CPE220 Electric Circuit
Analysis
Chapter 1: Introduction
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Chapter 1
1. Systems of Units
1
Electrical
circuit analysis deals with
measurable quantities. Hence, we must
give both a number and a unit. This
requires a standard language that virtually
all professionals can understand. Such an
international measurement language is the
International System of Units (SI). The SI
is built upon seven basic units: the meter,
kilogram, second, ampere, kelvin, mole, and
candela as shown in Table 1.1. Units for
other quantities such as volume, force,
energy, etc., are derived from these seven
base units.
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T a b l e
1.1
S I
b a s e
u n i t s
Base Quantity
Name
Symbol
length
meter
m
mass
kilogram
kg
time
second
s
electric current
ampere
A
thermodynamic
kelvin
K
temperature
amount of substance mole
mol
luminous intensity
cd
candela
Prefixes are often used to emphasize the
significant figures when the magnitudes of
the measurable quantities are too large or
too small. Common prefixes and their
corresponding factors are shown in Table
1.2.
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Factor
Prefix
Symbol
10-24
yocto
y
10-21
zepto
z
10-18
atto
a
10-15
femto
f
10-12
pico
p
10-9
nano
n
10-6
micro
m
10-3
milli
m
10-2
centi
c
10-1
deci
d
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Factor
Prefix
Symbol
10
deka
da
102
hecto
h
103
kilo
k
106
mega
M
109
giga
G
1012
tera
T
1015
peta
P
1018
exa
E
1021
zetta
Z
1024
yotta
Y
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1. Fundamental Quantities
2
We encounter so many electronic devices in
our daily life such as radio, television,
mobile phone, etc.. These electronic devices
are formed by many electric circuits which
are the interconnection of electrical
elements. Hence, it is important to know
the properties of the involved components
as well as the components are connected to
form the circuit.
To study and analyze electric circuits, we
need to determine the movement of the
electrically charged particles, "q" or simply
charges. The concept of charge is the
underlying principle for explaining all
electrical phenomena. The movement of
charge can be described by the "through"
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quantity called "current" and the "across"
quantity called "voltage".
1.2.1 Charge
Charge, q, is defined as an electrical
property of the atomic particles of which
matter consists. From basic physics, atoms
are fundamental building blocks of all
matter. Each atom is composed of
electrons, protons, and neutrons.
In the SI system, charge is measured in
coulombs (C). The charge on an electron is
-1.602 x 10-19 C. Hence, in 1 C of charge,
there are 1/(1.602 x 10-19) = 6.24 x 1018
electrons. The law of conservation of
charge states that charge can neither be
created nor destroyed, only transferred. In
other words,
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the algebraic sum of the electric
charges in a system does not change.
When you rub a comb with a woolen cloth,
a negative (corresponding to an electron)
charge is produced on the comb and a
positive (corresponding to a proton) charge
is produced on the cloth. (Benjamin
Franklin defined the charge on the comb as
negative). The comb acquires its negative
charge because some of the electrons on the
cloth are rubbed off onto the comb.
Charge in motion results in energy transfer.
Of special interest in the case where the
motion is confined to a definite closed path.
In this case, current "flows".
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1.2.2 Current
Current, i, is the rate at which charge is
transferred or "flows" through an
electronic device. Mathematically, the
current is defined as:
dq(t)
i(t) =
(1.1)
dt
where i = the current in amperes (A)
q = the charge in coulombs (C)
That is, we define the current, i, flowing in
an electronic device as the amount of
charge passing through that device per unit
of time. The unit of current is the ampere
(A) and
1 ampere is 1 coulomb per second.
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It is convenient to think of current as
the flowing motion of positive charge
although, physically, current flow in
metallic conductors results from
negative charge (electron) motion.
That is, the charge carriers are
negative and move in the opposite
direction.
++ + ++ +
i(t)
+ ++ +
i(t)
Figure 1.1 Charge/Current Relationship.
From Fig. 1.1, the charge and the current
flow through the conductor, passing from
one side of the imaginary surface to the
other.
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During the time interval from ta to tb the
charge passing through the imaginary
surface can be determined by Eq. 1.2 as
following:
tb
q(t) 

i(t)dt
(1.2)
ta
Current defined by Eq. 1.1 demonstrates
that current is a time-varying function.
That is, it is not a constant-valued function.
Practically, there are so many types of
current. However, the most common types
of current in electric circuit are shown in
Fig. 1.2 which are: direct current (dc),
alternating (sinusoidal) current (ac),
exponential current, damped sinusoidal
current, and transient current.
