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Module 1: DC Circuits
Contents: Electrical circuit elements (R, L and C), Concept of active and passive elements,
voltage and current sources, concept of linearity and linear network, unilateral and bilateral
elements, Kirchhoff‟s laws, Loop and nodal methods of analysis, Star-delta transformation,
Superposition theorem, Thevenin’s theorem, Norton’s theorem.
1. Electrical circuit elements (R, L and C): The interconnection of various electric
elements in a prescribed manner comprises as an electric circuit in order to perform a
desired function. The electric elements include controlled and uncontrolled source of
energy, resistors, capacitors, inductors, etc. Analysis of electric circuits refers to
computations required to determine the unknown quantities such as voltage, current
and power associated with one or more elements in the circuit. To contribute to the
solution of engineering problems one must acquire the basic knowledge of electric
circuit analysis and laws. We shall discuss briefly some of the basic circuit elements
and the laws that will help us to develop the background of subject.
a) Resistor: Resistor is a dissipative element, which converts electrical energy into heat
when the current flows through it in any direction. The law governing the current into
and voltage across a resistor is:
� = �. � (i)
The relationship is known as Ohm’s law.
But resistor can be regarded as linear only within the specified limits, outside
which the behavior becomes non-linear. The resistance of a resistor is
temperature dependent and rises with temperature.
Mathematically it can be represented as:
�� = �0(1 + ��) (ii)
Where �0 = Resistance at 0℃ and �� = Resistance at �℃
� = Temperature coefficient and it may be positive and negative both
� = Temperature in ℃
And power dissipated by resistor is � = �. �
�2
� = �2� = Watts
�
Resistor is represented by the symbol
Unit of Resistance is ohm ( )
b) Capacitor (C): It is a two terminal element that has the capability of energy storage in
electric field. The law governing the � − � relationship of capacitor is:
��
�=� (iii)
��
After integrating equation (iii),1 we� get
� = ∫ �. �� + � (0) (iv)
� 0 �
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Where ��(0) = Capacitor voltage at � = 0, for initially uncharged capacitor
��(0) = 0
1 �
Hence, � = ∫ �. �� (v)
� 0
The above expressions show that the voltage of a capacitor cannot change
instantaneously.
Energy stored in capacitor can be represented by
1
W = ∫ �. �� = ∫ �. �. �� = � ∫ �. �� = ��2 ����� (ix)
2
Capacitor is represented by the symbol
Unit of Capacitance is Farad (F)
c) Inductor (L): It is a two-terminal storage element in which energy is stored in the
magnetic field. The � − � relation of an inductance is:
��
�=� (vi)
��
After integrating expression (vi), we get
1 �
� = ∫ �. �� + � (0) (vii)
� 0 �
Where ��(0) = Inductor current at � = 0, for initially if current through inductor
��(0) = 0
1 �
Hence, � = ∫ �. �� (viii)
� 0
The above expressions show that the current through an inductor cannot change
instantaneously.
Energy stored in inductor can be represented by
1
W = ∫ �. �� = ∫ �. �. �� = � ∫ �. �� = ��2 ����� (ix)
2
Inductor is represented by the symbol
Unit of Inductance is Henry (H)
2. Concept of active and passive elements:
Electrical Network: Any possible combination of various electric elements
(Resistor, Inductor, Capacitor, Voltage source, Current source) connected in any
manner what so ever is called an electrical network. We may classify circuit
elements in two categories, passive and active elements.
Electrical Circuit: An electric circuit is a closed energized electric network. It
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means circuit must have closed path with energy sources. From the above
example, we can say that fig 1 and fig 2 are electric networks but only fig 2 is
electric circuit.
It means, electric circuit is always an electric network but electric network may or
may not be an electric circuit.
Passive Element: The element which receives energy (or absorbs energy) and
then either converts it into heat (R) or stored it in an electric (C) or magnetic (L )
field is called passive element, and the network containing these elements
without energy sources are known as passive network. Examples are resistor,
inductor, capacitor, transformer etc.
Active Element: The elements that supply energy to the circuit is called active
element and the network containing these sources together with other circuit
elements are known as active network. Examples of active elements include
voltage and current sources, generators, and electronic devices that require
power supplies. A transistor is an active circuit element, meaning that it can
amplify power of a signal.
3. Energy Sources (Voltage and Current Sources): There are two types of energy
sources namely Voltage Sources and Current Sources.
Energy Sources
Voltage Source Current Source
Independent Dependent Independent Dependent
Voltage Source Voltage Source Current Source Current Source
Voltage Dependent Current Dependent Voltage Dependent Current Dependent
Voltage Source Voltage Source Current Source Current Source
(VDVS) (CDVS) (VDCS) (CDCS)
Here, we shall study only about independent voltage source and independent current
source.
a) Independent Voltage Source: A hypothetical generator which maintains its
value of voltage independent of the output current. It can be represented as:
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Fig: Ideal DC Voltage Source Fig: Practical DC Voltage Source
If the value of internal resistance will be zero, then the voltage source is called as
ideal voltage source. The V-I characteristics for ideal and practical voltage source
is given below:
Fig: V-I Characteristic of Voltage Source
b) Independent Current Source: A generator which maintains its output current
independent of the voltage across its terminals. It can be represented as:
Fig: Ideal DC Current Source Fig: Practical DC Current Source
if the value of internal resistance will be infinity, then the current source is called
as ideal current source. The V-I characteristics for ideal and practical current
source is given below:
Fig: V-I Characteristic of Current Source
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Module 1: DC Circuits
Contents: Electrical circuit elements (R, L and C), Concept of active and passive elements,
voltage and current sources, concept of linearity and linear network, unilateral and bilateral
elements, Kirchhoff‟s laws, Loop and nodal methods of analysis, Star-delta transformation,
Superposition theorem, Thevenin’s theorem, Norton’s theorem.
