K. A. Saaifan, Jacobs University, Bremen
Basic Electrical Circuits Analysis
ECE 221
PhD. Khodr Saaifan
http://trsys.faculty.jacobs-university.de
k.saaifan@jacobs-university.de
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K. A. Saaifan, Jacobs University, Bremen
Reference:
Electric Circuits, 8th Edition
James W. Nilsson, and Susan A. Riedel
Lecturer:
Khodr Saaifan (k.saaifan@jacobs-university.de)
Research 1, Room 79
Office Phone 0421-200-3107
Office: M,W 1:30-3:30 pm
Class Meets: Tue (11:15 am-12:30 pm) and Th (9:45-11:00 am)
Grading:
Two In-class Exams
Laboratory/Practicum
Final Exam
Homework
50%
20%
20%
10%
K. A. Saaifan, Jacobs University, Bremen
Outline
1. Introduction
2. Basic Components and Electrical Circuits
1.Units and Scales
2.Circuit Variables
3.Voltage and Current Sources
4.Ohm's Law
3.Voltage and Current laws
1.Node, Branches, and loops
2.Kirchhoff's Current Law
3.Kirchhoff's Voltage Law
4.The Single-Loop Circuit
5.The Single-Node-Pair Circuit
6.Series and Parallel Connected Sources
7.Resistors Series and Parallel
8.Voltage and Current Division
K. A. Saaifan, Jacobs University, Bremen
4.Basic Nodal and Mesh Analysis
1.Units and Scales
2.The Supernode
3.Mesh Analysis
4.The Supermesh
5.Handy Circuit Analysis Techniques
1.Linearity and Superposition
2.Source Transformations
3.Thévenin and Norton Equivalent Circuits
4.Maximum Power Transfer
5.Delta-Wye Conversion
6.A Summary of Various Techniques
K. A. Saaifan, Jacobs University, Bremen
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1. Introduction
Electrical engineering is the field of engineering concerned with systems
that produce, transmit, and measure electric signals
Electrical circuits and systems are networks of electrical components used
to supply, transmit and use electric power
Circuit analysis is the process of studying and analyzing the various
electrical quantities, such as currents, voltages, or powers, associated with
each circuit's component
Basic electrical circuits analysis covers the following topics:
Linear circuit analysis
Transient analysis
Phasor domain circuit analysis
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K. A. Saaifan, Jacobs University, Bremen
2. Basic Components and Electrical Circuits
2.1 Units and Scales
The International System of Units (SI) defines 6 principal units from
which the units of all other physical quantities can be derived
Table 2.1 SI base units
Basic Quantity
Length
Mass
Time
Electric current
Thermodynamic temperature
Luminous intensity
Unit
meter
kilogram
second
ampere
kelvin
candela
Symbol
m
kg
s
A
K
cd
The SI unit of work or energy is the joule (J), which equals to a kg m2
s-2 in SI base units
The SI unit of power is equivalent to one joule per second
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K. A. Saaifan, Jacobs University, Bremen
The SI unit uses prefixes based on the power of 10 to relate larger and
smaller units to the basic unit
Table 2.2 SI prefixes
Basic
Quantity
1012
109
106
103
10-2
10-3
10-6
10-9
10-12
Name
Symbol
tera
giga
mega
kilo
centi
milli
micro
nano
pico
T
G
M
k
c
m
n
p
K. A. Saaifan, Jacobs University, Bremen
2.2 Circuit Variables
2.2.1 Charge
Electric charge is an electrical property of the atomic particles of
which matter consists, measured in coulombs (C)
The charge of an electron is -1.602 X 10-19 C
The coulomb is a large unit for charges such that in 1 C of charge,
there are 1/(1.602 X 10-19)=6.24 X 1018 electrons
The realistic or laboratory values of charges are on the order pC, nC,
uC
2.2.2 Current
Electric current is a flow of electric charge measured in ampere (A)
note that 1 ampere (A) is equal to 1 coulombs per second (C/s)
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K. A. Saaifan, Jacobs University, Bremen
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The total charge transferred between time t0 and t can be expressed
as
There are several different types of current
Direct current (dc)
Sinusoidal current (ac)
Exponential current
Damped sinusoidal currents
Representation of current in circuit analysis
K. A. Saaifan, Jacobs University, Bremen
Practice
In the wire of shown figure, electrons are moving left to right to
create a current of 1 mA. Determine and
Ans: the current is in the opposite direction to flow of electrons
2.2.3 Voltage
Voltage or potential difference measured in volts (V) is the energy
required to move a unit of charge through an element
note that 1 volt (V) is equal to 1 joule per coulombs (J/C)
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K. A. Saaifan, Jacobs University, Bremen
Representation of voltage in circuit analysis
The plus (+) and minus (-) signs at the points
a and b are used to define a reference
direction (the voltage polarity)
Similar to the electric current, a constant voltage is called a dc
voltage, whereas a sinusoidal voltage (time-varying) is called an ac
voltage
For practical purposes, the power and energy are important
measures in circuit analysis
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K. A. Saaifan, Jacobs University, Bremen
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2.2.4 Power
Measured in watts (W) to indicate the average absorbing energy by a
circuit element
The sign of power
+ sign: the power is absorbed by the element (resistor)
- sign: the power is supplied by the element (?)
