Chapter 37: A Numerical Method of Harmonic Analysis [18551]
37.1 Introduction
Harmonic analysis is the process of resolving a complex periodic waveform into a series of sinusoidal components of ascending order of magnitude [18556][18557][18562]. A Fourier series is merely a trigonometric series given in the form [18564]:
The Fourier coefficients require functions that can be integrated. However, irregular waveforms are generally not defined by mathematical equations, and these coefficients cannot be determined directly by integration [18566][18569][18571]. In such cases, approximate methods can be used to evaluate the Fourier coefficients [18574][18578].
Most waveforms to be analyzed are periodic. Using the trapezoidal rule, the area under the curve can be approximated, leading to formulas for the mean value and coefficients [18579][18587]. The mean value of the function over the range is a0 [18602]:
37.3 Complex Waveform Considerations
Sometimes it is possible to predict the harmonic content of a periodic waveform on inspection, particularly by identifying symmetries [18611][18613]. Here are some clues:
- If the area above the horizontal axis is equal to the area below, then the mean value is zero. Hence a0 = 0 [18617][18621][18622][18623].
- An even function is symmetrical about the vertical axis and contains no sine terms [18618][18625][18626].
- An odd function is symmetrical about the origin and contains no cosine terms [18628][18630].
- If f(x) = -f(x + π), the waveform has positive and negative half cycles that are identical in shape, and only odd harmonics are present [18636][18641].
Chapter 38: Magnetic Properties of Materials
Basic Magnetic Constants and Reluctance
For air and non-magnetic materials, the permeability of free space is given by [18661][18663]:
The relative permeability μr is the ratio of flux density in the material to the flux density in a vacuum, where μ = μ0μr [18668][18671][18674].
Reluctance (S) is the magnetic resistance of a magnetic circuit to the presence of magnetic flux [18689][18690]:
38.2 Magnetic Properties of Materials
There are three main types of magnetism: Diamagnetism, Paramagnetism, and Ferromagnetism [18707][18709]. Magnetism is caused by electrons orbiting around a nucleus and by the angular momentum of electron spins [18710][18711][18712].
- Diamagnetism: Materials have a relative permeability slightly less than 1. This occurs because the applied field induces an electron orbit change that opposes the applied flux [18723][18727][18742][18747].
- Paramagnetism: Materials have a relative permeability slightly greater than 1. The electron spins tend to line up with the applied field and strengthen it in that region [18758][18760][18777][18780].
- Ferromagnetism: Materials have a relative permeability considerably greater than 1, meaning their domains align permanently to create strong internal fields (e.g., iron, cobalt, nickel) [18785][18787][18795].
38.3 Hysteresis and Hysteresis Loss
If a ferromagnetic material is completely demagnetized and then subjected to an increasing magnetic field strength (H), the flux density (B) will reach saturation [18869][18872][18875]. When the field is removed, the domains tend to stay aligned, leaving a residual or remanent flux density [18893][18894].
A disturbance in the alignment of the domains in a ferromagnetic material causes energy to be expended in taking it through a cycle of magnetization [18929][18930][18932]. This effect is called hysteresis [18925].
The instantaneous emf induced in the windings is given by [18958]:
The net energy absorbed by the magnetic field over one cycle corresponds directly to the area of the hysteresis loop [18991][18994][18998].
Hysteresis Loss Equations
The hysteresis loss per cycle is measured as Area × α × β Joules per cubic meter, where α and β are the axis scaling factors for H and B [19002][19010].
The hysteresis loss per cycle was empirically found to be proportional to (Bm)n, where n is the Steinmetz index and can have a value between 1.6 and 3.0 depending on the ferromagnetic material [19024].
The total hysteresis power loss in Watts is [19027][19030]:
Where kh is a constant for a given specimen, v is the volume in cubic meters, f is the frequency in Hertz, and Bm is the maximum flux density [19032][19033][19060].
Problem Example
Given an area A = 12.5 cm2, with horizontal scale 1 cm = 500 A/m and vertical scale 1 cm = 0.2 T [19073][19074][19075]. The hysteresis loss per cycle is: