Fundamental Electrical and Electronic Principles, Third Edition

D.C. Transients 251Having confirmed the initial and final values for the transients, we nowneed to consider how they vary, with time, between these limits. It hasalready been stated that the variations will be non-linear (i.e. nota straight line graph). In fact the variations follow an exponential law.Any quantity that varies in an exponential fashion will have a graphlike that shown in Fig. 8.2(a) if it increases with time, and as inFig. 8.2(b) for a decreasing function.XXinitial rate of changeXX 00.632initial rate of change0.3680τ2τ 3τ 4τ 5τ t(s)(a)0τ2τ 3τ 4τ 5τ t(s)(b)Fig. 8.2In Fig. 8.2(a) , X represents the final steady state value of the variable x,and in Fig. 8.2(b) , X 0 represents the initial value of x . In each case thestraight line (tangent to the curve at time t 0) indicates the initial rateof change of x . The time interval shown as τ shown on both graphs isknown as the time constant, which is defined as follows:The time constant is the time that it would take the variable to reach itsfinal steady state if it continued to change at its initial rate.From the above Figures it can be seen that for an increasingexponential function, the variable will reach 63.2% of its final valueafter one time constant, and for a decreasing function it will fall to36.8% of its initial value after τ seconds.Note: Considering any point on the graph, it would take one timeconstant for the variable to reach its final steady value if it continued tochange at the same rate as at that point. Thus an exponential graph maybe considered as being formed from an infinite number of tangents,each of which represents the slope at a particular instant in time. Thisis illustrated in Fig. 8.3 .Also, theoretically, an exponential function can never actually reachits final steady state. However, for practical purposes it is assumedthat the final steady state is achieved after 5 time constants. This isjustifiable since the variable will be within 0.67% of the final valueafter 5 τ seconds. So for Fig. 8.2(a) , after 5 τ seconds, x 0.9973 X.