Electrical Machines

Electrical Machines

Chapter 1: Units

Units Introduction

Units

Over the years systems of units have been devised to meet the needs of commerce, industry, and science. A system of units may be described as one in which the units bear a direct numerical relationship to each other, usually expressed as a whole number. 

SI Units

SI Units

SI is the official International System of Units and its adoption by most countries of the world.

The SI possesses a number of remarkable features shared by no other system of units:

1. It is a decimal system

2. It employs many units commonly used in industry and commerce.

3. It is a coherent system that expresses with startling simplicity some of the most basic relationships       in electricity, mechanics, and heat.

Base Units

Base Units

The foundation of the International System of Units rests upon the seven base units listed in the Table.

Derived Units

Derived Units

From base units, we derive other units to express quantities such as area, power, force, magnetic flux, and so on. There is no limit to the number of units we can derive, but some occur so frequently that they have been given special names.

Multiples and sub-multiples of SI units

Multiples and Sub-multiples of SI units

Multiples and sub-multiples of SI units are generated by adding appropriate prefixes to the units. Thus prefixes such as kilo, mega, nano, and centi multiply the value of the unit by factors listed in the table above. For example,

1 kiloampere = 1000 amperes,

1 nanosecond = 10^-9 seconds,

1 megawatt = 10^6 watts.

Conversion Chats and their use

Conversion Chats and their use

Unfamiliar units can be converted to units we know well by using standard conversion tables. Conversion charts can show the relative size of a unit by the position it occupies. The largest unit is at the bottom, the smallest at the top, and intermediate units are ranked in between.

Example

Convert 2.5 yards to meters

2.5 yd = 2.5*0.914 meters = 2.285 m

The per-unit system of measurement

The per-unit system of measurement

In order to get a better idea of the size of something, we compare it to the size of something similar. In effect, we can create our own unit and specify the size of similar quantities compared to this arbitrary unit. This concept gives rise to the per-unit method of expressing the magnitude of a quantity.

Chapter 2: Fundamentals of Electricity, Magnetism, and Circuits

Conventional and electron current flow

Conventional and electron current flow

Consider the dry cell having one positive and one negative terminal. The difference of potential between them is due to an excess of electrons at the negative terminal compared to the positive terminal. If we connect a wire across the terminals, the potential difference causes an electric current to flow in the circuit. This current is composed of a steady stream of electrons that come out of the negative terminal, move along the wire, and reenter the cell by the positive terminal.

Distinction between sources and loads

Distinction between sources and loads

By definition a source delivers electrical power whereas a load absorbs it. Every electrical device that carries a current can be classified as either a source or a load.

A devise is a source whenever current flows out of the positive terminal. A device is a load whenever current flows into a positive terminal.

For example; a battery that delivers electrical power is a source, but when it is being charged, it acts as a load.

Sign Notation

Sign Notation

The symbols (+) and (-) indicate the direction of an electric current of a mechanical force of a rotational speed, voltage, etc.

For example; if the speed of a motor changes from +100r/min to -100r/min, it means that the direction of rotation has reversed. 

Graph of an alternating voltage

Graph of an alternating voltage

Alternating voltages may be represented by means of a graph. The vertical axis indicates the voltage at each instant, while the horizontal axis indicates the corresponding time. Voltages are positive when they are above the horizontal axis and negative when they are below.

Positive and Negative Currents

Positive and Negative Currents

Positive and negative signs are used to indicate the direction of current flow. The signs are allocated with respect to a reference direction given on the circuit diagram.

For example, the current in a resistor flow from X to Y. The positive direction is shown arbitrarily by means of an arrow. If a current of 2 A flows from X to Y, it flows in the positive direction    (+2 A).  Conversely, if current flows from Y to X (opposite direction to the arrow), it is designated by the symbol -2 A.

Sinusoidal Voltage

Sinusoidal Voltage

AC voltage generated by commercial alternators is a sine wave that can be expressed by equation on the left.

Converting cosine functions into sine functions

Converting cosine to sine function

We can convert a cosine function of voltage or current into a sine fuction by adding 90 degrees to the angel.

Converting sine to cosine function

Similarly, we can convert a sine function into a cosine function by subtracting 90 degrees from the angle.

Effective value of an ac voltage

Effective value of an ac voltage

Although the properties of an ac voltage are known when its frequency and pea value (Em) are specified, it is much more common to use the effect value (Eeff). For a voltage that varies sinusoidally, the relationship between Eeff and Em is given by the expression.

Eeff = Em/√2

Eeff is sometimes called RMS(root mean squared). The same remarks apple to the effective value of an AC current whose peak value is Im possesses an effective value Ieff.

