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Electric Circuits

James W. Nilsson, Susan A. Riedel

Chapter 8

Natural and Step Responses of RLC Circuits - all with Video Answers

Educators

Chapter Questions

05:23

Problem 1

The resistance, inductance, and capacitance in a Parallel $R L C$ circuit are $5000 \Omega, 1.25 \mathrm{H},$ and $8 \mathrm{nF}$ respectively.
a) Calculate the roots of the characteristic equation that describe the voltage response of the circuit.
b) Will the response be over, under, or critically damped? c) What value of $R$ will yield a damped frequency of $6 \mathrm{krad} / \mathrm{s} ?$
What are the roots of the characteristic equation for the value of $R$ found in $(\circ) ?$
e) What value of $R$ will result in a critically damped response?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:10

Problem 2

Suppose the capacitor in the circuit shown in Fig. 8.1 has a value of $0.05 \mu \mathrm{F}$ and an initial voltage of $15 \mathrm{V}$. The initial current in the inductor is zero. The resulting voltage response for $t \geq 0$ is
\[
v(t)=-5 e^{-5000 t}+20 e^{-20,000 t} \mathrm{V}
\]
a) Determine the numerical values of $R, L, \alpha$
and $\omega_{0}$
b) Calculate $i_{R}(t), i_{L}(t),$ and $i_{C}(t)$ for $t \geq 0^{+}$

Vishal Gupta
Vishal Gupta
Numerade Educator
02:18

Problem 3

The natural voltage response of the circuit in Fig. 8.1 is $v(t)=125 e^{-4000 t}(\cos 3000 t-2 \sin 3000 t) \mathrm{V}, \quad t \geq 0$
when the capacitor is $50 \mathrm{nF}$. Find $(\mathrm{a}) L ;(\mathrm{b}) R$
(c) $V_{0} ;(\mathrm{d}) I_{0} ;$ and $(\mathrm{e}) i_{L}(t)$

Salamat Ali
Salamat Ali
Numerade Educator
02:52

Problem 4

The voltage response for the circuit in Fig. 8.1 is known to be $$v(t)=D_{1} t e^{-4000 t}+D_{2} e^{-4000 t}, \quad t \geq 0$$ The initial current in the inductor $\left(I_{0}\right)$ is $5 \mathrm{mA}$, and the initial voltage on the capacitor $\left(V_{0}\right)$ is 25 V. The inductor has an inductance of $5 \mathrm{H}$
a) Find the value of $R, C, D_{1},$ and $D_{2}$
b) Find $i_{C}(t)$ for $t \geq 0^{+}$

James Chok
James Chok
Numerade Educator
02:52

Problem 5

The initial value of the voltage $v$ in the circuit in Fig. 8.1 is zero, and the initial value of the capacitor current, $i_{c}\left(0^{+}\right),$ is $15 \mathrm{mA}$. The expression for the capacitor current is known to be $$i_{c}(t)=A_{1} e^{-160 t}+A_{2} e^{-40 t}, \quad t \geq 0^{+}$$ when $R$ is $200 \Omega$. Find
a) the value of $\alpha, \omega_{0}, L, C, A_{1},$ and $A_{2}$ $$\left(\text { Hint: } \frac{d i_{c}(0)}{d t}=-\frac{d i_{L}(0)}{d t}-\frac{d i_{R}(0)}{d t}=\frac{v(0)}{L}-\frac{1}{R} \frac{i_{C}\left(0^{+}\right)}{C}\right)$$ b) the expression for $v(t), t \geq 0$
c) the expression for $i_{R}(t) \geq 0$
d) the expression for $i_{L}(t) \geq 0$

