Magnetic Circuits

 

21.1 Goals of the lesson

In this lesson, we shall acquaint the reader, primarily with the basic concepts of magnetic circuit and methods of solving it. Biot-Savart law for calculating magnetic field due to a known current distribution although fundamental and general in nature, requires an integration to be evaluated which sometimes become an uphill task. Fortunately, due to the specific nature of the problem, Ampere's circuital law (much easier to apply) is adopted for calculating field in the core of a magnetic circuit. You will also understand the importance of B-H curve of a magnetic material and its use. The concept and analysis of linear and non linear magnetic circuit will be explained. The lesson will conclude with some worked out examples.

Key Words: mmf, flux, flux density, mean length, permeability, reluctance.

fter going through this section students will be able to answer the following questions.

A

 

1. What is a magnetic circuit?

 

2. What are linear and non linear magnetic circuits?

 

3. What information about the core is necessary for solving linear magnetic circuit?

 

4. What information about the core is necessary for solving non linear magnetic circuit?

 

5. How to identify better magnetic material from the B-H characteristics of several materials?

 

6. What should be done in order to reverse the direction of the field within the core?

 

7. What assumption is made to assume that the flux density remains constant throughout the section of the core?

 

8. What is the expression for energy stored in the air gap of a magnetic circuit?

 

9. Enumerate applications of magnetic circuit.

 

10. Is the core of a magnetic material to be laminated when the exciting current is d.c?

 

They will also be able to do the following:

 

1. How to translate a given magnetic circuit into its electrical equivalent circuit.

 

2. How to draw B-H curve of a given material from the data supplied and how to use it for solving problem.

 

3. How to solve various kinds of problems involving magnetic circuits.

 

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21.2 Introduction

Before really starting, let us look at some magnetic circuits shown in the following figures.

All of them have a magnetic material of regular geometric shape called core. A coil having a number of turns (= N) of conducting material (say copper) are wound over the core. This coil is called the exciting coil. When no current flows through the coil, we don't expect any magnetic field or lines of forces to be present inside the core. However in presence of current in the coil, magnetic flux φ will be produced within the core. The strength of the flux, it will be shown, depends on the product of number of turns (N) of the coil and the current (i) it carries. The quantity Ni called mmf (magnetomotive force) can be thought as the cause in order to produce an effect in the form of flux φ within the core. Is it not somewhat similar to an electrical circuit problem where a voltage (emf) is applied (cause) and a current is produced (effect) in the circuit? Hence the term magnetic circuit is used in relation to producing flux in the core by applying mmf (= Ni). We shall see more similarities between an electrical circuit and a magnetic circuit in due course as we go along further. At this point you may just note that a magnetic circuit may be as simple as shown in figure 21.1 with a single core and a single coil or as complex as having different core materials, air gap and multiple exciting coils as in figure 21.2. After going through this lesson you will be able to do the following.

 

1. to distinguish between a linear and non linear magnetic circuit.

 

2. to draw the equivalent electrical circuit for a given magnetic circuit problem.

 

3. to calculate mean lengths of various flux paths.

 

4. to calculate the reluctances of the various flux paths for linear magnetic circuit problem.

 

5. to understand the importance of B-H characteristics of different materials.

 

6. how to deal with a non linear magnetic circuit problem using B-H characteristic of the materials.

 

21.3 Different laws for calculating magnetic field

21.3.1 Biot-Savart law

We know that any current carrying conductor produces a magnetic field. A magnetic field ℜ is characterized either byH→, the magnetic field intensity or byB→, the magnetic flux density vector. Figure 21.1: Figure 21.3: Figure 21.2:

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These two vectors are connected by a rather simple relation:0rB=μμH→→; where -704×10Hmμ=π is called the absolute permeability of free space and rμ, a dimensionless quantity called the relative permeability of a medium (or a material). For example the value of rμ is 1 for free space or could be several thousands in case of ferromagnetic materials.

Biot-Savart law is of fundamental in nature and tells us how to calculate or at a given point with position vector , due to an elemental current and is given by: dBdH→ridl→

34→→→×=0rμμidl rdBπr

If the shape and dimensions of the conductor carrying current is known then field at given point can be calculated by integrating the RHS of the above equation.

3 length4→→→×=∫0rμμidlrBπr

where, length indicates that the integration is to be carried out over the length of the conductor. However, it is often not easy to evaluate the integral for calculating field at any point due to any arbitrary shaped conductor. One gets a nice closed form solution for few cases such as:

 

1. Straight conductor carries current and to calculate field at a distance d from the conductor.

 

2. Circular coil carries current and to calculate field at a point situated on the axis of the coil.

 

 

1.3.2 Ampere's circuital law

 

2

This law states that line integral of the vector along any arbitrary closed path is equal to the current enclosed by the path. Mathematically: H→

ΗdlI→→⋅=∫􀁶

For certain problems particularly in magnetic circuit problems Ampere's circuital law is used to calculate field instead of the more fundamental Biot Savart law for reasons going to be explained below. Consider an infinite straight conductor carrying current i and we want to calculate field at a point situated at a distance d from the conductor. Now take the closed path to be a circle of radius d. At any point on the circle the magnitude of field strength will be constant and direction of the field will be tangential. Thus LHS of the above equation simply becomes H × 2πd. So field strength is

Am2IH=πd

It should be noted that in arriving at the final result no integration is required and it is obtained rather quickly. However, one has to choose a suitable path looking at the distribution of the current and arguing that the magnitude of the field remains constant through out the path before applying this law with advantage.

