EE101 CIRCUITS 1
1T SY 2014-2015
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Resistors and Resistive Circuits
Week 2
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Learning Outcomes
Solve for the total resistance, current through
an element, and voltage drop across an
element in a series, parallel, and series-parallel
connected dc network using voltage divider,
current divider, Ohms Law, and Kirchhoffs
Law.
Solve the total resistance, current through an
element, and voltage drop across an element
in a wye- or delta-connected dc network.
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NODES, BRANCHES, AND LOOPS
A branch represents a single element such as a
voltage source or a resistor.
A node is the point of connection between two or
more branches.
A loop is any closed path in a circuit.
A network with b branches, n nodes, and l
independent loops will satisfy the fundamental
theorem of network topology:
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NODES, BRANCHES, AND LOOPS
Original circuit
Equivalent circuit
How many branches,
nodes and loops are
there?
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NODES, BRANCHES, AND LOOPS
How many branches, nodes and loops are there?
Should we consider it as one
branch or two branches?
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NODES, BRANCHES, AND LOOPS
Two or more elements are in series if they exclusively
share a single node and consequently carry the same
current.
Two or more elements are in parallel if they are
connected to the same two nodes and consequently have
the same voltage across them.
Is 5- resistor in series or in parallel with 2- resistor?
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RESISTANCE
Materials in general have a characteristic behavior of
resisting the flow of electric charge. This physical property,
or ability to resist current, is known as resistance and is
represented by the symbol R.
The circuit element used to model the current-resisting
behavior of a material is the resistor.
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OHMS LAW
Georg Simon Ohm (1787-1854), a German physicist, is
credited with finding the relationship between current
and voltage for a resistor
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OHMS LAW
It states that the voltage across a resistor is directly
proportional to the current flowing through the resistor.
That is,
Ohm defined the constant of proportionality for a
resistor to be the resistance, R.
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KIRCHHOFFS LAWS
Kirchhoffs laws were first introduced in 1847 by the
German physicist Gustav Robert Kirchhoff (1824-1887).
These laws are formally known as Kirchhoffs current law
(KCL) and Kirchhoffs voltage law (KVL).
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KIRCHHOFFS CURRENT LAW (KCL)
Kirchhoffs current law (KCL) states that the
algebraic sum of currents entering a node (or a
closed boundary) is zero.
It is based on the law of conservation of charge, which requires
that the algebraic sum of charges within a system cannot
change.
The sum of the currents entering a node is equal to the sum of
the currents leaving the node.
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KIRCHHOFFS CURRENT LAW (KCL)
Mathematically,
Where
N = number of branches
connected to the node
= nth current entering (or leaving) the node
= 0
=1
Current entering a node is (+) and
current leaving a node is (-) or vice-
versa
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KIRCHHOFFS CURRENT LAW (KCL)
Current sources in parallel:
(a) Original circuit
(b) Equivalent circuit
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KIRCHHOFFS VOLTAGE LAW (KVL)
Kirchhoffs voltage law (KVL) states that the algebraic
sum of all voltages around a closed path (or loop) is
zero. It is based on the principle of conservation of energy.
The sum of voltage drops is equal to the sum of voltage rises.
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KIRCHHOFFS VOLTAGE LAW (KVL)
Mathematically,
Where
M = number of voltages in
the loop (or the number of
branches in the loop)
= nth voltage
= 0
=1
Sign on each voltage is the polarity
of the terminal encountered first as
we travel around the loop.
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KIRCHHOFFS VOLTAGE LAW (KVL)
Voltage sources in series:
(a) Original circuit
(b) Equivalent circuit
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ILLUSTRATIVE PROBLEM 1
Find the currents and voltages in the circuit shown.
= , = , = , = . , = , = .
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ILLUSTRATIVE PROBLEM 2
Calculate in the circuit below.
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ILLUSTRATIVE PROBLEM 3
Find and in the circuit below.
= =
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ILLUSTRATIVE PROBLEM 4
Given the circuit shown, determine the following:
a)
b) c) 30
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= = =
SERIES RESISTORS AND VOLTAGE
DIVISION
Recall: Two or more elements are in
series if they exclusively share a
single node and consequently carry
the same current.
Resistors in series behave as a
single resistor whose resistance is
equal to the sum of the
resistances of the individual
resistors.
For two resistors in series:
= 1 + 2 ++
=
=1
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SERIES RESISTORS AND VOLTAGE
DIVISION
The equivalent conductance of resistors connected in
series is:
Voltage Division
=
1 + 2 ++
1
= 1
1+ 1
2+ +
1
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SERIES RESISTORS AND VOLTAGE
DIVISION
1 = 1
1 + 2 + 3 1 =
11 + 2
Voltage Divider Circuit
2 = 2
1 + 2 + 3
3 = 3
1 + 2 + 3
2 = 21 + 2
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PARALLEL RESISTORS AND CURRENT DIVISION
Recall: Two or more elements are in
parallel if they are connected to the
same two nodes and consequently
have the same voltage across them.
