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systems analysis and design

The Analysis And Design Of Linear Circuits 8th Edition Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint - Solutions

Exercise Analyze inputoutput controllability for Gs s s ss Compute the zeros and poles plot the RGA as a function of frequency etc
Exercise 5.14 The model of an industrial robot arm is as follows Gs as s sas a s a where a Sketch the Bode plot for the two extreme values of a What kind of control performance do you expect Discuss how you may best represent
Exercise 5.13 A heat exchanger is used to exchange heat between two streams a coolant with owrate q kgs is used to cool a hot stream with inlet temperature T C to the outlet temperature T which should be C The measurement delay for T is s The main disturbance is on T The
Exercise 5.12 Let H s Kes G s Kes sTs and Gd sG s H s The measurement device for the output has transfer function Gm ses The unit for time is seconds The nominal parameter values are K s K s and T sa Assume all variables have
Exercise 5.11 What information about a plant is important for controller designor more specically in which frequency range is it important to know the model wellTo answer this problem you may think about the following subproblemsa Explain what information about the plant is used for
Exercise 5.10 Let d qin ms denote a owrate which acts a disturbance to the process We add a buer tank with liquid volume V m and use a slow level controller K such that the out ow d qout the new disturbance is smoother then the in ow qin the original disturbance The idea is
Exercise 5.9 a The eect of a concentration disturbance must be reduced by a factor of at the frequency radmin The disturbances should be dampened by use of buer tanks and the objective is to minimize the total volume How many tanks in series should one have What is the total residence
Exercise 5.8 Comparison of local feedback and cascade control Explain why a cascade control system with two measurements pH in each tank and only one manipulated input the base ow into the rst tank will not achieve as good performance as the control system in Figure where we use local
9–73 Butterworth Poles Steven Butterworth, a British engineer, 1885–1958, discovered a method of designing electric filters. He was quoted saying“An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted
9–72 Complex Differentiation Property The complex differentiation property of the Laplace transformation states thatUse this property to find the Laplace transforms of f ðtÞ = ftgðtÞguðtÞ when gðtÞ = e−αt . Repeat for gðtÞ = sin βt and gðtÞ = cos βt. If A{f(1)} = F(s)then {if(t)}
9–71 Solving State Variable Equations With zero input, a series RLC circuit can be described by the following coupled first-order equations in the inductor current iLðtÞ and capacitor voltage vCðtÞ.(a) Transform these equations into the s domain and solve for the transforms ILðsÞ and
9–70 Inverse Transform for Complex Poles In Section 9–4, we learned that complex poles occur in conjugate pairs and that for simple poles the partial fraction expansion of F s ð Þ will contain two terms of the formShow that when the complex conjugate residues are written in rectangular form
9–69 First-Order Circuit Step Response In Chapter 7, we found that the step response of a first-order circuit can be written aswhere f ð0Þ is the initial value, f ð∞Þ is the final value and TC is the time constant. Show that the corresponding transform has the formand relate the time-domain
9–68 The Dominant Pole Approximation When a transform FðsÞ has widely separated poles, then those closest to the j-axis tend to dominate the response because they have less damping. An approximation to the waveform can be obtained by ignoring the contributions of all except the dominant poles.
