ELECTRICAL ENGINEERING DEPARTMENT EE201: ELECTRIC CIRCUITS I SUMMER SESSION, 2020-2021 (203) WEEK 5 – LECTURE 24-25 CH. 7: RESPONSE OF FIRST-ORDER RL AND RC CIRCUITS COPYRIGHT © ALGARNI Wednesday , July 07, 2021 1 SUMMARY OF THE PREVIOUS LECTURE 7.4 A General Solution for Step and Natural Responses 𝒙 𝒕 = 𝒙𝒇 + 𝒙 − 𝒕−𝒕𝟎 𝒕𝟎 − 𝒙𝒇 𝒆 𝑟 𝑥 𝑡 : the unknown variable as a function of time 𝑥 𝑓: the final value of the variable. 𝑥 𝑡0 : the initial value of the variable 𝑟: time constant 𝑡0 : time of switching COPYRIGHT © ALGARNI Wednesday , July 07, 2021 2 SUMMARY OF THE PREVIOUS LECTURE 7.5 Sequential Switching • We can use the general form to find the unknow variable at first switching time 𝒕𝟎 𝒙 𝒕 = 𝒙𝒇 + 𝒙 • − 𝒕−𝒕𝟎 𝒕𝟎 − 𝒙𝒇 𝒆 𝑟 Then, at the next switching time 𝒕𝟏, we can also apply the general form again with initial value using the previous expression at 𝒕𝟏. 𝒚 𝒕 = 𝒚𝒇 + 𝒙 COPYRIGHT © ALGARNI − 𝒕−𝒕𝟏 𝒕𝟏 − 𝒚𝒇 𝒆 𝑟 Wednesday , July 07, 2021 3 CHAPTER 7: RESPONSE OF FIRST-ORDER RL AND RC CIRCUITS Chapter contents: 1. The Natural Response of an RL Circuit 2.The Natural Response of an RC Circuit 7.3 The Step Response of RL and RC Circuits 4. A General Solution for Step and Natural Responses 5. Sequential Switching 6. Unbounded Response COPYRIGHT © ALGARNI Wednesday , July 07, 2021 4 EXAMPLE 1: ANALYZING AN RL CIRCUIT THAT HAS SEQUENTIAL SWITCHING • The two switches in the circuit shown in Fig. 7.31 have been closed for a long time. At t = 0 switch 1 is opened. Then, 35 ms later, switch 2 is opened. a) b) c) Find 𝑖𝐿 𝑡 for 0 ≤ 𝑡 ≤ 35𝑚𝑠. Find 𝑖𝐿 𝑡 for 𝑡 ≥ 35𝑚𝑠. What percentage of the initial energy stored in the 150 mH inductor is dissipated in the 18Ω resistor? d) Repeat (c) for the 3Ω resistor. e) Repeat (c) for the 6Ω resistor. COPYRIGHT © ALGARNI Wednesday , July 07, 2021 5 EXAMPLE 2: ANALYZING AN RC CIRCUIT THAT HAS SEQUENTIAL SWITCHING • The uncharged capacitor in the circuit shown in Fig. 7.35 is initially switched to terminal a of the three-position switch. At t = 0 the switch is moved to position b, where it remains for 15 ms. After the 15 ms delay, the switch is moved to position c, where it remains indefinitely. a) b) c) d) Derive the numerical expression for the voltage across the capacitor. Plot the capacitor voltage versus time. When will the voltage on the capacitor equal 200 V? When will the voltage on the capacitor equal 350 V? COPYRIGHT © ALGARNI Wednesday , July 07, 2021 6 7.6 UNBOUNDED RESPONSE • A circuit response may grow, rather than decay, exponentially with time. This type of response called an unbounded response, is possible if the circuit contains dependent sources. • In that case, the Thévenin equivalent resistance with respect to the terminals of either an inductor or a capacitor may be negative. • This negative resistance generates a negative time constant, and the resulting currents and voltages increase without limit. COPYRIGHT © ALGARNI Wednesday , July 07, 2021 7 EXAMPLE 3: FINDING THE UNBOUNDED RESPONSE IN AN RC CIRCUIT a) b) When the switch is closed in the circuit shown in Fig. 7.37, the voltage on the capacitor is 10 V. Find the expression for 𝑣0 for 𝑡 ≥ 0. Assume that the capacitor short-circuits when its terminal voltage reaches 150 V. How many milliseconds elapse before the capacitor short circuits? COPYRIGHT © ALGARNI Wednesday , July 07, 2021 8