Magnetic Flux and Magnetic Flux Linkage

8 Key excerpts on "Magnetic Flux and Magnetic Flux Linkage"

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  Engineering Science

  • W. Bolton(Author)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  Chapter 14 Magnetic flux

  14.1 Introduction

  This chapter follows on from the discussion of magnetism in Chapter 10 and considers in more detail electromagnetic induction. Magnetic lines of force can be thought of as lines along which something flows, this being termed flux . When the magnetic flux linked by a coil changes then an e.m.f. is induced.

  We also look at the effect of the materials through which lines of magnetic flux pass. This is important since most devices employing magnetism involve the use of materials such as iron or steel in their construction. When any material is placed in a magnetic field, the extent to which the magnetic field permeates the medium when compared with what would happen in a vacuum is known as the relative permeability. For a material termed ferromagnetic, such as iron, there is a tendency for the lines of magnetic flux to crowd through it and it has a high relative permeability (Figure 14.1(a) ). An important consequence of this high permeability of iron is that an object surrounded by iron is almost completely screened from external magnetic fields as the magnetic flux lines crowd through the iron (Figure 14.1(b) ).

  Finally in this chapter we look at the forces experienced by current-carrying conductors when in magnetic fields, this being the basic principle behind d.c. motors.

  Figure 14.1 (a) A piece of iron in a magnetic field, (b) screening

  14.2 Electromagnetic induction

  We can represent Faraday’s law and Lenz’s law for electromagnetic induction (see Chapter 10 ) as:

  induced e.m.f. e ∝ – (rate of change of flux Φ with time t )

  The minus sign indicates that the induced e.m.f. is in such a direction as to oppose the change producing it. We can put the constant of proportionality as 1 and write rate of change of flux as dΦ /dt :

  The unit of flux is the weber (Wb). If the flux linked changes by 1 Wb/s then the induced e.m.f. is 1 V. For a coil with N turns, each turn will produce an induced e.m.f. and so the total e.m.f. will be the sum of those due to each turn and thus:

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  Electric Machines and Drives

  • Ned Mohan(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  From Equation 5.32, the total flux linkage of the coil can be written as λ ¼ ðL m þ L ‘ Þi (5.33) Hence, from Faraday’s Law in Equation 5.24, eðt Þ ¼ L ‘ di dt þ L m di dt |ffl{zffl} e m ðtÞ (5.34) This results in the circuit of Figure 5.12a. In Figure 5.12b, the voltage drop due to the leakage inductance can be shown separately so that the voltage induced in the coil is solely due to the magnetizing flux. The coil resistance R can then be added in series to complete the representation of the coil. 5.7.1 Mutual Inductances Most magnetic circuits, such as those encountered in electric machines and transformers, consist of multiple coils. In such circuits, the flux established by the current in one coil partially links the other coil or coils. This phenomenon can be described mathematically by means of mutual inductances, as examined in circuit theory courses. Mutual induc- tances are also needed to develop mathematical models for dynamic analysis of electric machines. Since it is not the objective of this book, we will not elaborate any further on the topic of mutual inductances. Rather, we will use simpler and more intuitive means to accomplish the task at hand. v(t) + − R L l i(t) e m (t) e(t) + − + − di L l dt e m (t) e(t) + − + − + − i(t) L m L l (a) (b) φ m FIGURE 5.12 (a) Circuit representation; (b) Leakage inductance separated from the core. 82 Electric Machines and Drives: A First Course 5.8 TRANSFORMERS Electric machines consist of several mutually-coupled coils where a portion of the flux produced by one coil (winding) links other windings. A transformer consists of two or more tightly-coupled windings where almost all of the flux produced by one winding links the other windings. Transformers are essential for transmission and distribution of electric power. They also facilitate the understanding of ac motors and generators very effectively.