James Chok
James Chok
Numerade Educator
02:06

Problem 6

The circuit elements in the circuit in Fig. 8.1 are $R=2 \mathrm{k} \Omega, C=10 \mathrm{nF},$ and $L=250 \mathrm{mH}$. The ini-
tial inductor current is $-30 \mathrm{mA}$, and the initial capacitor voltage is $90 \mathrm{V}$
a) Calculate the initial current in each branch of the circuit.
b) $\operatorname{Find} v(t)$ for $t \geq 0$
c) Find $i_{L}(t)$ for $t \geq 0$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:02

Problem 7

The resistance in Problem 8.6 is increased $2.5 \mathrm{k} \Omega .$ Find the expression for $v(t)$ for $t \geq 0$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
02:08

Problem 8

The resistance in Problem 8.6 is increased $12,500 / 3 \Omega$. Find the expression for $v(t)$ for $t \geq 0$.

Rachel B.
Rachel B.
Numerade Educator
01:54

Problem 9

The natural response for the circuit shown in Fig.8. is known to be
\[
v(t)=-12\left(e^{-200 t}+e^{-1800 t}\right) \mathrm{V}, \quad t \geq 0
\]
If $C=18 \mu \mathrm{F},$ find $i_{L}\left(0^{+}\right)$ in milliamperes

Kajal Gautam
Kajal Gautam
Numerade Educator
05:23

Problem 10

In the circuit shown in Fig. $8.1,$ a $5 \mathrm{H}$ inductor is shunted by a 8 nF capacitor, the resisto $R$ is adjusted for critical damping, $V_{0}=-25$ $I_{0}=-1 \mathrm{mA}$
a) Calculate the numerical value of $R$
b) Calculate $v(t)$ for $t \geq 0$
c) Find $v(t)$ when $i_{C}(t)=0$
d) What percentage of the initially stored energy remains stored in the circuit at the instant $i_{C}(t)$ is $0 ?$

Keshav Singh
Keshav Singh
Numerade Educator
04:17

Problem 11

In the circuit in Fig. $8.1, R=2 \Omega, L=0.4 \mathrm{H}$ $C=0.25 \mathrm{F}, V_{0}=0 \mathrm{V},$ and $I_{0}=-3 \mathrm{A}$
a) Find $v(t)$ for $t \geq 0$
b) Find the first three values of $t$ for which $d v / d$ is zero. Let these values of $t$ be denoted $t_{1}, t_{2}$ and $t_{3}$
c) Show that $t_{3}-t_{1}=T_{d}$
d) Show that $t_{2}-t_{1}=T_{d} / 2$
e) Calculate $v\left(t_{1}\right), v\left(t_{2}\right),$ and $v\left(t_{3}\right)$
f) Sketch $v(t)$ versus $t$ for $0 \leq t \leq t_{2}$

Kajal Gautam
Kajal Gautam
Numerade Educator
01:54

Problem 12

a) Find $v(t)$ for $t \geq 0$ in the circuit in Problem 8 . if the $2 \Omega$ resistor is removed from the circuit.
b) Calculate the frequency of $v(t)$ in hertz.
c) Calculate the maximum amplitude of $v(t)$ in vols

Sheh Lit Chang
Sheh Lit Chang
University of Washington
05:28

Problem 13

Assume the underdamped voltage response circuit in Fig. 8.1 is written as $$v(t)=\left(A_{1}+A_{2}\right) e^{-\alpha t} \cos \omega_{d} t+j\left(A_{1}-A_{2}\right) e^{-\alpha t} \sin \omega$$ The initial value of the inductor current is 10 the initial value of the capacitor voltage is $V_{0}$. Show that $A_{2}$ is the conjugate of $A_{1}$. (Hint: Use the sam Process as outlined in the text to find $A_{1}$ and $A_{2}$.

Narayan Hari
Narayan Hari
Numerade Educator
04:37

Problem 14

Show that the results obtained from Problem 8.13 that is, the expressions for $A_{1}$ and $A_{2}-$ are consis tent with Eqs. 8.30 and 8.31 in the text.

