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 twoterminal resistors may be linear or nonlinear time invariant or time varying The equations describing the v i characteristics are called element eq latiorzsor 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 drivingpoint 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 connected 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 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 G 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 0n gort Figure 2 2 Two nonlinear resistors connected in series together with the rest of the circuit V The KVL equation for the node sequence v v v 0 0 0 0 leads to 2 3 Combining Eqs 3 1 2 2 and 2 3 we obtain 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 G i v A C i where 3 5 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 R i 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 9zwith the terminals of 9 turned around o 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 pninrcharacteristic 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 b e c a s ethe 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 to deliver power to sin external circuit we usually prefer to use the opposite of the associated reference direction when the battery is connected to an external circuit The characteristic plotted on the i t v plane where i i is shown in Fig 2 5 together with the external circuit Example 2 ideal diode circuit Consider the series connection of a real battery and an ideal diode as shown in Fig 2 6 Since the ideal diode is not a current controlled resistor we cannot use Eq 2 5 to …
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