Tellegen’s-Substitution-Reciprocity-Theorem.ppt

This Presentation gives complete idea about Tellegen's theorem, substitution theorem and Reciprocity Theorem.

1. 1
Tellegen’s Theorem
Dr. K Hussain

2. 2
Recap
In the previous lecture, we have discussed
•Substitution Theorem
2
Dr. K Hussain

3. 3
Objectives
On completion of this lecture, you will be able to
•State Tellegen’s theorem
•Explain Tellegen’s theorem
•Apply Tellegen’s theorem to a given circuit
Dr. K Hussain

4. 4
Known To Unknown
We have already discussed that
•Many laws and theorems are available for solving networks
•Some of them are, Kirchhoff’s laws, Maxwell’s Loop Current
theorems, Superposition theorem, Thevini’s and Norton’s
theorem, etc.,
•Tellegen’s Theorem is also one of such theorems
Dr. K Hussain

5. 5
Tellegen’sTheorem
Statement:
“Tellegen’sTheoremstatesthatthesummationofinstantaneous
powersforthennumberofbranchesinanelectricalnetworkis
zero.
OR
“Tellegen’sTheoremcanalsobestatedinanothersentenceas,in
anylinear,nonlinear,passive,active,time-variantortime-invariant
network,thesummationofpower(instantaneousorcomplexpower
ofsources)iszero”
Dr. K Hussain

6. 6
Tellegen’sTheorem
•Tellegen’stheorem is independent of the network elements.
Thus, it is applicable for any lump system that has linear, active,
passive and time-variant elements.
•Also, the theorem is convenient for the network which
followsKirchhoff’s current law and Kirchhoff’s voltage law.
•It is mainly applicable for designing the filters in signal processing.
•It is also used in complex operating systems for regulating
stability.
•It is mostly used in the chemical and biological system and for
finding the dynamic behaviour of the physical network.
Dr. K Hussain

7. 7
Explanation of Tellegen’s Theorem
•Tellegen’sTheorem can also be stated in another word as, in any
linear, nonlinear, passive, active, time-variant or time-invariant
network the summation of power (instantaneous or complex
power of sources) is zero.
•Thus, for the K
th
branch, this theorem states that:
•Where,
n is the number of branches; v
Kis the voltage in the branch
i
Kis the current flowing through the branch.
Dr. K Hussain

8. Dr. K Hussain 8
Equation (1) shows the K
th
branch through current
v
K
is the voltage drop in branch K and is given as:
We have
Let,
Where v
p
and v
q
are the respective node voltage at p and q nodes.

9. Dr. K Hussain 9
Also
Obviously
Summing the above two equations (2) and (3), we get
Such equations can be written for every branch of the network.
Assuming n branches, the equation will be:

10. Dr. K Hussain 10
However,accordingtotheKirchhoff’scurrentlaw(KCL),the
algebraicsumofcurrentsateachnodeisequaltozero.
Therefore,
Thus,fromtheaboveequation(4)finally,weobtain
Thus,ithasbeenobservedthatthesumofpowerdeliveredtoa
closednetworkiszero.
ThisprovestheTellegen’stheoremandalsoprovesthe
conservationofpowerinanyelectricalnetwork.

11. •It is also evident that the sum of power delivered to
the network by an independent source is equal to the
sum of power absorbed by all passive elements of
the network.
Note:
•It depends on voltage and current product of an
element but not on the type of element.
•While verifying Tellegen’stheorem do not disturb
original network.
Dr. K Hussain 11

12. Dr. K Hussain 12
Step2–Findthecorrespondingbranchcurrentsusing
conventionalanalysismethods.
Step1–Inordertojustifythistheoreminanelectrical
network,thefirststepistofindthebranchvoltagedrops.
Steps for Solving Networks Using Tellegen’s Theorem
Thefollowingstepsaregivenbelowtosolveany
electricalnetworkbyTellegen’stheorem:
Step3–Tellegen’stheoremcanthenbejustifiedbysumming
theproductsofallbranchvoltagesandcurrents.

13. •Now, if the set of voltages and currents is taken, corresponding
the two different instants of time, t
1and t
2, then Tellegen’s
theorem is also applicable where we get the equation as shown
below:
Dr. K Hussain 13
•For example, if a network having some branches “b”
then:

14. Dr. K Hussain 14
Example:
Verify the Tellegen’stheorem for the given circuit.
Solution:
If current flows from + to –then treat it as
power absorption.
If current flow from –to + then treat it as
power delivering.
∴P
10V= V. I = 10 ×1 = 10 watt (P
absorbed)
P
2A= V. I = 10 ×2 = 20 watt (P
delivered)
P
10Ω= I
2
. R = 1 ×10 = 10 watt (P
absorbed)
∴P
delivered= P
absorbed= 20 watt
Hence Tellegen’stheorem is verified.

