Norton’s theorem states that a

linear two-terminal circuit

can be replaced by

an equivalent circuit

consisting of a current source

IN

in parallel with a resistor

RN

, where

IN

is the short-circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the independent sources are turned off.

Thevenin's Theorem

Thevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh, where VTh is the open-circuit voltage at the terminals and RTh is the input or equivalent resistance at the terminals when the independent sources are turned off.

**Electric Circuit**

**Chapter 4: Circuit Theorem**

LINEARITY PROPERTY

-describing a linear relationship between cause and effect.

-The property is a combination of both the homogeneity (scaling) property and the additivity property.

For a resistor, for example, Ohm’s law relates the input i to the output v:

v = iR

If the current is increased by a constant k,

k v = k iR

A

linear circuit

is one whose output is linearly related (or

directly proportional

) to its input.

Therefore, the theorems covered in

this chapter are not applicable to power

.

Superposition

The superposition principle states that the

voltage across (or current through) an element in a linear circuit

is the

algebraic sum of the voltages across (or currents through) that element

due to

each independent source acting alone.

However

, to apply the superposition principle, we must keep two things in mind:

1. We consider

one independent source at a time while all other independent sources are turned off

. This implies that we

replace every voltage source by 0 V (or a short circuit), and every current source by 0 A (or an open circuit)

.

2.

Dependent sources are left

intact because they are controlled by circuit variables.

Steps to Apply Superposition Principle:

1.

Turn off all independent sources except one source

. Find the

output (voltage or current) due to that active source using

nodal or mesh analysis

.

2.

Repeat step 1 for each of the other independent sources

.

3. Find the

total contribution by adding algebraically

all the

contributions due to the independent sources.

Source transformation also applies to dependent sources, provided we carefully handle the dependent variable. A dependent voltage source in series with a resistor can be transformed to a dependent current source in parallel with the resistor or vice versa.

Case 1

If the network has no dependent sources, we turn off all independent sources. RTh is the input resistance of the network looking between terminals a and b.

Case 2

If the network has dependent sources, we turn off all independent sources. As with superposition, dependent sources are not to be turned off because they are controlled by circuit variables. We apply a voltage source Vo at terminals a and b and determine the resulting

current .

Thevenin’s theorem is very important in circuit analysis. It helps simplify a circuit. A large circuit may be replaced by a single independent voltage source and a single resistor. This replacement technique is a powerful tool in circuit design.

As mentioned earlier, a linear circuit with a variable load can be replaced by the Thevenin equivalent, exclusive of the load. The equivalent

network behaves the same way externally as the original circuit. Consider a linear circuit terminated by a load RL. The current IL through the load and the voltage VL across the load are easily determined once the Thevenin equivalent of the circuit at the load’s terminals is obtained.

This is essentially source transformation. For this reason, source transformation is often called

Thevenin-Norton transformation.

**The Thevenin and Norton equivalent circuits are related by a source transformation.

We can calculate any two of the three using the method that takes the

least effort and use them to get the third using

Ohm’s law

.

In many practical situations, a circuit is designed to provide power to a load. There are applications in areas such as communications where it is desirable to maximize the power delivered to a load. We now address the problem of

delivering the maximum power to a load

when given a system with known

internal losses

. It should be noted that this will result in significant internal losses greater than or equal to the power delivered to the load.

The

Thevenin equivalent

is useful in finding the

maximum power a linear circuit can deliver to a load

. We assume that we can adjust the load resistance If the entire circuit is replaced by its Thevenin equivalent except for the load. the power delivered to the load is :

To prove the maximum power transfer theorem:

Therefore,

Maximum Power Transfer

Note

that since (making it a quadratic function rather than a linear one), the relationship between power and voltage (or current) is nonlinear.

The growth in areas of application of electric circuits has led to an evolution from simple to complex circuits. To handle the complexity, engineers over the years have developed some theorems to simplify circuit analysis. Such theorems include

Thevenin’s and Norton’s theorems

.

Since these theorems are applicable to

linear circuits

, we first discuss the concept of

circuit linearity.

In addition to circuit theorems, we discuss the concepts of

superposition, source transformation, and maximum power transfer

in this chapter.

Introduction

Source Transformation

Source transformation is

another tool for simplifying circuits

. Basic to these tools is t

he concept of equivalence

. We recall that an

equivalent circuit is one whose v-i characteristics are identical

with the original circuit.

Example 1:

Example 2:

Example 3:

Example 1:

Example 2:

Example 3:

Example 1:

Example 2:

Example 3:

Example 1:

Example 2:

Example 3: