The complete response is the sum of the natural response and the forced responseX = Xn+XfNatural response of a first-order circuitNatural response of a second-order circuitCASEOverdampedCritically dampedUnderdampedazS1 =FORCING FUNCTIONKKtK₁²K sin cotKe atForced response of a first-order, or a second-order circuit+ A₁NATURAL FREQUENCIES$1,52 = -α± √√² - 0²$1,$₂=-αS1, S2 = -x±j√√/0²-a² -α±jooddxdtd²xdt²Solution of the Second-Order Differential Equationd²xx(t) = xn(t) + xf (t)a2dt²+ a。x = f(t)+ A1dxdt"xn(t)" = AestXn(t) = Ke-t/t+ a。x = 0-a₁ + √²-4a₂a0242-Xn(t) = A₁e³₁ª + A₂e³₂tASSUMED RESPONSEAAt + BAt² +Bt+CA sin cot + B cos cotAe-atNATURAL RESPONSE, XnAle+Azer(A₁+A₂t)e-at(A, cos coat+A₂ sin coat)exn(t) = ?(a₂s² + a₁s+ao) = 0-α₁ - √²-4a₂a0242NATURAL RESPONSE, XnA₁ est + A₂e21-α1(A₁+A₂t)e(A₁ cos wat+A₂ sin wat)ext • Problem 3. (c) The Circuit C shown is at the steady state, before the switch opens at time t = 2s.t = 2sI6 V 2μFVin mR₁U10-Brief catalog of the Operational Amplifier circuits(Based on the ideal operational amplifier)(a) Inverting amplifierR₁U20-W10ΚΩwwVout=R₂:UnwR₂wwR₁+R₁Ovout R₁wwR₁v(t) 12 VR₁R₁-R₁1+R₂₂+VoutR₁Vin(d) Summing amplifier(5) Determine the capacitor voltage, v(t), for t≥ 2sVin -(c) Circuit CR₁2R₁vout= (1 + R ₁) vin(b) Noninverting amplifierR₂K₁V10-WR₂K₂U20-WRJK3U30MR₂/(1-(K₁ + K₂ + K3))R(K4-1)Vin OI(c) Voltage follower (buffer amplifier)RpOvout VinOvout=K4(K₁v1 + K₂v2 + K303)(e) Noninverting summing amplifier

Answered: Image /qna-images/answer/4207ebe1-fe5b-4f28-b0ce-66d1ec80f6ad.jpg

dt² $1 a2 dt d²x dt² + a₁ dx dt "Xn(t)" = Aest + aox = 0 -a₁ + √²-4a₂ao 2a2 =n(t) = A₁e³₁t + A₂eszt xn(t) = ? (a₂s² + a₁s + α₁) = 0 $2 a1 - √a²-4a2a0 2a2 NATURAL RESPONSE, Xn Alest + A₂e21 (A₁+A₂t)e (A₁ cos wat+A₂ sin wat)e -al

dt² $1 a2 dt d²x dt² + a₁ dx dt "Xn(t)" = Aest + aox = 0 -a₁ + √²-4a₂ao 2a2 =n(t) = A₁e³₁t + A₂eszt xn(t) = ? (a₂s² + a₁s + α₁) = 0 $2 a1 - √a²-4a2a0 2a2 NATURAL RESPONSE, Xn Alest + A₂e21 (A₁+A₂t)e (A₁ cos wat+A₂ sin wat)e -al

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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Question

The complete response is the sum of the natural response and the forced response
X = Xn+Xf
Natural response of a first-order circuit
Natural response of a second-order circuit
CASE
Overdamped
Critically damped
Underdamped
az
S1 =
FORCING FUNCTION
K
Kt
K₁²
K sin cot
Ke at
Forced response of a first-order, or a second-order circuit
+ A₁
NATURAL FREQUENCIES
$1,52 = -α± √√² - 0²
$1,$₂=-α
S1, S2 = -x±j√√/0²-a² -α±jood
dx
dt
d²x
dt²
Solution of the Second-Order Differential Equation
d²x
x(t) = xn(t) + xf (t)
a2
dt²
+ a。x = f(t)
+ A1
dx
dt
"xn(t)" = Aest
Xn(t) = Ke-t/t
+ a。x = 0
-a₁ + √²-4a₂a0
242
-
Xn(t) = A₁e³₁ª + A₂e³₂t
ASSUMED RESPONSE
A
At + B
At² +Bt+C
A sin cot + B cos cot
Ae-at
NATURAL RESPONSE, Xn
Ale+Azer
(A₁+A₂t)e-at
(A, cos coat+A₂ sin coat)e
xn(t) = ?
(a₂s² + a₁s+ao) = 0
-α₁ - √²-4a₂a0
242
NATURAL RESPONSE, Xn
A₁ est + A₂e21
-α1
(A₁+A₂t)e
(A₁ cos wat+A₂ sin wat)ext

• Problem 3. (c) The Circuit C shown is at the steady state, before the switch opens at time t = 2s.
t = 2s
I
6 V 2μF
Vin m
R₁
U10-
Brief catalog of the Operational Amplifier circuits
(Based on the ideal operational amplifier)
(a) Inverting amplifier
R₁
U20-W
10ΚΩ
ww
Vout=
R₂
:
Unw
R₂
ww
R₁
+
R₁
Ovout R₁
ww
R₁
v(t) 12 V
R₁
R₁
-R₁1+R₂₂+
Vout
R₁
Vin
(d) Summing amplifier
(5) Determine the capacitor voltage, v(t), for t≥ 2s
Vin -
(c) Circuit C
R₁2
R₁
vout
= (1 + R ₁) vin
(b) Noninverting amplifier
R₂K₁
V10-W
R₂K₂
U20-W
RJK3
U30M
R₂/(1-(K₁ + K₂ + K3))
R(K4-1)
Vin O
I
(c) Voltage follower (buffer amplifier)
Rp
Ovout Vin
Ovout=K4(K₁v1 + K₂v2 + K303)
(e) Noninverting summing amplifier

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