Introduction
Conventional and electron current flow
In a dry cell having one positive terminal (+) and one negative terminal (-), the potential difference between them cause an electron current to flow in the circuit; from negative to positive terminal. This is due to electron excession at the negative terminal compared to the positive terminal. Therefore, when we connect a wire between the two terminals, electrons will move along the wire from the negative terminal and reenter the cell by the positive terminal.
Meanwhile, back in the 17th century before the electron theory was fully understood, the scientists decided that electric current flows from the positive terminal to the negative terminal. This 'conventional current flow' is still used today and is accepted direction of current flow in electrical machines topics.
Distinction between sources and loads
We can simply distinguish between the source and the load by observing circuit elements; a source delivers electrical power, where the load absorbs it.
Sign notation
We use symbols (+) and (-) to indicates the direction of electric current, mechanical force, or rotational speed. If the sign changes, it means that the direction is reversed.
Double-subscript notation for voltages
Consider a source having a positive terminal A and a negative terminal B ;
The potential difference and the relative polarities of the terminals A and B can be designated by the double-subscript notation;
➡ The volatge between A and B is 100 V, and A is positive with respect to B.
➡The voltage between A and B is 100 V, and B is negative with respect to A.
Sign notation for voltages
Although we can represent the value and the polarity of volatges by the double-script notation, we can often prefer to use the sign notation. It consises of designating the voltage by a symbol and identifying one of the terminals by a positive (+) sign.
If E₂₁ = - 100V, terminal 2 is negative with respect to terminal 1.
Sign notation to designate a voltage.
Graph of an alternating voltage
The figure above shows a graph of an alternating voltage having a peak of 100 V.
In the following contents, we will be dealing with sources whose voltages change polarity periodically. Such alternating voltages may be represented by means of graph (Fig. 1.) 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.
Starting from zero, E₂₁ gradually increases, attaining +100 V after 0.5 seconds. It then gradually falls to zero at the end of one second. During this one-second interval, terminal 2 is positive with respect to terminal 1.
Positive and negative currents
The signs for current flow are allocated with respect to a reference direction given on the circuit diagram.
The figure shown is a circuit element showing positive direction of current flow. The positive direction is shown arbitrarily by means of an arrow.
Sinusoidal voltage
The ac voltage generated by alternators can be expressed by the equation
e = Eₘcos(2πft + θ)
where
e = instantaneous voltage [V]
Eₘ = peak valus of the sinusoidal voltage [V]
f = frequency [Hz]
t = time [s]
θ = a fixed angle [rad]
The expression 2πft and θ are angles, expressed in radians. However, it is often more convenient to express the angle in degrees, as follows;
e = Eₘcos(360ft + θ)
or
e = Eₘcos(Φ + θ)
where Φ (= 360 ft) is expressed in degrees along with θ.
Converting cosine functions into sine functions
We can convert a cosine fuction of voltage or current into a sine function and vise versa.
Converting cosine to sine by adding 90° to θ.
Converting sine to cosine by subtracting 90° from θ.
Effective value of an AC voltage
Although the peak value Eₘ are specified, it is much more common to use the effective value.
For a voltage that varies sinusoidally, the relationship between the effective value and Eₘ is given by the expression
The effective value of an ac voltage is sometimes called the RMS (root mean square) value of the voltage. It is measured of the heating effect of as compared to that of an equivalent dc voltage.
The same remarks apply to the effective value of an ac current. Thus a current that varies sinusoidally and whose peak value is Im possesses an efective value Ieff given by
Phasor representation
- length = i/e
- angle = electrical phase angle between i and e
- in phase ➡ two vectors are parallel, pointing in the same direction, angle = 0.
- out of phase ➡ two vectors point in diffrent way.
Harmonics
The voltages and currents in a power circuit are frequently not pure sine waves.
Energy in an inductor
A coil stores energy in its magnetic field when it carries a current I. The energy is given by
where
W = energy stored in the coil [J]
L = inductance of the coil [H]
I = current [A]
If the current varies, the stored energy rises and falls in step with the current. Thus, whenever the current increases, the coil absorbs energy and whenever the current falls, energy is released.
