### Circuit Theory 3a - Electrical Networks - Classification of Networks, Voltage and current sources,Network theorems Contents:

### 3. Network theorems: Superposition theorem, Thevenin’s theorem

1)  Classification of a network

### Active and Passive Elements

Elements which supply the energy to the circuit are known as active elements. A network which contains active elements is known as active networks. Ex: batteries, generators, transistors etc. Elements which absorb the energy are known as passive elements. A network containing only passive elements is known as passive networks. Ex: resistors, inductors and capacitors.

### Unilateral/Bilateral Elements

An element whose operational behaviour is dependent on the direction of flow of current through is known as unilateral elements. Elements like semiconductor diode, which allow the current to pass through them only in one direction.
An element whose behaviour is same irrespective of the direction of flow of current through it is known as bilateral element. Passive elements that allow the current to pass through them in both directions are known as bilateral elements.

### Lumped and Distributed Networks

Networks consisting of elements which can be physically separated are known as lumped networks. Most of the networks we deal with, are lumped in nature and consists of R, L,C and sources. Networks, like transmission lines, having inseparable elements are known as distributed networks.

### Linear and Non-Linear Elements

A linear element is one which has linear output/input relation and always follows superposition and homogeneity principles. Ohm’s can be applied to such networks.
The element that which does not follow these is known as a nonlinear element. Ohm’s law cannot be applied to such networks.

### 2) Voltage and current sources:

There are two principal types of source, voltage source and current source. Sources can be either independent or dependent upon some other source quantities.

### Dependent sources:

VCVS: voltage controlled voltage source: voltage source depends on a voltage value of V1.
VCCS: voltage controlled current source: current source depends on a voltage value of V1.
CCVS: current controlled voltage source: voltage source depends on a current value of I1.
CCCS: current controlled current source: current source depends on a current value of I1.

### Superposition Theorem: It states that in an active, linear, bilateral network consisting of active and passive elements with more than one source, the overall response (voltage or current) is equal to the sum of the responses due to each of the sources acting independently.

Statement:
In any linear network having number of voltage or current sources and resistances, the voltage across or the current through any branch is given by algebraic sum of all the individual voltages or currents caused by each independent source acting alone, with
all other independent voltage sources replaced by short circuits and all other independent current sources replaced by open circuit.

Number of networks to be analyzed = Number of independent sources
Note: the total power delivered to a resistance element must be determined using the total current through or the total voltage across the element and cannot be determined by a simple sum of the power levels established by each source.

### Thevenin’s Theorem (or Helmholtz’s Theorem)

Statement: Any two-terminal, linear bilateral network can be replaced by an equivalent network having a single voltage source called Thevenin’s voltage (V Th) and a single series resistance called Thevenin’s resistance (RTh). The Thevenin’s equivalent circuit provides equivalence at the terminals only—the internal construction and characteristics of the original network and the Thevenin’s equivalent are usually quite different.

For finding the thevenin’s resistance, the equivalent resistance across the terminals have to be found with all the independent sources being zero.
Here are kind of the problems which have been solved in the tutorial document :

• Verifying the superposition theorem for given networks.
• Gain an understanding of how to apply Kirchoff laws for voltage and current, KVL and KCL, and solve mesh equations using them
• Reduce circuits to simpler forms using the theorems above
• Finding current in a link using the superposition theorem.
• Finding the current in a resistor in a given circuit using the Thevenin Theorem.
• Using superposition theorem to find the current through a link that is to be  connected between two given terminals a-b, assuming the link resistance to be zero.
• In a given network, finding the current through a given resistor utilising Thevenin’s theorem.
• Finding the current through a given resistance, given a network of voltage, current sources and resistors.

### Complete Tutorial with Solved Problems :

Related Tutorials ( Introduction to Electrical Circuits - DC ) :

 Circuit Theory 1a - Introduction to Electrical Engineering, DC Circuits, 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, 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.