CHAP. 7]

FIRST-ORDER CIRCUITS

133

Fig. 7-8

that is, at t ¼  the function is 36.8 percent of the initial value. It may also be said that the function has undergone 63.2 percent of the change from f ð0þ Þ to f ð1Þ. At t ¼ 5, the function has the value 0.0067A, which is less than 1 percent of the initial value. From a practical standpoint, the transient is often regarded as over after t ¼ 5. The tangent to the exponential curve at t ¼ 0þ can be used to estimate the time constant. In fact, since slope ¼ f 0 ð0þ Þ ¼ 

A 

the tangent line must cut the horizontal axis at t ¼  (see Fig. 7-9). More generally, the tangent at t ¼ t0 has horizontal intercept t0 þ . Thus, if the two values f ðt0 Þ and f 0 ðt0 Þ are known, the entire curve can be constructed.

Fig. 7-9

At times a transient is only partially displayed (on chart paper or on the face of an oscilloscope), and the simultaneous values of function and slope needed in the preceding method are not available. In that case, any pair of data points, perhaps read from instruments, may be used to ﬁnd the equation of the transient. Thus, referring to Fig. 7-10, f1 ¼ Aet1 =

f2 ¼ Aet2 =

which may be solved simultaneously to give ¼ and then A in terms of  and either f1 or f2 .

t2  t1 ln f1  ln f2

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An