zen barcode ssrs WAVEFORMS AND SIGNALS in Software

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WAVEFORMS AND SIGNALS
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Fig. 6-12 0.3682 0:135, respectively, also belong to the curve. a rather good approximation (see Fig. 6-12). Using the preceding indicators, the curve may be drawn with
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EXAMPLE 6.21 (a) Show that the rate of change with respect to time of an exponential function v Aest is at any moment proportional to the value of the function at that moment. (b) Show that any linear combination of an exponential function and its n derivatives is proportional to the function itself. Find the coe cient of proportionality. (a) The rate of change of a function is equal to the derivative of the function, which, for the given exponential function, is dv sAest sv dt (b) Using the result of (a) we get d nv sn Aest sn v dtn a0 v a1 where dv d nv an n a0 a1 s an sn v Hv dt dt H a0 a1 s an sn 35 (36)
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Specifying and Plotting f t Ae at B We often encounter the function f t Ae at B This function is completely speci ed by the three numbers A, B, and a de ned as A initial value final value or, in another form, Initial value f 0 A B Final value f 1 B Time constant 1=a B final value a inverse of the time constant 37
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EXAMPLE 6.22 Find a function v t which decays exponentially from 5 V at t 0 to 1 V at t 1 with a time constant of 3 s. Plot v t using the technique of Example 6.20. From (37) we have v t Ae t= B. Now v 0 A B 5, v 1 B 1, A 4, and  3. Thus v t 4e t=3 1 The preceding result can be generalized in the following form: v t initial value final value e t= final value The plot is shown in Fig. 6-13.
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WAVEFORMS AND SIGNALS
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Fig. 6-13 EXAMPLE 6.23 The voltage v V0 e jtj= ,  > 0, is connected to a capacitor. Find the current i in the capacitor. Sketch v and i for V0 10 V, C 1 mF, and  1 ms. Using i C dv=dt, for t < 0; for t > 0; v V0 et= v V0 e t= and and i I0 et= i I0 e t= Graphs of v and i are shown in Figs. 6-14(a) and
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where I0 CV0 =. For V0 10 V, C 1 mF, and  10 3 s, we get I0 10 mA. (b), respectively.
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Fig. 6-14
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DAMPED SINUSOIDS A damped sinusoid, with its amplitude decaying exponentially has the form v t Ae at cos !t  38
This function will be discussed in more detail in 8.
EXAMPLE 6.24 The current i I0 e at cos !t passes through a series RL circuit. (a) Find vRL , the voltage across this combination. (b) Compute vRL for I0 3 A, a 2, ! 40 rad/s, R 5  and L 0:1 H. Sketch i as a function of time.
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WAVEFORMS AND SIGNALS
(a) We have vR Ri RI0 e at cos !t di vL L LI0 e at a cos !t ! sin !t dt vRL vR vL I0 e at R La cos !t L! sin !t V0 e at cos !t  q V0 I0 R La 2 L2 !2 and  tan 1 L!= R La
where
(39)
(b) Substituting the given data into (39), V0 18:75 V and  39:88. Current i and voltage vRL are then given by i 3e 2t cos 40t The current i is graphed in Fig. 6-15. and vRL 18:75e 2t cos 40t 39:88
Fig. 6-15
RANDOM SIGNALS
So far we have dealt with signals which are completely speci ed. For example, the values of a sinusoidal waveform, such as the line voltage, can be determined for all times if its amplitude, frequency, and phase are known. Such signals are called deterministic. There exists another class of signals which can be speci ed only partly through their time averages, such as their mean, rms value, and frequency range. These are called random signals. Random signals can carry information and should not be mistaken with noise, which normally corrupts the information contents of the signal. The voltage recorded at the terminals of a microphone due to speech utterance and the signals picked up by an antenna tuned to a radio or TV station are examples of random signals. The future course and values of such signals can be predicted only in average and not precisely. Other examples of random signals are the binary waveforms in digital computers, image intensities over the area of a picture, and the speech or music which modulates the amplitude of carrier waves in an AM system. It may not seem useful to discuss signals whose values are speci ed only in average. However, through harmonic analysis we can still nd much about the average e ect of such signals in electric circuits.
EXAMPLE 6.25 Samples from a random signal x t are recorded every 1 ms and designated by x n . imate the mean and rms values of x t from samples given in Table 6-2. Approx-
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