FIGURE 2.8 Ideal independent current source circle which also denotes this as an ideal independent source. The voltage across the element will be determined by the circuit that is attached to the terminals of this source Numerous functional forms are useful in describing the source variation with time. These were discussed in ection 2.1-the step, impulse, ramp, sinusoidal, and dc functions. For example, an ideal independent dc voltag source is described by vs(t)=Vs where Vs is a constant. An ideal independent sinusoidal current source is described by is(o=Issin(ot +) or i(0=I,cos(ot +o), where Is is a constant, 0= 2Tfwithf the frequency in hertz and o is a phase angle. Ideal sources may be used to model actual sources such as temperature transducers, phonograph cartridges, and electric power generators. Thus usually the time form of the output cannot generally be described with a simple, basic function such as dc, sinusoidal, ramp, step, or impulse waveforms. We often, however, represent the more complicated waveforms as a linear combination of more basic functions Practical Sources The preceding ideal independent sources constrain the terminal voltage or current to a known function of time dependent of the circuit that may be placed across its terminals. Practical sources, such as batteries, have their terminal voltage(current)dependent upon the terminal current(voltage)caused by the circuit attached to the ource terminals. A simple example of this is an automobile storage battery. The battery's terminal voltage is approximately 12 V when no load is connected across its terminals. When the battery is applied across the terminals of the starter by activating the ignition switch, a large current is drawn from its terminals. During arting, its terminal voltage drops as illustrated in Fig. 2.9(a). How shall we construct a circuit model using the ideal elements discussed thus far to model this nonideal behavior? A model is shown in Fig. 2.9(b)and consists of the series connection of an ideal resistor, R and an ideal independent voltage source, Vs=12 V. To determine the terminal voltage-current relation, we sum Kirchhoffs voltage law around the loop to give V=VS -RSi This equation is plotted in Fig. 2.9(b)and approximates that of the actual battery. The equation gives line with slope-Rs that intersects the v axis (i=0)at v= Vg The resistance Rs is said to be the internal of this nonideal source model. It is a fictitious resistance but the model nevertheless gives an equivalent Although we have derived an approximate model of an actual source, another equivalent form may be obtained. This alternative form is shown in Fig. 2.9(c)and consists of the parallel combination of an ideal independent current source, Is=VRs and the same resistance, Rs used in the previous model. Although it may seem strange to model an automobile battery using a current source, the model is completely equivalent to the series voltage source-resistor model of Fig. 2.9(b)at the output terminals ab. This is shown by writing e 2000 by CRC Press LLC© 2000 by CRC Press LLC circle which also denotes this as an ideal independent source. The voltage across the element will be determined by the circuit that is attached to the terminals of this source. Numerous functional forms are useful in describing the source variation with time. These were discussed in Section 2.1—the step, impulse, ramp, sinusoidal, and dc functions. For example, an ideal independent dc voltage source is described by vS(t) = VS, where VS is a constant. An ideal independent sinusoidal current source is described by iS(t) = I S sin(wt + f) or iS(t) = I S cos(wt + f), where IS is a constant, w = 2pf with f the frequency in hertz and f is a phase angle. Ideal sources may be used to model actual sources such as temperature transducers, phonograph cartridges, and electric power generators. Thus usually the time form of the output cannot generally be described with a simple, basic function such as dc, sinusoidal, ramp, step, or impulse waveforms. We often, however, represent the more complicated waveforms as a linear combination of more basic functions. Practical Sources The preceding ideal independent sources constrain the terminal voltage or current to a known function of time independent of the circuit that may be placed across its terminals. Practical sources, such as batteries, have their terminal voltage (current) dependent upon the terminal current (voltage) caused by the circuit attached to the source terminals. A simple example of this is an automobile storage battery. The battery’s terminal voltage is approximately 12 V when no load is connected across its terminals. When the battery is applied across the terminals of the starter by activating the ignition switch, a large current is drawn from its terminals. During starting, its terminal voltage drops as illustrated in Fig. 2.9(a). How shall we construct a circuit model using the ideal elements discussed thus far to model this nonideal behavior? A model is shown in Fig. 2.9(b) and consists of the series connection of an ideal resistor, RS, and an ideal independent voltage source,VS = 12 V. To determine the terminal voltage–current relation, we sum Kirchhoff’s voltage law around the loop to give (2.1) This equation is plotted in Fig. 2.9(b) and approximates that of the actual battery. The equation gives a straight line with slope –RS that intersects the v axis (i = 0) at v = VS . The resistance RS is said to be the internal resistance of this nonideal source model. It is a fictitious resistance but the model nevertheless gives an equivalent terminal behavior. Although we have derived an approximate model of an actual source, another equivalent form may be obtained. This alternative form is shown in Fig. 2.9(c) and consists of the parallel combination of an ideal independent current source, IS = VS /RS , and the same resistance, RS , used in the previous model. Although it may seem strange to model an automobile battery using a current source, the model is completely equivalent to the series voltage source–resistor model of Fig. 2.9(b) at the output terminals a–b. This is shown by writing Kirchhoff’s current law at the upper node to give FIGURE 2.8 Ideal independent current source. v(t) + – b a i(t) = i S (t) i S (t) i S(t) t v V R i S S = -