Ajay Singhal
Ajay Singhal
Numerade Educator
04:48

Problem 15

The resistor in the circuit in Example 8.4 is changed
$104000 / \sqrt{2} \mathrm{S} 2$
a) Find the numerical expression for $v(t)$ when $t \geq 0$
b) Plot
$v(t)$ versus $t$ for the time interval $0 \leq t \leq 7 \mathrm{ms}$. Compare this response with the one in Example $8.4 \quad(R=20 \mathrm{k} \Omega)$ and Example $8.5(R=4 \mathrm{k} \Omega) .$ In particular, compare peak values of $v(t)$ and the times when these peak values occur.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:35

Problem 16

The switch in the circuit of Fig. $P 8.16$ has been in position a for a long time. At $t=0$ the switch moves instantancously to position b. Find $v_{o}(t)$ for $t \geq 0$.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:29

Problem 17

The capacitor in the circuit of Fig. $P 8.16$ is decreased to $1 \mathrm{nF}$ and the inductor is increased to 10 H. Find $v_{c}(t)$ for $t \geq 0$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:05

Problem 18

The capacitor in the circuit of Fig. $\mathrm{P} 8.16$ is decreased to $800 \mathrm{pF}$ and the inductor is increased to $12.5 \mathrm{H}$. Find $v_{o}(t)$ for $t \geq 0$

Keshav Singh
Keshav Singh
Numerade Educator
02:16

Problem 19

The two switches in the circuit seen in Fig. P8.19 operate synchronously. When switch 1 is in position a, switch 2 is in position d. When switch 1 moves to position b, switch 2 moves to position c. Switch 1 has been in position a for a long time. At $t=0$, the switches move to their alternate positions. Find $v_{o}(t)$ for $t \geq 0$.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:06

Problem 20

The resistor in the circuit of Fig. $P 8.19$ is increased from $1.6 \mathrm{k} \Omega$ to $2 \mathrm{k} \Omega$ and the inductor is decreased from 1 H to 640 mH. Find $v_{o}(t)$ for $t>0$ .

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:06

Problem 21

The resistor in the circuit of Fig. $\mathrm{P} 8.19$ is decreased from $1.6 \mathrm{k} \Omega$ to $800 \Omega,$ and the inductor is decreased from 1 H to $160 \mathrm{mH}$. Find $v_{o}(t)$ for $t \geq 0$.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
05:34

Problem 22

For the circuit in Example $8.6,$ find, for $t \geq 0$
(a) $v(t) ;$ (b) $i_{R}(t) ;$ and
(c) $i_{c}(t)$.

Kajal Gautam
Kajal Gautam
Numerade Educator
05:34

Problem 23

For the circuit in Example $8.7,$ find, for $t \geq 0$
(a) $v(t)$ and $(b) i_{c}(t)$.

Kajal Gautam
Kajal Gautam
Numerade Educator
04:12

Problem 24

For the circuit in Example $8.8,$ find $v(t)$ for $t \geq 0$.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:29

Problem 25

Assume that at the instant the $15 \mathrm{mA}$ dc current source is applied to the circuit in Fig. $\mathrm{P} 8.25$, the initial current in the $20 \mathrm{H}$ inductor is $-30 \mathrm{mA}$, and the initial voltage on the capacitor is $60 \mathrm{V}$ (positive at the upper terminal). Find the expression for $i_{L}(t)$ for $t \geq 0$ if $R$ equals $800 \Omega$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:48

Problem 26

The resistance in the circuit in Fig. P8.25 is changed to $1250 \Omega$. Find $i_{L}(t)$ for $t \geq 0$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:48

Problem 27

The resistance in the circuit in Fig. P8.25 is changed to $1000 \Omega$. Find $i_{L}(t)$ for $t \geq 0$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:06

Problem 28

The switch in the circuit in Fig. P8.28 has been open for a long time before closing at $t=0$. Find $v_{0}(t)$ for $t \geq 0$.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:11

