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
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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
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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
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− 𝒕−𝒕𝟏
𝒕𝟏 − 𝒚𝒇 𝒆 𝑟
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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
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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.
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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?
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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.
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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?
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