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Fig. 6-20
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Let v designate the voltage across the parallel RC combination. The current in R is iR v=R 10 6 v. During the pulse, iR remains negligible because v cannot exceed 1 V and iR remains under 1 mA. Therefore, it is reasonable to assume that during the pulse, iC 1 A and consequently v 0 1 V. For t > 0, from application of KVL around the RC loop we get v dv 0; dt v 0 1 V 41
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The only solution to (41) is v e t for t > 0 or v t e t u t for all t. For all practical purposes, is can be considered an impulse of size 10 6 A, and then v e t u t (V) is called the response of the RC combination to the current impulse.
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Plot the function v t which varies exponentially from 5 V at t 0 to 12 V at t 1 with a time constant of 2 s. Write the equation for v t .
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Identify the initial point A (t 0 and v 5 and the asymptote v 12 in Fig. 6-21. The tangent at A intersects the asymptote at t 2, which is point B on the line. Draw the tangent line AB. Identify point C belonging to the curve at t 2. For a more accurate plot, identify point D at t 4. Draw the curve as shown. The equation is v t Ae t=2 B. From the initial and nal conditions, we get v 0 A B 5 and v 1 B 12 or A 7, and v t 7e t=2 12.
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Fig. 6-21
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The voltage v V0 e ajtj for a > 0 is connected across a parallel combination of a resistor and a capacitor as shown in Fig. 6-22(a). (a) Find currents iC , iR , and i iC iR . (b) Compute and graph v, iC , iR , and i for V0 10 V, C 1 mF, R 1 M, and a 1.
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(a) See (a) in Table 6-3 for the required currents. (b) See (b) in Table 6-3. Figures 6-22(b) (e) show the plots of v, iC , iR , and i, respectively, for the given data. During t > 0, i 0, and the voltage source does not supply any current to the RC combination. The resistor current needed to sustain the exponential voltage across it is supplied by the capacitor.
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WAVEFORMS AND SIGNALS
Fig. 6-22
WAVEFORMS AND SIGNALS
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Table 6-3 Time (a) (b) t<0 t>0 t<0 t>0 v v V0 e v V0 e at v 10et v 10e t
iC C dv=dt iC CV0 ae iC CV0 ae at iC 10 5 et iC 10 5 e t
iR v=R iR V0 =R e iR V0 =R e at iR 10 5 et iR 10 5 e t
i iC iR i v0 Ca 1=R eat i V0 Ca 1=R e at i 2 10 5 et i 0
Supplementary Problems
6.22 Let v1 8 sin 100t and v2 6 sin 99t. Show that v v1 v2 is periodic. Find the period, and the p maximum, average, and e ective values of v. Ans: T 2; Vmax 14; Vavg 0; Veff 5 2 Find period, frequency, phase angle in degrees, and maximum, minimum, average, and e ective values of v t 2 6 cos 10t =6 . p Ans: T 0:2 s; f 5 Hz; phase 308; Vmax 8; Vmin 4; Vavg 2; Veff 22 Reduce v t 2 cos !t 308 3 cos !t to v t A sin !t  . Ans: A 4:84;  1028
6.24 6.25
Find V2;avg and V2;eff in the graph of Fig. 6-1(b) for V1 V2 3, and T 4T1 =3. Ans: V2;avg 1:5; V2;eff 3 Repeat Problem 6.25 for V1 0, V2 4, and T 2T1 . Ans: p V2;avg 2; V2;eff 2 2
6.26 6.27
Find V3;avg and V3;eff in the graph of Fig. 6-1(c) for V0 2 and T 200T1 . Ans: V3;avg 0; V3;eff 0:1 The waveform in Fig. 6-23 is sinusoidal. Express it in the form v A p sin !t  and nd its mean and B rms values. Ans: v t 1 6 sin t=12 1208 ; Vavg 1; Veff 19
Fig. 6-23
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