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ECE 3301
General Electrical Engineering
Section 01
Fundamental Facts of Circuit Theory
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Why study circuit theory?
Circuit theory is necessary for the understanding of all
electrical systems.
Circuit theory is an excellent exercise in system
analysis.
Circuit theory is an excellent exercise in problem
solving.
A course in circuit theory will improve your algebra
and trigonometry skills.
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Obstacles To Overcome
Unfamiliarity.
– Words likes volts and amps and watts are vaguely
familiar, but not understood.
Phenomenon outside everyday experience.
– No intuitive concept of voltage and current.
Lack of mathematical skill.
– Enough said.
Student dodges.
– Trying to memorize an equation, only as needed to get
by, rather than developing a deeper understanding.
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Definition of Electricity
Electricity (and magnetism) is a characteristic of the
universe that may be used to transmit (and manipulate)
energy and to transmit (and manipulate) information.
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Physical Circuits and Models
Resistance
I
120 Ω
12 V
Model
Physical Circuit
DC Voltage Source
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Models and Physical Systems
The model is never a perfect representation of the
physical electrical system. However, if the idealized
components in the model are chosen judiciously, the
results gained from analyzing the model are sufficient
to build a useful electrical system.
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Models and Physical Systems
The appropriate model must be chosen to reflect the
realities of the physical system.
Any idealized circuit will at best be a good
approximation of a physical system.
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Basic SI Units
Quantity
Length
Mass
Time
Electrical
Current
Thermodynamic
temperature
Luminous
Intensity
Basic Unit
meter
kilogram
second
ampere
Symbol
m
kg
s
A
kelvin
K
candella
cd
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SI Multipliers
Multiplier
Prefix
Symbol
1018
15
10
12
10
109
6
10
103
102
1
10
exa
peta
tera
giga
mega
kilo
hecto
deca
E
P
T
G
M
k
h
da
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SI Multipliers
Multiplier
Prefix
Symbol
10 –1
–2
10
–3
10
10 –6
–9
10
10 –12
10 –15
–18
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deci
centi
milli
micro
nano
pico
fempto
atto
d
c
m
μ
n
p
f
a
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Fundamental Quantities
Charge: Electric charge is a characteristic of
electrons and protons.
In circuit theory, current is defined as the flow of
charged particles.
Each electron has an electric charge of
- 1.6022  10 – 19 coulombs (C).
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Electrical Charge
Protons have a positive charge of the same magnitude.
All charge exists in integer multiples of these values.
Since each electron posses such a small value of
charge, it takes about 6.24  10 18 electrons to
accumulate –1 C of charge.
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Electrical Charge
As an extension of the law of the conservation of
matter, charge can be neither created nor destroyed. It
can only be transferred from one location in a circuit
to another.
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Fundamental Quantities
Voltage: Voltage is a measure of the work (energy)
required to move one coulomb of charge between two
points (nodes) in a network.
Voltage is a measure of the electrical potential energy
difference between the two points.
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Electrical Voltage
The Voltage difference between points a and b is
defined by:
𝑑𝑤
J
𝑣𝑎𝑏 =
joules per coulomb,
𝑑𝑞
C
Where
𝑣𝑎𝑏 = voltage difference between
points 𝑎 and 𝑏 volts, V
𝑤 = energy joules, J
𝑞 = charge (coulombs, C)
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Electrical Voltage
Voltage is always measured across a circuit element .
a
𝑣𝑎𝑏 = 𝑣𝑎 − 𝑣𝑏
Circuit Element
b
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Electrical Voltage
The voltage at some point in a network is always
measured with respect to the voltage at some other
point in the network.
This voltage is the measure of the potential difference
between those two points.
This may be measured across a single circuit element,
or may be measured with respect to a reference point
(ground.)
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Electrical Voltage
Each voltage in a network has a direction, or polarity
associated with it. This is indicated by the “+” and “–
” signs shown on the circuit diagram.
a
b
𝑣𝑎𝑏
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Electrical Voltage
The value of the voltage may be positive or negative.
a
a
𝑣𝑎𝑏 = 3 V
b
Point a is 3 volts
higher than point b.