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  Electrical Machines and Their Applications

  • Ali Mehrizi-Sani, Turan Gonen(Authors)
  • 2024(Publication Date)
  • CRC Press

    (Publisher)

  λ (lambda) and expressed as

  λ = N × Φ   Wb

  (3.17)

  _________________ 1 Older units of magnetic flux density (i.e., the flux per unit area) that are still in use include lines/in.2 , kilolines/in.2 , and gausses (G). Note that 1 G = 1 Mx/cm2 and 1 T = 10 kG = 104 G. Therefore, if a flux density is given in lines/in.2 , it must be multiplied by

  1.55 ×

  10

  − 5

  to convert it to Wb/m2 or T.

  2 Permeability, based on Equation 3.8, can be defined as the ratio of change in magnetic flux density to the corresponding change in magnetic field intensity. Therefore, in a sense, permeability is not a constant parameter but depends on the flux density or on the applied mmf that is used to energize the magnetic circuit.

  3 The SI unit for magnetic flux is Webers (Wb). The older unit of flux was the line or Maxwell. Thus, 1 Wb = 108 Mx = 108 lines = 105 kilolines.

  3.4 Magnetic Circuits

  Consider the simple magnetic core shown in Figure 3.6a , by substituting Equation 3.5 into 3.15,

  Φ = B × A =

  μ N I A

  l

  =

  N I

  l / μ A

  (3.18)

  =

  N × I

  R

  =

  F R

  (3.19)

  from which

  F = Φ × R

  (3.20)

  where R is the reluctance of the magnetic path, and therefore

  R =

  l

  μ A

  (3.21)

  for uniform permeability μ, cross-sectional area A, and mean path length l of the magnetic circuit. The reluctance1 can also be expressed as

  R =

  F Φ

  (3.22)

  _________________ 1 In the SI system, no specific name is given to the dimension of reluctance except to refer to it as so many units of reluctance. One can observe from Equation 3.22 that its real dimensions are ampere-turns/weber. In some old literature, the word rels has been used as the unit of reluctance.

  The reciprocal of the reluctance is called the permeance of the magnetic circuit and is expressed as

  P =

  1 R

  (3.23)

  Therefore, the flux given by Equation 3.19 can be expressed as

  Φ = F × R

  (3.24)

  In many aspects, the electric and magnetic circuits are analogous. For example, notice the analogy between the electric circuit shown in Figure 3.6b and the magnetic circuit shown in Figure 3.6c . The flux in a magnetic circuit acts like the current in an electric circuit. The reluctance in the magnetic circuit can be treated like the resistance1 in the electric circuit, and the mmf in the magnetic circuit can be treated like the emf in the electric circuit. Equation 3.20 is often referred to as Ohm's law of the magnetic circuit. However, electric and magnetic circuits are not analogous in all respects. For instance, energy must be continuously provided when a direct current is flowing in an electric circuit, whereas in the case of a magnetic circuit, once the flux is established, it remains constant. Similarly, there are no magnetic insulators, only electric insulators

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  Circuit Analysis with PSpice

  A Simplified Approach

  • Nassir H. Sabah(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  It is independent of which coil carries the current. 1 1 2 2 2 0° A 1 0° A FIGURE 9.6 Figure for Primal Exercise 9.2. – + v SRC1 Coil 1 Coil 2 R v 2 + – FIGURE 9.7 Figure for Primal Exercise 9.3. Linear Transformer 241 That the flux linkage per unit current does not depend on which of two magnetically coupled coils carries the current is a consequence of M 12 and M 21 being equal. The definition of mutual inductance is seen to be a general- ization of the definition of the inductance of a single coil, this being the flux linkage in the coil per unit current in the coil itself. In the case of two magnetically coupled coils, the mutual inductance is the flux linkage in one coil per unit current in the other coil. To distinguish the two types of inductance when both are present, the inductance L of a coil associated with current in the coil itself is referred to as the self-inductance. Just as L is an inherently positive quantity, so is M. 9.2.1 Coupling Coefficient When two coils are magnetically coupled, not all the flux that links one coil links the other coil. This is par- ticularly true for two coils in a nonmagnetic medium (Figure 9.8a). The flux due to i 1 in coil 1 extends in three dimensions around this coil but only a relatively small fraction of this three-dimensional flux links coil 2. Only this flux is shown in Figure 9.8a and in its two- dimensional representation in Figure 9.8b. In contrast, when the coils are coupled through a core of high permeability (Figure 9.9a), practically all the flux origi- nating from one coil links the other coil. The flux that links a current-carrying coil, but not another coil coupled to it, is the leakage flux of the current-carrying coil. It is seen that the leakage flux is relatively large in the case of coupling through a medium of low permeability (Figure 9.8a) and is rela- tively small for coupling through a core of high perme- ability (Figure 9.9a), where the leakage flux of coil 1 is denoted as ϕ 11leak .

