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Electric Circuits, 9th Edition P25. Designed for use in a one or two-semester Introductory Circuit Analysis or Circuit Theory Course taught in Electrical or Computer Engineering Departments. Electric Circuits 9/e is the most widely used introductory circuits textbook of the past 25 years. As this book has evolved over the years to meet the changing learning styles of students, importantly, the underlying teaching approaches and philosophies remain unchanged. | 216 Response of First-Order RL and RC Circuits We derive the power dissipated in the resistor from any of the following expressions n 1 2 p vi p rR or p 7.11 R Whichever form is used the resulting expression can be reduced to p llRe 2 R L t t 0 . 7.12 The energy delivered to the resistor during any interval of time after the switch has been opened is w pdx llRe 2WL xdx Jo Jo - e 2WL lZ 1 - e 2 R L t . t 0. 7.13 Note from Eq. 7.13 that as t becomes infinite the energy dissipated in the resistor approaches the initial energy stored in the inductor. The Significance of the Time Constant The expressions for i t Eq. 7.7 and v t Eq. 7.8 include a term of the form The coefficient of t namely R L determines the rate at which the current or voltage approaches zero. The reciprocal of this ratio is the time constant of the circuit denoted Time constant for RL circuit . L t time constant . 7.14 Using the time-constant concept we write the expressions for current voltage power and energy as Zoe 7 t 0 7-15 u 0 lQRe T t 0 7.16 p ïlRe 2 lr t 0 7.17 w - e 2 7 t 0. 7.18 The time constant is an important parameter for first-order circuits so mentioning several of its characteristics is worthwhile. First it is convenient to think of the time elapsed after switching in terms of integral multiples of r. Thus one time constant after the inductor has begun to release its stored energy to the resistor the current has been reduced to e-1 or approximately 0.37 of its initial value. 7.1 The Natural Response of an RL Circuit 217 Table 7.1 gives the value of e f T for integral multiples of r from 1 to 10. Note that when the elapsed time exceeds five time constants the current is less than 1 of its initial value. Thus we sometimes say that five time constants after switching has occurred the currents and voltages have for most practical purposes reached their final values. For single time-constant circuits first-order circuits with 1 accuracy the phrase a long time implies that five or more time
