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MISCELLANEOUS NETWORK THEOREMS

INTRODUCTION
TELLEGEN'S THEOREM
RECIPROCITY THEOREM
SUBSTITUTION THEOREM
COMPENSATION THEOREM
MILLMAN'S THEOREM
MILLER'S THEOREM
SUMMARY


INTRODUCTION

p06Theorems

This page describes briefly the other theorems, which are listed above.  We start with Tellegens' theorem.

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TELLEGEN'S THEOREM

p06Tellegen1

Tellegen's theorem is based on the fundamental law of conservation of energy and is a logical outcome of Kirchoff's laws.  It is a general and useful theorem.  It states that the algebraic sum of power absorbed by all elements in a circuit is zero at any instant.  Note that power absorbed by a resistor is always positive, whereas a source may deliver power.  Then in this case, the power associated with the source is negative. This theorem is useful to validate the solution of a problem.

ole44

The proof of this theorem can be derived using either Kirchoff's voltage law, or, Kirchoff's current law.  Here Kirchoff's current law is used to prove Tellegen's theorem.   Equation (70) states the Kirchoffs current law at a node.  If the node voltage is vk, then we get equation (71).  Equation (71) is true for every node and hence Tellegen's theorem is proved.

F42Tellegen

A circuit is shown in Fig. 42 for illustrating Tellegen's theorem.  The values of components used are presented below.

ole45

From the values of components, the values of currents I1, I2, and I3, are obtained first and their values are displayed above.  Verification of solution using Tellegen's theorem is shown next.

ole46

It is seen that the algebraic sum of power in each element in the circuit is zero.  If current leaves the positive terminal of a source, then power associated with that source is negative, in accordance with the passive sign convention.  Passive sign convention is to be used for computing power in each element, and then only the algebraic  sum of power in each element in the circuit is zero.  It is verified here that the solution obtained for the loop currents is correct.

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RECIPROCITY THEOREM

F43RecTh

Reciprocity is an important property of networks, especially two port networks. This theorem is illustrated by an example.  Let current through resistor R3 be I3 due to voltage source VS.  This source is in series with resistor R1.  Remove this source and connect it in series with resistor R3 .  Let current through resistor be I1.  According to reciprocity theorem, current I1 will be equal to current I3. Reciprocity is valid for almost all passive networks.

ole47

For the circuit in Fig. 43, expressions for currents, I1 and I3, can be obtained and they are the same, as expressed by equation (72).   This theorem can be extended to more complex passive circuits and it can be found to be valid.  Other than reflecting an important property of passive networks, the reciprocity theorem is not of much use for simplifying a problem and obtaining a solution. However, a lot more attention is paid to the reciprocal property of networks, when the two-port networks are presented and analysed.

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SUBSTITUTION THEOREM

F44Sub

Substitution theorem states that components can be interchanged, so long as the terminal current and voltage are maintained.  For example, a five ampere current source with a terminal voltage of 20 Volts, can be replaced by a voltage source of 20 volts, delivering 5 amperes, as shown by the circuit in Fig. 45.  This theorem is used sparingly.

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COMPENSATION THEOREM

F45Comp1

Compensation theorem is a very useful theorem.  It reflects the changes that can occur due to incremental changes in the value of a component.  This theorem is illustrated by the circuit in Fig. 45.  Let the resistance change by a small amount.  Its effect can be computed as shown.  The effect can be computed by the response due to a voltage source VC , the value of which is obtained as shown below.

ole48

From equation (73), cancel the common term on both sides of equation.  Then we obtain equation (74).  The value of compensating voltage VC is obtained as shown by equation (75).  The change in current is also obtained, as shown.

F46Comp2

Compensation theorem is illustrated by an example.  For the circuit in Fig. 46, obtain current I3.  Let the value of resistor R3 change by a small value.  Compute the change in current I3.

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The values of components are specified.  The value of current I3 is obtained as shown.  The value of compensating source VC is then obtained and the change in current I3, denoted by DI3, is obtained and its value has been shown above.

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MILLMAN'S THEOREM

F47Millman

Millman's theorem is useful in some applications.  It is illustrated by the circuit in Fig. 47.  The value of output voltage is obtained, as shown below.

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Equation (76) is obtained by applying the Kirchoff's voltage law to the output node.  By re-arranging the terms, the output voltage is obtained as shown by equation (77).

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MILLER'S THEROEM

F48Miller

Miller's theorem is used mostly in connection with electronic circuits.  Given a circuit with gain, the resistor connected between the input port and the output port can be replaced by two resistors, one at each port.  The values of these resistors are obtained as shown below.

ole51

Equation (78) shows the relationship between the input and output voltages.  Expression for current I1 is obtained as shown.  Hence the value of impedance Z1 is obtained, as shown by equation (79).  Similarly the value of Z2 is obtained, as shown by equation (80).  These two impedances or resistors can be connected across the input and output ports, as shown in Fig. 48.  

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SUMMARY

This chapter has described network theorems. The merit of theroems lies in the insight they offer into the behaviour of circuits. The superposition theorem defines the behaviour of linear circuits. A circuit is linear only if it behaves in accorddance with the superposition theorem. This theorem states that the linear responses in a circuit can be obtained as the algebraic sum of responses, due to each of the sources acting alone. Thevenin's thoerem shows how a network containing one or more independent sources and other linear components, can be represented by a single voltage source and an equivalnet resistance, from the viewpoint of a load resistor. Norton's thoerem is the dual of Thevenin's theorem and it shows how a network containing one or more independent sources and other linear components, can be represented by a single current source and an equivalent resistance, from the viewpoint of a load resistor.

The maximum power transfer theorem states the condition for maximum power transfer to the load. The Tellegen's theorem highlights the fact that there is power balance in a circuit. Reciprocity theorem highlights the reciprocal property of a passive network. The substitution theorem states how components can be interchanged, when equivalence is maintained. Compensation theorem shows how the change in the value of a component affects the response of circuit. Millman's theorem uses Kirchoff's voltage law and is useful in particular instances. Miller's theorem is used for analyzing circuits with feedback.

A few interactive examples have been presented to illustrate the application of theorems. The next topic is the study of ac circuits.

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