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Direct Current (dc)
i
t
1
Alternating/Sinusoidal
Current (ac)
0.5
0
-0.5
-1
0
5
10
15
20
1
Exponential Current
0.8
0.6
0.4
0.2
0
0
5
10
15
20
1
Damped Sinusoidal
Current
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
5
10
15
20
0
5
10
15
20
5
4
3
2
Transient Current
1
0
-1
-2
-3
Figure 1.2 Types of current: Direct Current (dc),
Alternating/Sinusoidal Current (ac),
Exponential Current, Damped Sinusoidal
Current and Transient Current.
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The two most common types of current
used with electronic devices or apparatus
are direct current and alternating current.
From Fig. 1.2, a direct current (dc) is a
current that is constant in time. By
convention the symbol "I" is used to
represent such a constant current. An
alternating current (ac), on the other hand,
is a current that varies sinusodially with
time as shown in Fig. 1.3.
i(t)
Acos(q)
t
Tp
Figure 1.3 A sinusoidal (alternating) current.
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Such current is usually called a "sinusoidal
current" and can be represented as a cosine
function:
i(t) = Acos(wt + q)
  t  
(1.3)
Where
w = the angular frequency (rad/sec)
= 2pf
f = the frequency (cycles/sec)
Tp = the fundamental period (sec)
= 1
f
q = the phase
The sinusoidal current described by Eq. 1.3
has these following properties:
1. Periodical: i.e.
i(t + Tp) = i(t)
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(1.4)
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2. Distinction: sinusoidal currents with
different frequencies are themselves
distinct.
3. The rate of oscillation of the current
will be increased if the frequency "f" is
increased.
In defining current, both the direction of
the arrow and the value are required as
illustrated in Fig. 1.4.
(a)
(b)
Figure 1.4 Two different methods of labeling the
same current.
A negative current of -3 A flowing in one
direction as shown in Fig. 1.4(b) is the same
as a current of +3 A flowing in the opposite
direction.
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As previously mentioned, the direction of
current flow is conventionally taken as the
direction of positive charge movement.
Hence, the direction of the arrow and the
positive value of current indicate the
direction of a net positive charge. For
example, in Fig. 1.4(a), the direction of the
arrow and the value 3 A indicate that a net
positive charge of 3 C/s is moving to the
right or that a net negative charge of -3 C/s
is moving to the left each second.
To define the current i1(t) having only
either the value of the current or the arrow
is an in complete, improper, and incorrect
definition as shown in Fig. 1.5(a) and (b),
respectively. Fig. 1.5(c) is the correct
definition of a current.
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Figure 1.5 (a,b) Incomplete, improper, and incorrect
definitions of a current. © the correct
definition of i1(t).
Example 1.1: DC current.
Find the charge passing through a point in
a conductor if the current passing through
the conductor has the waveform indicated.
i
2A
t
Solution:
The current does not change with time.
Hence, the charge 2 C is passing through
the point each second.
Ans.
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Example 1.2: For the given current, which
is given in graphical form, find the charge
passing through a given point in the time
interval 0-4 s.
i(t)
10 A
0
1
2
3
4
t (sec)
Solution:
For the time interval 0-4 s, the charge
passing through a given point on the
conductor that carries the current is:
4
q(t) 

 i(t)dt
0
1
2
30
1
 10tdt  10dt 
4
 (10t  30)dt   (10t  30)dt
2
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= 5 + 10 -25 +35 C
= 25 C
Ans.
1.2.3 Voltage
Charges in motion yield an energy transfer.
The separation of charge creates an electric
force, which was recognized by the 18th
century Italian physicist Allessandro
Antonio Volta. This force is known as
"voltage" or "potential difference". Voltage
is defined as the energy required to move a
unit charge through an element, measured
in volts (V). In the other words, voltage is
the energy per unit charge that is created
by the charge separation.
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i
i
circuit
element
...
a
+
v
...
b
-
Figure 1.6 Current i flowing from point a to point b
induces the voltage across point a and b.
In Fig. 1.6, the current I flowing through a
circuit element from point "a" to point "b"
requires an energy to move the charges.
Voltage between points "a" and "b" is
defined as:
dw
vab =
dq
(1.5)
where vab = the voltage between a and b,
w = the energy in joules (J),
q = the charge in coulombs (C).
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1 volt is 1 joule per coulomb.
Traditionally, vab has two meanings:
 The potential at point a is assumed
higher than the potential at point b.
 vab is the potential at point a with
respect to point b.
