Chapter 37: A Numerical Method of Harmonic Analysis

Date: 13/06/2020

37.1 Introduction

Harmonic analysis is the process of resolving a periodic, non-sinusoidal quantity into a series of sinusoidal components of ascending order of frequency. A Fourier series is merely a long series given in the form:

f(x) = a₀ + a₁cos(x) + a₂cos(2x) + … + b₁sin(x) + b₂sin(2x) + …

The Fourier functions require functions that can be integrated to produce the exact Fourier coefficients. However, irregular waveforms are usually defined by a graphic representation, and these cannot be determined by integration. In such cases, approximate methods can be used to evaluate the Fourier coefficients.

Most waveforms to be analysed are periodic. If the period of the wave is 2π and it is divided into p equal parts, the width of each interval is 2π/p. If the ordinates be labelled y₀ to y_p, the trapezoidal rule states:

Area ≈ (width of interval) × [ (1/2)(y₀ + y_p) + ∑ y_k ]

Hence, the mean value a_#theta; is the mean value of f(x) in the range theta to 2π:

a₀ = (1/p) ∑ y_k

Similarly, the coefficients a_n and b_n are approximated by:

a_n = (2/p) ∑ y_k cos(nx_k)
b_n = (2/p) ∑ y_k sin(nx_k)

37.3 Complex Waveform Considerations

Sometimes it’s possible to predict the harmonic content of a waveform on inspection, particularly observing its symmetrical characteristics. Here are some clues:

If the periodic waveform’s area above the horizontal axis is equal to the area below, then the mean value is zero. Hence C = 0.
An even function is symmetrical about the vertical axis and contains no sine terms.
An odd function is symmetrical about the origin and contains no cosine terms.
If f(x) = f(x+π), the waveform repeats exactly after half a cycle, and only even harmonics are present.
If f(x) = -f(x+π), the positive and negative half cycles are identical in shape, and only odd harmonics are present.

An odd function that only contains sine terms and odd harmonics must be symmetrical about the origin and have identical positive and negative half cycles.

Chapter 38: Magnetic Materials

Date: 4/8/2020

38.2 Magnetic Properties of Materials

There are three types of magnetism discussed here: Diamagnetism, Paramagnetism, and Ferromagnetism.

Magnetic Causes:

By electrons orbiting around a nucleus in a certain path.
By the angular momentum of electrons about their own axis (electron spin).

Diamagnetism

This is the cause of materials having a relative permeability of less than 1. It occurs because the magnetic moment due to its orbital electrons is zero. The orbit, which consists of an electron going around a closed loop, acts like a current loop. When a magnetic field is applied, it will induce an EMF to oppose the applied flux. As a result, the flux density within the material becomes less than that in a vacuum.

Paramagnetism

This causes relative permeability (\mu_r) to be slightly greater than 1. In these materials, electron orbit magnetic moments may not cancel each other out. Electron spins tend to pair up and cancel each other out; however, if the atom contains an odd number of electrons, a permanent “dipole moment” will exist. When a field is applied, they tend to line up and strengthen the flux in the region.

Ferromagnetic Materials

These materials have a \mu_r considerably greater than 1, and this varies with flux density. There are only four elements which are ferromagnetic at room temperature: Iron, Cobalt, Nickel, and Gadolinium. They are metals or alloys comprised of one or more of these elements.

Magnetism in Ferromagnetic Materials: This is a result of electron spinning. In Iron, 14 electrons in the M shell consist of 9 of them spinning in one direction and 5 in the other. The resultant flux is there due to 4 electrons. In Cobalt, it is 3; in Nickel, it is 2. The resultant flux in Iron, Cobalt, and Nickel results in “domains” which tend to align and permanently magnetize. Heating the material breaks these domains. For iron, heating above the Curie temperature (1040K) causes it to lose its ferromagnetic properties and behave as a paramagnetic material.

38.3 Hysteresis and Hysteresis Loss

A ferromagnetic material is completely demagnetized at point O. Subjecting it to an increasing magnetic field H, the flux density B will increase until it reaches a saturation point, past which the flux density will not increase any further. This is because most of the domains will be aligned. The curve follows O-a-b.

When the field H is removed, the domains will tend to stay aligned. The flux density remaining is called the remanent flux density.

To completely demagnetize the material again, a negative field H must be applied. The negative field required to remove the residual flux is called the coercive force. Continuing this cycle of applied positive and negative fields forms a closed loop known as the Hysteresis Loop.

A disturbance in the alignment of the domains in a ferromagnetic material causes energy to be expended in taking a specimen through a cycle of magnetization. This energy is dissipated as heat and is called hysteresis loss.

The net hysteresis loss for one cycle is given by the area of the hysteresis loop in units of Joules per cubic meter (J/m^3).

Hysteresis Loss per cycle = A × α × β Joules/m³

Where A is the area of the loop, lpha is the scale of H (Amperes per meter), and eta is the scale of B (Tesla).

Steinmetz found that the hysteresis loss per cycle was proportional to (B_m)^n, where B_m is the maximum flux density and n is the Steinmetz index (usually between 1.6 and 3.0 depending on the material). The equation for hysteresis power loss is:

P_h = k_h × v × f × (B_m)ⁿ Watts

Where P_h is the hysteresis loss in Watts, k_h is a constant for the given material, v is the volume in cubic meters, and f is the frequency in Hertz.