). B =( dq / dt ). dl. B = i. dl. B
For the general case: dF r^ = i ( dl rΛ B r)≡ dl ( l rΛ B r)
And for a conductor of length ‘ L ’
F i ( L B ) L ( l B r ) r r r r = Λ ≡ Λ
Magnetic Moment = p r^ m = lA^ r
where A = a vector, normal to the plane at the loop in the direction of the magnetic field and its magnitude is given by the area out the loop.
For an ‘n’ turn coil: p (^) m NiA r r =
The current-carrying coil when placed in a uniform magnetic field of strength B^ r^ ,
develops a torque which turns into alignment with vector B^ r^. The torque is given by:
T pm B
r (^) r r = Λ T r^ = Nl ( A rΛ B r )
The direction l is given by the right hand screw rule.
In the above equation, it seems that the direction of vector A^ r^ is given by the direction of the field produced by the current carrying loop.
Then the direction of A^ r^ depends on that of i. Therefore the equation can be rewritten as:
T r^ = NA ( i rΛ B r ) The above equation shows that torque is developed in a machine by the interaction of the field produced by a current-carrying conductor and the magnetic field in which this conductor is immersed. The torque is proportional to the cross product of the current and existing field vectors. This equation forms the basis of vector analysis in machines. Example A closely packed 100 turn coil of diameter 10cm is placed in a magnetic field of flux density 0.1T. The coil carries a current of 2A. Determine the torque developed by the coil in the position where the flux lines are perpendicular to the diameter of the coil. Solution
Magnetic moment of coil pm is given by:
pm = NAl = 100 Π( 0. 12 / 4 ) 2 T pm B pmB k
r (^) r r r r r = Λ = sin θ. T r^ = p r mB r k^ r T r^ = p r mB^ r= 100 Π( 0. 12 / 4 ) 2 × 0. 1 Nm
4. MAGNETIC FLUX AND FLUX DENSITY
Magnetic fluxΦ , is the integration of flux density B^ r^ through a closed path as in Fig. 6.
Figure 6: Magnetic Flux Density through a Closed Path
∴ Φ =∫ AB dA
r r .
This expression is the surface integral of the scalar product of flux density and area over any surface bounded by the closed path.
r is normal to the elementary area dA.
- Vector B^ r^ is the flux density through dA when B^ r^ is perpendicular to d A
r . Φ= BA
5. MAGNETIC FIELD INTENSITY Magnetic flux may be produced by: a. Electric Currents b. Permanent Magnets For magnetic flux produced by electric currents, there exists a physical property intermediate between current and flux density. This property is called magnetic field intensity ‘ H ’. The relationship is as follows:
- Current i produce magnetic field intensity H
- and H produces flux density B. In a vacuum, the relationship between B and H is given by:
Then applying:
∫ H^ dl =∫ AJd^ A
r (^). r r. r
H ( 2 Π r )= i
∴ H = i / 2 Π r
6. INDUCED ELECTROMOTIVE FORCE AND INDUCTANCE
Faraday’s Law: If the flux linking a coil changes, then an emf is induced in the coil. The flux linking the coil may be changed by: a. Moving the coil in the magnetic field b. Changing the size of the coil or c. Change the flux density
N.B.: Flux linkage – flux passing through the coil. The flux linking the coil may be as a result of a current flowing through the coil due to an applied potential difference. If the current i flowing through the coil is increased by adjusting the source, the flux linking the coil will increase and consequently an emf will be induced in the coil. By Lenz’s Law, the direction of the induced emf will oppose the increase of coil current.
∴ e = d Φ/ dt
The above equation refers to a single turn coil. For an ‘N’ turns coil. e = N ( d Φ/ dt )
Flux Linkage λ = N Φ
∴ e = dλ / dt
