EENG223: CIRCUIT THEORY I
DC Circuits:
Basic Laws
Dr. Hasan Demirel
EENG223: CIRCUIT THEORY I
Resistance and Ohm’s Law
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All materials resist the flow of current..
Resistance R of an element denotes its ability to resist the flow of electric
current, which is measured in ohms (Ω).
A cylindrical material of length l and cross-sectional area A has the
following resistance:
l
R
A
R= resistance of the element in ohms (Ω)
ρ= resistivity of the material in ohm-meters (Ω-m)
l = lenghth of the cylindrical material in meters (m)
A=Cross sectional area of the material in meters2 (m2)
EENG223: CIRCUIT THEORY I
Resistance and Ohm’s Law
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Resistance: Basic Concepts and Assumptions:
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Conductors (e.g. Wires) have very low resistance (<0.1 Ω), which is usually be
neglected (i.e. We will assume that wires have zero resistance).
Insulators (e.g. air) have very large resistance (>50 MΩ) that can be usually
ignored ( omitted from circuit for analysis).
Resistors have a medium range of resistance and must be accounted for the
circuit analysis.
Conceptually, a light bulb is similar to the resistor.
Properties of the bulb control how much current flows and how much power is
dissipated (absorbed then emitted as light and heat).
As with the circuit elements, we need to know the current through and voltage
across the device are ralated.
The relationship between the current and voltage can be linear or nonlinear.
EENG223: CIRCUIT THEORY I
Resistance and Ohm’s Law
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Ohm’s Law states that the voltage v accross a resistor is directly
proportional to the current i flowing through the resistor.
υ  i R
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υ = voltage in volts (V),
i = current in (A),
R = resistance in (Ω).
Sign ± is determined by passive sign convention (PSC).
Materials with linear relationships between the current and the voltage
satisfy the Ohm’s Law.
Resistor symbol
Ohm’law applies
•
Resistance R is equal to the slope m in (a).
Ohm’s Law does Not apply
EENG223: CIRCUIT THEORY I
Resistance and Ohm’s Law
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Resistors and Passive Sign Convention (PSC)
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
Note that the the relationship between current and voltage are sign sensitive.
PSC is satisfied if the current enters the positive terminal of an element:
 İf PSC is satisfied : υ =iR
 İf PSC is not satisfied : υ =−iR
PSC satisfied
(υ =iR)
PSC not satisfied
(υ =−iR)
PSC not satisfied
(υ =−iR)
PSC satisfied
(υ =iR)
EENG223: CIRCUIT THEORY I
Equations derived from Ohm’s Law
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Ohm’s Law:
υ
R
i
υ  iR
υ
i
R
(1 Ω = 1 V/A)
υ
p  υi  i R 
R
2
2
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Recall:
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Resistors cannot produce power , so the power absorbed by a resistors will
always be positive.
•
1 Ω = 1 V/A
EENG223: CIRCUIT THEORY I
Short Circuit and Open Circuit
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Short Circuit as Zero resistance:
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An element (or wire) with R =0 is called a short circuit.
Short circuit is just drawn as a wire (line).
EENG223: CIRCUIT THEORY I
Short Circuit and Open Circuit
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Short Circuit as Voltage Source (0V):
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An ideal voltage sorce with Vs=0 V is equivalent to a short circuit.
Since υ =iR and R =0, υ =0 regardless of i.
You could draw a source with Vs=0 V, but it is not done in practice.
You cannot connect a voltage source to a short circuit.
İf connected, usually wire wins and the voltage source melts (smoke
comes out) if not protected.
EENG223: CIRCUIT THEORY I
Short Circuit and Open Circuit
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An element (or wire) with R=∞ is called an open circuit.
Such an element is just omitted.
EENG223: CIRCUIT THEORY I
Short Circuit and Open Circuit
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Open Circuit as Current Source (0 A):
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An ideal current source with I=0 A is equivalent to a open circuit.
Since υ =iR and if I=0 A, then R = ∞.
You could draw a source with I=0 A, but it is not done in practice.
You cannot connect a current source to an open circuit.
İf connected, usually you blow the current source (smoke comes
out) if not protected.
The insulator (air) wins. Else, sparks fly.
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EENG223: CIRCUIT THEORY I
Conductance: inverse of resistance
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Conductance is the ability of an element to conduct electric
current . Conductance is the inverse of resistance.
1
i
G

