Download Module 2_Network Theorems

USE OF ICT IN EDUCATION FOR ONLINE AND BLENDED LEARNING-IIT BOMBAY
BIRLA INSTITUTE OF TECHNOLOGY
MESRA, RANCHI
ASSIGNMENT( MODULE NETWORK THEOREMS)
Submitted by:
RC_1331_Birla Institute of Technology, Mesra
Dr. Deepak Kumar(Group Leader)
Dr. Vikas Kumar Gupta
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Statement of superposition theorem
• The response (voltage or current) in any branch of a bilateral
linear circuit having more than one independent source equals
the algebraic sum of the responses caused by each independent
source acting alone, where all the other independent sources
are replaced by their internal impedances.
• Superposition theorem can be explained through a simple
resistive network as shown in fig. 1.9 .
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Superposition Theorem
• Procedure for using the superposition theorem
• Step-1: Retain one source at a time in the circuit and replace
all other sources with their internal resistances.
• Step-2: Determine the output (current or voltage) due to the
single source acting alone using the techniques discussed in
previous lecture.
• Step-3: Repeat steps 1 and 2 for each of the other independent
sources.
• Step-4: Find the total contribution by adding algebraically all
the contributions due to the independent sources.
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Example: Using the superposition theorem, determine the voltage drop and current
across the resistor 3.3K as shown in figure below.
step:1
step:2
Step 1: Remove the 8V power supply from the original circuit, such that the new circuit becomes as the following
and then measure voltage across resistor.
Here 3.3K and 2K are in parallel, therefore resultant resistance will be 1.245K.Using voltage divider rule voltage
across 1.245K will be V1= [1.245/(1.245+4.7)]*5 = 1.047V
Step 2: Remove the 5V power supply from the original circuit such that the new circuit becomes as the following
and then measure voltage across resistor.
Here 3.3K and 4.7K are in parallel, therefore resultant resistance will be 1.938K.
Using voltage divider rule voltage across 1.938K will be
V2= [1.938/(1.938+2)]*8 = 3.9377V
Therefore voltage drop across 3.3K resistor is V1+V2 = 1.047+3.9377=4.9847
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Limitations of superposition
Theorem
• Superposition theorem doesn’t work for power calculation.
• Power calculations involve either the product of voltage and
current, the square of current or the square of the voltage.
•
They are not linear operations.
• This statement can be explained with a simple example .
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Thevenin’s Theorem
• Any linear electrical network with voltage and current sources
and only resistances can be replaced at terminals A-B by an
equivalent voltage source Vth in series connection with an
equivalent resistance Rth.
• The equivalent voltage Vth is the voltage obtained at terminals
A-B of the network with terminals A-B open circuited.
• The equivalent resistance Rth is the resistance that the circuit
between terminals A and B would have if all ideal voltage
sources in the circuit were replaced by a short circuit and all
ideal current sources were replaced by an open circuit.
• If terminals A and B are connected to one another, the current
flowing from A to B will be Vth/Rth.
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Thevenin’s Theorem
Fig. 1.10(a) Circuit for Thevenin’s Theorem
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Thevenin’s Theorem
Fig. 1.10(b) Circuit for Thevenin’s Theorem
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Thevenin’s Theorem
Fig. 1.10(c) Circuit for Thevenin’s Theorem
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Thevenin’s Theorem
Fig. 1.10(d) Circuit for Thevenin’s Theorem
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Thevenin’s Theorem
Fig. 1.10(e) Circuit for Thevenin’s Theorem
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Thevenin’s Theorem
…..(1.9)
…(1.10)
….(1.11)
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The procedure for applying
Thevenin’s theorem
• To find a current through the load resistance (as shown in fig. 1.10(a))
using Thevenin’s theorem, the following steps are followed:
• Step-1: Disconnect the load resistance from the circuit, as indicated in fig.
1.10(b)
• Step-2: Calculate the open-circuit voltage (shown in fig. 1.10(b)) at the
load terminals (A & B). In general, one can apply any of the techniques
(mesh-current, node-voltage and superposition method) learnt in earlier
lessons to compute.
• Step-3: Redraw the circuit ( shown in fig. 1.10 (b)) with each practical
source replaced by its internal resistance as shown in fig. 1.10(c) (note,
voltage sources should be short-circuited and current sources should be
open-circuited .
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Thevenin’s Theorem
• Step-4: Look backward into the resulting circuit from the load terminals (A
&B) , as suggested by the eye in fig.1.10(c). Calculate the resistance that
would exist between the load terminals .The resistance is described in the
statement of Thevenin’s theorem.
• Step-5: Place in Thevenin resistance series with Thevenin voltage to form
the Thevenin’s equivalent circuit as shown in fig. 1.10(d)
• Step-6: Reconnect the original load to the Thevenin voltage circuit as
shown in fig 1.10(e) ; the load’s voltage, current and power may be
calculated by a simple arithmetic operation only.
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Thevenin’s Theorem
(i) One great advantage of Thevenin’s theorem is this: once the
Thevenin equivalent circuit has been formed, it can be reused
in calculating load current , load voltage and load power for
different loads using the equations .
(ii) Also one can find the choice of load resistance that results in
the maximum power transfer to the load..