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21.3.3 Application of Ampere's circuital law in magnetic circuit

 

Ampere's circuital law is quite handy in determining field strength within a core of a magnetic material. Due to application of mmf, the tiny dipole magnets of the core are aligned one after the other in a somewhat disciplined manner. The contour of the lines of force resembles the shape the material. The situation is somewhat similar to flow of water through an arbitrary shaped pipe. Flow path is constrained to be the shape of the bent pipe. For an example, look at the sectional view (figure 21.4 & 21.5) of a toroidal magnetic circuit with N number of turns wound uniformly as shown below. When the coil carries a current i, magnetic lines of forces will be created and they will be confined within the core as the permeability of the core is many (order of thousands) times more than air.

Take the chosen path to be a circle of radius r. Note that the value of H will remain same at any point on this path and directions will be always tangential to the path. Hence by applying Ampere's circuital law to the path we get the value of H to be2NIπr. If r is increased from a to be b the value of H decreases with r. a and b are respectively the inner and outer radius of the toroidal core.

Assumptions

 

1. Leakage flux & Fringing effect

Strictly speaking all the flux produced by the mmf will not be confined to the core. There will be some flux lines which will complete their paths largely through the air as depicted in figure 21.6. Since the reluctance (discussed in the following section) or air is much higher compared to the reluctance offered by the core, the leakage flux produced is rather small. In our discussion here, we shall neglect leakage flux and assume all the flux produced will be confined to the core only.

 

 

φ • • • • • • • • • • • • • • • • • • • • • • X X X X X X X X X X Figure 21.5: r a b • • • • • • • • • • • • • • • • • • • • • • X X X X X X X X X X Figure 21.4: Flux line Coil section

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In the magnetic circuit of figure 21.6 an air gap is present. For an exciting current, the flux lines produced are shown. These flux lines cross the air gap from the top surface of the core to the bottom surface of the core. So the upper surface behaves like a north pole and the bottom surface like a south pole. Thus all the flux lines will not be vertical and confined to the core face area alone. Some lines of force in fact will reach the bottom surface via bulged out curved paths outside the face area of the core. These flux which follow these curved paths are called fringing flux and the phenomenon is called fringing effect. Obviously the effect of fringing will be smaller if the air gap is quite small. Effect of fringing will be appreciable if the air gap length is more. In short the effect of fringing is to make flux density in the air gap a bit less than in the core as in the air same amount of flux is spread over an area which is greater than the core sectional area. Unless otherwise specified, we shall neglect the fringing effect in our following discussion. Effect of fringing sometimes taken into account by considering the effective area in air to be about 10 to 12% higher than the core area.

 

2. In the practical magnetic circuit (as in figure 21.5), the thickness (over which the lines of forces are spread = b-a) are much smaller compared to the overall dimensions (a or b) of the core. Under this condition we shall not make great mistake if we calculate H at 2(b-a)mr= and take this to be H every where within the core. The length of the flux path corresponding to the mean radius i.e., 2ml=πr is called the mean length. This assumption allows us to calculate the total flux φ produced within the core rather easily as enumerated below:

 

 

• Calculate the mean length l_m of the flux path from the given geometry of the magnetic circuit.

 

• Apply Ampere's circuital law to calculate H = mNIl

 

• Note, this H may be assumed to be same every where in the core.

 

• Calculate the magnitude of the flux density B from the relation B = μ_oμ_rH.

 

• Total flux within the core is φ = BA, where A is the cross sectional area of the core.

leakage fluxair gap Fringing effectN S Figure 21.6:

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21.3.4 Reluctance & permeance

 

Let us now try to derive a relationship between flux produced φ and mmf Ni applied for linear case.

linear relationship between and . putting the expression for . arranging the terms.ororor = BA = μμHABHNi= μμAHlNi = 1lμμAφ⎛⎞⎜⎟⎝⎠∵

Now defining Ni = mmf and 01ℜrlμμA=, the above equation can be written in the following handy form

0 = ==1ℜrNiNimmflReluctanceμμAφ

This equation resembles the familiar current voltage relationship of an electric circuit which is produced below for immediate comparison between the two:

ρlavVVoltagei===RResistance

The expression in the denominator is called resistance which impedes the flow of the current. 01ℜrlμμA= is known as reluctance of the magnetic circuit and permeance (similar to admittance in electric circuit) is defined as the reciprocal of reluctance i.e., 1ρ=ℜ.

 

21.4 B-H Characteristics

 

A magnetic material is identified and characterized by its B - H characteristic. In free space or in air the relationship between the two is linear and the constant of proportionality is the permeability μ₀. If B is plotted against H, it will be straight a line. However, for most of the materials the relationship is not linear and is as shown in figure 21.7. A brief outline for experimental determination of B-H characteristic of a given material is given now. First of all a sample magnetic circuit (with the given material) is fabricated with known dimensions and number of turns. Make a circuit arrangement such as shown in Figure 21.8, to increase the current from 0 to some safe maximum value. Apart from ammeter reading one should record the amount of flux produced in the core by using a flux meter-let us not bother how this meter works!

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Let,

Number of turns  Mean length of the flux path  Cross sectional area  Reading of the ammeter Reading of the flux meter=  =  =  =  = N  l in m.  a in m2  I in A  φ in Wb