Electrical Circuit - Network Theorems

 Network Theorem (D.C. Circuits)

The applicability of these theorems in different types of d.c. networks (with independent sources as well as with controlled sources). A thorough understanding of each theorem is important because analyzing electrical and electronic circuits needs these theorems very much.

Thevenin's theorem

This theorem is possibly the most extensively used network theorem. It is applicable where it is desired to determine the current through or voltage across any one element in a network without going through the rigorous method of solving a set of network equations.

Statement of Thevenin's Theorem:- 

Any two terminal bilateral linear d.c. the circuit can be replaced by an equivalent circuit consisting of a voltage source and a series resistor.

Thevenin's Theorem

Equivalent Resistance and Voltage

Equivalent Circuit

Norton's theorem

Norton theorem is converse of Thevenin's theorem. It consists of an equivalent voltage source as done in Thevenin's theorem. The determination of internal resistance of the source network is identical in both the theorems. However, in the final stage i.e. in the Norton equivalent circuit, the current generator is placed in parallel to the internal resistance unlike that in Thevenin's theorem where the equivalent voltage source was placed in series with the internal resistance.

Statement of Norton Theorem:-

A linear active network consisting of independent and or dependent voltage and current sources and linear bilateral network elements can be replaced by an equivalent circuit consisting of a current source in parallel with a resistance, the current source being the short-circuited current across the load terminal and the resistance being the internal resistance of the source network looking through the open-circuited load terminals.

DC Network - Norton Theorem

Conversion between Thevenin and Norton 

Norton Equation 

Norton Equivalent Circuit

Superposition Theorem

This theorem finds use in solving a network where two or more sources are present and connected not in series or parallel.

Statement of Superposition Theorem:-

If a number of voltage or current sources are acting simultaneously in a linear network, the resultant current in any branch is the algebraic sum of the currents that would be produced in it, when each source acts alone replacing all other independent sources by their internal resistances.

Circuit Diagram

Superposition Equation

                                         

                                                               I3' =  I2' -  I1' 

Maximum Power Transfer Theorem

This theorem is used to find the value of load resistance for which there would be the maximum amount of power transfer from source to load.

Statement of Maximum Power Transfer Theorem:-

A resistance load, being connected to a dc network, receives maximum power when the load resistance is equal to the internal resistance (Thevenin's equivalent resistance) of the source network as seen from the load terminals.

Maximum Power Circuit

Maximum Power Transfer Equation

P and Pmax Equation

During maximum power transfer, the Efficiency η becomes.

                                                                 η = Pmax/P *100

Millman's Theorem

The utility of this theorem is that any number of parallel voltage sources can be reduced to one equivalent one. 

Statement of Millman's Theorem:-

When a number of voltage sources (V1, V2......Vn) are in parallel having internal resistance (R1, R2......Rn) respectively, the arrangement can be replaced by an equivalent voltage source V in series with an equivalent series resistance R.

a). A number parallel of voltage sources fading power to a load resistance.
b). Equivalent voltage and resistance of the source network following Millman Theorem

Equivalent Equation

Circuit Diagram

Reciprocity Theorem

Statement of Reciprocity Theorem:-

In any branch of a network, the current (I) due to a single source of voltage (V) elsewhere in the network is equal to the current through the branch in which the source was originally placed when the source is placed in the branch in which the current (I) was originally obtained. 

Reciprocity Theorem

Substitution Theorem

The theorem, in its simplest form, tells that for branch equivalence, the terminal voltage and current must be the same.

Statement of Substitution Theorem:-

The voltage across the current through any branch of a dc bilateral network being known, this branch can be replaced by any combination of elements that will make the same voltage across and current through the chosen branch.

Compensation Theorem

Statement of Compensation Theorem:-

In a linear time-invariant network when the resistance (R) of  an uncoupled branch, carrying a current (I) is changed by (🛆R), the currents in all the branches would change and can be obtained by assuming that an ideal voltage source of (Vc) has been connected {such that Vc = I(🛆R)} in series with 

(R+ 🛆R).

Compensation Theorem

Tellegen's Theorem

This theorem is one of the most general theorems in network analysis. Regardless of the type and nature, Tellegen's theorem is applicable to any network made up of lumped two terminals elements.

Statement of Tellegen's Theorem:-

For any given time, the sum of power delivered to each branch of any electric network is zero. 

Thus for the Kth branch, this theorem states that ΣVk Ik = 0, n being the branches, Vk the drop in the branch, and Ik the through current.

Tellegen's Theorem

Analysis of Network

Kirchhoff's Laws:-

A German physicist Gustav Kirchoff developed two laws enabling easier analysis of interconnection of any number of circuit elements. The first law deals with the flow of current and is properly known as Kirchoff's current law (KCL) while the second one deals with voltage drop in closed networks and is known as Kirchhoff's voltage law (KVL).

Kirchoff's Voltage Law

Kirchoff's Current Law

 

Nodal and Mesh Analysis of Electric Circuits

There is another method of solution of the network i.e. by nodal analysis where it is essential to compute all branch currents. In writing the current expression the assumption is made that the node potential is always higher than the other voltages appearing in the equation. In case it turns out not to be so, a negative value for the current would result.

In the nodal method, the number of independent node-pair equations needed is less than the number of junctions in the network. That is, if "n" denotes the number of independent node equations and "j" the number of junctions. 

n =j-1

 There is one more method of analyzing an electrical network - the mesh analysis, the name being derived from the similarities in appearance between the closed loops of a network and a physical "fence" or mesh. In this method,