Aco π+26 FIGURE 2.4 The sinusoid A cos(ot 0)with -rt/2<0<0 Conversely, the first derivative of r(t) with respect to t is equal to u( t), except at t=0, where the derivative of r(t is not defined. Sinusoidal function The sinusoid is a continuous-time signal: A cos(ot 8) Here A is the amplitude, o is the frequency in radians per second (rad/s), and e is the phase in radians. The frequency f in cycles per second, or hertz(Hz), is f=(/2. The sinusoid is a periodic signal with period 2T/o he sinusoid is plotted in Fig. 2.4 Decaying Exponential In general, an exponentially decaying quantity(Fig. 2.5) can be expressed as where a instantaneous value A= amplitude or maximum value e= base of natural logarithms = 2.718 0.368 t= time constant in second t time in seconds The current of a discharging capacitor can be approxi mated by a decaying exponential function of time. Time Constant FIGURE 2.5 The decaying exponential Since the exponential factor only approaches zero as t increases without limit, such functions theoretically last forever. In the same sense, all radioactive disintegrations last forever. In the case of an exponentially decaying current, it is convenient to use the value of time that makes the exponent -l. When t=t= the time constant, e value In other words, after a time equal to the time constant, the exponential factor is reduced to approximatly 37% of its initial value e 2000 by CRC Press LLC© 2000 by CRC Press LLC Conversely, the first derivative of r(t) with respect to t is equal to u(t), except at t = 0, where the derivative of r(t) is not defined. Sinusoidal Function The sinusoid is a continuous-time signal: A cos(wt + q). Here A is the amplitude, w is the frequency in radians per second (rad/s), and q is the phase in radians. The frequency f in cycles per second, or hertz (Hz), is f = w/2p. The sinusoid is a periodic signal with period 2p/w. The sinusoid is plotted in Fig. 2.4. Decaying Exponential In general, an exponentially decaying quantity (Fig. 2.5) can be expressed as a = A e –t/t where a = instantaneous value A = amplitude or maximum value e = base of natural logarithms = 2.718 … t = time constant in seconds t = time in seconds The current of a discharging capacitor can be approxi￾mated by a decaying exponential function of time. Time Constant Since the exponential factor only approaches zero as t increases without limit, such functions theoretically last forever. In the same sense, all radioactive disintegrations last forever. In the case of an exponentially decaying current, it is convenient to use the value of time that makes the exponent –1. When t = t = the time constant, the value of the exponential factor is In other words, after a time equal to the time constant, the exponential factor is reduced to approximatly 37% of its initial value. FIGURE 2.4 The sinusoid A cos(wt + q) with –p/2 < q < 0. p + 2q 2w p - 2q 2w 3p - 2q 2w 3p + 2q 2w q w A cos(wt + q) 0 –A A t FIGURE 2.5 The decaying exponential. e e e - - t = = = = t 1 1 1 2 718 0 368 .