R
υ
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Units: siemens (S) or mho (
)
υ  Ri
2
υ
p  υi  i 2 R 
R
i  G
2
i
p  υi  υ 2G 
G
EENG223: CIRCUIT THEORY I
Circuit Building Blocks: Nodes, Branches and Loops:
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In circuit analysis, we need a common language and
framework for describing circuits.
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In this course, circuits are modelled to be the same as
networks.
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Networks are composed of nodes, braches and loops.
EENG223: CIRCUIT THEORY I
Circuit Building Blocks: Nodes, Branches and Loops:
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A branch represents a single element such as voltage
source, resistor or current source.
There are 5 branches
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How many branches?
Branch: a single two-terminal circuit element.
Wire segments are not counted as branches.
Examples: voltage source/current source/resistors.
EENG223: CIRCUIT THEORY I
Circuit Building Blocks: Nodes, Branches and Loops:
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A node is a point of connection between two or more
branches.
There are 3 nodes (a, b and c)
2 essential nodes (b and c)
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How many nodes?
Node: a connection point between two or more branches.
May include a portion of circuit (more than a single point).
Essential Node: the point of connection between three or
more brances.
EENG223: CIRCUIT THEORY I
Circuit Building Blocks: Nodes, Branches and Loops:
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A loop is a closed path in a circuit.
There are 6 loops.
There are 3 independent loops.
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How loops?
Loop: a closed path in a circuit.
Independent Loop: A loop is independent if it contains at least
one branch which is not a part of any other independent loop.
EENG223: CIRCUIT THEORY I
Kirchhoff’s Laws: Overview
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Ohm’s Law is not sufficient to analyze circuits alone.
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Kirchhoff’s Laws help Ohm’s law to form the foundation for
circuit analysis:
 The defining equations for circuit elements (Ohm’s Law).
 Kirchhoff’s Current Law (KCL).
 Kirchhoff’s Voltage Law (KVL).
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The defining equations (from Ohm’s law) tell us how the
voltage and current within a circuit element are related.
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Kirchhoff’s laws tell us how the voltages and currents in
different branches are related.
EENG223: CIRCUIT THEORY I
Kirchhoff’s Current Law
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Kirchhoff’s current law (KCL) states that the algebraic sum of
currents entering a node (or a closed boundary) is zero.
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The sum of currents entering a node is equal to the sum of the
currents leaving the node.
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KCL also applies to a closed boundary:
EENG223: CIRCUIT THEORY I
Kirchhoff’s Current Law
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Applying KCL to node a:
I1  I T  I 2  I 3  0
I T  I 2  I1  I 3
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Equivalent circuit can be generated as follows:
I T  I1  I 2  I 3
EENG223: CIRCUIT THEORY I
Kirchhoff’s Current Law: for Closed Boundaries
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KCL applies to a closed boundaries:
I1  I 2  I 3  I 4  0
I1  I 3  I 2  I 4
Closed boundary
EENG223: CIRCUIT THEORY I
Kirchhoff’s Current Law: Example
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Apply KCL to the each essential node in the circuit.
i1
i2
i3
i4
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Essential node 1: i1  i2  i3  0

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Essential node 2: 5 mA  i3  i4  0
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Essential node 3: i2  i4  i1  5 mA  0
i1  i3  i2

i3  i4  5 mA
 i2  i4  i1  5 mA
EENG223: CIRCUIT THEORY I
Ideal Current Sources: Series
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Ideal currtent sources cannot be connected in series.
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Recall: ideal current sources guarantee the current flowing
through source is at specified value.
Recall: the current entering a circuit element must be equal to
the current leaving the circuit element: Iin = Iout.
Ideal current sources do not exist.
Tecnically allowed if : I1 = I2 , but it is a bad idea.
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EENG223: CIRCUIT THEORY I
Kirchhoff’s Voltage Law: KVL
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Kirchhoff’s voltage law (KVL) states that the algebraic sum of
voltages around a closed path (or loop) is zero.
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or
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sum of voltage drops = sum of voltage rises
EENG223: CIRCUIT THEORY I
Kirchhoff’s Voltage Law: KVL
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Apply KVL to each loop in the following circuit:
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Loop 1:
Loop 2:
Loop 3:
Loop 4:
Loop 5:
Loop 6:
EENG223: CIRCUIT THEORY I
Kirchhoff’s Voltage Law (KVL) Example:
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Calculate V2, V6 and VI.
 10  V2  V6  VI  0
3

