Network Theorems - 2 | GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) PDF Download

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 R_L without having to apply a method such as mesh analysis, nodal analysis, superposition, and so on.

Basically, three steps are included in its application
Step 1: Convert all voltage sources to current sources.

Step 2: Convert the parallel current source into resulting network below as

I_T = I₁+I₂+I₃   &    R_T = R₁+R₂+R₃

Step 3: Convert the resulting current source to a voltage source, and the desired single-source network is obtained as

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

In terms of the resistance values,

and

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.

Example: verify reciprocity theorem.

The Total resistance is  

and

with

Interchanging the location of E and I of last figure to demonstrate the validity of the reciprocity theorem.

and

so that

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 v_kj_k 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
  
  

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. 

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. 

• Through the use of the substitution theorem, a number of equivalent branches are
  
  

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. 

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 (R_AB, R_BC, R_CA) 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.

Between A & C terminals:

Between C & B terminals:

Between B & A terminals:

By combining above three equations, one can write an expression as given below

• on solving above equations we get for star network resistances
  

Conversion from Star or Wye (Y) to Delta (Δ):

• To convert a Wye (Y) to a Delta (Δ), the relationships R_AB, R_BC & R₃ must be obtained in terms of the Wye (Y) resistances R_A R_B and R_C Considering the Y connected network, we can write the current expression through R_A resistor as 

Appling KCL at ‘ N ’ for Y connected network (assume A, B, C terminals having higher potential than the terminal N) we have,


Few Δ-network
Current entering at terminal A = Current leaving the terminal ‘ A ’

Using the V_N expression in the above equation, we get

• Equating the coefficients of VABand VAC in both sides we obtained the following relationship.
  
• similarly we can obtain for R_BC for equivalent delta configuration
  
• Observations: In order to note the symmetry of the transformation equations, the Wye (Y ) and Delta (Δ) networks have been superimposed on each other.
  
  
• 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.