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Norton’s Theorem
Norton’s theorem states that:
• Any linear electrical network with voltage and current sources and only
resistances can be replaced at terminals A-B by an equivalent current
source INO in parallel connection with an equivalent resistance RNO.
• This equivalent current INO is the current obtained at terminals A-B of the
network with terminals A-B short circuited.
• This equivalent resistance RNO is the resistance obtained at terminals A-B
of the network with all its voltage sources short circuited and all its current
sources open circuited.
• Norton’s theorem is a dual of Thevenin’s theorem.
• To find a current through the load resistance using Norton’s theorem, the
following steps are followed:
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Norton’s Theorem
Fig. 1.11(a) Circuit for Norton’s Theorem
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Norton’s Theorem
• Step-1: Short the output terminal after disconnecting the load resistance
from the terminals A&B and then calculate the short circuit current .
• Step-2: Redraw the circuit with each practical sources replaced by its
internal resistance ,voltage sources should be short-circuited and current
sources should be open- circuited .
• Step-3: Calculate the resistance that would exist between the load terminals
A&B Looking backward into the resulting circuit from the load terminals
(A&B).
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Norton’s Theorem
Fig. 1.11(b) Circuit for Norton’s Theorem
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Norton’s Theorem
Fig. 1.11(c ) Circuit for Norton’s Theorem
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Norton’s Theorem
• Step-4: form the Norton’s equivalent circuit (replacing the imaginary
fencing portion or fixed part of the circuit with an equivalent practical
current source) .
• Step-5: After Reconnecting the original load; the load’s voltage, current
and power may be calculated by a simple arithmetic operation only.
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Norton’s Theorem
Fig. 1.11 (d) Circuit for Norton’s Theorem
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Norton’s Theorem
(i)
Norton’s theorem has also a similar advantage like the thevenin’s theorem
over the normal circuit reduction technique.
(ii) With help of either Norton’s theorem or Thevenin’s theorem one can find
the choice of load resistance that results in the maximum power transfer to
the load.
(iii) Norton’s current source may be replaced by an equivalent Thevenin’s
voltage source .
In other words, a source transformation converts a Thevenin equivalent
circuit into a Norton equivalent circuit or vice-versa.
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Maximum Power Transfer Theorem
• The maximum power transfer theorem states that, to obtain
maximum external power from a source with a finite internal
resistance, the resistance of the load must equal the resistance
of the source as viewed from its output terminals.
• The theorem states how to choose (so as to maximize power
transfer) the load resistance, once the source resistance is
given.
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Maximum Power Transfer Theorem
• Let us consider an electric network as shown in fig.1.12(a), the problem is
to find the choice of the resistance so that the network delivers maximum
power to the load .
• This problem can be solved using nodal or mesh current analysis to obtain
an expression for the power absorbed by Load resistance
•
The derivative of this expression with respect to load resistance will
establish the condition under what circumstances the maximum power
transfer occurs.
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Maximum Power Transfer Theorem
Fig. 1.12(a) Circuit for Maximum Power Transfer Theorem
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Circuit for Maximum Power
Transfer
Fig. 1.12(b) Circuit for Maximum Power Transfer Theorem
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Maximum Power Transfer Theorem
….(1.12)
….(1.13)
..(1.14)
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….(1.15)
….(1.16)
….(1.17)
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Thank You
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EE 1x01 BASIC ELECTRICAL ENGINEERING
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L-T-P : 3-0-3 Credit : 5
1. Introduction : Electrical Elements and their Classification, KCL, KVL equation and node voltage method, D.C
circuits- steady state analysis with independent and dependent sources, Series and parallel circuits, Stardelta conversion, Superposition Theorem, Thevenin’s Theorem, Norton’s theorem, Maximum Power
Transfer Theorem.
Lecture : 12
2 A.C Circuits : Common signals and their waveform, R.M.S and Average value, form factor and Peak factor of
sinusoidal wave. Impedance of series and parallel circuits, Phasor diagram, Power, Power factor, Power
triangle, Coupled circuits. Resonance ; series, parallel, Q-factor, Superposition, Thevenin’s and Norton’s,
Maximum power transfer theorem for A.C circuits
Lecture : 12
3 Three phase A.C Circuits : Star delta connection, line and phase relation, Power relation, Analysis of
balanced and unbalanced
3-phase circuits.
Lecture : 8
4 Magnetic circuits : Introduction; Flux, MMF, flux density, reluctance, Series & Parallel magnetic circuits.
Analysis of Linear and non linear magnetic circuits. Energy storage, A.C. excitation. Eddy current and
hysteresis losses. Lecture : 5
5 Basic indicating instruments for measurements Current Voltage, Power, Energy Insulation resistance.
Lecture : 5
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Text Book:
1.D.P. Kothari, I.J.Nagrath, Basic Electrical Engineering , 2nd Edition, TMH , New Delhi ,, 2010.
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Reference Books:Tata Fitzerald
1. Fitzerald, et al ,Basic Electrical Engineering by, Tata McGraw Hill, 2005
2. R. Prasad, Fundamentals of Electrical Engg., PHI Publication
3. Leonard S. Bobrow, Fundamental of Electrical Engg., Oxford
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