Problem 29

a) For the circuit in Fig. $\mathrm{P} 8.28$, find $t_{0}$ for $t \geq 0$
b) Show that your solution for $i_{0}$ is consistent with the solution for $v_{c}$ in Problem 8.28

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:06

Problem 30

There is no chergy stored in the circuit in Fig. P8.30 when the switch is closed at $t=0 .$ Find $v_{o}(t)$ for
$t \geq 0$

Kajal Gautam
Kajal Gautam
Numerade Educator
04:17

Problem 31

a) For the circuit in Fig. $\mathrm{P} 8.30,$ find $i_{c}$ for $t \geq 0$
b) Show that your solution for $i_{o}$ is consistent with the solution for $v_{o}$ in Problem 8.30

Kajal Gautam
Kajal Gautam
Numerade Educator
03:06

Problem 32

The switch in the circuit in Fig. $P 8.32$ has been open a long time before closing at $t=0 .$ At the time the switch closes, the capacitor has no stored energy. Find $v_{o}$ for $t \geq 0$.

Kajal Gautam
Kajal Gautam
Numerade Educator
03:06

Problem 33

The switch in the circuit in Fig, $P 8.33$ has been open a long time before closing at $t=0 .$ Find
a) $v_{o}(t)$ for $t \geq 0^{+}$
b) $i_{L}(t)$ for $t \geq 0$.

Kajal Gautam
Kajal Gautam
Numerade Educator
04:32

Problem 34

Use the circuit in Fig. $\mathrm{P} 8.33$
a) Find the total energy delivered to the inductor
b) Find the total energy delivered to the equivalent resistor.
c) Find the total energy delivered to the cafacitor
d) Find the total energy delivered by the equiva lent current source.
e) Check the results of parts (a) through (d) against the conservation of energy princitic

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:06

Problem 35

The switch in the circuit in Fig. $\mathrm{P} 8.35$ has bech open a long time before closing at $t=0 .$ Find $i_{L}(1)$ for $t \geq 0$.

Kajal Gautam
Kajal Gautam
Numerade Educator
04:30

Problem 36

Switches 1 and 2 in the circuit in Fig. $P 8.36$ are $\$$ chronized. When switch 1 is opened, switch 2 close and vice versa. Switch 1 has been open a long tins before closing at $t=0 .$ Find $i_{L}(t)$ for $t \geq 0$.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
04:21

Problem 37

The initial energy stored in the 50 nF capacitor in the circuit in Fig. P8.37 is $90 \mu$ J. The initial energy stored in the inductor is zero. The roots of the char. acteristic equation that describes the natural behavior of the current i are $-1000 \mathrm{s}^{-1}$ and $-4000 \mathrm{s}^{-1}$
a) Find the numerical values of $R$ and $L$
b) Find the numerical values of $i(0)$ and $d i(0) / d t$ immediately after the switch has been closed
c) Find $i(t)$ for $t \geq 0$
d) How many microseconds after the switch closes does the current reach its maximum value?
e) What is the maximum value of i in milliamperes?
f) Find $v_{L}(t)$ for $t \geq 0$

Keshav Singh
Keshav Singh
Numerade Educator
05:12

Problem 38

The current in the circuit in Fig. 8.3 is known to be $$i=B_{1} e^{-800 t} \cos 600 t+B_{2} e^{-800 t} \sin 600 t, \quad t \geq 0$$ The capacitor has a value of $500 \mu \mathrm{F}$; the initial value of the current is zero; and the initial voltage on the capacitor is 12 V. Find the values of $R, L, B_{1},$ and $B_{2}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:01

Problem 39

Find the voltage across the $500 \mu \mathrm{F}$ capacitor for the circuit described in Problem 8.38 . Assume the reference polarity for the capacitor voltage is positive at the upper terminal.