15. Dr. K Hussain 15
For this network, we will assume a set of branch
voltages satisfy the Kirchhoff voltage law and a set
of branch current satisfy Kirchhoff current law at
each node.
We will then show that these arbitrary
assumed voltages and currents satisfy
the equation.
And it is the condition of Tellegen’s
theorem.
In the network shown in the figure,
let v
1, v
2and v
3be 7, 2 and 3 volts respectively.
Applying Kirchhoff Voltage Law around loop ABCDEA.
We see that v
4= 2 volt is required. Around loop CDFC, v
5is required
to be 3 volt and around loop DFED, v
6is required to be 2.
Example:

16. Dr. K Hussain 16
We next apply Kirchhoff’s Current Law successively
to nodes B, C and D.
At node B let i
i= 5 A, then it is required that i
2= –5 A.
At node C let i
3= 3 A and then i
5is required to be –8.
At node D assume i
4to be 4 then i
6is required to be –9.
Carrying out the operation of equation,
We get,
Hence Tellegen’stheoremis verified.

17. Dr. K Hussain 17
Application of Tellegen’s Theorem
•It is used in the digital signal processing system for designing
filters.
•In the area of the biological and chemical process.
•In topology and structure of reaction network analysis.
•The theorem is used in chemical plants and oil industries to
determine the stability of any complex systems.

18. THANK YOU
Dr. K Hussain 18

19. 19
Substitution Theorem
Dr. K Hussain

20. 20
Recap
In the previous lecture, we have discussed
•Maximum Power Transfer Theorem
•Explanation of Maximum Power Transfer Theorem
•Application of Maximum Power Transfer Theorem to the circuit
20
Dr. K Hussain

21. 21
Objectives
On completion of this lecture, you will be able to
•State Substitution theorem
•Explain Substitution theorem
•Apply Substitution theorem to a given circuit
Dr. K Hussain

22. 22
Known To Unknown
We have already discussed that
•Many laws and theorems are available for solving networks
•Some of them are, Kirchhoff’s laws, Maxwell’s Loop Current
theorems, Superposition theorem, Thevini’s and Norton’s
theorem, etc.,
•Substitution Theorem is also one of such theorems
Dr. K Hussain

23. Dr. K Hussain 23
Substitutiontheoremstatesthatifanelementina
networkisreplacedbyavoltagesourcewhosevoltageatany
instantoftimeisequalstothevoltageacrosstheelementinthe
previousnetworkthentheinitialconditionintherestofthe
networkwillbeunaltered.
Substitution Theorem
Alternatelyifanelementinanetworkisreplacedbyacurrent
sourcewhosecurrentatanyinstantoftimeisequaltothecurrent
throughtheelementinthepreviousnetworkthentheinitial
conditionintherestofthenetworkwillbeunaltered.
or

24. Dr. K Hussain 24
Explanation of Substitution Theorem
ThecurrentI,isflowingthroughthecircuit,whichisdividedinto
currentI
1flowingthroughtheresistanceR
1andthecurrent
I
2flowingthroughtheresistanceR
2.V
1,V
2andV
3arethevoltage
dropacrosstheresistanceR
1,R
2andR
3respectively.

25. Dr. K Hussain 25
Now if the resistance R
3is substituted by the voltage
source V
3as shown in the circuit diagram below:
In the circuit diagram shown below the resistance, R
3is replaced
by the current flowing through that element, i.e. I
1

26. Dr. K Hussain 26
In both the cases shown above if the element is substituted by the
voltage source or the current source, then also, the initial
conditions of the circuit does not alter.
This means that the voltage across the resistance and current
flowing through the resistance unaltered even if they are
substituted by other sources.

27. Dr. K Hussain 27
Example
Let us take a circuit as shown in fig –d.
For more efficient and clear understanding let us go
through a simple practical example:
As per voltage division rule voltage across
3Ω and 2Ω resistance are

28. Dr. K Hussain 28
If we replace the 3Ω resistance with a voltage source of 6 V as
shown in fig –e, then
According to Ohm’s law the voltage across 2Ω
resistance and current through the circuit is
Alternately if we replace 3Ω resistance with a current source
of 2A as shown in fig –f, then
Voltage across 2Ω is V
2Ω= 10 –3×2 = 4 V
and
voltage across 2A current source is V
2A= 10 –4 = 6 V
We can see the voltage across 2Ω resistance and current through the circuit is
unaltered
i.e., all initial condition of the circuit is intact.

