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

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Tellegen’s Theorem
Dr. K Hussain

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

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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

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Tellegen’s Theorem
Statement:
“Tellegen’s Theorem states that the summation of instantaneous
powers for the n number of branches in an electrical network is
zero.
OR
“Tellegen’s Theorem can also be stated in another sentence 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”
Dr. K Hussain

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Tellegen’s Theorem
• Tellegen’s theorem 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
follows Kirchhoff’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

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Explanation of Tellegen’s Theorem
• Tellegen’s Theorem 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 Kth branch, this theorem states that:
• Where,
n is the number of branches; vK is the voltage in the branch
iK is the current flowing through the branch.
Dr. K Hussain

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

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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:

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However, according to the Kirchhoff’s current law (KCL), the
algebraic sum of currents at each node is equal to zero.
Therefore,
Thus, from the above equation (4) finally, we obtain
Thus, it has been observed that the sum of power delivered to a
closed network is zero.
This proves the Tellegen’s theorem and also proves the
conservation of power in any electrical network.

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’s theorem do not disturb
original network.
Dr. K Hussain 11

12. Dr. K Hussain 12
Step 2 – Find the corresponding branch currents using
conventional analysis methods.
Step 1 – In order to justify this theorem in an electrical
network, the first step is to find the branch voltage drops.
Steps for Solving Networks Using Tellegen’s Theorem
The following steps are given below to solve any
electrical network by Tellegen’s theorem:
Step 3 – Tellegen’s theorem can then be justified by summing
the products of all branch voltages and currents.

13. • Now, if the set of voltages and currents is taken, corresponding
the two different instants of time, t1 and t2, 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:

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Example:
Verify the Tellegen’s theorem 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.
∴ P10V = V. I = 10 × 1 = 10 watt (Pabsorbed)
P2A = V. I = 10 × 2 = 20 watt (Pdelivered)
P10Ω = I2. R = 1 × 10 = 10 watt (Pabsorbed)
∴ Pdelivered = Pabsorbed = 20 watt
Hence Tellegen’s theorem is verified.

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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 v1, v2 and v3 be 7, 2 and 3 volts respectively.
Applying Kirchhoff Voltage Law around loop ABCDEA.
We see that v4 = 2 volt is required. Around loop CDFC, v5 is required
to be 3 volt and around loop DFED, v6 is required to be 2.
Example:

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We next apply Kirchhoff’s Current Law successively
to nodes B, C and D.
At node B let ii = 5 A, then it is required that i2 = – 5 A.
At node C let i3 = 3 A and then i5 is required to be – 8.
At node D assume i4 to be 4 then i6 is required to be – 9.
Carrying out the operation of equation,
We get,
Hence Tellegen’s theorem is verified.

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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

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Substitution Theorem
Dr. K Hussain

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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
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Dr. K Hussain

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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

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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
Substitution theorem states that if an element in a
network is replaced by a voltage source whose voltage at any
instant of time is equals to the voltage across the element in the
previous network then the initial condition in the rest of the
network will be unaltered.
Substitution Theorem
Alternately if an element in a network is replaced by a current
source whose current at any instant of time is equal to the current
through the element in the previous network then the initial
condition in the rest of the network will be unaltered.
or

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Explanation of Substitution Theorem
The current I, is flowing through the circuit, which is divided into
current I1 flowing through the resistance R1 and the current
I2 flowing through the resistance R2. V1, V2 and V3 are the voltage
drop across the resistance R1, R2 and R3 respectively.

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Now if the resistance R3 is substituted by the voltage
source V3 as shown in the circuit diagram below:
In the circuit diagram shown below the resistance, R3 is replaced
by the current flowing through that element, i.e. I1

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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.

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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

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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 V2Ω = 10 – 3× 2 = 4 V
and
voltage across 2A current source is V2A = 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.

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THANK YOU

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Reciprocity Theorem

31. 31
Recap
In the previous lecture, we have discussed
• Norton’s theorem and source transformation
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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
The reciprocity theorem states that if an emf E in one branch
of a reciprocal network produces a current I in another, then
if the emf E is moved from the first to the second branch, it
will cause the same current in the first branch, where the emf
has been replaced by a short circuit.
Reciprocity theorem states that if we consider two loops A
and B of a reciprocal network N, and if an ideal voltage
source, E, in loop A, produces a current I in loop B, then an
identical source in loop B will produce the same current I in
loop A.
Reciprocity Theorem
Statement:
OR

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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.
The reciprocal circuit may be a simple or complex network. But
every complex reciprocal passive network can be simplified into
a simple network.
The ratio of V and I is called the transfer resistance.
As per reciprocity theorem, in a linear passive network, supply
voltage V and output current I are mutually transferable.

35. Steps for Solving a Network Utilizing Reciprocity Theorem
Step 1 – Firstly, select the branches between which
reciprocity has to be established.
Dr. K Hussain 35
Step 5 – Now, it is seen that the current obtained in the previous
connection, i.e., in step 2 and the current which is calculated
when the source is interchanged, i.e., in step 4 are identical to
each other.
Step 4 – The current in the branch where the voltage source was
existing earlier is calculated.
Step 3 – The voltage source is interchanged between the branch
which is selected.
Step 2 – The current in the branch is obtained using any
conventional network analysis method.

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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, (IL/VS) = (IS/VL)

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

38. • It is clear from the figure above that the voltage
source and current sources are interchanged for
solving the network with the help of Reciprocity
Theorem.
• The limitation of this theorem is that it is applicable only to
single-source networks and not in the multi-source network.
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• The network where reciprocity theorem is applied should be
linear and consist of resistors, inductors, capacitors and
coupled circuits. The circuit should not have any time-varying
elements.

39. THANK YOU
Dr. K Hussain 39

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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

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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

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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

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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

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

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THANK YOU
Dr. K Hussain