9–67 The MATLAB function limit can be used to take the limit of a symbolic expression. Use MATLAB and the initial and final value properties to find the initial and final values of the waveforms corresponding to the following transforms.If either property is not applicable, explain why. Use
9–65 Use the initial and final value properties to find the initial and final values of the waveforms corresponding to the following transforms. If either property is not applicable, explain why. (a) Fi(s): s(s+6) 2+68+9
9–64 Use the initial and final value properties to find the initial and final values of the waveforms corresponding to the transforms below. If either property is not applicable, explain why. (a) F(s). (b) F(s)= 50s (s + 7s+10) (s+2)(s+6)(s+24) 125 (s+10x+40) $($2-625)
9–63 Use the initial and final value properties to find the initial and final values of the waveforms corresponding to the transforms below. If either property is not applicable, explain why. (a) F(s) 400- (b) F(8)= S (s+100)+2002 50(3s4+10s+4) s(s+1)(s+4)
9–62 Use the initial and final value properties to find the initial and final values of the waveforms corresponding to the transforms below. If either property is not applicable, explain why 30s (a) Fi(s)= (b) F(s)= (s+1)(s2+13s+12) s+1000 s(s+10)(s+100)
9–61 Use the initial and final value properties to find the initial and final values of the waveforms corresponding to the transforms below. If either property is not applicable, explain why (a) F(s)= (b) F(s)= 100(s+1) (s+2)(s+3) s-20 s(s+5)
9–60 For the inverting OP AMP circuit shown in Figure P9–60, write a differential equation for vOðtÞ in terms of the elements and vSðtÞ. Assume vCð0Þ = 0: Then let vSðtÞ = e−10t uðtÞ V, R1 =1 kΩ, R2 =10 kΩ, and C =1 μF and using Laplace techniques, find the output vOðtÞ Vs(1) R
9–59 Find vCðtÞ for t ≥ 0 when the input to the RC circuit shown in Figure P9–59 is vSðtÞ =VArðtÞ V. Assume vCð0 – Þ = 0 V. Vs(t) = Var(f) R www + C=vc(t) FIGURE P9-59
9–58 Repeat Problem 9–57 when an exponential source, vSðtÞ = 15 1 − e−500t V, is suddenly connected to the circuit.
9–57 The RLC circuit in Figure P9–57 is in the zero state when at t = 0 an exponential source, vSðtÞ =VAe−αt V, is suddenly connected to it.(a) Find the circuit integrodifferential equation that describes the behavior of the current in the circuit.(b) If R=82 Ω, L= 100 mH, C =15 μF, VA =
9–56 The switch in Figure P9–55 has been open for a long time and is closed at t =0.(a) Find the circuit differential equation in iLðtÞ and the initial conditions iLð0Þ and vCð0Þ .(b) The circuit parameters are L=50H, C =0:25 μF, R1 =10 kΩ, R2 =10 Ω, and vS =10u t ð Þ V. Use Laplace
9–55 The switch in Figure P9–55 has been closed for a long time and is opened at t =0(a) Find the circuit differential equation in vCðtÞ and the initial conditions iLð0Þ and vCð0Þ(b) The circuit parameters are L=50H, C =0:25 μF, R1 =10 kΩ, R2 =20 kΩ, and vS =10u t ð Þ V.V. Use
9–54 The switch in Figure P9–53 has been open for a long time and is closed at t = 0. The circuit parameters are R= 500 Ω, L=2:5H, C =2:5 μF, and VA = 500 V.(a) Find the circuit differential equation in vCðtÞ and the initial conditions iLð0Þ and vCð0Þ.(b) Use Laplace transforms to solve
9–53 The switch in Figure P9–53 has been open for a long time and is closed at t = 0. The circuit parameters are R=50 Ω, L= 250 mH, C =0:5 μF, and VA = 1000 V(a) Find the circuit differential equation in iLðtÞ and the initial conditions iLð0Þ and vCð0Þ.(b) Use Laplace transforms to
9–52 Use the Laplace transformation to find the vðtÞ that satisfies the following second-order differential equation: dr dv(0-) d-v(t) dv(t) +40 dt 400 v(t) =0, v(0-)=0 and 500 V/s dt
9–51 Use the Laplace transformation to find the vðtÞ that satisfies the following second-order differential equation: dv(t) dr dv(0-) =0. dt +20 dv(r) == + 1000 v(t) = 0, v(0-)=20 V and dt
9–50 Repeat Problem 9–48 for the input waveform vSðtÞ = 10 e−2000tuðtÞ V. Use MATLAB to plot the result.Validate your results by simulating the circuit in Multisim.
9–49 Repeat Problem 9–48 for the input waveform vSðtÞ = 169 ½cos 377t uðtÞ V. Use MATLAB to plot the result. Validate your results by simulating the circuit in Multisim.