29. Dr. K Hussain 29
THANK YOU

30. Dr. K Hussain 30
Reciprocity Theorem

31. 31
Recap
In the previous lecture, we have discussed
•Norton’s theorem and source transformation
31
Dr. K Hussain

32. 32
Objectives
On completion of this lecture, you will be able to
•State Reciprocity Theorem
•Explain Reciprocity Theorem
•Apply Reciprocity Theorem to a given circuit
Dr. K Hussain

33. Dr. K Hussain 33
ThereciprocitytheoremstatesthatifanemfEinonebranch
ofareciprocalnetworkproducesacurrentIinanother,then
iftheemfEismovedfromthefirsttothesecondbranch,it
willcausethesamecurrentinthefirstbranch,wheretheemf
hasbeenreplacedbyashortcircuit.
ReciprocitytheoremstatesthatifweconsidertwoloopsA
andBofareciprocalnetworkN,andifanidealvoltage
source,E,inloopA,producesacurrentIinloopB,thenan
identicalsourceinloopBwillproducethesamecurrentIin
loopA.
Reciprocity Theorem
Statement:
OR

34. Dr. K Hussain 34
Explanation of Reciprocity Theorem
The voltage source and the ammeter used in this
theorem must be ideal. That means the internal
resistance of both the voltage source and ammeter
must be zero.
Thereciprocalcircuitmaybeasimpleorcomplexnetwork.But
everycomplexreciprocalpassivenetworkcanbesimplifiedinto
asimplenetwork.
The ratio of V and I is called the transfer resistance.
Asperreciprocitytheorem,inalinearpassivenetwork,supply
voltageVandoutputcurrentIaremutuallytransferable.

35. Steps for Solving a Network Utilizing Reciprocity Theorem
Step1–Firstly,selectthebranchesbetweenwhich
reciprocityhastobeestablished.
Dr. K Hussain 35
Step5–Now,itisseenthatthecurrentobtainedintheprevious
connection,i.e.,instep2andthecurrentwhichiscalculated
whenthesourceisinterchanged,i.e.,instep4areidenticalto
eachother.
Step4–Thecurrentinthebranchwherethevoltagesourcewas
existingearlieriscalculated.
Step3–Thevoltagesourceisinterchangedbetweenthebranch
whichisselected.
Step2–Thecurrentinthebranchisobtainedusingany
conventionalnetworkanalysismethod.

36. Dr. K Hussain 36
The theorem can easily be understood by
this following example.
In a network if we interchange the position of response and
excitation then the ratio of response to excitation is constant.
Then, (I
L/V
S) = (I
S/V
L)

37. Dr. K Hussain 37
Find the value of I in the given network?
Example:
Solution:
By reciprocity theorem
5 = − I
10 30
I = -15A

38. •Itisclearfromthefigureabovethatthevoltage
sourceandcurrentsourcesareinterchangedfor
solvingthenetworkwiththehelpofReciprocity
Theorem.
•Thelimitationofthistheoremisthatitisapplicableonlyto
single-sourcenetworksandnotinthemulti-sourcenetwork.
Dr. K Hussain 38
•Thenetworkwherereciprocitytheoremisappliedshouldbe
linearandconsistofresistors,inductors,capacitorsand
coupledcircuits.Thecircuitshouldnothaveanytime-varying
elements.

39. THANK YOU
Dr. K Hussain 39

40. 40
Summary
In this period, we have discussed
•Statement of Norton’s theorem
•Explanation of Norton’s theorem
•Application of Norton’s theorem to a given circuit
Dr. K Hussain

41. 41
Quiz
1. What is the internal resistance for an ideal current
source?
a)Infinite
b)Zero
c)Depends on other factors of the circuit
d)None
Dr. K Hussain

42. 42
2. Norton’s equivalent circuit will consist of
a)A current source in parallel with resistance
b)A voltage source in series with resistance
c)None
Quiz
Dr. K Hussain

43. 43
Frequently Asked Questions
1.State & explain Norton’s theorem.
2.Explain the steps to obtain the Norton’s equivalent circuit
for a given network.
Dr. K Hussain

44. 44
1. Determine current supplied by the battery, using Norton’s
theorem.
Assignment
Dr. K Hussain

45. 45
THANK YOU
Dr. K Hussain