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38 CHAPTER ONE the circuit abstractioniSlope = 1 R --FIGURE 1.40 i–v plot for aresistor.vIt is appropriate at this point to check experimentally the validity of themodels developed here by plotting their v i characteristics. The v i characteristicscan be plotted using the setup shown in Figure 1.33. First, use a 100-1/10-W resistor as the ‘‘unknown’’ two-terminal device. If the oscillator voltageis a few volts, a straight line passing through the origin with slope 1/Rwill appear on the screen (see Figure 1.40), showing that Ohm’s law applies.However, if the voltage is increased so that v D is 5 or 10 V, then the 1/10-Wresistor will heat up, and its value will change. If the oscillator is set to a verylow frequency, say 1 Hz, the resistor heats up and cools down in the sourceof each cycle, so the trace is decidedly nonlinear. If the oscillator is in the midaudiorange, say 500 Hz, thermal inertia prevents the resistor from changingtemperature rapidly, so some average temperature is reached. Thus the line willremain straight, but its slope will change as a function of the amplitude of theapplied signal.Resistor self-heating, with the associated change in value, is obviouslyundesirable in most circuit applications. For this reason manufacturers providepower ratings for resistors, to indicate maximum power dissipation(p max ) without significant value change or burnout. The power dissipated in aresistor isp = vi, (1.22)which is the hyperbola in v i space, as indicated in Figure 1.41. Our idealresistormodel ohmic with constant value matches the actual resistorbehavior only in the region between the hyperboli.The plot on the oscilloscope face will also deviate from a straight line if theoscillator frequency is made high enough. Under this condition, capacitive andinductive effects in the circuit will generate an elliptical pattern. These will bediscussed in later <strong>chapter</strong>s.
1.7 Modeling Physical Elements CHAPTER ONE 39iip maxSlope = --- 1 Rvp max0Vv tFIGURE 1.41 Power constraint for a resistor in the i – v plane.FIGURE 1.42 i–v characteristic of a battery at low currentlevels.Now plot the i v characteristic of a battery. At low current levels, thecurve appears as a vertical line in i v space (see Figure 1.42). But if the oscillatoramplitude is increased so that substantial currents are flowing, and we makean appropriate change in scope vertical sensitivity, the line remains straight,but now has a definite tip, as suggested by Figure 1.43a, indicating a nonzeroseries resistance. If the battery terminal voltage and current are defined as ini+iSlope = 1 R --Vv tShort-circuitcurrent0(a)Vv tOpen-circuitvoltage+V-R(b)(c)-i+v t-FIGURE 1.43 Battery characteristic,larger current scale.
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