𝑣𝑎𝑏 = −3 V
b
Point a is −3 volts
higher than point b.
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Fundamental Quantities
Current: Current is a measure of the time rate of flow
of electrical charge through a circuit element.
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Electrical Current
Current is defined by:
𝑑𝑞
C
𝑖=
coulombs per second,
𝑑𝑡
s
Where
𝑖 = current amperes, A
𝑞 = charge coulombs, C
𝑡 = time (seconds, s)
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Electrical Current
Current is measured through a circuit element. Each
circuit element will have an electrical current flowing
through it.
Current is a measure of the electrons in motion due to
the electric field that results from the potential
difference (voltage) across the circuit element.
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Electrical Current
Positive
Current
Flow
i
-
-
-
Electron
Motion
Voltage
-
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Electrical Current
Current is defined as the flow of positive charges
through a circuit element.
Current flows in the opposite direction of electron
flow.
Current only flows in closed loops.
This is an extension of the Law of Conservation of
Matter.
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Electrical Current
Each current in a network has a direction associated
with it.
𝑖1
𝑖2
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Electrical Current
In addition to a direction, the value of current may be
positive or negative.
𝑖 =3A
𝑖 = −3 A
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Total Charge and Current
Since the definition of electrical current is
𝑑𝑞
𝑖=
𝑑𝑡
The total charge transferred in a given time interval is
𝑡1
𝑞=
𝑖 𝑑𝑡
𝑡0
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Fundamental Quantities
Power: The time rate at which energy is dissipated in
a circuit element.
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Power
Power is defined:
𝑑𝑤
J
𝑝=
joules per second,
𝑑𝑡
s
Where
𝑝 = power watts, W
𝑤 = energy joules, J
𝑡 = time (seconds, s)
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Power
Using the chain rule:
𝑑𝑤 𝑑𝑞
𝑝=
×
𝑑𝑞 𝑑𝑡
And the definitions of voltage and current
𝑝 = 𝑣𝑖
This describes the instantaneous power delivered to,
dissipated in, absorbed by a circuit element.
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Fundamental Quantities
In each network, some devices will be sources of
power, others will be sinks of power.
In all instances, the total power delivered in the
network will equal the total power absorbed.
This is a statement of the Law of Conservation of
Energy.
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Fundamental Quantities
Energy: The energy dissipated by a circuit element in
a given time interval is given by
𝑡1
𝑤=
𝑝 𝑑𝑡 (joules)
𝑡0
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Energy
The electric utility company uses a more convenient
measure, kilowatt-hours.
𝑡1
𝑤=
𝑝 kW 𝑑𝑡 (hours) (kWH)
𝑡0
A 5000 Watt clothes dryer operated for 45 minutes
consumes 5(.75) = 3.75 kWH
At 10cents/kWH it costs about 38 cents.
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Energy
Batteries are rated in amp-hours (the voltage is
constant).
A 3.6 Volt, 800 mAH battery has 3.6(0.8)(60)(60) =
10368 Joules of energy
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Fundamental Circuit Elements
Resistance: The voltage across a resistance is directly
proportional to the current through the resistance.
The constant of proportionality is called the resistance
R, measured in ohms.
The relationship between the current through a
resistance and the voltage across a resistance is
expressed by Ohm’s Law.
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Resistance
i
𝑣 = 𝑖𝑅
R
v
𝑣
𝑖=
𝑅
Positive current enters the positive terminal of the
resistance.
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Resistance
There is a difference between a “resistor” and a
“resistance.”
A “resistor” is a physical device with voltage, current
and power and frequency limitations.
It also has inductive, capacitive and thermal effects.
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Resistance
“Resistance” is the idealized model of a resistor.
Provided a resistor is applied within its limitations, it
can be very well modeled as an ideal resistance.
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Resistance
i
𝑣 = −𝑖𝑅
R
v
𝑣
𝑖=−
𝑅
Positive current enters the negative terminal of the
resistance.
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Resistance
The behavior of a resistor may be expressed
graphically as follows.
I
I
R
V
1
R
V
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Resistance
I=?
R=5Ω
V = 10 V
𝑉
𝐼=
𝑅
A 5 ohm resistor with 10 V placed across it results in a
current flow of 2 amps through the resistor.