V2  (5 10 ).(2 10 )  10V
3
V6  (5 10 3 ).(6 103 )  30V
VI  10  10  30  50V
VI  10  V2  V6
EENG223: CIRCUIT THEORY I
Applying Basic Laws: Example
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Calculate vo and i.
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Apply KVL around the loop:
 12  4i  2v0  4  6i  0
 12  4i  12i  4  6i  0
 2i  16
 i  8A

v0  6i
v0  48V
EENG223: CIRCUIT THEORY I
Applying Basic Laws: Example
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Calculate I2, I3, I7, V3, and VI .
EENG223: CIRCUIT THEORY I
Series Resistors and Voltage Division
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The equivalent resistance of any number of resistors
connected in series is the sum of individual resistance.
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Consider the following circuit:
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Applying KVL Vs  R1I s  R2 I s  R3 I s  R4 I s
 I s ( R1  R2  R3  R4 )  I s Req
Req  R1  R2  R3  R4
1
1
1
1
1




Geq G1 G2 G3 G4
(Resistors in series add)
EENG223: CIRCUIT THEORY I
Series Resistors and Voltage Division
(Applying KVL)
(voltage divider circuit)
• voltage accross each resistor, is proportional to its resistance. Larger the
resistance, larger the voltage drop on that resistor:
• voltage division principle
EENG223: CIRCUIT THEORY I
Parallel Resistors and Current Division
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The equivalent resistance of N resistors number of resistors
can be calculated by:
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Consider the following circuit:
I s  I1  I 2  I 3  I 4
 Vs (
1
1
1
1

  )
R1 R2 R3 R4
Vs

Req
1
1
1
1
1
 
 
Req R1 R2 R3 R4
 Req 
1
1
1
1
1

 
R1 R2 R3 R4
EENG223: CIRCUIT THEORY I
Parallel Resistors and Current Division
1
1
1
1
1
 
 
Req R1 R2 R3 R4
Geq  G1  G2  G3  G4
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Resistors in parallel have more complicated relationship
It is easier to express in conductance
EENG223: CIRCUIT THEORY I
Parallel Resistors and Current Division
(Applying KCL at node a)
(current divider circuit)
• Given the total current i entering to node a the current is shared by the
resistors by inverse proportion to their resistance:
• Current division principle
EENG223: CIRCUIT THEORY I
Resistor Networks and Equivalent Resistance
• Example: Calculate the Req for the
following circuit.
EENG223: CIRCUIT THEORY I
Resistor Networks and Equivalent Resistance
• Example: Calculate the Req for the
following circuit.
EENG223: CIRCUIT THEORY I
Resistor Networks and Equivalent Resistance
• Example: Calculate the Rab for the
following circuit.
EENG223: CIRCUIT THEORY I
Resistor Networks and Equivalent Resistance
• Example: Calculate the Geq for the following
circuit.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations
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There are cases where the resistors are neither in parallel nor
in series.
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More tools are needed.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations
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Three-terminal equivalent networks can be used to simplify the
analysis of the circuits.
There are two types of three-terminal networks:
 Wye (Y) Networks
Two equivalent forms of
(Y) Network
 Delta(Δ) Networks
Two equivalent forms of
(Δ) Network
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations
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Every Wye (Y) network is functionally equivalent to a Delta (Δ)
network (vice versa).
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations
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Example: Convert the following D network to a Y network.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations
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Example: Convert the following Ynetwork to a D network.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations
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Example: Calculate Req and Power delivered by the source.
EENG223: CIRCUIT THEORY I
Resistor Networks: Wye-Delta Transformations
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Example: Calculate Rab and and use it to calculate i.