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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)
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 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
Transcribed Image Text:• 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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