Simple ways to Learn Network Analysis Using Top 7 Circuit Theorems

Most electrical networks are very complex. To analyze them, the need of circuit theorems, techniques and laws are really important. Superposition Theorem, Thevenin Theorem, Norton Theorem, Maximum Power Transfer Theorem, Substitution Theorem, Millman Theorem and Reciprocity Theorem are the fundamental laws used for network analysis. All these circuit theorems helps to find the basic electrical parameters in a simpler way. The top 7 circuit theorems for network analysis  are discussed below:

Superposition Theorem

Superposition Theorem states that “In any linear circuit containing multiple independent sources, the current or voltage at any point in the network may be calculated as algebraic sum of the individual contributions of each source acting alone.”

To solve an electrical network or to perform network analysis using Superposition Theorem, below steps must be followed:

  • Step1:Ensure the electrical circuit to be a linear circuit.
  • Step2:Replace all voltage and current sources except one of them.
  • Step3:Replace the voltage source with a short and a current source with an open.
  • Step4:Either using Kirchhoff’s Law or Ohms Law find the voltage drop or current in each branch.
  • Step5:Repeat steps 2 to 4 for every source the circuit has.
  • Step6:Finally, add the voltage results of the circuit due to all sources and also add the current results of each source.

Example: Find current through a 6R using Superposition theorem.

Superposition Theorem:An Electrical circuitStep 2 to 4 : Superposition TheoremStep 5 to 6: Superposition Theorem

Thevenin Theorem

This circuit theorem states that “The current flowing through a load resistance connected across any two terminals a and b of a linear and active bilateral network is given by Voc|| (Rt +RL) where VOC is the open circuit voltage and Rt is the internal resistance of the network as viewed back into the open circuited network from terminals a and b with all voltage sources replaced by their internal resistance and current sources by infinite resistance.”

Simple Steps to analyze an electric circuit using Thevein’s theorem:

  • Step1: Open the load resistor.
  • Step2: Calculate the Open Circuit Voltage. This is the Thevenin Voltage   (Voc).
  • Step3: Open Current Sources and Short Voltage Sources.
  • Step4: Calculate the Open Circuit Resistance. This is the Thevenin Resistance (Rt).
  • Step5: Redraw the circuit with Voc as the voltage Source and Rt as the resistance and connect the load resistor which we had removed in Step (1). This is the Equivalent Thevenin Circuit.
  • Step6: Now find the total current flowing through load resistor by using the Ohms Law, i = Voc/ (Rt + RL).

 

Example: Find current through a 2R using Thevenin theorem in the above electrical circuit.

Norton Theorem

Norton’s Theorem for network analysis states that “Any linear circuit containing several energy sources and resistances can be replaced by a single constant current source in parallel with a single resistor“.

Simple Steps to analyze an electric circuit using Norton Theorem.

  • Step1: Short the load resistor.
  • Step2: Calculate the short circuit current. This is the Norton Current (isc).
  • Step3: Open Current Sources, Short Voltage Sources and Open Load Resistor.
  • Step4: Calculate the open circuit resistance. This is the Norton Resistance (RN).
  • Step5: Redraw the circuit with isc as current source and RN as a parallel resistance and connect the load resistor which has been removed in the step1. This is the Equivalent Norton Circuit.
  • Step6: Find the load current and load voltage across load resistor by using the equation IL = isc / (RN / (RN+ RL)).

 

Example: Find the Norton Equivalent Circuit for the electrical circuit given above.

Maximum Power Transfer Theorem

Moritz von Jacobi states the maximum power transfer theorem as, the load resistance must be equal to the source resistance when seen from the output terminals. This circuit theorem was published in the year 1840 for network analysis  and is also known as Jacobi's law.

Proof: Consider the linear circuit given below. Here RL is the load resistance, Rt is the Thevenin Resistance and i is the current flowing through it.

Power delivered to the load is given by,

 

Here Vt is the voltage source. Now by, differentiating p w.r.t RL and equating to zero, we get, RL=Rt.

 

Now the maximum power given to the load resistor is found out by substituting Rt=RL.

Reciprocity Theorem

Reciprocity Theorem states that in a linear and a single source circuit, the ratio of excitation to response is constant when the positions of excitation and response are interchanged. Now consider the circuit given below:

Applying Kirchoff's Voltage Law to mesh1, mesh2 and mesh3 we get,

KVL for mesh1: 6.375 I1-2I2-3I3 =0

KVL for mesh2: -2 I1+14 I2-10 I3=0

KVL for mesh3:-3 I1-10 I2+14 I3=-10

By solving the above equations we get, the current I1=-2A.

Now we can verify the reciprocity theorem by following the below process.

Verifying Reciprocity Theorem

Consider the redrawn circuit and the mesh equations are given below. Applying the Kirchoff's Voltage Law to mesh1, mesh2, mesh3.

KVL for mesh1: 6.375 I1-2I2+3I3 =10

KVL for mesh2: -2 I1+14 I2+10 I3=0

KVL for mesh3: 3 I1+10 I2+14 I3=0

By solving the above equations we get, I’3=-2A.

Since I1= I’3=-2A.

Hence, the reciprocity theorem is verified.

Millman Theorem

Millman's theorem is basically used to compute the voltage at the ends of a circuit made up of only branches in parallel.

Consider the circuit shown in the figure below:

Let im the current generators and Vk be the voltage generators.

Let Rk be the resistance on the branch with voltage generators.

Let Ri be the resistance on the branch with no generator.

Let Rm be the resistances on the branches with current generators. Then,

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