Circuit Theory 3b - More network theorems, solved problems. Maximum Power Transfer theorem, Compensation Theorem, Tellegen Theorem


Electrical DC Networks
 
Contents 

Here's a quick look at the network theorems which will be introduced in this tutorial.

Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE Main or Advanced/AIEEE, and anyone else who needs this Tutorial as a reference!


More Network Theorems : Norton’s theorem, Maximum power transfer theorem, Substitution theorem, Reciprocity theorem, Millman’s theorem, Compensation Theorem, Tellegen’s theorem, Star-Delta transformation
We will also go through a number of problems based on these network theorems.

Norton’s Theorem:

In a linear, active, bilateral network consisting of active sources, passive elements and a load resistor RL, the circuit can be replaced by a single current source of magnitude IN and a resistor RN parallel to the load, where IN is the short circuit current through the points where the load is connected and RN is the equivalent resistance as seen from the terminal where the load is connected. For finding the Norton’s resistance, the equivalent resistance across the terminals have to be found with all the independent sources being zero. Here the battery source E is removed and shorted. This is equal to thevenin’s resistance.

Maximum Power Transfer Theorem:

Statement: this theorem states that in an active, linear, bilateral network, maximum power is delivered to the load when the load resistance is equal to the equivalent resistance looking back into the network from the terminals where the load is connected. The value of the maximum power is given by V2/4RL. In simpler terms, Maximum power is delivered from a source to a load when the load resistance is equal to the source resistance, assuming that the load resistance is a variable. Consider a network with a source of emf E and internal resistance r connected to a load resistance RL . The power delivered to the load resistance is maximum when the load resistance is equal to the internal resistance of the source. The power versus resistance curve is parabolic in nature. And the maximum power delivered to the load is only half the power generated by the source or the maximum power transfer efficiency is 50%. The remaining 50% power is lost across the internal resistance of the source.

Substitution theorem:

In any linear bilateral active network, any branch within a circuit may be placed by an equivalent branch, provided the replacement branch has the same current through it and the voltage across it as the original branch.

Reciprocity Theorem:

The reciprocity theorem tells us that in a linear passive bilateral network and the corresponding response may be interchanged. Statement: The ratio of excitation remains in a reciprocal network with respect to an interchange between the points of application of excitation and measurement of response.

Millman’s Theorem:

Millman’s theorem is used to simplify circuits having several parallel voltage sources. In fact other theorems explained in this tutorial will work in many cases, but millman’s
theorem provides a much simpler and more direct equivalent. And certain kinds of circuits can be replaced with single, simple equivalent circuit

Compensation Theorem:

In any linear bilateral active network, if any branch carrying a current I has its impedance Z changed by an amount Delta(Z), the resulting changes that occur in the other branches are the same as those which would have been caused by the injection of a voltage source of ( –I x Delta(Z) ) in the modified branch.

Tellegen's theorem:

This theorem states that the sum o f the power in the elements in a circuit is zero at any instant of time.
Tellegen’s theorem is simply about the power balance in a circuit and is based on the fundamental law of conservation of energy and Kirchhoff’s laws. The power absorbed
or dissipated by a resistor is always considered positive, whereas as the source is delivering deliver power, the power associated with the source is negative. The proof of this theorem can be derived using either Kirchhoff’s voltage law, or, Kirchhoff’s current law.

Delta to Star and Star to Delta Network Conversions

In a delta connection, three branches are so connected, such that they form a closed triangular loop. As these three branches are connected nose to tail, they form a closed loop, called the delta connection. On the other hand, when either terminal of three branches is connected to a common point to form a Y like pattern - this is the star connection. But we can transform between these two connections in a manner such that the potential difference between the pairs of points remains the same. For simplifying complex network, delta to star or star to delta conversion or transformation formula is often required.

Electrical DC Networks

Complete tutorial with solved problems :




Related Tutorials ( Introduction to Electrical Circuits - DC ) :

Circuit Theory 1a - Introduction to Electrical Engineering, DC Circuits, Resistance and Capacitance, Kirchoff Law

 Resistors, Capacitors, problems related to these.   Circuit Theory 1b - More solved problems related to DC Circuits with Resistance and Capacitance

 Capacitors, computing capacitance, RC Circuits, time constant of decay, computing voltage and electrostatic energy across a capacitance  Circuit Theory 2a - Introducing Inductors


 Inductors, inductance, computing self-inductance, flux-linkages, computing energy stored as a magnetic field in a coil,  mutual inductance, dot convention,
introduction to RL Circuits and decay of an inductor. 
 Circuit Theory 2b - Problems related to RL, LC, RLC circuits

 Introducing the concept of oscillations. Solving problems related to RL, LC and RLC circuits using calculus based techniques.  Circuit Theory 3a - Electrical Networks and Network Theorems


 Different kind of network elements: Active and passive, linear and non-linear, lumped and distributed. Voltage and current sources. Superposition theorem, Thevenin (or Helmholtz) theorem and problems based on these.  Circuit Theory 3b - More network theorems, solved problems


 More solved problems and examples related to electrical networks. Star and Delta network transformations, maximum power transfer theorem, Compensation theorem and Tellegen's Theorem and examples related to these. 

 
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