Integration

 

 

 

 

 

 

 

 

 

 

 

Summary

(1) The average rate of change of a function over a certain interval is
the same as the gradient of the chord drawn on the graph of the
function. This chord gradient, for a function, y = f (x), is given by

(2) If the chord is very short then the gradient of the chord is approximately
the gradient of the tangent to the graph at a particular spot that is, the slope of the graph at that point. The slope of the graph
gives the instantaneous rate of change of the function with respect to
its independent variable, known as its derivative. This is represented
by dy/dx. Then we have the definition

 

Trigonometric functions

 

 

 

 

 

Electromagnetism

When current flows through a conductor a magnetic field is
produced around the conductor. The magnetic field is made up
of lines of flux, just like a natural magnet. The size and strength
of the magnetic field will increase and decrease as the current
flow strength increases and decreases

Left-Hand Rule for Conductors

A definite relationship exists between the direction of current
flow and the direction of the magnetic field. The left-hand rule
for conductors demonstrates this relationship. If a current carrying
conductor is grasped with the left hand with the thumb
pointing in the direction of electron flow, the fingers will point in
the direction of the magnetic lines of flux

In the following illustration it can be seen that when the
electron flow is away from the viewer (indicated by the plus
sign) the lines of flux flow in a counter clock wise direction
around the conductor. When the electron flow reverses and
current flow is towards the viewer (indicated by the dot) the
lines of flux reverse direction and flow in a clockwise direction

Electromagnet

An electromagnet can be made by winding the conductor into
a coil and applying a DC voltage. The lines of flux, formed by
current flow through the conductor, combine to produce a larger
and stronger magnetic field. The center of the coil is known as
the core. In this simple electromagnet the core is air

Adding an Iron Core

Iron is a better conductor of flux than air. The air core of an
electromagnet can be replaced by a piece of soft iron. When a
piece of iron is placed in the centre of the coil more lines of flux
can flow and the magnetic field is strengthened

Number of Turns

The strength of the magnetic field in the DC electromagnet can
be increased by increasing the number of turns in the coil. The
greater the number of turns the stronger the magnetic field

Changing Polarity

The magnetic field of an electromagnet has the same
characteristics as a natural magnet, including a north and south
pole. However, when the direction of current flow through
the electromagnet changes, the polarity of the electromagnet
changes. The polarity of an electromagnet connected to an AC
source will change at the same frequency as the frequency
of the AC source. This can be demonstrated in the following
illustration.At Time 1 current flow is at zero. There is no magnetic field produced around the electromagnet At Time 2 current is
flowing in a positive direction. A magnetic field builds up around
the electromagnet. The electromagnet assumes a polarity with
the south pole on the top and the north pole on the bottom. At
Time 3 current flow is at its peak positive value. The strength
of the electromagnetic field is at its greatest value. At Time
4 current flow decreases and the magnetic field begins to
collapse, until Time 5 when current flow and magnetic field are
at zero. Current immediately begins to increase in the opposite
direction. At Time 6 current is increasing in a negative direction.
The polarity of the electromagnetic field has changed. The
north pole is now on top and the south pole is on the bottom.
The negative half of the cycle continues through Times 7 and 8,
returning to zero at Time 9. This process will repeat 60 times a
second with a 60 Hz AC power supply

Magnetism

The principles of magnetism play an important role in the
operation of an AC motor. All magnets have two characteristics.
They attract and hold metal objects like steel and iron. If free
to move, like the compass needle, the magnet will assume
roughly a north-south position.

Magnetic Lines of Flux

We know that a magnet attracts an iron or steel object by an
invisible force. The magnet’s invisible force is called lines of flux.
These lines of flux make up an invisible magnetic field. Every
magnet has two poles, one north pole and one south pole.
Invisible magnetic lines of flux leave the north pole and enter
the south pole. While the lines of flux are invisible, the effects
of magnetic fields can be made visible. When a sheet of paper
is placed on a magnet and iron filings loosely scattered over it,
the filings will arrange themselves along the invisible lines of
flux

By drawing lines the way the iron filings have arranged
themselves, the following illustration is obtained. Magnetic lines
of flux always form closed loops, leaving the north pole and
entering the south pole. They return to the north pole through
the magnet

                               Unlike Poles Attract

                      Like Poles Repel

Three-Phase Transformers

Delta Connections

Three-phase transformers are used when three-phase power
is required for larger loads such as industrial motors. There
are two basic three-phase transformer connections, delta and
wye. Delta transformers are used where the distance from
the supply to the load is short. A delta is like three single phase
transformers connected together. The secondary of a
delta transformer is illustrated below. For simplicity, only the
secondary of a three-phase transformer is shown. The voltages
shown on the illustration are secondary voltages available
to the load. Delta transformers are schematically drawn in a
triangle. The voltages across each winding of the delta triangle
represents one phase of a three phase system. The voltage is
always the same between any two wires. A single phase (L1 to
L2) can be used to supply single phase loads. All three phases
are used to supply three phase loads

Balanced Delta Current

When current is the same in all three coils, it is said to be
balanced. In each phase, current has two paths to follow. For
example, current flowing from L1 to the connection point at
the top of the delta can flow down through one coil to L2, and
down through another coil to L3. When current is balanced, coil
current is 58% of the line current measured on each phase. If
the line current is 50 amps on each phase, coil current would be
29 amps

Unbalanced Delta Current

When current is different in all three coils, it is unbalanced. The
following diagram depicts an unbalanced system

Though current is usually measured with an ammeter, line current
of an unbalanced delta transformer can be calculated with
the following formulas

Wye Connections

The wye connection is also known as a star connection. Three
transformers are connected to form a “Y” shape. The wye
transformer secondary, (shown below) has four leads, three
phase connectors, and one neutral. The voltage across any
phase (line-to-neutral) will always be less than the line-to-line
voltage. The line-to-line voltage is 1.732 times the line-to-neutral
voltage. In the circuit below, line-to-neutral voltage is 277 volts.
Line-to-line voltage will be 480 volts (277 x 1.732)