Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

Networking Theory

Created by: Machine Experts

Electrical Engineering (EE) : Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

The document Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Networking Theory.
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)

1. MILLMAN’S THEOREM
Through the application of Millman’s theorem, any number of parallel voltage sources can be reduced to one. In the given figure below, for example, the three voltage sources can be reduced to one. This permits finding the current through or voltage across RL without having to apply a method such as mesh analysis, nodal analysis, superposition, and so on.
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Basically, three steps are included in its application
Step 1: Convert all voltage sources to current sources.
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Step 2: Convert the parallel current source into resulting network below as
IT = I1+I2+I3   &    GT = G1+G2+G3
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Step 3: Convert the resulting current source to a voltage source, and the desired single-source network is obtained as
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
The plus-and-minus signs appear in last equation to include those cases where the sources may not be supplying energy in the same direction.The equivalent resistance is
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
In terms of the resistance values,
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
and Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

2. RECIPROCITY THEOREM
The reciprocity theorem is applicable only to single-source networks. The theorem states that the current I in any branch of a network due to a single voltage source E anywhere else in the network will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current I was originally measured.
In other words, the location of the voltage source and the resulting current may be interchanged without a change in current. The theorem requires that the polarity of the voltage source have the same correspondence with the direction of the branch current in each position.
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Example: verify reciprocity theorem.
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
The Total resistance is  
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
and Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
with Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Interchanging the location of E and I of last figure to demonstrate the validity of the reciprocity theorem.
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
and Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
so that Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
with agrees with the above.

3. TELLEGEN'S THEOREM

  • Tellegen’s theorem is based on two Kirchhoff’s laws and is applicable for any lumped network having elements which are linear or non-linear, active or passive, time-varying or time-invariant. 
  • For a lumped network whose element assigned by associate reference direction for branch voltage vk and branch current jk.The product vkjk is the power delivered at time t by the network to the element k. 
  • If all branch voltages and branch currents satisfy KVL and KCL then
    Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

Application of Tellegen's Theorem:
As seen from last equation Tellegen’s Theorem implies the law of energy conservation.“The sum of power delivered by the independent sources to the network is equal to the sum of the power absorbed by all branches of the network”. so the application of Tellegen's theorem can be classified as

  • Conservation of energy 
  • Conservation of complex power 
  • The real part and phase of driving point impedance 
  • Driving point impedance

Example: Find all branch currents and voltages for both networks N1, N2 Then verify Tellegen’s theorem. 

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

4. SUBSTITUTION THEOREM
The substitution theorem states that If the voltage across and the current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.
More simply, the theorem states that for branch equivalence, the terminal voltage and current must be the same. Consider the circuit in in which the voltage across and current through the branch a-b are determined. Through the use of the substitution theorem, a number of equivalent a-a′ branches are shown Note that for each equivalent, the terminal voltage and current are the same. 

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

  • Through the use of the substitution theorem, a number of equivalent branches are
    Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

Note that for each equivalent, the terminal voltage and current are the same.a known potential difference and current in a network can be replaced by an ideal voltage source and current source, respectively 

  • Example: 

The current source equivalence where a known current is replaced by an ideal current source, permitting the isolation of R4 and R5

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

Recall from the discussion of bridge networks that V = 0 and I = 0 were replaced by a short-circuit and an open circuit, respectively. This substitution is a very specific application of the substitution theorem.

5. STAR-DELTA TRANSFORMATION 

  • A part of a larger circuit that is configured with three-terminal network Y (orΔ) to convert into an equivalent Δ (or Y) through transformations. 
  • Application of these transformations will be studied by solving resistive circuits. 

Delta (Δ) – Wye (Y) conversion: 
Let us consider the network shown below and assumed the resistances (RAB, RBC, RCA) in Δ network are known. Our problem is to find the values of in Wye (Y) network that will produce the same resistance when measured between similar pairs of terminals. We can write the equivalence resistance between any two terminals in the following form.

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Between A & C terminals:
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Between C & B terminals:
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Between B & A terminals:

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
By combining above three equations, one can write an expression as given below
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

  • on solving above equations we get for star network resistances
    Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

Conversion from Star or Wye (Y) to Delta (Δ):
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

  • To convert a Wye (Y) to a Delta (Δ), the relationships RAB, RBC & R3 must be obtained in terms of the Wye (Y) resistances RA RB and RC Considering the Y connected network, we can write the current expression through RA resistor as 

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Appling KCL at ‘ N ’ for Y connected network (assume A, B, C terminals having higher potential than the terminal N) we have,
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Few Δ-network
Current entering at terminal A = Current leaving the terminal ‘ A ’

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Using the VN expression in the above equation, we get

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

  • Equating the coefficients of VABand VAC in both sides we obtained the following relationship.
    Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
  • similarly we can obtain for RBC for equivalent delta configuration
    Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
  • Observations: In order to note the symmetry of the transformation equations, the Wye (Y ) and Delta (Δ) networks have been superimposed on each other.
    Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev
  • The equivalent star (Wye) resistance connected to a given terminal is equal to the product of the two Delta (Δ) resistances connected to the same terminal divided by the sum of the Delta (Δ) resistances. 
  • The equivalent Delta (Δ) resistance between two-terminals is the sum of the two star (Wye) resistances connected to those terminals plus the product of the same two star (Wye) resistances divided by the third-star resistance.
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!
30 videos|20 docs|27 tests

Up next >

Dynamic Test

Content Category

Related Searches

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

,

Extra Questions

,

MCQs

,

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

,

practice quizzes

,

Previous Year Questions with Solutions

,

Summary

,

Semester Notes

,

Sample Paper

,

Exam

,

pdf

,

Free

,

mock tests for examination

,

ppt

,

Viva Questions

,

Objective type Questions

,

Network Theorems (Part - 2) Electrical Engineering (EE) Notes | EduRev

,

video lectures

,

study material

,

past year papers

,

Important questions

,

shortcuts and tricks

;