Energy in a capacitor
A capacitor stores energy in its electric field whenever a voltage E appears across its terminals. The energy is given by
where
W = energy stored in the capacitor [J]
C = capacitance of the capacitor [F]
E = voltage [V]
[Table] Impedance of some common AC circuits
Circuit diagram | Impedance |
Equivalent Circuit
Below are the equivalent circuit for Electric and Magnetic;
ELECTRIC |
MAGNETIC |
E = Volatage I = Current R = Resistance where R = ρl/A |
mmf = Magnetic Force Φ = Magnetic Flux R = Reluctance where R = l/μA |
Electromagnetism
Magnetic field intensity H and flux density B
Whenever a magnetic flux Φ exists in a body or component, it's due to the presence of a magnetic filed intensity H, given by
where
B = flux density [T or Wb/m²]
Φ = flux in the component [Wb]
A = cross section of the component [m²]
where
H = magnetic field intensity [A/m]
U = magnetomotive force acting on the component [A] (or ampere turn)
l = length of the component [m]
There is a definite relationship between the flux density (b) and the magnetic filed intensity (H) of any material. This relationship is usually expressed graphically by the B-H curve of the material.
B-H curve of vacuum
In vacuum, the magnetic flux density B is expressed by the equation
B = μ₀ × H
where
B = blux density [T]
H = magnetic field intensity [A/m]
μ₀ = magnetic constant or permeability of vacuum [= 4π ×10⁻⁷]
The figure above shows the B-H curve of vacuum and non magnetic materials.
The B-H curve of vacuum is a straight line. A vacuum never saturates, no matter how great the flux density maybe.
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
The value of μᵣ is not constant but varies with the flux density in the material, therefore the relationship between B and H is not linear.
Determining the relative permeability
The relative permeability μᵣ of a material is the ratio of flux density in the material to the flux density that would be produced in vacuum, under the same magnetic field intensity H.
Faraday's law of electromagnetic induction
It revealed a fundamental relationship between the voltage and flux in a circuit. Faraday;s law states:
- If the flux linking loop (ot turn) varies as a function of time, a voltage is induced between its terminals.
- The value of the induced voltage is proportional to the rate of change of flux.
Consequently, if the flux varies inside a coil of N turns, the voltage induced is given by
where
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]
Voltage induced in a conductor
In many motors and generators, the coils move with respect to a flux that is fixed in space. The relative motion produces a change in the lux linking the coils, and consequently, a voltage is induced according to Faraday's Law. However, it is easier to calculate the induced voltage with reference to the conductors, rather than with reference to the coil itself. In effect, whenever a conductor cuts a magenetic field, a voltage is induced across its terminals. The value of the induced voltage is given by
where
E = induced voltage [V]
B = flux density [T]
l = active length of the conductor in the magnetic field [m]
v = relative speed of the conductor [m/s]
Lorentz force on a conductor
When a current-carrying conductor is placed in a magnetic field, it is subjected to an electromagnetic force, or Lorentz force. It constitutes of the force depends upon the orientation of the conductor with respect to the direction of the field. The force is greatest when the conductor is perpendicular to the field and zero when it is parallel to it.
The maximum force acting on a straight conductor is given by
where
F = force acting on the conductor [N]
B = flux density of the field [T]
l = active length of the conductor [m]
I = current in the conductor [A]
where F = Lorentz force on a conductor
The upward direction for F can be obtained by changing the polar of the voltage source.
Direction of the force acting on a straight conductor
When there exists current in a conductor, the conductor will be surrounded by a magnetic field.
a. Magnetic field due to magnet and conductor
b. Resulting magnetic field pushes the conductor downward.
It can be observed that the lines of force created respectively by the conductor and the permanent magnet act in the same direction above the conductor and in opposite directions below it. Therefore, the number of lines above the conductor must be greater than the number below.
Residual flux density and coercive force
A current source, connected to the coil, produces a current whose value and direction can be changed at will. When I is gradually increased, B and H increased.
The maximum I will generates maximum H. When I is decreasing, there will be flux residue in the magnetic core. The value depends on μᵣ, if μᵣ is high, there will be more Bᵣ left after we discharge the circuit.When the graph of current is negative, there occurs negative maximum current which effect this graph of magnetization curve.
Here, if we let the current run in the circuit for a period of time then we open the circuit, there will be Bᵣ left in the material, the value depending on μᵣ.
Hysteresis Loop
The area shaded under the Bᵣ curve is the amount of coercive force, that reduces I to zero. This coercive force acts like one kind of energy.
The direction of the loop will be from a to e respectively.
xx
Hysteresis losses caused by rotation
Hysteresis losses occur when iron cores in an AC generator are subject to effects from a magnetic field. The magnetic domains of the cores are held in alignment with the field in varying numbers, dependent upon field strength.
The magnetic domains rotate, with respect to the domains not held in alignment, one complete turn during each rotation of the rotor. This rotation of magnetic domains in the iron causes friction and heat.
The heat produced by this friction is called magnetic hysteresis loss. To reduce hysteresis losses, most AC armatures are constructed of heat-treated silicon steel, which has an inherently low hysteresis loss. After the heat-treated silicon steel is formed to the desired shape, the lamination are heated to a dull red and then allowed to cool. This process, known as annealing, reduces hysteresis losses to a very low value.
Hysteresis losses caused by rotation
Hysteresis losses occur when iron cores in an AC generator are subject to effects from a magnetic field. The magnetic domains of the cores are held in alignment with the field in varying numbers, dependent upon field strength.
The magnetic domains rotate, with respect to the domains not held in alignment, one complete turn during each rotation of the rotor. This rotation of magnetic domains in the iron causes friction and heat.
The heat produced by this friction is called magnetic hysteresis loss. To reduce hysteresis losses, most AC armatures are constructed of heat-treated silicon steel, which has an inherently low hysteresis loss. After the heat-treated silicon steel is formed to the desired shape, the lamination are heated to a dull red and then allowed to cool. This process, known as annealing, reduces hysteresis losses to a very low value.
Eddy currents
Consider an ac flux Φ that links a rectangular-shaped conductor. According to Faraday's law, an ac voltage E is induced across its terminals.
If the conductor is short-circuited, a substantial alternating current I₁ will flow, causing the conductor is placed inside the first, a smaller voltage is induced because it links a smaller flux. Consequently, the short-circuit current I₂ is less than I₁ and so, too, is the power dissipated in this loop.
Here, we have North and South pole of magnet to generate magnetic flux. Then if the open circuit are shorted, there will be a current called eddy current from the magnetic induction.
The thin mica layer will block the current, with very small amount of current in a particular plate, it will minimize the eddy current.
Eddy currents in a stationary iron core
We can reduce the losses by splitting the core in two along its length, insulating the two sections from each other. The voltage induced in each section is one half of what it was before, with the result that the eddy currents, and the corresponding losses, are considerably reduced.
If we continue to subdivide the core, we find that the losses decrease progressively. Furthermore, a small amount of silicon is alloyed with the steel to increase its resistivity, thereby reducing the losses still more.
Eddy-current losses in a revolving core
The stationary field in direct-current motors and generators produces a constant dc flux. This constant flux induces eddy current in the revolving armature. Consider a solid cylindrical iron core that revolves between the poles of a magnet. As it turns, the core cuts the flux lines and, according to Faraday's law, a voltage is induced along its length having the polarities shown. These eddy currents produce large power losses which are immediately converted into heat, in which the power loss is proportional to the square of the speed and the square of the flux density.
To reduce the eddy-current losses, we laminate the armature using thin circular laminations that are insulated from each other.
Current in an inductor
In an inductive circuit, the voltage and current are related by the equation
where
e = instantaneous voltage induced in the circuit [V]
L = inductance of the circuit [H]
Δi/Δt = rate of change of current [A/s]
This equation enables us to calculate the instantaneous voltage e, when we know the rate of change of current, However, it often happens that e is known and we want to calculate the resulting current I.
Question [1.1]
- The peak voltage is 138 V.
- The peak voltage is 219 V.
- The peak voltage is 339 V.
- The peak voltage is 457 V.
- The peak current is 2.75 A.
- The peak current is 5.17 A.
- The peak current is 7.93 A.
- The peak current is 14.1 A.
Question [1.2]
- Phasor E is said to lead phasor I.
- Phasor E is said to lag phasor I.
Question [1.3]
- F = 100 N
- F = 200 N
- F = 300 N
- F = 400 N