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NORTONS’ THEOREM AND SOURCE TRANSFORMATION

PRESENTATION OF NORTONíS THEOREM
PROOF OF NORTON’S THEOREM
APPLICATION OF NORTON’S THEOREM
LINKING THEVENIN’S AND NORTON’S THEOREMS
SOURCE TRANSFORMATION
SUMMARY


PRESENTATION OF NORTON’S THEOREM

Let a linear network consisting of one or more independent sources and linear elements be feeding power to a load. Norton’s theorem says that the linear network consisting of one or more independent sources and linear elements can be represented by a current source and an equivalent resistance in parallel with the current source. To find the value of the current source, replace the load by a short-circuit. The value of the current source is the current that flows through the short-circuit. The resistance is the resistance of the network as viewed from the load terminals, with the independent sources removed. This resistance is known as Thevenin’s resistance. The current source is known as the Norton’s current source.

F28Nort01

Let a circuit be represented by a box, as shown in Figure 28. Its output characteristic is also displayed. As the load resistor is varied, the load current varies. The load current is bounded between two limits, zero and ISC, and the load voltage is bounded between two limits, VTh and zero volts. When the load resistor is infinite, it is an open circuit. In this case, the load voltage is at its highest, specified as VTh and the load current is zero. This is the point at which the output characteristic intersects with the Y axis. When the load resistor is of zero value, there is a short circuit across the output terminals of the circuit and in this instance, the load current is maximum, specified as ISCand the load voltage is zero. It is the point at which the output characteristic intersects with the X axis.

F29Nort02

The circuit in Fig. 29 has an output current of ISC, when the load voltage is zero. Hence the model of the circuit can have a current source of ISCamperes. When the output terminals are open circuited, the load current is zero and it can be stated that the source current passes through the internal resistance, leading to a terminal voltage of VTh. This means that the internal resistance of the circuit, called as RTh, has a value, as shown by the equation displayed. Hence the circuit consists of a current source of value ISCamperes and a resistor of value equal to RTh. This resistor is the resistance of the circuit, as viewed from the load terminals, after removal of the independent sources in the circuit. A voltage source is replaced by a short-circuit, and a current source is replaced by an open circuit. The internal resistance of each of these sources is left in the circuit, as it is, where it is. A non-ideal voltage source consists of a voltage source with a source resistance, connected in series with the source. A non-ideal current source consists of a current source with a source resistance, connected across the source.
F30Nort03

The circuit in Fig. 30, which has already been presented earlier, is shown here for illustrating Nortons theorem. The short circuit current and the load current can be obtained, as shown by the simplified circuits. We can describe the operation of the circuitby equations described below.

ole34

Equation (54) presents the resistance seen by the source. Equation (55) shows how the current drawn from the source can be obtained. The source current flows into resistors, R2 and R3, connected in parallel. Equation (56) expresses the current through resistor R3.

ole35


For the circuit in Fig. 30, let R3 be considered the load resistor. Then the load current can be expressed by equation (57). This equation is obtained by replacing the source current in equation (56) in terms of the source voltage and the resistances. From equation (57), expressions for the short circuit current, and Thevenin’s resistance can be obtained, and equation (58) presents the expressions . It is seen that the load current can be expressed in terms of Norton/s current and Thevenin’s resistance.

F31Nort04

Based on the example, Norton’s theorem can be stated as follows.Let a network with one or more sources supply power to load resistor as shown in Fig. 31. Norton’s theorem states that the network can be replaced by a single equivalent current source, marked as ISCand a resistor marked as RTh, as shown in Fig. 31.