Ieff = Im/√2

Phasor representation

Phasor representation

The basic purpose of phasor diagrams is to show the magnitudes and phases angles between voltages and currents. A phasor is similar to a vector in the sense that it bears an arrow, and its length is proportional to the effective value of the voltage or current it represents. The angle between two phasors is equal to the electrical phase angle between the quantities.

The following rules apply to phasors:

1. Two phasors are said to be in phase when they are parallel to each other.

2. Two phasors are said to be out of phase when they point in different directions.

In phase

The current phasor I and voltage phasor E are in phase.

Out of phase

Phasor I lags behind phasor E by an angle of θ degrees.

Harmonics

Harmonics

The voltages and currents in a power circuit are frequently not pure sine waves. The line voltages usually have a satisfactory wave shape but the currents are sometimes badly distorted as shown in the figure on the left.

The distorted sine waves for currents may be caused by magnetic saturation in the cores of transformer or by other defects.

Distorting effect

Consider two sinusoidal sources e1 and e2 connected in series. Their frequencies and peak voltage amplitudes are shown in the figure above.

Since the sources are in series, the terminal voltage e3 is equal to the sum of the instantaneous voltages produced by each source. The resulting terminal voltage is a flat-topped wave.

Energy in an inductor

Energy in an inductor

A coil stores energy in its magnetic field when it carries a current I. The energy is given by 

W=(1/2)*L*I^2

W = Energy stored in the coil [J]

L = Inductance of the coil [H]

I = Current [A]

Energy in a capacitor

Energy in a capacitor

A capacitor stores its energy in an electric field whenever a voltage E appears across its terminals. The energy is given by

W = (1/2)*C*E^2

W = Energy stored in the capacitor [J]

C = Capacitance of capacitor [F]

E = Voltage [V]

Magnetic field intensity H and flux density B

Magnetic field intensity and flux density

Whenever a magnetic flux exists in a body or component, it is duo to the presence of magnetic field intensity H given by

H = U/l

H = magnetic field intensity [A/m]

U = magnetomotive force acting on the component [A]

l = Length of the component [m]

The resulting magnetic flux density is given by

B = Φ/A

B = Flux density [T]

Φ = Flux in the component [Wb]

A = cross section of the component [m^2]

B-H curve of vacuum

B-H curve of vacuum

In vacuum, the magnetic flux is directly proportional to the magnetic field intensity, and is expressed by the equation

B = (µ0)*H

B = flux density [T]

H = magnetic field intensity [A/m]

µ0 = magnetic constant [4π × 10−7 H·m^−1]

B-H curve of a magnetic material

B-H curve of a magnetic material

The flux density in a magnetic material also depends upon the magnetic field intensity to which it is subjected. Its value is given by

B = μ0μrH

μr = Relative permeability of the material

The value μr is not constant but varies with the flux density in the material. Consequently, the relationship between B and H is not linear. We prefer to show the relationship by means of a B-H saturation curve.

Determining the relative permeability

Relative permeability

The relative permeability (μr) of a material is the ratio of the flux density in the material to the flux density that would be produced in vacuum, under the same magnetic field intensity H.

Given the saturation curve of a magnetic material, it is easy to calculate the relative permeability using the approximate equation

μr ≈ 800,000 B/H

B = Flux density in the magnetic material [T]

H = Corresponding magnetic field intensity [A/m]

Ex. Determine the permeability of silicon iron at a flux density of 1.4 T.

Referring to the figure above, a flux density of 1.4 T requires a magnetic field intensity of 1000 A/m.

μr ≈ 800,000 B/H

μr ≈ (800,000 *1.4)/1000 = 1120

At 1.4 T flux density, silicon iron is 1120 times more permeable than vacuum (air).

Faraday's law of electromagnetic induction

Faraday's law of electromagnetic induction

Faraday's law of electromagnetic induction revealed a fundamental relationship between the voltage and flux in a circuit. Faraday's law states:

1. If the flux linking a loop (or turn) varies as a function of time, a voltage is induced between its terminals.

2. The value of the induced voltage is proportional to the rate of change of flux.

If the flux varies inside a coil of N turns, the voltage induced is given by

ENΔΦ/Δt

E = Induced voltage [V]

N = number of turns in the coil

ΔΦ = Change of flux inside the coil [Wb]

Δt = Time interval during which the flux changes [s]

Ex. A coil of 2000 turns surrounds a flux of 5 mWb produced by a permanent magnet. The magnet is suddenly withdrawn causing the flux inside the coil to drop uniformly to 2 mWb in 1/10 of a second. What is the voltage induced?

Flux change: ΔΦ = 5 - 2 = 3 mWb

E = NΔΦ/Δt = 2000*3/(1000*1/10) = 60 V

Note: The induced voltage falls to zero as soon as the flux ceases to change.