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Unformatted text preview: 1/3/2007 Copyright 2005 S.D. Sudhoff Page 4 1 / Magnetic Systems and Magnetic Equivalent Circuits The purpose of this chapter is set forth the basic background needed to analyze electromagnetic and electromechanical systems. The first step to doing this is to set forth the notion of magnetic equivalent circuit – an approximate but often fairly accurate technique for the analysis of magnetic systems. Using this technique, we will see how to translate a geometrical description of a magnetic device consisting of coils of wire, magnetic conductor such as iron, and permanent magnet material into an electrical circuit description. It should be noted that throughout this work, SI units will be strictly adhered to. Thus, flux density is measured in Tesla and field intensity in A/m. The reader is referred to  for a discussion of other unit systems. As an additional note, with regard to notation, vector and matrix quantities will appear in a bold, non-italicized font throughout this work. Scaler quantities will appear in a non-bold, italicized font. Thus, for quantity such a flux density, Β refers to a flux density vector, whereas B refers to a scalar representing the flux density in some particular direction (such as along a specified path) 1.1 MMF AND KIRCHOFF’S VOLTAGE LAW FOR MAGNETIC CIRCUITS We begin our analysis of magnetic system by developing the concept of Kirchoff’s voltage law for magnetic systems – in particular the idea that the sum of the magneto-motive force drops around any closed loop is equal to the sum of the magneto-motive force sources. To formalize this idea, we begin with Ampere’s law, which states that the line integral of the field intensity is equal to the current enclosed by that path. This is illustrated in Fig. 1.1-1 and may be stated mathematically as x x i d = ⋅ ∫ l H (1.1-1) In (1.1-1), H is the field intensity (a vector field), l d is an incremental segment in the path (again a vector), x denotes a path, and x i is the total current enclosed by that path, where the sign is determined in accordance with the right-hand rule. The total enclosed current for the path is the total magneto-motive force source x F for the path, i.e. x x i F = (1.1-2) 1/3/2007 Copyright 2005 S.D. Sudhoff Page 5 H l d start end x i Figure 1.1-1. Illustration of Ampere’s Law. It will often be convenient to break the total enclosed current into a number of separate components, i.e. ∑ = = J j j x x i i 1 , (1.1-3) where J is the number of components. Each component can be viewed as a magneto- motive force source component j x j x i F . , = (1.1-4) Comparing (1.1-2)-(1.1-4) ∑ = = J j j x x F F 1 , (1.1-5) It will be also be convenient to break the path integral in (1.1-1) into a number of sub line-integrals as depicted in Fig. 1.1-2. If the path integral is broken into K segments we have x l l K k l l F d d K k k = ⋅ + ⋅ ∫ ∑ ∫ − = + 1 1 1 1 l H l H (1.1-6) where at this point it is convenient to define a magneto-motive force drop as...
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This note was uploaded on 09/22/2011 for the course ECE 321 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.
- Spring '08