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Resistance
I =3A
R=2Ω
V=?
𝑉 = 𝐼𝑅
A 3 amp current through a 2 ohm resistor results in a
voltage of 6 volts across the resistor.
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Resistance
I=5A
R=?
V = 20 V
𝑉
𝑅=
𝐼
A 5 amp current through a resistor with a 20 V drop
across it requires a 4 Ohm resistance.
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Resistance
Power dissipated in a resistance: The power
dissipated in a resistor is given by the equation
𝑝 = 𝑣𝑖
recall 𝑣 = 𝑖𝑅
𝑝 = 𝑖𝑅𝑖
𝑝 = 𝑖2𝑅
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Resistance
This is often referred to as the “i-squared-R” power
loss.
𝑝 = 𝑖2𝑅
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Resistance
Power dissipated by a resistance: The power
dissipated in a resistor is given by the equation
𝑝 = 𝑣𝑖
recall 𝑖 =
𝑣
𝑅
𝑣 𝑣2
𝑝=𝑣 =
𝑅
𝑅
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Conductance
Conductance: On some occasions, instead of using
resistance, the conductance of a device may be used,
where
1
𝐺=
siemens S (or mhos)
𝑅
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Fundamental Circuit Elements
Inductance: The voltage across an inductance is
proportional to the time rate-of-change of the current
through it.
The constant of proportionality is called the
inductance L, measured in henries.
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Inductance
𝑖 𝑡
L
𝑑𝑖
𝑣=𝐿
𝑑𝑡
𝑣 𝑡
1
𝑖=
𝐿
𝑡0
𝑣 𝑑𝑡
−∞
The limits of integration are from –  to the present
time. Thus the inductance has “memory.”
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Inductance
An “inductor” is a physical device that may be
modeled as an “inductance.”
An inductor also has voltage, current, power and
frequency limitations because of resistive, capacitive
and thermal effects.
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Fundamental Circuit Elements
Capacitance: The current through a capacitance is
proportional to the time rate-of-change of the voltage
across it.
The constant of proportionality is call capacitance, C,
measured in farads.
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Capacitance
𝑖 𝑡
C
𝑑𝑣
𝑖=𝐶
𝑑𝑡
𝑣 𝑡
1
𝑣=
𝐶
𝑡0
𝑖 𝑑𝑡
−∞
The limits of integration are from –  to the present
time. Thus the capacitance has “memory.”
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Capacitance
A “capacitor” is a physical device that may be
modeled as a “capacitance.”
A capacitor also has voltage, current, power and
frequency limitations because of resistive, inductive
and thermal effects.
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Fundamental Laws of Circuit Theory
Kirchhoff’s Current Law: The algebraic sum of all
currents entering any node equals zero.
Consider the current into a node as positive, and the
current leaving a node as negative.
This Law is an extension of the Law of Conservation
of Matter.
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Kirchhoff’s Current Law
𝑖1
Node
𝑖3
𝑖4
𝑖2
𝑖5
𝑖1 + 𝑖2 − 𝑖3 − 𝑖4 − 𝑖5 = 0
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Kirchhoff’s Current Law
𝑖1
Node
𝑖3
𝑖4
𝑖2
𝑖5
𝑖1 + 𝑖2 = 𝑖3 + 𝑖4 + 𝑖5
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Fundamental Laws of Circuit Theory
Kirchhoff’s Voltage Law: The algebraic sum of
voltage differences around any closed loop equals
zero.
Consider moving from a “–” to a “+” sign a positive
voltage and moving form a “+” to a “–” sign a
negative voltage.
This Law is an extension of the Law of the
Conservation of Energy.