Since the energy can neither be created or dissipated (only
transferred), the algebraic sum of powers in a circuit, at any instant
of time, must be zero
K. A. Saaifan, Jacobs University, Bremen
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Determine p1
Ans:
2.2.5 Energy
The energy absorbed or supplied by an element from time 0 to t is
Electricity bills:: The electric power utility companies measure energy
in kilowatt-hours (kWh), where 1 kWh = 3600 kJ
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K. A. Saaifan, Jacobs University, Bremen
2.3 Voltage and Current Sources
There are two types of circuit elements:
Active elements (supplying energies), e.g., electric generator, batteries
Passive elements (absorbing energy), e.g., resistors, capacitors, and
inductors
The passive elements can be classified according to the relationship
of the current through it to the voltage
Resistor,
Capacitor,
Inductor,
Voltage and current sources:
Voltage sources provides the circuit with a specified voltage
Current source provides the circuit with a specified current
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K. A. Saaifan, Jacobs University, Bremen
Independent voltage source
The source is characterized by a terminal voltage which is completely
independent of the current through it
dc voltage source
ac voltage source
Independent current source
The current through the element is completely independent of the
voltage across it
dc current source
ac current source
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K. A. Saaifan, Jacobs University, Bremen
Dependent sources
The value of dependent sources depends on a voltage or currents of
some other elements
There are 4 different types of dependent sources
current-controlled
current source
voltage-controlled
current source
Find vL
Ans:
voltage-controlled
voltage source
current-controlled
voltage source
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K. A. Saaifan, Jacobs University, Bremen
Dependent sources
The value of dependent sources depends on a voltage or currents of
some other elements
There are 4 different types of dependent sources
current-controlled
current source
voltage-controlled
current source
voltage-controlled
voltage source
current-controlled
voltage source
Find the power absorbed by each element in the circuit
Ans:
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K. A. Saaifan, Jacobs University, Bremen
2.4 Ohm's Law
Ohm's law states the voltage across conducting materials is directly
proportional to the current flowing through the material, or
where R is the resistance
slope=R (V/A)
The unit of the resistance is Ohm (Ω)
The Resistor has a linear relation between the applied voltage and
the current
The current goes from a higher potential to a lower potential
The power absorbed by the resistor can be expressed as
The resistor is a passive element that cannot deliver or store energy
Find i and R, if v=-10 V and R is absorbing 0.1 W
Ans:
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K. A. Saaifan, Jacobs University, Bremen
The resistance of any cylindrical object is given as
l
Material with resistivity r
A
For a linear resistor, the ratio of the current to the voltage is called
the conductance
The SI unit of the electrical conductance G is siemens (S)
Homework Assignments
P2.11, P2.12, P2.15, P2.17, P2.20, P2.22, P2.23, P2.26, P2.31, P2.32, P2.33,
and P2.35
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K. A. Saaifan, Jacobs University, Bremen
3. Voltage and Current laws
3.1 Node, Branches, and loops
A branch represents a single element such as a voltage source or a
resistor
A node is the point of the connection between two or more
elements (branches)
It is usually indicated by a dot in a circuit
If a connecting wire (short circuit) connects two nodes, the two nodes
constitute a single nodes
A loop is any closed path in a circuit
A closed path is formed by starting at a node, passing through a set of
nodes and returning to the start node without passing through any
node more than once
branch
loop
K. A. Saaifan, Jacobs University, Bremen
3.2 Kirchhoff's Current Law
Kirchhoff's Current Law (KCL) is based on the law of conservation
of charge
The algebraic sum of the currents entering any node is zero
An alternative form of KCL is “the current entering any node =
the current leaving that node”
KCL can be applied to any closed boundary (closed path)
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K. A. Saaifan, Jacobs University, Bremen
3.3 Kirchhoff's Voltage Law
Kirchhoff's voltage Law (KVL) is based on the law of conservation of
energy
The algebraic sum of the voltages around any closed path is zero
KVL
When voltage sources are connected in series, KVL can be applied to
obtain the total voltage
a
a
=
b
b
K. A. Saaifan, Jacobs University, Bremen
Determine vx in the circuit
Ans:
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K. A. Saaifan, Jacobs University, Bremen
Determine vx in the circuit
Ans:
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K. A. Saaifan, Jacobs University, Bremen
3.4 The Single-Loop Circuit
Single-loop circuits
Elements are connected in series
All elements carry the same current
We shall determine
The current through each element
The voltage across each element
The power absorbed by each element