1. Electrical circuit elements (R, L and C): The interconnection of various electric
elements in a prescribed manner comprises as an electric circuit in order to perform a
desired function. The electric elements include controlled and uncontrolled source of
energy, resistors, capacitors, inductors, etc. Analysis of electric circuits refers to
computations required to determine the unknown quantities such as voltage, current
and power associated with one or more elements in the circuit. To contribute to the
solution of engineering problems one must acquire the basic knowledge of electric
circuit analysis and laws. We shall discuss briefly some of the basic circuit elements
and the laws that will help us to develop the background of subject.
a) Resistor: Resistor is a dissipative element, which converts electrical energy into heat
when the current flows through it in any direction. The law governing the current into
and voltage across a resistor is:
� = �. � (i)
The relationship is known as Ohm’s law.
But resistor can be regarded as linear only within the specified limits, outside
which the behavior becomes non-linear. The resistance of a resistor is
temperature dependent and rises with temperature.
Mathematically it can be represented as:
�� = �0(1 + ��) (ii)
Where �0 = Resistance at 0℃ and �� = Resistance at �℃
� = Temperature coefficient and it may be positive and negative both
� = Temperature in ℃
And power dissipated by resistor is � = �. �
�2
� = �2� = Watts
�
Resistor is represented by the symbol
Unit of Resistance is ohm ( )
b) Capacitor (C): It is a two terminal element that has the capability of energy storage in
electric field. The law governing the � − � relationship of capacitor is:
��
�=� (iii)
��
After integrating equation (iii),1 we� get
� = ∫ �. �� + � (0) (iv)
� 0 �
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Where ��(0) = Capacitor voltage at � = 0, for initially uncharged capacitor
��(0) = 0
1 �
Hence, � = ∫ �. �� (v)
� 0
The above expressions show that the voltage of a capacitor cannot change
instantaneously.
Energy stored in capacitor can be represented by
1
W = ∫ �. �� = ∫ �. �. �� = � ∫ �. �� = ��2 ����� (ix)
2
Capacitor is represented by the symbol
Unit of Capacitance is Farad (F)
c) Inductor (L): It is a two-terminal storage element in which energy is stored in the
magnetic field. The � − � relation of an inductance is:
��
�=� (vi)
��
After integrating expression (vi), we get
1 �
� = ∫ �. �� + � (0) (vii)
� 0 �
Where ��(0) = Inductor current at � = 0, for initially if current through inductor
��(0) = 0
1 �
Hence, � = ∫ �. �� (viii)
� 0
The above expressions show that the current through an inductor cannot change
instantaneously.
Energy stored in inductor can be represented by
1
W = ∫ �. �� = ∫ �. �. �� = � ∫ �. �� = ��2 ����� (ix)
2
Inductor is represented by the symbol
Unit of Inductance is Henry (H)
2. Concept of active and passive elements:
Electrical Network: Any possible combination of various electric elements
(Resistor, Inductor, Capacitor, Voltage source, Current source) connected in any
manner what so ever is called an electrical network. We may classify circuit
elements in two categories, passive and active elements.
Electrical Circuit: An electric circuit is a closed energized electric network. It
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2
,Basic Electrical Engineering www.aktutor.in
means circuit must have closed path with energy sources. From the above
example, we can say that fig 1 and fig 2 are electric networks but only fig 2 is
electric circuit.
It means, electric circuit is always an electric network but electric network may or
may not be an electric circuit.
Passive Element: The element which receives energy (or absorbs energy) and
then either converts it into heat (R) or stored it in an electric (C) or magnetic (L )
field is called passive element, and the network containing these elements
without energy sources are known as passive network. Examples are resistor,
inductor, capacitor, transformer etc.
Active Element: The elements that supply energy to the circuit is called active
element and the network containing these sources together with other circuit
elements are known as active network. Examples of active elements include
voltage and current sources, generators, and electronic devices that require
power supplies. A transistor is an active circuit element, meaning that it can
amplify power of a signal.
3. Energy Sources (Voltage and Current Sources): There are two types of energy
sources namely Voltage Sources and Current Sources.
Energy Sources
Voltage Source Current Source
Independent Dependent Independent Dependent
Voltage Source Voltage Source Current Source Current Source
Voltage Dependent Current Dependent Voltage Dependent Current Dependent
Voltage Source Voltage Source Current Source Current Source
(VDVS) (CDVS) (VDCS) (CDCS)
Here, we shall study only about independent voltage source and independent current
source.
a) Independent Voltage Source: A hypothetical generator which maintains its
value of voltage independent of the output current. It can be represented as:
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, www.aktutor.in
Fig: Ideal DC Voltage Source Fig: Practical DC Voltage Source
If the value of internal resistance will be zero, then the voltage source is called as
ideal voltage source. The V-I characteristics for ideal and practical voltage source
is given below:
Fig: V-I Characteristic of Voltage Source
b) Independent Current Source: A generator which maintains its output current
independent of the voltage across its terminals. It can be represented as:
Fig: Ideal DC Current Source Fig: Practical DC Current Source
if the value of internal resistance will be infinity, then the current source is called
as ideal current source. The V-I characteristics for ideal and practical current
source is given below:
Fig: V-I Characteristic of Current Source
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