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  Fundamentals of Electrical Engineering, Part 1

  • S. B. Lal Seksena, Kaustuv Dasgupta(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  Prerequisite knowledge Basic concept of electric circuit Magnetostatic 6.1 Introduction Our discussion on electrical engineering has been so far pivoted upon two major aspects. First, the electrical and magnetic field and the correlation between them. Second, the different types of electrical circuits through which a continuous flow of electric charge can be established. We by this time have acquired the knowledge that the electric and magnetic fields complement each other. They often co-exist in a material. The most obvious supposition from the above discussion is the analogy between magnetic and electric field knowing the facts of electrical circuit we can imagine the existence of magnetic circuit following the similar laws to electric circuit laws. In the first half of seventeenth century the idea of magnetic circuit was developed. When two magnetic coils are placed in close vicinity to each other, the coils get magnetically coupled. The same flux is linked with both coils. Thus, there must be a magnetic connection between two magnetically coupled coils. The connection between the coils is certainly not via an electrical circuit. The magnetic connection is made by another circuit. There is no material link between two magnetically coupled coils. So no electron flow is practically possible. We can only think of a circuit through which magnetic flux can be flowed and linked the two coils. This circuit is named as magnetic circuit. In this chapter, we shall discuss more on different phenomena and laws of magnetic circuit. 6.2 Concept of Magnetic Flux Flow We have known so far that in an electric circuit we need to have at least one active source and at least on passive element. A potential difference is created across the passive element by the active source due to which an electric current flows through the passive component. Let us depict such a basic 6 MAGNETIC CIRCUIT Fig. 6.1: Basic electric circuit E R I

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  Basic Electrotechnology

  • R A Ashen(Author)
  • 2016(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  The product of these two quantities is termed mag-netomotive force (m.m.f.), for reasons which are explained later. The unit of m.m.f. is strictly A, but it is commonly stated as AT to indicate that turns are also involved in addition to current. The symbol for m.m.f. is F. The effect of the m.m.f. depends on the length of magnetic circuit over which it acts. The m.m.f./unit length is defined as magnetic field strength, which is given the symbol H, and has the unit A/m or AT/m. H is a vector quantity, having the same direction as the magnetic flux set up by the associated m.m.f. N turns Magnetic circuit parameters and laws 75 Magnetic flux density, with symbol B, is a measure of how closely magnetic flux lines are packed together. It is defined as flux/unit cross-sectional area and is also a vector quantity. The unit of B is evidently W b / m 2 which is given the name Tesla (T). For the purposes of magnetic-circuit calculations, materials may be classified into one or two categories, depending on their relation-ship between magnetic field strength and magnetic flux density. Members of the first category include vacuum, air, and most other materials, excluding iron. For these, there is a linear relationship be-tween H and B which is expressed by the equation The constant of proportionality, fi Q , is called the permeability of free space. From the definition of inductance, its unit must be H/m and the value of jU Q follows from the definition of the Ampere as 4 7 c x l O -7 H / m . Members of the second category include iron, most steels, and a few other materials, and for this reason they are termed ferromag-netics. They are characterised by highly non-linear relationships between B and H. Hence, for these materials jU r is termed the relative permeability of the ferromagnetic material, and is evidently dimensionless. Its value varies typically from some hundreds to several thousands.