Hence,
vab = -vba
(1.6)
Voltage polarity indicates the relative
potential between two points. Usually, the
plus "+" sign is assigned to a higher
potential point and minus "-" sign is
assigned to a lower potential point.
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Figure 1.7 Point B is 5 V positive with respect to
point A in (a) and (b). Point A is 5 V
positive with respect to point B in (c) and
(d).
Actual direction and polarity will be
governed by the sign of the value. Hence,
during the analysis, direction and polarity
can be arbitrarily assigned on circuit
diagram. In Fig. 1.7 (a), the “+” sign is
placed at the point “A” but the numerical
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value of vAB is -5 V. That is, conventionally,
point “A” is -5 V positive with respect to
point “B” or, actually, point “B” is 5 V
positive with respect to point “A” as
illustrated in Fig. 1.7(b). Similarly, the
representation of vAB in Fig. 1.7(c) and (d)
have the same meaning which is point “A”
is 5 V positive with respect to point “B”.
Therefore, the definition of voltage requires
both the numerical value of voltage and the
“plus-minus” sign pair.
Like electric current, a constant voltage is
called a “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 usually
produced by a battery and an ac voltage is
produced by an electric generator.
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1.2.4 Power and Energy
Power
Two other important quantities that are
frequently used in describing electrical
system are “power, p” and “energy, w”.
Power, p, is defined as the rate at
which energy, w, is expended or
absorbed, measured in watts (W).
That is, the relationship between the power
and the energy can be expressed as:
p =
where
dw
dt
(1.7)
p = the power in watts (W),
w = the energy in joules (J),
t = the time in seconds (sec).
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We can rewrite Eq. 1.7 as:
p =
dw . dq
dq
dt
= vi
(1.8)
Thus, the power absorbed or supplied by an
element is the product of the voltage across
the element and the current through it. If
the numerical value of power is positive (+),
power is being delivered to or absorbed by
the element. On the other hand, if the
numerical value of power is negative (-),
power is being supplied by the element.
Passive Sign Convention
Current direction and voltage polarity
determine the sign of power. The passive
sign convention is satisfied when the
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current enters through the positive(plusmarked) terminal of an element as
shown in Fig. 1.8.
i
...
+
v
i
circuit
element
–
Figure 1.8 The passive sign convention.
When this passive sign convention is being
used:
 If v i > 0, then the circuit element
is absorbing power.
 If v i < 0, then the circuit element
is supplying power.
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In general,
Power absorbed = -Power supplied
Power Conservation Theorem
The sum of the powers absorbed by
all the elements in a circuit equals
zero.
Example 1.3
Verify the power conservation theorem for
the following circuit.
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+ 240 V –
2
+
300 V 1
–
2 A 1.5 A
2A
0.5 A
+
3 60 V
–
+
4 60 V
–
Solution:
p1 = -(300)(2) = -600 W
p2 = (240)(2) = 480 W
p3 = (60)(1.5) =
90 W
p4 = (60)(0.5) =
30 W
 supplied
 absorbed
 absorbed
 absorbed
Σpi = -600 +480 + 90 + 30 W
= 0W
Ans.
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Energy
The power p in Eq. 1.8 is a time-varying
and is called the “instantaneous power”.
The energy absorbed or supplied by an
element from time t0 to time t is determined
by:
t
w(t) 
 p(t) dt
(1.9)
t0
t

 v(t)i(t) dt
t0
That is, energy is the capacity to do work,
measured in joules (J).
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Source-Load Circuit
A simple model for many circuits consists of
one source and one load. The source
produces the power and the load consumes
the power.
There is a voltage rise at the source and a
voltage drop at the load (when proceeding
in the direction of the reference current).
i
+
Source
V
Load
–
The source get its
energy through this
connection.
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Voltage rise in
the direction of
the current:
power supplied.
36
Voltage drop in
the direction of
the current:
power
consumed.
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Source-Load Circuit: Hydraulic
Analog
At the source (pump) there is a pressure
rise in the direction of the flow.
At the turbine (load) there is a pressure
drop in the direction the flow.
High pressure
Pump
Turbine
Low pressure
The pump get its
energy through this
connection.
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1. Voltage and Current Sources
3
Thus far voltage, current, power and
energy have been defined. In this section
an element which is the basic building block
of a circuit will be discussed. In general,
the basic circuit elements can be classified
by the relationship of the current through
the element to the voltage across the
element. For example, the voltage across an
element can be linearly proportional to the
current through it (a resistor) or the voltage
across an element can be proportional to
the derivative (a inductor) or the integral (a
capacitor) of the current with respect to
time.