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Kirchhoff’s Voltage Law
𝑣𝑐𝑎
𝑣𝑏𝑎
a
𝑣𝑎
𝑣𝑐𝑏
b
c
𝑣𝑏
𝑣𝑐
0
+ 𝑣𝑎 + 𝑣𝑏𝑎 − 𝑣𝑏 = 0
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Kirchhoff’s Voltage Law
𝑣𝑐𝑎
𝑣𝑏𝑎
a
𝑣𝑎
𝑣𝑐𝑏
b
c
𝑣𝑏
𝑣𝑐
0
𝑣𝑎 + 𝑣𝑏𝑎 = 𝑣𝑏
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Kirchhoff’s Voltage Law
𝑣𝑐𝑎
𝑣𝑏𝑎
a
𝑣𝑎
𝑣𝑐𝑏
b
c
𝑣𝑏
𝑣𝑐
0
+ 𝑣𝑎 + 𝑣𝑐𝑎 − 𝑣𝑐𝑏 − 𝑣𝑏 = 0
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Kirchhoff’s Voltage Law
𝑣𝑐𝑎
𝑣𝑏𝑎
a
𝑣𝑎
𝑣𝑐𝑏
b
c
𝑣𝑏
𝑣𝑐
0
𝑣𝑎 + 𝑣𝑐𝑎 = 𝑣𝑐𝑏 + 𝑣𝑏
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Kirchhoff’s Voltage Law
𝑣𝑐𝑎
𝑣𝑏𝑎
a
𝑣𝑎
𝑣𝑐𝑏
b
c
𝑣𝑏
𝑣𝑐
0
𝑣𝑎 + 𝑣𝑐𝑎 = 𝑣𝑐𝑏 + 𝑣𝑏 = 𝑣𝑐
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Basic Assumptions
Linear Elements: There is a linear relationship
between the voltage across all circuit elements and the
current through the circuit elements.
This assumption is largely true for passive elements
(resistors, inductors and capacitors) as long as they are
operated within their voltage, current frequency and
thermal limitations.
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Basic Assumptions
Time-Invariant Elements: The characteristics of the
circuit elements do not change over time. That is, the
resistance, inductance and capacitance are constants
and not functions of time.
In reality, all physical devices age and their electrical
characteristics change.
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Basic Assumptions
Bi-lateral, Two-terminal Elements: The elements
have two terminals.
The electrical characteristics of the circuit elements
are the same regardless of the polarity of the voltage
across the element and regardless of the direction of
the current through the circuit element.
The same current flows through both terminals.
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Basic Assumptions
Lumped Elements: The circuit elements operate at a
point within the electrical network.
From another point of view, the electrical network is
small compared to the wavelength of the highest
frequency of interest.
Since waveforms move through electrical networks at
roughly the speed of light, the relevant dimensions of
a network can be approximated.
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Basic Assumptions
Approximate Wavelengths at Various Frequencies.
Frequency
60 Hz

5106 m  3100 miles
/10
310 miles
20 KHz
100 MHz
600 MHz
1 GHz
15103 m  9 miles
3 m  10 feet
.5 m  20 inches
.3 m  12 inches
.9 miles
12 inches
2 inches
1.2 inches
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Basic Assumptions
Idealized Components: Idealized models are used
for all circuit elements. These idealized models are
accurate under many conditions. For other conditions
a combination of idealized components may be used to
represent real devices
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Conventions for Variables
Constant (time invariant) variables are denoted by
upper case variables such as
R, C, L, V, I.
Variables that are time varying are designated using
lower case variables
v, i
or explicitly
v(t), i(t)
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Conventions for Variables
Complex variables (those with real and imaginary
parts) will be indicated by upper case bold face.
𝑽, 𝑰
When hand-written, complex variables are written in
upper case with a bar above the variable symbol.
𝑉, 𝐼
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Reference Node (Ground)
The voltage at any point in an electrical network is not
absolute. It is always measured with respect to the
voltage at some other point in the universe.
This demands we establish a reference point and
assign that point a voltage. This reference point is
traditionally called “ground” and is assigned the
voltage of zero volts.
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Reference Node (Ground)
In some cases, this reference point involves a physical
connection to the earth.
In most instances, the reference point is the metallic
case or chassis containing the circuit.
The reference node in circuits is typically indicated by
the “ground” symbol.
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Other things you need to know
Algebra
Trigonometry
– Sinusoidal
Functions
– Exponential
Functions
Complex Numbers
– Rectangular Form
– Polar Form
Linear Algebra
– Matrices and
determinates
– Solving sets of
simultaneous
equations
– Cramer’s Rule
Differential and
Integral calculus
Differential Equations
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