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  Electromechanical Motion Devices

  Rotating Magnetic Field-Based Analysis with Online Animations

  • Paul C. Krause, Oleg Wasynczuk, Steven D. Pekarek, Timothy O'Connell(Authors)
  • 2020(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  Simi- larly, all of the magnetizing flux of one winding may not link all of the turns of the other winding. To acknowledge this practical aspect of the magnetic system, i 1 i 2 Φ m2 Φ l2 Φ l1 Φ m1 N 1 v 1 N 2 v 2 + – + – Figure 1.5-1 Magnetically coupled circuits. 1.5 STATIONARY MAGNETICALLY COUPLED CIRCUITS 19 N 1 and N 2 are often considered to be the equivalent number of turns rather than the actual number. The voltage equations may be expressed as v 1 = r 1 i 1 + dλ 1 dt (1.5-3) v 2 = r 2 i 2 + dλ 2 dt (1.5-4) In matrix form, v 1 v 2 = r 1 0 0 r 2 i 1 i 2 + d dt λ 1 λ 2 (1.5-5) The resistances r 1 and r 2 and the flux linkages λ 1 and λ 2 are related to wind- ings 1 and 2, respectively. Since it is assumed that Φ 1 links the equivalent turns of winding 1 (N 1 ) and Φ 2 links the equivalent turns of winding 2 N 2 , the flux lin- kages may be written as λ 1 = N 1 Φ 1 (1.5-6) λ 2 = N 2 Φ 2 (1.5-7) where Φ 1 and Φ 2 are given by (1.5-1) and (1.5-2), respectively. If we assume that the magnetic system is linear, we may apply Ohm’s law for magnetic circuits to express the fluxes. Thus, the fluxes may be written as Φ l1 = N 1 i 1 li (1.5-8) Φ m1 = N 1 i 1 m (1.5-9) Φ l2 = N 2 i 2 l2 (1.5-10) Φ m2 = N 2 i 2 m (1.5-11) where l1 and l2 are the reluctances of the leakage paths, and m is the reluc- tance of the path of magnetizing fluxes. Typically, the reluctances associated with leakage paths are much larger than the reluctance of the magnetizing path. The reluctance associated with an individual leakage path is difficult to determine exactly, and it is usually approximated from test data or by using the computer to solve the field equations numerically. On the other hand, the reluctance of the magnetizing path of the core shown in Fig. 1.5-1 may be computed with suf- ficient accuracy as in Example 1B.

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  Electromagnetism for Engineers

  • Andrew J. Flewitt(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  The concept of the magnetic circuit allows us to quantitatively design magnetic systems. For example, if we wanted to generate a particular magnetic flux density in the toroid of Figure 4.6, then we could do this using Eq. (4.43), if we knew the relative permeability of the magnetic material. We could handle a similar situation for a nonlinear magnetic material if the relationship between B and H is known. Figure 4.8 is a graph showing the relationship between these quantities for cast steel, which is a soft ferromagnetic material. In this case, if we wanted to generate a magnetic flux density of 1.5 T inside the toroid, then this equates to a magnetic field of 1650 A m –1 . Equation (4.42) could then be used to determine the required magnetomotive force nI to produce this. A particularly common problem is one where a magnetic material is being used to gen- erate a particular magnetic flux density in a small volume of free space. This could be produced by having a small air gap of length l g in a toroid of cross-sectional radius r and toroidal radius R, as shown in Figure 4.9. In this case, let us imagine that we are using a hard ferromagnetic material, such as 35% cobalt steel, as a permanent magnet, so that no current-carrying coil is required. Hard ferromagnetic materials are clearly nonlinear. Therefore, one approach to solving such a problem is to use a load line, where an equation Magnetic Fields in Materials 67 Figure 4.9 A toroid of a hard ferromagnetic material that is permanently magnetized produces a magnetic field in a small air gap of length l g . 2R 2r B l g is generated which relates B and H in the magnetic material given that there is a linear rela- tion between these two quantities in the air gap – this is called the load line. The load line is then plotted on top of the nonlinear characteristics of the magnetic material and the inter- section defines the operating point of the system.

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