The equivalent resistance of a
circuit with N resistors in parallel
is:
For 2 resistors in parallel:
= 121 + 2
1
= 1
1+ 1
2+ +
1
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PARALLEL RESISTORS AND CURRENT
DIVISION
The equivalent conductance of resistors connected in
parallel is:
Current Division
=
=
= 1 + 2 ++
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PARALLEL RESISTORS AND CURRENT DIVISION
Current Divider Circuit
1 = 21 + 2
2 = 11 + 2
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EQUIVALENT RESISTANCE
In general, it is often convenient and possible to combine
resistors in series and parallel and reduce a resistive
network to a single equivalent resistance.
Such an equivalent resistance is the resistance between
the designated terminals of the network and must exhibit
the same i-v characteristics as the original network at the
terminals.
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POWER IN RESISTIVE NETWORKS
Total power is the sum of the power dissipated by each
resistor in the circuit:
= 1 + 2+ . . . +
= = 2
= 2
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Consider the figure below. What are the corresponding
currents? What about equivalent resistances?
? ?
? ?
= 0
= 1
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MORE ILLUSTRATIVE PROBLEMS
1. Two 115-V incandescent lamps A and B are connected
in series across 230-V source. If lamp A is rated 75
watts and lamp B is rated 50 watts, determine the
current drawn by the series connection.
2. R resistor is connected in series with two resistances
1 = 2 and 2 = 4 . The series combination is connected across a 36-V source. A voltmeter, placed
across 2, reads 12 V. Find the value of the resistor R.
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MORE ILLUSTRATIVE PROBLEMS
3. Two resistances of 10 and 5 ohms are connected in
parallel and the combination is connected in series with
a 10-ohm resistance. If these are connected across a 48-
V battery, determine the current through the 5-ohm
resistance.
4. The equivalent resistance of three resistors A, B and C
connected in parallel is 1.714 ohms. If A is twice of B
and C is half as much as B, find the equivalent resistance
when the three of them are connected in series.
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MORE ILLUSTRATIVE PROBLEMS
5. Three resistors R1, R2, and R3 are connected in series-
parallel with R1 in series with the parallel combination
of R2 and R3. The whole combination is connected
across a 120-V DC source. R1, R2, and R3 take 750 W,
250 W, and 200 W, respectively. Calculate the resistance
R2.
6. Resistances X, Y, and Z are connected in series. The
voltage across X and Y is 21 V, across Y and Z is 24 V,
and across X and Z is 27 V. Find the value of resistance
Y when the current is 1 A.
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MORE ILLUSTRATIVE PROBLEMS
7. A 16- resistor is connected in series with a parallel combination of two resistors, one of which has an
ohmic value of 48 and the other is unknown. What is the resistance of R if the power taken by the 16- resistor is equal to the power taken by the parallel-
connected pair of resistors?
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MORE ILLUSTRATIVE PROBLEMS
8. Calculate .
= 11.2
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MORE ILLUSTRATIVE PROBLEMS
9. Find the equivalent resistance at terminals a-b.
Rab = R
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MORE ILLUSTRATIVE PROBLEMS
10. Find the equivalent resistance at terminals a-b.
Rab = 54
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MORE ILLUSTRATIVE PROBLEMS
11. Calculate Vo and Io in the circuit below.
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VO = 8 V IO = 0.2 A
MORE ILLUSTRATIVE PROBLEMS
12. Determine V in the circuit below.
Confused if resistors are series- or parallel-connected?
They are neither in series nor in parallel. So, how do we combine them?
They can be simplified by using three-terminal equivalent network!
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WYE (TEE)-CONNECTED
NETWORK
Two forms of the same network: (a) Y, (b) T
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DELTA (PI)-CONNECTED NETWORK
Two forms of the same network: (a) , (b)
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DELTA-WYE CONVERSION
Each resistor in the Y network is
the product of the resistors in the
two adjacent branches, divided by the sum of the three resistors.
Mathematical expression:
1
2
3
b c
a b c
a c
a b c
a b
a b c
R RR
R R R
R RR
R R R
R RR
R R R
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WYE-DELTA CONVERSION
Mathematical expression: Each resistor in the network is the sum of all possible products of
Y resistors taken two at a time,
divided by the opposite Y resistor.
1 2 2 3 3 1
1
1 2 2 3 3 1
2
1 2 2 3 3 1
3
a
b
c
R R R R R RR
R
R R R R R RR
R
R R R R R RR
R
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ILLUSTRATIVE PROBLEM 12
Determine V in the circuit below.
42.18 V
Can we now answer this?
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ILLUSTRATIVE PROBLEM 13
Determine the current I as indicated in the circuit
shown below.
I = 0.375 A
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ILLUSTRATIVE PROBLEM 14
Find in the four-way power divider circuit in the figure below. Assume each element is 1 .
2
References
Please refer to course syllabus.
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