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az + A1 dt² — dt "xn(t)" = Aest S1 = + a。x = 0 −a₁ + √√a² − 4a₂a0 2a2 n(t) = A₁e³₁t + A₂e³₂t xn(t) = ? (a₂s² + a₁s + α₁) = 0 $2 -α₁ - √a² - 4a₂ª0 242 NATURAL RESPONSE, Xn Alest+A₂est (A₁+A₂t)e (A₁ cos wat+A₂ sin wat)e¯¤t

az + A1 dt² — dt "xn(t)" = Aest S1 = + a。x = 0 −a₁ + √√a² − 4a₂a0 2a2 n(t) = A₁e³₁t + A₂e³₂t xn(t) = ? (a₂s² + a₁s + α₁) = 0 $2 -α₁ - √a² - 4a₂ª0 242 NATURAL RESPONSE, Xn Alest+A₂est (A₁+A₂t)e (A₁ cos wat+A₂ sin wat)e¯¤t

Introductory Circuit Analysis (13th Edition)
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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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
Transcribed Image Text: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 2. Please solve the following circuit problems, as directed in the questions. (a) The current i(t) of the current source is shown in the figure (a). i(t) + i(t) ( 84 3H v(t) 4A 0 (a) Circuit A ■ Vin m Brief catalog of the Operational Amplifier circuits (Based on the ideal operational amplifier) R₁ V10M (a) Inverting amplifier R₁ U₂0-M -M R₁ Un m R₁ R₁ Ovout= R₁ www R₁ R₁ R₁ R₂2+ Vout=₁1+1 Vout R₁ Vin (d) Summing amplifier (1) What is the voltage of the inductor for t>0 ? where v(t) = 0, for t<0, and t > 8s t(s) 8s (2) How much is the energy stored in the capacitor for t>0? for Energy(t = 0) = 0 J Vin O R₁ R₁ -out (b) Noninverting amplifier R₂K1 U10-W R₂K₂ U20-W Ra/K3 R₁ + Vin R1 V30-W R₂/(1-(K₁ + K₂ + K3)) R(K4-1) Vin (c) Voltage follower (buffer amplifier) R₂ Ovout Vin Ovout K4(K₁v1 + K₂V2 + K303) (e) Noninverting summing amplifier
Transcribed Image Text:• Problem 2. Please solve the following circuit problems, as directed in the questions. (a) The current i(t) of the current source is shown in the figure (a). i(t) + i(t) ( 84 3H v(t) 4A 0 (a) Circuit A ■ Vin m Brief catalog of the Operational Amplifier circuits (Based on the ideal operational amplifier) R₁ V10M (a) Inverting amplifier R₁ U₂0-M -M R₁ Un m R₁ R₁ Ovout= R₁ www R₁ R₁ R₁ R₂2+ Vout=₁1+1 Vout R₁ Vin (d) Summing amplifier (1) What is the voltage of the inductor for t>0 ? where v(t) = 0, for t<0, and t > 8s t(s) 8s (2) How much is the energy stored in the capacitor for t>0? for Energy(t = 0) = 0 J Vin O R₁ R₁ -out (b) Noninverting amplifier R₂K1 U10-W R₂K₂ U20-W Ra/K3 R₁ + Vin R1 V30-W R₂/(1-(K₁ + K₂ + K3)) R(K4-1) Vin (c) Voltage follower (buffer amplifier) R₂ Ovout Vin Ovout K4(K₁v1 + K₂V2 + K303) (e) Noninverting summing amplifier
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