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K. A. Saaifan, Jacobs University, Bremen
3.4 The Single-Loop Circuit
Single-loop circuits
Elements are connected in series
All elements carry the same current
We shall determine
The current through each element
The voltage across each element
The power absorbed by each element
We apply the following steps
1) Assign a reference direction for the unknown current
2) Assign voltage references to the elements
3) Apply KVL to the closed loop path
4) Use Ohm's law where needed to get an equation in “i”
5) Solve for i
KVL
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K. A. Saaifan, Jacobs University, Bremen
Find i and p for all elements in the circuit
Ans:
KVL
1) Assign a reference direction for the unknown current
2) Assign voltage references to the elements (note that vA=-v2)
3) Apply KVL to the closed loop path
4) Use Ohms law where needed to get an equation in “i”
5) Solve for i
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K. A. Saaifan, Jacobs University, Bremen
Find i and p for all elements in the circuit
Ans:
KVL
Computing the power absorbed by each element
The total power absorbed by all elements
K. A. Saaifan, Jacobs University, Bremen
3.5 The Single-Node-Pair Circuit
Single-node-pair circuits
Elements are connected in parallel
All elements have a common voltage
We shall determine
The current through each element
The voltage across each element
The power absorbed by each element
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K. A. Saaifan, Jacobs University, Bremen
3.5 The Single-Node-Pair Circuit
Single-node-pair circuits
Elements are connected in parallel
All elements have a common voltage
We shall determine
The current through each element
The voltage across each element
The power absorbed by each element
We apply the following steps
1) Define the voltage v and arbitrary select its polarity
2) Use passive sign convention to determine the currents directions
3) Apply KCL at the node
4) Use Ohm's law where needed to get an equation in “v”
5) Solve for v
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K. A. Saaifan, Jacobs University, Bremen
Find v and p supplied by the independent source
Ans:
1) Assign an arbitrary sign for the unknown voltage
2) passive sign convention to find the currents directions (note that ix=-i2)
3) Apply KCL to the nodes
4) Use Ohm's law where needed to get an equation in “v”
5) Solve for v
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K. A. Saaifan, Jacobs University, Bremen
HW: Find i1, i2, i3, and i4
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K. A. Saaifan, Jacobs University, Bremen
3.6 Series and Parallel Connected Sources
Series-connected voltage sources can be replaced by a single
source
Parallel current sources can be replaced by a single source
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K. A. Saaifan, Jacobs University, Bremen
3.7 Resistors Series and Parallel
Series connection
KVL
Parallel connection
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K. A. Saaifan, Jacobs University, Bremen
Find the voltage and the power of the independent source
1) Apply KCL at the top node
2) Use Ohm's law for (i1=vx/6) and (vx=3i3)
3) Solve i3 and vx
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K. A. Saaifan, Jacobs University, Bremen
3.8 Voltage and Current Division
Voltage divider: is a passive linear circuit that produces an output
voltage (vout) that is a fraction of its input voltage (vin)
Easily solved with KCL, KVL, & equivalent
resistances
Then,
Generally, assume we have
The voltage vN can be given as
Easy to find the other voltages, too
K. A. Saaifan, Jacobs University, Bremen
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K. A. Saaifan, Jacobs University, Bremen
Current divider: is a simple linear circuit that produces an output
current (iout) that is a fraction of its input current (iin)
Easily solved with
Since
For n=2, we have
The circuit divider reduces to
K. A. Saaifan, Jacobs University, Bremen
Use resistance combination methods and current division
to find i1 and i2 and vx
Ans:
We note i1 goes to the following equivalent resistor
Use current divider, we have
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K. A. Saaifan, Jacobs University, Bremen
We note i2 goes to the following equivalent resistor
Use current divider, we have
HW: Solve vx
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K. A. Saaifan, Jacobs University, Bremen
We note i2 goes to the following equivalent resistor
Use current divider, we have
HW: Solve vx
Homework Assignments
P3.6, P3.7, P3.13, P3.15, P3.16, P3.19, P3.20, P3.21, P3.30, P3.31, P3.35,
P3.39, P3.73, P3.75 and P3.82
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K. A. Saaifan, Jacobs University, Bremen
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4. Basic Nodal and Mesh Analysis
This chapter introduces two basic circuit analysis techniques named nodal
analysis and mesh analysis
4.1 Nodal Analysis
For a simple circuit with two nodes, we often have one unknown
“voltage between two nodes”
To solve the unknown, applying KCL at this node gives
Adding a node should provide an additional unknown, three-node
circuit has 2 unknown
N-node circuit has (N-1) voltages with (N-1) equations.
