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Berkeley ELENG 100 - SERIES AND PARALLEL CONNECTIONS

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Thus a sinusoidal voltage with angular frequency w, applied to a linear time-varying resistor generates. in addition to a sinusoidal current with the same angular frequency W,. two sinusoids at angular frequencies w + w, and w - W,. This property is the basis of several modulation schemes in communication systems. With linear time-invariant resistors. a sinusoidal input can only generate a sinusoidal response at the same frequency. 2 SERIES AND PARALLEL CONSECTIONS In Chap. 1 we considered general circuits with arbitrary circuit elements. The primary objective was to learn Krrchhoff's current law (KCL) and Kirchhoffs voltage law (KVL). KCL and KVL do not depend on the nature of the circuit elements. They lead to tlvo sets of linear algebraic equations in terms of tivo sets of pertinent circuit variables: the branch crtrrents and the branch ~,oirages. These equations depend on the topology of the circuit, i.e.. howr the circuit elements are connected to one another. The branch currents and branch , voltages are in turn related according to the characteristics of the circuit elements. As seen in the previous section, these characteristics for two- terminal resistors may be linear or nonlinear, time-invariant or time-varying. The equations describing the v-i characteristics are called element eq~latiorzs or branch equations. Together with ihe equations from KCL and KVL. the]; give a complete specification of the circuit. The purpose of circuit analysis is to write down the complete specification of any circuit and to obtain pertinent solutions of interest. In this section we will consider a special but very important class of circuits: circuits formed by series and parallel connections of two-terminal resistors. First, we wish to generalize the concept of the v-i characteristic of a resistor to that of a two-terminal circuit made of two-terminal resistors. or more succinctly a resistive one-port. We will demonstrate that the series and parallel connections of two-terminal resistors will yield a one-port whose v-i characteristic is again that of a resistor. We say that two resistive one-ports are eqriivalent iff their v-i characteristics are the same. When we talk about resistive one-ports, we naturally use port volrage and port current as the pertinent variables. The v-i characteristic of a one-port in terms of its port voltage and port current is often referred to as the driving- point characteristic of the one-port. The reason we call it the driving-point characteristic is that we may consider the one-port as being driven by an independent voltage source v, or an independent current source is as shown in Fig. 2.1. In the former. the input is v, = v, the port voltage; and the response is the port current i. In the latter, the input is is = i, the port current; and the response is the port voltage v. In the following subsections we will discuss the driving-point characteristics of one-ports made of two-terminal resistors con- nected in series, connected in parallel, and connected in series-parallel.Figure 2.1 X one-port N driven (a) by an independent voltage source and (b) by an independent current source. 2.1 Series Connection of Resistors From physics we know that the series connection of finear resistors yields a linear resistor whose resistance is the sum of the resistances of each Iinear resistor. Let us extend this simple result to the series connections of resistors in general. G Consider the circuit shown in Fig. 2.2 where two nonfinear resistors 9, and 3, are connected at node @. Nodes (0 and are connected to the rest of the cirfuit. which is designated by N. Looking toward the right from nodes (0 and B. we have a one-port which is formed by the series connection of rwo resistors 2, and 3L. For our present purposes the nature of JV" is irrelevant. We are interested in obtaining the driving-point characteristic of the one-port with port voltage v and port current i. Let us assume that both resistors are current-controlfed, i.e., U, = C, (i,) and v, = Cz(iz) (2.1) These are the two branch equations. Next, we consider the circuit topology and write the equations using KCL and KVL. KCL applied to nodes @ and @ gives I "/ Figure 2.2 Two nonlinear resistors connected in series 0n;gort together with the rest of the circuit ;V.The KVL equation for the node sequence 0-0-0-0 leads to v = v, + v, (2.3) Combining Eqs. (3.1). /2.2), and (2.3). we obtain v = G,(i) + fi,(i) (2.4) which is the v-i characteristic of the one-port. It states that the driving-point characteristic of the one-port is again a current-controlled resistor v = ;(i) (3.5~) A where C) = (i) + (1) for all i (2.5b) Exercise If the two terminals of the nonlinear resistor 2, in Fig. 2.2 are turned around as shown in Fig. 2.3, show that the series connection gives a one-port which has a driving-point characteristic: Example 1 (a battery model) A battery is a physical device which can be modeled by the series connection of a linear resistor and a dc voltage source, as shown in Fig. 2.4. Since both the independent voltage source and the linear resistor are current-controlled resistors, this is a special case of the circuit in Fig. 2.2. The branch equations are v, =Ri, and v,= E (2.6) Adding v, and u2 and setting i, = i, we obtain v=Ri+E Or, we can add the characteristics graphically to obtain the driving-point characteristic of the one-port shown in the i-v plane in the figure? The l 0 + - Figure 2.3 Series connection of 9, and 9z with the terminals of 9, o turned around. 3 We use the i-v plane to facilitate the addition of voltages.Figure 2.4 (a) Series connection of a linear resistor and a dc voltage source. and (b) its dr~iing-pninr characteristic heavy line in Fis. 2.4b gives the characteristic of a battery with an internal resistance R. Usually R is small. thus the characteristic in the i-v plane is reasonably flat. However, it should be clear that 2 real battery does not behave like an ~ndependent voltage source beca~se the port voltage v depends on the current i. If we connect the real battery to an external load, e.g.. a linear resistor with resistance R, the actual voltage across the load will ie El?. ::.me a battery is used


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Berkeley ELENG 100 - SERIES AND PARALLEL CONNECTIONS

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