Chapter 1: magnetic circuits and magnetic materials

Chapter 1: Magnetic Circuits and Magnetic Materials

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

• The objective of this course is to study the devices used in the interconversion of electric and mechanical energy, with emphasis placed on electromagnetic rotating machinery.
• The transformer, although not an electromechanical-energy-conversion device, is an important component of the overall energy-conversion process.
• Practically all transformers and electric machinery use ferro-magnetic material for shaping and directing the magnetic fields that acts as the medium for transferring and converting energy. Permanent-magnet materials are also widely used.
• The ability to analyze and describe systems containing magnetic materials is essential for designing and understanding electromechanical-energy-conversion devices.
• The techniques of magnetic-circuit analysis, which represent algebraic approximations to exact field-theory solutions, are widely used in the study of electromechanical-energy-conversion devices.

§1.1 Introduction to Magnetic Circuits

Assume the frequencies and sizes involved are such that the displacement-current term in Maxwell’s equations, which accounts for magnetic fields being produced in space by time-varying electric fields and is associated with electromagnetic radiations, can be neglected.

• H : magnetic field intensity, amperes/m, A/m, A-turn/m, A-t/m
• B : magnetic flux density, webers/m2, Wb/m2, tesla (T)
• 1 Wb = ${\text{10}}^8$ lines (maxwells); 1 T = ${\text{10}}^4$ gauss
• (1.1)From (1.1), we see that the source of H is the current density J .The line integral of the tangential component of the magnetic field intensity H around a closed contour C is equal to the total current passing through any surface S linking that contour.

${\oint }_{c}H\text{.}\text{dl}={\int }_{s}J\text{.}\text{da}$

• (1.2)Equation (1.2) states that the magnetic flux density B is conserved. No net flux enters or leaves a closed surface.There exists no monopole charge sources of magnetic fields.

${\oint }_{s}B\text{.}\text{da}=0$

• A magnetic circuit consists of a structure composed for the most part of high-permeability magnetic material. The presence of high-permeability material tends to cause magnetic flux to be confined to the paths defined by the structure.

Figure 1.1Simple magnetic circuit.

• In Fig. 1.1, the source of the magnetic field in the core is the ampere-turn product N i , the magnetomotive force (mmf) F acting on the magnetic circuit.
• The magnetic flux $\phi$ (in weber, Wb) crossing a surface S is the surface integral of the normal component B :

$\phi ={\oint }_{s}B\text{.}\text{da}$ (1.3)

• ${\phi }_c$ : flux in core, $B_c$ : flux density in core

${\phi }_c=B_c{A}_c$ (1.4)

• $H_c$ : average magnitude H in the core. The direction of $H_c$ can be found from the RHR.

$\begin{array}{}F=\text{Ni}=\oint \text{Hdl}\\ F=\text{Ni}=H_c{l}_c\end{array}$ (1.5)

• The relationship between the magnetic field intensity H and the magnetic flux density B:

$$ $B=\mathrm{\mu H}$ (1.6)

• Linear relationship?
• $\mu ={\mu }_r{\mu }_o$ , $\mu$ : magnetic permeability, Wb/A-t-m = H/m
• ${\mu }_o=\mathrm{4\pi }\text{.}{\text{10}}^{-7}$ : the permeability of free space
• ${\mu }_r$ : relative permeability, typical values: 2000-80,000