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PROOF OF NORTON’S THEOREM

F32Nort05

The circuit in Fig. 32 can be used to prove Norton’s theorem. Equation (1) expresses current source IY connected to the load terminals, as a function of voltage VYacross the source and some coefficients. Coefficient k1 reflects conductance of the circuit as seen by external current source IY, and coefficientk2reflects the contribution to terminal voltage by internal sources and components of the circuit. It is valid to do so, since we are dealing with a linear circuit. Each independent internal source within the circuit contributes its part to terminal voltage and coefficientk2is the algebraic sum of contributions of internal sources. There is a negative sign in front of k1in equation (1). Let voltage VYbe a a positive value. Then as current IYincreases, voltage VYtends to decrease. By assigning a negative sign in front of k1, we get apostivie value for k1. Adjust external current source such that voltage VYbecomes zero. As shown by equation (2), coefficientk2is Norton’s current. To determine Thevenin’s resistance, set external source current to zero. Then in this case, the voltage across the load terminals is the open circuit voltage or Thevenin’s voltage. Coefficient k1is the reciprocal of Thevenin’s resistance. This concludes the proof of Norton’s theorem.

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APPLICATION OF NORTON’S THEOREM


The application of Norton’s theorem is similar to the application of Thevenin’s theorem. It can be illustrated with the help of a sketch.

F33Nort06

Steps involved in obtaining Nortons equivalent circuit are outlined. For the generic block diagram in Fig. 33, Obtain the short circuit current first, as described.
p05Nort01

Thevenin’s resistance is obtained as described above. These steps have been described earlier, for getting Thevenin’s resistance for Thevenin’s theorem.

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LINKING THEVENIN’S AND NORTON’S THEOREMS

Norton’s Theorem and Thevenin’s Theorem are equivalent, as we will see now. Their equivalence leads to source transformation.
F35Nort07

The relationship between the Thevenin’s theorem, and, Norton’s theorem can be established using the equivalent circuits in Fig. 34.

ole36

From Fig. 34, the load current, obtained using Thevenin’s equivalent circuit, is expressed by equation (59). The load current obtained using Norton’s equivalent circuit is expressed by equation (60). Both the expressions in equations, (59) and (60) yield the same value, when equation (61) is valid. Equation (61) describes the relationship between the Thevenin’s theorem, and the Norton’s theorem. The Thevenin’s voltage equals the product of Norton’s current and the Thevenin’s resistance. We can use the equivalence between Thevenin’s equivalent circuit, and Norton’s equivalent circuit to show how source transformation can be brought about.

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SOURCE TRANSFORMATION

F35SrceTr

What is meant by source transformation is that a voltage source, with a resistor in series, can be effectively replaced by a current source, and, a resistor in parallel. The voltage source equals the product of the value of the current source, and, the resistance in series with the voltage source. What is being stated is, that Thevenin’s equivalent circuit can be replaced by the Norton’s equivalent circuit, and, vice versa. The source transformation theorem is a direct consequence of Thevenin’s and Norton’s theorems. Figure 35 illustrates the process of source transformation. Source transformation can be used for simplification of some circuits. In some instances, we can derive Thevenin’s and Norton’s equivalent circuits using source transformation , as illustrated by the example presented below.

F36SrPr03

A problem is presented now by the circuit in Fig. 36. The task is to obtain the Thevenin’s equivalent circuit, and, the Norton’s equivalent circuit by using source transformation. The first step is displayed next.

F37Pr3St1

The current source of 2 Amperes, and, 3 W resistor can be replaced by a voltage source of 6 Volts, with the resistor connected in series. The resulting circuit is displayed in Fig. 37.

F38Pr3St2

The two voltage sources can be combined into a single voltage source, with the resistance in series being the sum of 3 W and 5 W . The resultant circuit is displayed by circuit in Fig. 38.

F39Pr3St3

The circuit in Fig. 39 displays the next step. The voltage source of 16 Volts, and, the 8 W resistor can be replaced by a current source of 2 Amperes, with the 8 W connected across it. The two current sources in parallel can be replaced by a single current source, and, the two resistors in parallel can be replaced by the equivalent parallel value.

F40Pr3St4

The circuits in Figure 40 display the Thevenin’s and the Norton’s equivalent circuits. A few interactive examples are presented in the subsequent pages.

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SUMMARY

This page has presented Norton’s theorem and source transformation. It is seen that Norton’s theorem and Thevenin’s theorem are linked, and they form the basis for source transformation. A few interactive examples are presented in the subsequent pages.After that, maximum power transfer theorem is described.

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