There are two types of basic electrical
elements: passive elements and active
elements. An passive element is one that
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never supplies energy while an active
element is capable of generating energy. In
this section, only the active elements will be
discussed.
The most important active elements are
voltage and current sources which generally
deliver power to the circuit connected to
them. The voltage and current sources can
be classified as either independent or
dependent sources.
An ideal independent source is an
active element that can supply a
specified voltage or current that is
completely independent of a current
or voltage elsewhere in the circuit.
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An ideal dependent source, on the other
hand, is an active element in which the
source voltage or current depends on
another voltage or current in the circuit.
1.3.1 Independent Voltage Sources
i
i
vs
vs
(a)
vs
(b)
(c)
Figure 1.9 Circuit symbol of the independent voltage
source.
An ideal dependent source in Fig. 1.9a will
deliver to the circuit whatever current is
necessary to maintain its terminal voltage
vs. The current arrow labeled "i" in Fig.
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1.9b indicates that this independent voltage
source deliver the power p = vsi to the
circuit. On the other hand, the independent
voltage source in Fig. 1.9c absorb the power
p = vsi from the circuit.
The lower-case vs indicates a time-varying
terminal voltage of the independent voltage
source. As mentioned earlier, the most two
common current types are direct current
(dc) and alternating current (ac). Hence, if
vs is constant in time, such the independent
voltage source is termed "an independent
dc voltage source" represented by either of
the symbols shown in Fig. 1.10a and b.
In the case that vs varies sinusodially with
time the independent voltage source is
termed "an independent ac voltage source"
represented by the symbols shown in Fig.
1.10c.
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vs
V
Vs
(a)
(b)
(c)
Figure 1.10 (a) and (b) DC independent voltage source
symbols. (c) AC independent voltage
source symbol.
1.3.2 Independent Current Sources
Similarly, an ideal independent current
source provides a specified current
completely independent of the voltage
across the source. That is, the current
source delivers to the circuit whatever
voltage is necessary to maintain the
designated current.
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i
I
(a)
(b)
Figure 1.11 (a) A time-varying independent current
source. (b) A constant (DC) independent
current source.
Fig. 1.11 illustrates the symbols of
independent current source where the
arrow indicates the direction of current i or
I. In Fig. 1.11a, lower-case "i" indicates a
time-varying independent current source
which is normally a sinusoidal function (ac
current). In Fig. 1.11b, on the other hand,
upper-case "I" indicates an independent dc
current source.
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1.3.3 Dependent Sources
The dependent, or controlled, source is a
source in which the source voltage or
current is determined by another voltage or
current elsewhere in the circuit being
analyzed. It is usually represented by
diamond-shaped symbol, as shown in Fig.
1.12.
v
i
(a)
(b)
Figure 1.12 (a) A dependent voltage source. (b) A
dependent current source.
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Since the dependent source can be
controlled by either voltage or current of
some other element in the circuit, we can
categorize the dependent source into four
types, as shown in Fig. 1.13, which are:
1. A voltage-controlled voltage source (VCVS),
2. A current-controlled voltage source (CCVS),
3. A voltage-controlled current source (VCCS),
4. A current-controlled current source (CCCS).
VCVS:
i
v = b vx
v = b vx
i = whatever
vx is somewhere (not
shown) and b is a
constant.
(a)
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CCVS:
i
v = b ix
v = b ix
i = whatever
ix is somewhere (not
shown) and b is a
constant.
(b)
VCCS:
+
i = b vx
i = b vx
v
_
v = whatever
vx is somewhere (not
shown) and b is a
constant.
(c)
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CCCS:
+
i = b ix
v
_
i = b ix
v = whatever
ix is somewhere (not
shown) and b is a
constant.
(d)
Figure 1.13 Four possible types of dependent sources:
(a) a voltage-controlled voltage source
(VCVS), (b) a current-controlled voltage
source (CCVS), (c) a voltage-controlled
current source (VCCS), (d) a currentcontrolled current source (CCCS).
Controlled sources are used primarily to
model electronic devices. For example, the
junction field-effect transistor (JFET) and
the bipolar junction transistor (BJT) are
modeled as shown in Fig 1.14. These devices
are used to construct electronic circuits
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such as amplifiers and digital computers.
Without dependent sources we would not
be able to model these important electrical
components.
D
D
G +
G
gmvGS
vGS
S
ro
S
_
JFET
iB
C
B
B
biB
ro
E
D
ro
S
BJT
Figure 1.14 The electronic devices, JFET and BJT, can
be modeled by controlled sources.
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