EEE212: Applied Electricity II
Course outline:
Concept of Vectors, Complex operators.
Phasors.
Definition of Impedance, Resistance and Reactance.
Definition of Admittance, Conductance and Susceptance.
Phasor diagrams for RLC circuits.
Resonance.
Power Circuits.
Introduction to Electrical Installations.
Introduction
Vectors
A vector is any physical quantity that has both magnitude and direction. Geometrically, we can
picture a vector as a directed line segment with an arrow, whose length is the magnitude of the
vector and its arrow indicating the direction. The direction of the vector ⃗⃗⃗⃗⃗ is from its tail to
its head . Two vectors are the same if they have the same magnitude and direction. This
means that if we take a vector and translate it to a new position (without rotating it), then the
vector we obtain at the end of this process is the same vector we had in the beginning.
B
A
Fig. 1.1: Vector⃗⃗⃗⃗⃗ .
There are various forms of representing vector quantities all of which enable those operations
which are carried out graphically in phasor diagrams to be performed analytically. These forms
of representing vectors include rectangular form, trigonometric form, exponential form and
polar form.
Rectangular form.
A vector in rectangular form is specified in terms of its X-component and Y-component as
shown in the fig. 1.2
Y
E
X
, Fig.1.2: Graphical representation of vector E.
Can be written in rectangular form as
……………………………………1
where is the horizontal component and is the vertical component. j is an operator and it
indicates that is perpendicular to . In Mathematics the imaginary operator is denoted
by but in electrical engineering is adopted because letter is reserved for representing
current.
Trigonometric form.
Applying trigonometric ratios on fig.1.2, we can deduce that the X-component of is
and the Y-component is . Hence we can represent the vector E in the form
. This is equivalent to because and .
In general
……………………………………2
Where √ and
Exponential Form.
It has been proven that
This equation is known as Euler’s equation. The derivation of this equation will be done
when Maclaurin series is studied later on in your program. Now, substituting the Euler’s
equation in eqn. 2, we can write that , hence
……………………………………3
Polar form
The expression is written in simplified form as . In this expression E
represent the magnitude of the vector E and is its inclination in clockwise direction with
the X-axis. For anti-clockwise direction the expression becomes . In general, the
expresson is written as . Therefore the polar form of the vector E is
……………………………………4
The complex operator j
The operator j is used to indicate the counter clockwise rotation of a vector through . It has a
value of √ . When j is operated on vector , the new vector becomes which is displaced
in a counter clockwise direction from . Further application of j will give
( √ .
It can be summarized from the above that;
√ ( ccw rotation)
√ ( ccw rotation)
√ √ ( ccw rotation)
, √ √ ( ccw rotation)
Fig. 1.3
It should also be noted that (Rationalize the denominator).
Example 1: Two vectors and are represented by and , write the
various equivalent forms of the vectors and illustrate by means of a vector diagram the
magnitude and position of the vectors.
Solution:
√
√
( )
Trigonometric form, √
Exponential form, √
Polar form, √
Y
√
X
√
√
Course outline:
Concept of Vectors, Complex operators.
Phasors.
Definition of Impedance, Resistance and Reactance.
Definition of Admittance, Conductance and Susceptance.
Phasor diagrams for RLC circuits.
Resonance.
Power Circuits.
Introduction to Electrical Installations.
Introduction
Vectors
A vector is any physical quantity that has both magnitude and direction. Geometrically, we can
picture a vector as a directed line segment with an arrow, whose length is the magnitude of the
vector and its arrow indicating the direction. The direction of the vector ⃗⃗⃗⃗⃗ is from its tail to
its head . Two vectors are the same if they have the same magnitude and direction. This
means that if we take a vector and translate it to a new position (without rotating it), then the
vector we obtain at the end of this process is the same vector we had in the beginning.
B
A
Fig. 1.1: Vector⃗⃗⃗⃗⃗ .
There are various forms of representing vector quantities all of which enable those operations
which are carried out graphically in phasor diagrams to be performed analytically. These forms
of representing vectors include rectangular form, trigonometric form, exponential form and
polar form.
Rectangular form.
A vector in rectangular form is specified in terms of its X-component and Y-component as
shown in the fig. 1.2
Y
E
X
, Fig.1.2: Graphical representation of vector E.
Can be written in rectangular form as
……………………………………1
where is the horizontal component and is the vertical component. j is an operator and it
indicates that is perpendicular to . In Mathematics the imaginary operator is denoted
by but in electrical engineering is adopted because letter is reserved for representing
current.
Trigonometric form.
Applying trigonometric ratios on fig.1.2, we can deduce that the X-component of is
and the Y-component is . Hence we can represent the vector E in the form
. This is equivalent to because and .
In general
……………………………………2
Where √ and
Exponential Form.
It has been proven that
This equation is known as Euler’s equation. The derivation of this equation will be done
when Maclaurin series is studied later on in your program. Now, substituting the Euler’s
equation in eqn. 2, we can write that , hence
……………………………………3
Polar form
The expression is written in simplified form as . In this expression E
represent the magnitude of the vector E and is its inclination in clockwise direction with
the X-axis. For anti-clockwise direction the expression becomes . In general, the
expresson is written as . Therefore the polar form of the vector E is
……………………………………4
The complex operator j
The operator j is used to indicate the counter clockwise rotation of a vector through . It has a
value of √ . When j is operated on vector , the new vector becomes which is displaced
in a counter clockwise direction from . Further application of j will give
( √ .
It can be summarized from the above that;
√ ( ccw rotation)
√ ( ccw rotation)
√ √ ( ccw rotation)
, √ √ ( ccw rotation)
Fig. 1.3
It should also be noted that (Rationalize the denominator).
Example 1: Two vectors and are represented by and , write the
various equivalent forms of the vectors and illustrate by means of a vector diagram the
magnitude and position of the vectors.
Solution:
√
√
( )
Trigonometric form, √
Exponential form, √
Polar form, √
Y
√
X
√
√