K. A. Saaifan, Jacobs University, Bremen
Nodal technique applies the following step
1- Count the number of nodes (N)
2- Designate a reference node
3- Label the nodal voltages (we have N-1 voltages)
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K. A. Saaifan, Jacobs University, Bremen
4- Write KCL equations for the non-reference nodes (currents in
= currents out)
5- Express any additional unknowns in terms of nodal voltages
6- Organize the equations
(1)
(2)
7- Solve the system of equations for the nodal voltages
K. A. Saaifan, Jacobs University, Bremen
7- Solve the system of equations for the nodal voltages
using a Cramer's rule and determinants, we have
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K. A. Saaifan, Jacobs University, Bremen
Compute the voltages at each node
Ans:
Write KCL equations for the three nodes
Organize the equations
(1)
(2)
(3)
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K. A. Saaifan, Jacobs University, Bremen
Compute the voltage at each node
Ans:
Solve the system of equations for the nodal voltages
HW: use a Cramer's rule and
determinants to solve the system
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K. A. Saaifan, Jacobs University, Bremen
4.2 Nodal Analysis with Supernode
A supernode is formed when a voltage source is the only element
connected between two essential nodes
1- Define a current through the source and
write KCL equations for the two nodes
2- We note that there is no need to determine ivs to solve the circuit
(1)
3- Apply KVL between the two nodes
(2)
Thus, the KCL at the supernode is directly given
by
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K. A. Saaifan, Jacobs University, Bremen
Determines the node-to reference voltages
.
Node 1 to reference is supernode
Node 2
Node 3 & node 4
Express vx=v2-v1 and vy=v4-v1 in terms of
nodal voltages and organize the equations
(1)
(2)
(3)
Solve to get
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K. A. Saaifan, Jacobs University, Bremen
4.3 Mesh Analysis
In nodal analysis, circuit variables are node voltages
Nodal analysis applies KCL to find unknown voltages
In mesh analysis, circuit variables are mesh currents
Mesh analysis applies KVL to find unknown currents
Both methods result in a system of linear equations
Mesh analysis is only applicable to a circuit that is planar
Planar vs. Non-planar Circuits
Planar circuit: it can be drawn on a plane surface where no branch cross
any other branch (element)
Non-planar circuit there is no way to redraw it and avoid the branches
crossing
Planar circuit
Non planar circuit
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K. A. Saaifan, Jacobs University, Bremen
Mesh & mesh current
A mesh is a property of a planar circuit and it is defined a loop that does
not contain any other loops within it
The current through a mesh is known as a mesh current
mesh
mesh
K. A. Saaifan, Jacobs University, Bremen
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4.3 Mesh Analysis
1. Determine if the circuit is a planar circuit. If not, perform nodal analysis instead.
2. Count the number of meshes (M)
3. Label each of the M mesh currents (defining all mesh currents to flow clockwise
results in a simpler analysis)
4. Write a KVL equation around each mesh
For mesh 1, we have
or
(1)
For mesh 2, we have
or
The solution is easily obtained
(2)
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K. A. Saaifan, Jacobs University, Bremen
Determine the power supplied by the 2 V source
.
i1
We first define two clockwise mesh currents
For mesh 1, we write the following KVL equation
The same for mesh 2, we write
i2
K. A. Saaifan, Jacobs University, Bremen
Rearranging and grouping terms, we have
and
Solve the both equation yields
i1=1.132 A and i2=-0.1053 A
The 2 v source supplies (2)(i1-i2)=2.4 W
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K. A. Saaifan, Jacobs University, Bremen
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4.4 The Supermesh
Similar to the supernode in a node voltage analysis
A supermesh is formed when a current source is the only element connected
between two meshes
1- Define a voltage across the source and
write KVL equations for the two meshes
and
2- We do not need to evaluate vcs to solve the circuit
3- This leads us to create a supermesh whose interior is that of mesh 1 and mesh 2
4- Finally, the source current is related to the mesh currents,
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K. A. Saaifan, Jacobs University, Bremen
Determine the three mesh currents
.
i1
i2
i1
i2
i1-i2
i1-i2
i3-i2
i3
i3-i2
i3
i1-i3
The 7 A independent current source forms a supermesh between mesh 1 and
mesh 3
Applying KVL over the supermesh gives
or
KVL for mesh 2
or
K. A. Saaifan, Jacobs University, Bremen
Homework Assignments 3
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