Andrew Duncan
Andrew Duncan
Numerade Educator
03:06

Problem 40

The switch in the circuit shown in Fig. $P 8.40$ has been closed for a long time. The switch opens at $t=0 .$ Find
a) $i_{o}(t)$ for $t \geq 0$
b) $v_{o}(t)$ for $t=0$

Kajal Gautam
Kajal Gautam
Numerade Educator
03:18

Problem 41

In the circuit in Fig. $\mathrm{P} 8.41$, the resistor is adjusted for critical damping. The initial capacitor voltage is $90 \mathrm{V},$ and the initial inductor current is $24 \mathrm{mA}$
a) Find the numerical value of $R$
b) Find the numerical values of $i$ and $d i / d t$ immediately after the switch is closed.
c) Find $v_{C}(t)$ for $t \geq 0$

Zachary Warner
Zachary Warner
Numerade Educator
03:35

Problem 42

The switch in the circuit in Fig. $\mathrm{P} 8.42$ has been in position a for a long time. At $t=0,$ the switch moves instantaneously to position b.
a) What is the initial value of $v_{a} ?$
b) What is the initial value of $d v_{a} / d t ?$
c) What is the numerical expression for $v_{a}(t)$ for $t \geq 0 ?$

Kajal Gautam
Kajal Gautam
Numerade Educator
03:35

Problem 43

The make-before-break switch in the circuit shown in Fig. $P 8.43$ has been in position a for a long time. At $t=0,$ the switch is moved instantaneously to position b. Find $i(t)$ for $t \geq 0$

Kajal Gautam
Kajal Gautam
Numerade Educator
03:06

Problem 44

The switch in the circuit shown in Fig. P8.44 has been closed for a long time. The switch opens at $t=0 .$ Find $v_{o}(t)$ for $t \geq 0$.

Kajal Gautam
Kajal Gautam
Numerade Educator
01:56

Problem 45

The initial energy stored in the circuit in Fig. P8.45 is zero. Find $v_{o}(t)$ for $t \geq 0$.

Arpit Gupta
Arpit Gupta
Numerade Educator
02:19

Problem 46

The capacitor in the circuit shown in Fig. P8.45 is changed to 100 nF. The initial energy stored is still zero. Find $v_{o}(t)$ for $t \geq 0$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:36

Problem 47

The capacitor in the circuit shown in Fig. P8.45 is changed to $156.25 \mathrm{nF}$, The initial energy stored is still zero. Find $v_{c}(t)$ for $t \geq 0$.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:19

Problem 48

The switch in the circuit shown in Fig. P8.48 has been closed for a long time before it is opened at $t=0 .$ Assume that the circuit parameters are such that the response is underdamped
a) Derive the expression for $v_{o}(t)$ as a function $v_{g}, \alpha, \omega_{d}, C,$ and $R$ for $t \geq 0$
b) Derive the expression for the value of $t$ when the magnitude of $v_{o}$ is maximum.

Dominador Tan
Dominador Tan
Numerade Educator
06:04

Problem 49

The circuit parameters in the circuit of $\operatorname{Eg} .$ P8. .8 are $\quad R=120 \Omega, \quad L=5 \mathrm{mH}, \quad C=500 \mathrm{nF},$ and
$v_{g}=-600 \mathrm{V}$
a) Express $v_{o}(t)$ numerically for $t \geq 0$
b) How many microseconds after the switch opens is the inductor voltage maximum?
c) What is the maximum value of the inductor voltage?
d) Repeat (a)-(c) with $R$ reduced to $12 \Omega$

Vishal Gupta
Vishal Gupta
Numerade Educator
07:17

Problem 50

The circuit shown in Fig. P8.50 has been in opera tion for a long time. At $t=0$, the source voltage suddenly drops to 100 V. Find $v_{o}(t)$ for $t \geq 0$ .