9–48 The switch in Figure P9–48 has been open for a long time.At t = 0 the switch is closed.(a) Find the differential equation for the capacitor voltage and initial condition vCð0Þ.(b) Find vOðtÞ using the Laplace transformation for vSðtÞ = 100 uðtÞ V. Vs(f) 1+ 100 w 50 ww 1000 pF -
9–47 The switch in Figure P9–46 has been closed for a long time and is opened at t = 0. The circuit parameters are R = 50Ω, L = 200 mH, and VA = 30V.(a) Find the differential equation for the inductor current iLðtÞ and initial condition iLð0Þ.(b) Transform the equation into the s
9–46 The switch in Figure P9–46 has been open for a long time and is closed at t = 0. The circuit parameters are R = 10kΩ, L = 100 mH, and VA = 24V.(a) Find the differential equation for the inductor current iLðtÞ and initial condition iLð0Þ.(b) Transform the equation into the s domain.(c)
9–45 Use the Laplace transformation to find the iðtÞ that satisfies the following first-order differential equation: di(t) di +500 i(t) = [0.100e-100]u(t), i(0-) = 0 A
9–44 Use the Laplace transformation to find the vðtÞ that satisfies the following first-order differential equations: dv(t) (a) 250- +2500v(t)=0, v(0-)=100 V dt dv(t) (b) +300v(t)=300 u(t), v(0-)=-150 V dt
9–43 Find the transform F s ð Þ from the pole-zero diagram of Figure P9–43. K is 500. * -100-50 jo kj50 s-plane *-j50 50 FIGURE P9-43
9–42 Find the transform F s ð Þ from the pole-zero diagram of Figure P9–42. K is 5 × 106. Use MATLAB to find the corresponding waveform fðtÞ. jw 10 at co s-plane j500 -100-50 250 (2) -250 -j500 FIGURE P9-42
9–41 Find the transform F s ð Þ from the pole-zero diagram of Figure P9–41. K is 3. jo 10 at co s-plane 7.5 -10 -j7.5 FIGURE P9-41
9–40 Use MATLAB to find the inverse transform and plot the poles and zeros of the following function:(Hint: Refer to Web Appendix D for examples on how to use MATLAB’s function pzplot to find poles and zeros of transfer functions.) F(s) = 500 (3 + 2s+s+2) s(+48 +48 +16)
9–39 Use MATLAB to find the inverse transform and plot the poles and zeros of the following function:(Hint: Refer to Web Appendix D for examples on how to use MATLAB’s function pzplot to find poles and zeros of transfer functions.) 500s (s +30s +400) F(s)= (s+20) (s3+6s2+16s+16)
9–38 Find the inverse transforms of the following functions: (a) Fi(s)= (b) F(s)= (c) F3(s)= e-10s (s+100) (s+10)(s+1000) se-10s+100 (s+10)(s +1000) s+100e-10x (s+10)(s+1000)
9–37 Find the inverse transforms of the following functions: (a) Fi(s)= (b) F(s)= (s+1) (s+5) (s+1000)2 (s+10000)2
9–36 Find the inverse transforms of the following functions:(a) F1ðsÞ =sðs + 10Þðs + 100Þðs + 1Þðs + 1000Þðs + 10000Þ(b) F2ðsÞ =ðs + 1000Þðs + 100000Þðs + 10000Þ
9–35 A certain transform F s ð Þ=Ks + γs + α has a simple pole at s = −50, a simple zero at s = −γ, and a scale factor of K = 1.Select values for γ so that the inverse transform is(a) f ðtÞ = δðtÞ − 5e−50t(b) f ðtÞ = δðtÞ(c) f ðtÞ = δðtÞ + 5e−50t
9–34 Find the inverse Laplace transforms of the following functions using MATLAB:(a) F1ðsÞ =ðs + 100Þ3ðs + 50Þ2ðs + 200Þ2(b) F2ðsÞ =ðs + 50Þ3ðs + 100Þ2ðs + 200Þ2
9–33 Find the inverse transforms of the following functions:(a) F1ðsÞ =16 s2 + 256 sðs2 + 8s + 32Þ(b) F2ðsÞ =3 s2 +20s + 400 sðs2 + 50s + 400Þ
