Network Theorems

Network theorems are fundamental principles used in electrical engineering to analyze and simplify complex electrical circuits. These theorems help in determining the voltage, current, and power across various elements of the circuit. Here are some of the key network theorems:

1. Thevenin's Theorem

• Any two-terminal linear bilateral network can be replaced by a voltage source in series with an impedance. 
• The voltage source is the voltage across the open-circuited terminals and series impedance is the equivalent impedance seen across the terminals by replacing all the sources in-network with their internal impedances.
   
  V_th → Thevenin's voltage = V_oc across AB.
  R_th → Thevenin's equivalent impedance.

Example: Consider the circuit shown below. Find the equivalent Thevenin's voltage between nodes A and B. 

Solution: Thevenin's voltage is equal to the open-circuit voltage across the terminals AB that is across 12Ω resistors. 

Now by voltage divider rule, 

V_TH = 10×12/(12+2) = 10×12/14 = 8.57V. 

2. Norton's Theorem

• Any two-terminal linear bilateral network containing active and passive elements can be replaced by an equivalent current source in parallel to an equivalent impedance current source is equal to the short-circuited current through the given terminals and equivalent impedance is the impedance seen across the terminals by replacing all the sources with their internal impedance.
   
  I_no → Norton's equivalent current
  R_no → Norton's equivalent impedance.
  R_no= Z_th

Example: Find the current flowing between terminals A and B of the circuit shown below.Solution: The magnitude of the current in Norton's equivalent circuit is equal to the current passing through the short-circuited terminals that are 

Let us first calculate R_TH,

R_N=R_TH=(5x10)/(5+10) = 10/3 ohm

Voltage drop across AB = V_AB =(20x10)/(10+5) = 40/3 V

I_N=(40/3)/(10/3)=4A. 

3. Superposition Theorem 

• This theorem states that if a no. of voltage sources or current sources are acting together in a bilateral network then the resultant current voltage in any branch is an algebraic sum of current/voltage due to individual sources by replacing them with their internal impedance.
• The superposition theorem is applicable to linear time varying and LTI(Linear Time Invariant) networks.
• It is also applicable for circuits having the initial conditions. It is applicable for the linear networks.

Example: For the given circuit, the current I is:

(a) 2 A
(b) 5A
(c) 7 A
(d) 9A
Solution: 
(c) 7A

Using superposition theorem:
Due to current source:

 
I₁ = (2/2+2).4 = 2A 

Due to voltage source:

Current I = I₁ + I₂
⇒ I = 2 + 5
⇒ I = 7A 

4. Reciprocity Theorem

• This theorem states that in a linear bilateral reciprocal network if I is the current in branch X due to a voltage source V in-branch Y then same current will flow in branch Y when the same voltage source is connected in branch X.
• The initial conditions are assumed to be zero in reciprocity theorem.
• There should not be any extra dependent or independent sources in network.

Example: In the circuit given below, the current in the 4-ohm resistor is __________
 

5. Maximum Power Transfer Theorem

• The following maximum power transfer theorems determine the values of the load impedance which results in maximum power transfer across the terminals of an active network. 
• We consider a series combination of source and fixed complex impedance delivering power to a load consisting of a variable resistance or a variable complex impedance.

Case 1: Load is variable resistance R_L

Power delivered to R_L is:

To determine the value of R for maximum power transferred to the load,

We arrive at the condition:

Hence, with a variable pure resistance load the maximum power is delivered across the terminals of the active network if the load resistance is made equal to the absolute value of the active network impedance.
If the reactive component of the impedance in series with source is zero i.e., X_g = 0, then the maximum power is transferred to the load when the load and source resistances are equal i.e., R_L = R_g.

Case 2: Load Impedance Z_L with variable resistance and variable reactance.

For maximum power transfer: R_L= R_g, X_L = -X_g i.e. Z_L = Z_g

With the load impedance consisting of variable resistance and variable reactance, maximum power transfer across the terminals of the active network occurs when the load impedance Z₁ is equal to the complex conjugate of the network impedance Z_g.

Case 3: Load impedance, Z_L with variable resistance, and fixed reactance.

For maximum power transfer.

6. Compensation Theorem

• The compensation theorem is useful in determining the current and voltage changes in a circuit element when the value of its impedance is changed.

• This application occurs in bridge and potentiometer circuits where a slight change in one impedance results in shift from the null condition.
• In a linear time-invariant bilateral network if the resistance R of a branch is changed by  ΔR then the currents in all branches of the network will change and this change in current can be found by connecting a voltage source V_C such that V_C = i ΔR  in series with resistance (R +ΔR ).
• Here the current i represent current through R before change and all sources are replaced by their internal impedance to find out the change in current.

Example: Find the current through 3Ω resistors in the circuit shown below.
Solution: Total resistance in the circuit = 2+[3||(2+2||2)] = 3.5Ω. The total current drawn by the circuit =10/(4+6||3) = 1.67A. Current through 3Ω resistor = 1.11A ≅1A.

7. Tellegen's Theorem

• This theorem states that the algebraic sum of power consumed and delivered by each branch of a network is always zero.
   
  n= no. of branches of the network, v_K =voltage across K^th branch. 

8. Millman's Theorem

• Millman's Theorem states that - when a number of voltage sources (V₁, V₂, V₃......... V_n) are in parallel having internal resistance (R₁, R₂, R₃.............R_n) respectively, the arrangement can replace by a single equivalent voltage source V in series with an equivalent series resistance R. 
• In other words; it determines the voltage across the parallel branches of the circuit, which have more than one voltage source, i.e.,  reduces the complexity of the electrical circuit.
• This Theorem is given by Jacob Millman. The utility of Millman's Theorem is that the number of parallel voltage sources can be reduced to one equivalent source. It is applicable only to solve the parallel branch with one resistance connected to one voltage source or current source. It is also used in solving networks having an unbalanced bridge circuit.
  
• As per Millman's Theorem:
  

9. Substitution Theorem 

• A known voltage in a branch can be replaced by an ideal voltage source.
• This theorem applies to any network, linear or non-linear, time varying or time-invariant.
• It is applicable both in the time domain and in the 's' domain.

Explanation of Substitution Theorem

• Let us take a circuit as shown in fig - a,
  
• Let, V is supplied voltage and Z₁, Z₂ and Z₃ is different circuit impedances. V₁, V₂ and V₃ are the voltages across Z₁, Z₂ and Z₃ impedance respectively and I is the supplied current whose I₁ part is flowing through the Z₁ impedance whereas I₂ part is flowing through the Z₂ and Z₃ impedance.
• Now if we replace Z₃ impedance with V₃ voltage source as shown in fig-b or with I₂ current source as shown in fig-c then according to Substitution Theorem all initial conditions through other impedances and source will remain unchanged.
  
• i.e. - current through source will be I, voltage across Z₁ impedance will be V₁, the current through Z₂ will be I₂ etc.