Kajal Gautam
Kajal Gautam
Numerade Educator
03:35

Problem 51

The switch in the circuit of Fig. P8.51 has been in position a for a long time, At $t=0$ the swict moves instantaneously to position b. Find
a) $v_{o}\left(0^{+}\right)$
b) $d v_{o}\left(0^{t}\right) / d t$
c) $v_{o}(t)$ for $t \geq 0$

Kajal Gautam
Kajal Gautam
Numerade Educator
02:34

Problem 52

The two switches in the circuit seen in Fig. $P 8.52$ operate synchronously. When switch 1 is in position a, switch 2 is closed. When switch 1 is in position $b$ switch 2 is open. Switch 1 has been in position a for a long time. At $t=0$, it moves instantaneously to position b. Find $v_{c}(t)$ for $t \geq 0$

Shoukat Ali
Shoukat Ali
Other Schools
08:30

Problem 53

Assume that the capacitor voltage in the circuit of Fig. 8.15 is underdamped. Also assume that no energy is stored in the circuit elements when the switch is closed.
a) Show that $d v_{C} / d t=\left(\omega_{0}^{2} / \omega_{d}\right) V e^{-\alpha t} \sin \omega_{d} t$
b) Show that $d v_{C} / d t=0$ when $t=n \pi / \omega_{d},$ where $n=0,1,2, \ldots$
c) Let $t_{n}=n \pi / \omega_{d},$ and show that $v_{C}\left(t_{n}\right)$ $=V-V(-1)^{n} e^{-\alpha n \pi / \omega_{d}}$
d) Show that $$\alpha=\frac{1}{T_{d}} \ln \frac{v_{C}\left(t_{1}\right)-V}{v_{C}\left(t_{3}\right)-V}$$ $$\text { where } T_{d}=t_{3}-t_{1}$$

Vishal Gupta
Vishal Gupta
Numerade Educator
10:52

Problem 54

The voltage across a $200 \mathrm{nF}$ capacitor in the circuit of Fig. 8.15 is described as follows: After the switch has been closed for several seconds, the voltage is constant at $50 \mathrm{V}$. The first time the voltage exceeds $50 \mathrm{V},$ it reaches a peak of $63.505 \mathrm{V}$. This occurs $\pi / 12 \mathrm{ms}$ after the switch has been closed. The second time the voltage exceeds $50 \mathrm{V}$, it reaches a peak of $50.985 \mathrm{V}$. This second peak occurs $\pi / 4 \mathrm{ms}$ after the switch has been closed. At the time when the switch is closed, there is no energy stored in either the capacitor or the inductor. Find the numerical values of $R$ and $L$. (Hint: Work Problem 8.53 first.)

Vishal Gupta
Vishal Gupta
Numerade Educator
01:07

Problem 55

Show that, if no energy is stored in the circuit shown in Fig. 8.19 at the instant $v_{g}$ jumps in value, then $d v_{o} / d t$ equals zero at $t=0$.

Amit Srivastava
Amit Srivastava
Numerade Educator
04:17

Problem 56

a) Find the equation for $v_{o}(t)$ for $0 \leq t \leq t_{\text {sat }}$ in the circuit shown in Fig. 8.19 if $v_{o 1}(0)=5 \mathrm{V}$ and $v_{o}(0)=8 \mathrm{V}$
b) How long does the circuit take to reach saturation?

Kajal Gautam
Kajal Gautam
Numerade Educator
07:55

Problem 57

a) Rework Example 8.14 with feedback resistors $R_{1}$ and $R_{2}$ removed.
b) Rework Example 8.14 with $v_{o 1}(0)=-2 \mathrm{V}$ and $v_{o}(0)=4 \mathrm{V}$

Netra Sharma
Netra Sharma
University of Wisconsin - Milwaukee
03:34

Problem 58

The voltage signal of Fig. $P 8.58(a)$ is applied to the cascaded integrating amplifiers shown in Fig. $P 8.58(b) .$ There is no energy stored in the capacitors at the instant the signal is applied.
a) Derive the numerical expressions for $v_{o}(t)$ and $v_{o 1}(t)$ for the time intervals $0 \leq t \leq 0.2$ s and $0.2 \mathrm{s} \leq t \leq t_{s a t}$
b) Compute the value of $t_{s a l}$

