110_1_Desoer&Kuh_Chap9

110_1_Desoer&Kuh_Chap9 - s and Tellegen's Theorem...

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/ sand Tellegen's Theorem In the first part of this book we encountered most of the important concepts and properties of circuits. To help us to understand them, we illustrated them only with simple circuits. With the exception of Chap. 7, the typical circuits we con· sidered contained only a few elements and were described by first· or second·order differential equations. In the second part of this book we want to develop sys· tematic procedures to analyze and establish properties of any networ~ of any complexity. Note that we use the word network, which has the same meaning as the word circuit (i.e., an interconnection of elements); however, the word "net· work" usually carries the idea of complexity (a network is a circuit with many elements). In practice some networks may be very complicated and may contain several dozens of elements. An additional reason for our need to develop systematic procedures is that the engineering world has been completely changed by the computer. There are now on the market several computers that multiply two 8·digit numbers in less than a microsecond. For engineers, such capability means that it is now feasible and economical to perform complicated analyses and undertake designs that require a great deal of computation, say, 10 6 times as many as 15 years ago. It is there· fore important to learn systematic procedures so that we can handle any network, however complicated. As is usual in science and engineering, the first step is a process of abstraction. Since KVL and KCL do not make any assumption whatsoever concerning the nature of the elements of a network, it is natural to overlook the nature of the elements in order to reduce the network to a graph. The first section of this chapter is de· voted to developing the concept of a graph. Graph·theoretic ideas are then used to precisely formulate KVL and KCL. Then, as an illustration of the power of the abstract concept of a graph, we derive Tellegen's theorem, which allows us to prove extremely easily several very general properties of networks. The Concept of a Graph - Consider any physical network, for example, an 80-element lumped delay line or a telemetering repeater. Suppose we consider only those fre- quencies which permit us to model the physical network as a connection of lumped elements, namely resistors, capacitors, inductors, coupled induc- 381
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110_1_Desoer&Kuh_Chap9 - s and Tellegen's Theorem...

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