ECC 102: CIRCUIT THEORY
1. INTRODUCTION
Electric circuit theory and electromagnetic theory are the two fundamental theories upon
which all branches of electrical engineering are built. Many branches of electrical
engineering, such as power, electric machines, control, electronics, communications, and
instrumentation, are based on electric circuit theory.
Primary contributions in circuit theory document Alessandro Antonio Volta (1745 – 1827)
and Andre-Marie Ampere (1775 – 1836). Volta, an Italian physicist, invented the electric
battery – which provided the first continuous flow of electricity – and the capacitor. The unit
of voltage measurement, the Volt (V), is named in his honour.
Ampere, a French mathematician and physicist, on the other hand, laid the foundation of
electromagnetics. He defined the electric current and developed a way to measure it. The unit
of current measurement, the Ampere or Amp (A) is named after him.
Therefore, the basic circuit theory course is the most important course for an electrical
engineering student, and always an excellent starting point for a beginning student in
electrical engineering education and allied courses.
1.1 Fundamentals
An electric circuit is an interconnection of electrical elements, that is, voltage/current source,
load and connecting wires. Electrical engineers deal with measurable quantities. The
International System of Units (SI) is adopted. The basic six SI units are given in Table 1
below.
Table 1:
: Common SI units
Quantity Basic unit Symbol
Length meter m
Mass kilogram kg
Time second s
Electric current ampere A
Voltage volt V
Power watt W
Capacitance farad F
Inductance henry H
One great advantage of the SI unit is that it uses prefixes based on the power of 10 to relate
larger and smaller units to the basic unit. Table 2 shows the SI prefixes and their symbols.
1
, Table 2: The SI prefixes
Multiplier Prefix Symbol Multiplier Prefix Symbol
1018 exa E 101 deci d
1015 peta P 102 centi c
1012 tera T 103 milli m
109 giga G 106 micro
106 mega M 109 nano n
103 kilo k 1012 pico p
102 hecto h 1015 femto f
10 deka da 1018 atto a
1.2 Charge and Current
1.2.1 Charge
Charge is an electrical property of the atomic particles of which matter consists; measured in
coulombs (C). Matter consists of atoms which are electrically neutral. On the other hand, an
atom consists of a nucleus and electrons. The nucleus, situated at the centre, consists of
protons (positively charged) and neutrons (neutral). The nucleus is surrounded by orbiting
electrons which are electrically negative.
We know that the charge e on an electron is negative and equal in magnitude to
1.602 1019 C , while a proton carries a positive charge of the same magnitude as the
electron. The presence of equal numbers of protons and electrons leaves the atom neutrally
charged.
Consequently,
1) The coulomb is a large unit of charges. In 1 C of charge, there are
1/ 1.602 1019 6.24 1018 electrons. Thus realistic or laboratory values of charges
are on the order of pC , nC , or C .
2) According to experimental observations, the only charges that occur in nature are
integral multiples of the electronic charge e 1.602 1019 C
3) The law of conservation of charge states that charge can neither be created nor
destroyed only transferred. Thus the algebraic sum of the electric charges in a system
does not change.
2
,1.2.2 Current
Current is flow of electric charges. From atomic theory, valence electrons are loosely bound
in the atom and may break off to form an electron cloud in the atomic structure of materials
classified as metals. These charges are mobile. When a conducting wire is connected to a
battery, the charges are compelled to move; positive charges move in one direction while
negative charges move in the opposite direction. This motion of charges constitutes the
electric current. It is conventional to take the current flow as the movement of positive
charges, that is, opposite to the flow of negative charges.
Thus, electric current is the time rate of change of charge, measured in amperes (A).
Mathematically, the relationship between current i , charge q , and time t is
dq
i (1.1)
dt
where current is measured in amperes (A), and
1 ampere = 1 coulomb/second
The charge transferred between time t0 and t is obtained by integrating both sides of Eq.
(1.1). We obtain
t
q idt (1.2)
t0
If the current does not change with time, but remains constant, we call it direct current (dc).
Thus, a direct current (dc) is a current that remains constant with time (Fig. 1.2). By
convention the symbol I is used to represent such a constant current.
I
0
t
Fig. 1.1: Direct current (dc)
A time-varying current is represented by the symbol i . A common form of a time-varying
current is the sinusoidal current or alternating current (ac). Thus, an alternating current (ac)
is a current that varies sinusoidally with time (Fig. 1.2).
3
, 1
0.5
current (i)
0
-0.5
-1
0 2 4 6 8 10 12 14 16 18 20
time (t)
Fig. 1.2: Alternating current (ac)
1.3 Voltage
To move an electron in a conductor in a particular direction requires some work or energy
transfer. This work is performed by an external electromotive force (emf). This emf is also
called voltage or potential difference. The voltage vab between two points a and b in an
electric circuit is the energy (or work) needed to move a unit charge from a to b ; that is,
dw
vab (1.3)
dq
where w is energy in joules (J) and q is charge in coulombs (C). The voltage vab or simply
v is measured in volts (V). From Eq. (1.3), it is evident that
1 volt = 1 joule/coulomb
Thus, voltage is the energy required to move a unit charge through an element, measured in
volts (V) (Fig.1.3).
+
v ab
-
Fig. 1.3: Voltage
The plus (+) and minus (-) signs are used to define reference direction or voltage polarity. In
Fig. 1.3, point a is at a potential of vab volts higher than point b or the potential at point a
with respect to point b is vab . It follows logically that in general
vab vba (1.4)
Like electric current, a constant voltage is called dc voltage and is represented by V, whereas
a sinusoidally time-varying voltage is called an ac voltage and is represented by v . A dc
voltage is commonly produced by a battery; ac voltage is produced by a generator.
4
1. INTRODUCTION
Electric circuit theory and electromagnetic theory are the two fundamental theories upon
which all branches of electrical engineering are built. Many branches of electrical
engineering, such as power, electric machines, control, electronics, communications, and
instrumentation, are based on electric circuit theory.
Primary contributions in circuit theory document Alessandro Antonio Volta (1745 – 1827)
and Andre-Marie Ampere (1775 – 1836). Volta, an Italian physicist, invented the electric
battery – which provided the first continuous flow of electricity – and the capacitor. The unit
of voltage measurement, the Volt (V), is named in his honour.
Ampere, a French mathematician and physicist, on the other hand, laid the foundation of
electromagnetics. He defined the electric current and developed a way to measure it. The unit
of current measurement, the Ampere or Amp (A) is named after him.
Therefore, the basic circuit theory course is the most important course for an electrical
engineering student, and always an excellent starting point for a beginning student in
electrical engineering education and allied courses.
1.1 Fundamentals
An electric circuit is an interconnection of electrical elements, that is, voltage/current source,
load and connecting wires. Electrical engineers deal with measurable quantities. The
International System of Units (SI) is adopted. The basic six SI units are given in Table 1
below.
Table 1:
: Common SI units
Quantity Basic unit Symbol
Length meter m
Mass kilogram kg
Time second s
Electric current ampere A
Voltage volt V
Power watt W
Capacitance farad F
Inductance henry H
One great advantage of the SI unit is that it uses prefixes based on the power of 10 to relate
larger and smaller units to the basic unit. Table 2 shows the SI prefixes and their symbols.
1
, Table 2: The SI prefixes
Multiplier Prefix Symbol Multiplier Prefix Symbol
1018 exa E 101 deci d
1015 peta P 102 centi c
1012 tera T 103 milli m
109 giga G 106 micro
106 mega M 109 nano n
103 kilo k 1012 pico p
102 hecto h 1015 femto f
10 deka da 1018 atto a
1.2 Charge and Current
1.2.1 Charge
Charge is an electrical property of the atomic particles of which matter consists; measured in
coulombs (C). Matter consists of atoms which are electrically neutral. On the other hand, an
atom consists of a nucleus and electrons. The nucleus, situated at the centre, consists of
protons (positively charged) and neutrons (neutral). The nucleus is surrounded by orbiting
electrons which are electrically negative.
We know that the charge e on an electron is negative and equal in magnitude to
1.602 1019 C , while a proton carries a positive charge of the same magnitude as the
electron. The presence of equal numbers of protons and electrons leaves the atom neutrally
charged.
Consequently,
1) The coulomb is a large unit of charges. In 1 C of charge, there are
1/ 1.602 1019 6.24 1018 electrons. Thus realistic or laboratory values of charges
are on the order of pC , nC , or C .
2) According to experimental observations, the only charges that occur in nature are
integral multiples of the electronic charge e 1.602 1019 C
3) The law of conservation of charge states that charge can neither be created nor
destroyed only transferred. Thus the algebraic sum of the electric charges in a system
does not change.
2
,1.2.2 Current
Current is flow of electric charges. From atomic theory, valence electrons are loosely bound
in the atom and may break off to form an electron cloud in the atomic structure of materials
classified as metals. These charges are mobile. When a conducting wire is connected to a
battery, the charges are compelled to move; positive charges move in one direction while
negative charges move in the opposite direction. This motion of charges constitutes the
electric current. It is conventional to take the current flow as the movement of positive
charges, that is, opposite to the flow of negative charges.
Thus, electric current is the time rate of change of charge, measured in amperes (A).
Mathematically, the relationship between current i , charge q , and time t is
dq
i (1.1)
dt
where current is measured in amperes (A), and
1 ampere = 1 coulomb/second
The charge transferred between time t0 and t is obtained by integrating both sides of Eq.
(1.1). We obtain
t
q idt (1.2)
t0
If the current does not change with time, but remains constant, we call it direct current (dc).
Thus, a direct current (dc) is a current that remains constant with time (Fig. 1.2). By
convention the symbol I is used to represent such a constant current.
I
0
t
Fig. 1.1: Direct current (dc)
A time-varying current is represented by the symbol i . A common form of a time-varying
current is the sinusoidal current or alternating current (ac). Thus, an alternating current (ac)
is a current that varies sinusoidally with time (Fig. 1.2).
3
, 1
0.5
current (i)
0
-0.5
-1
0 2 4 6 8 10 12 14 16 18 20
time (t)
Fig. 1.2: Alternating current (ac)
1.3 Voltage
To move an electron in a conductor in a particular direction requires some work or energy
transfer. This work is performed by an external electromotive force (emf). This emf is also
called voltage or potential difference. The voltage vab between two points a and b in an
electric circuit is the energy (or work) needed to move a unit charge from a to b ; that is,
dw
vab (1.3)
dq
where w is energy in joules (J) and q is charge in coulombs (C). The voltage vab or simply
v is measured in volts (V). From Eq. (1.3), it is evident that
1 volt = 1 joule/coulomb
Thus, voltage is the energy required to move a unit charge through an element, measured in
volts (V) (Fig.1.3).
+
v ab
-
Fig. 1.3: Voltage
The plus (+) and minus (-) signs are used to define reference direction or voltage polarity. In
Fig. 1.3, point a is at a potential of vab volts higher than point b or the potential at point a
with respect to point b is vab . It follows logically that in general
vab vba (1.4)
Like electric current, a constant voltage is called dc voltage and is represented by V, whereas
a sinusoidally time-varying voltage is called an ac voltage and is represented by v . A dc
voltage is commonly produced by a battery; ac voltage is produced by a generator.
4