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:45

Problem 59

The circuit in Fig. $P 8.58(b)$ is modified by adding a $250 \mathrm{k} \Omega$ resistor in parallel with the $2 \mu \mathrm{F}$ capacitor and a $250 \mathrm{k} \Omega$ resistor in parallel with the $4 \mu \mathrm{F}$ capacitor. As in Problem 8.58 , there is no energy stored in the capacitors at the time the signal is applied. Derive the numerical expressions for $v_{o}(t)$ and $v_{o 1}(t)$ for the time intervals $0 \leq t \leq 0.2$ s and
$t \geq 0.2 \mathrm{s}$

Dominador Tan
Dominador Tan
Numerade Educator
01:41

Problem 60

a) Derive the differential equation that relates the output voltage to the input voltage for the circuit shown in Fig. P8.60
b) Compare the result with Eq. 8.75 when $R_{1} C_{1}=R_{2} C_{2}=R C$ in Fig. 8.18
c) What is the advantage of the circuit shown in Fig. P8.60?

Narayan Hari
Narayan Hari
Numerade Educator
05:32

Problem 61

We now wish to illustrate how several op amp circuits can be interconnected to solve a differential equation.
a) Derive the differential equation for the spring. mass system shown in Fig. P8.61 (a). (See page $329 .$ ) Assume that the force exerted by the spring is directly proportional to the spring displacement, that the mass is constant, and that the frictional force is directly proportional to the velocity of the moving mass,
b) Rewrite the differential equation derived in $(a)$ so that the highest order derivative is expressed as a function of all the other terms in the equa tion. Now assume that a voltage equal to $d^{2} x / d t^{2}$ is available and by successive integrations gen. erates $d x / d t$ and $x$. We can synthesize the coeffi. cients in the equations by scaling amplifiers, and we can combine the terms required to generate $d^{2} x / d t^{2}$ by using a summing amplifier. With these ideas in mind, analyze the interconnection shown in Fig. P8.61(b). In particular, describe the purpose of each shaded area in the circuit and describe the signal at the points labeled C, D, E, and $F$, assuming the signal at A repre. sents $d^{2} x / d t^{2}$. Also discuss the parameters $R ; R_{1}$ $C_{1} ; R_{2}, C_{2} ; R_{3}, R_{4} ; R_{5}, R_{6} ;$ and $R_{7}, R_{8}$ in terms
of the coefficients in the differential equation.

Mike Gaerlan
Mike Gaerlan
Numerade Educator
01:23

Problem 62

a) Derive Eq. 8.92
b) Derive Eq. 8.93
c) Derive Eq. 8.97

Manik Pulyani
Manik Pulyani
Numerade Educator
03:11

Problem 63

Derive Eq. 8.99

Narayan Hari
Narayan Hari
Numerade Educator
01:29

Problem 64

a) Using the same numerical values used in the Practical Perspective example in the text, find the instant of time when the voltage across the capacitor is maximum.
b) Find the maximum value of $v_{c}$
c) Compare the values obtained in (a) and (b) with $t_{\max }$ and $v_{c}\left(t_{\max }\right)$

Dominador Tan
Dominador Tan
Numerade Educator
08:30

Problem 65

The values of the parameters in the circuit in Fig. 8.21 are $R=3 \Omega ; L=5 \mathrm{mH} ; C=0.25 \mu \mathrm{F}$
$V_{\mathrm{dc}}=12 \mathrm{V} ;$ and $a=50 .$ Assume the switch opens when the primary winding current is $4 \mathrm{A}$
a) How much energy is stored in the circuit at $t=0^{+} ?$
b) Assume the spark plug does not fire. What is the maximum voltage available at the spark plug?
c) What is the voltage across the capacitor when the voltage across the spark plug is at its $\mathrm{m}^{2}$ mum value?

Vishal Gupta
Vishal Gupta
Numerade Educator