An exponential signal is an energy signal, as illustrated by an example in the previous page. The base of natural logarithm is the irrational number called e and it is equal to 2.718281.., and an exponential signal occurs repeatedly in electrical engineering and mathematics and is defined to be:
In equation (5.1), a is the neper frequency and it is preferable to express equation (5.1) as follows, where t, the time constant with the unit of second, equals 1/a. Then
The Laplace transform expression of the exponential function has a pole at s = - a. It can be seen that a pole on the negative real axis of s-plane corresponds to an exponential function. In this case, the multiplicity of pole at s = - a is unity and it needs to be mentioned that a practical system generating a signal has no poles in the right-half of s-plane. The poles may lie on either the jw-axis or on the left-half of s-plane. The plot of equation (5.2) is obtained with the help of a Matlab program and is shown in Fig. 30. It is seen that
Fig. 30: An Exponential Function
An exponentially decaying waveform decreases to less than 1% of its original value when the time elapsed equals five time constants. The plot of derivative is shown in Fig. 31.
Fig. 31: Derivative of an Exponential Function
Figure 32 illustrates one of the properties of an exponential waveform. In Fig. 32, f(t) is plotted as a function of (t/t). If a tangent to the exponential curve is drawn at t = 0, the tangent intersects the x-axis at t = t, after a time interval equaling its time constant. If a tangent is drawn at t = t, then the tangent intersects the x-axis at t = 2t, again after a time interval equaling its time constant. The tangent drawn at any instant intersects the x-axis after a time interval equaling its time constant. This aspect can be understood from equation (5.3).
This page has presented what an exponential signal is. The next page is on periodic signals.