9–32 Find the inverse transforms of the following functions:(a) F1ðsÞ =300ðs + 50Þs2ðs2 + 40s + 300Þ(b) F2ðsÞ =1000s2ðs + 5Þðs2 + 4s + 8Þ
9–31 Find the inverse transforms of the following functions: (a) Fi(s)= (s+10)(s+107) s(s+105) (s+10%) 5($4 +10s+4) (b) F2(s). s(s+1)(s+4)
9–30 Find the inverse Laplace transforms of the following functions and then validate your answers using MATLAB:(a) F1ðsÞ =16sðs + 3Þðs2 + 21s + 20Þ(b) F2ðsÞ =60 s2 +16 sðs2 + 36Þ
9–29 Use the sum of residues to find the unknown residue in the following expansion. Then find the inverse transform of the completed expansion. Finally, validate your answer using MATLAB. 5000 5000(s+1000) k F(s)= + (s+500)(s+5000) s+500 s+500 s+5000
9–28 Use the sum of residues to find the unknown residues in the following expansions: (a) Fi(s)=- (b) F(s)= 600 4 (s+10)(s+20)(s+30) s+10 s+20s+30 2(s+10) (s+15) (s+20) s+15 k + s+20 3. k 3 + +
9–27 Find the inverse Laplace transforms of the following functions:(a) F1ðsÞ =α2 s2ðs + αÞ(b) F2ðsÞ =α2 sðs + αÞ2
9–26 Find the inverse Laplace transforms of the following functions and sketch their waveforms for β > 0: (a) Fi(s) A- (b) F(s)= B (s+B) s(s +) As (s+B) 5+B
9–25 Find the inverse Laplace transforms of the following functions:(a) F1ðsÞ =9000ðs + 10Þ2 +302(b) F2ðsÞ =5ðs + 10Þðs + 10Þ2 +302
9–24 Find the inverse Laplace transforms of the following functions.Validate your answers using MATLAB. (a) F(s)= (b) F(s)=- s(s+10)(s+20) (s+5)(s+50) (s+1) (s+10)
9–23 Find the inverse Laplace transforms of the following functions:(a) F1ðsÞ =50 ðs + 1000Þðs + 2000Þðs + 500Þðs + 5000Þ(b) F2ðsÞ =50s2ðs + 100Þðs + 500Þ
9–22 Find the inverse Laplace transforms of the following functions.Validate your answers using MATLAB.(a) F1ðsÞ =sðs + 10Þðs + 40Þ(b) F2ðsÞ =ðs + 1Þðs + 10Þsðs + 100Þðs + 1000Þ
9–21 Find the inverse Laplace transforms of the following functions:(a) F1ðsÞ =10 sðs + 50Þ(b) F2ðsÞ =s+2ðs + 3Þðs + 4Þ
9–20 Consider the waveform in Figure P9–20.(a) Write an expression for the waveform f t ð Þ in Figure P9–20 using a delayed exponential.(b) Use the time-domain translation property to find the Laplace transform of the waveform f t ð Þ found in part (a).(c) Verify the Laplace transform
9–19 For the following waveform:f t ð Þ= 500 + 100e−500t t cos 1000t u t ð Þ(a) Find the Laplace transform of the waveform. Locate the poles and zeros of FðsÞ.(b) Validate your result using MATLAB.
9–18 Consider the waveform in Figure P9–18.(a) Write an expression for the waveform f t ð Þ using step functions.(b) Use the time-domain translation property to find the Laplace transform of the waveform f t ð Þ found in part (a).(c) Verify the Laplace transform found in (b) by applying the
9–17 Consider the waveform in Figure P9–17.(a) Write an expression for the waveform f t ð Þ using step and ramp functions.(b) Use the time-domain translation property to find the Laplace transform of the waveform f t ð Þ found in part (a).(c) Verify the Laplace transform found in (b) by
9–16 Find the Laplace transforms of the following waveforms.(a) f1ðtÞ =d dt 150e−1000tcos 20kt u t ð Þ(b) f2ðtÞ =Z t 020e−10xdx + 10 uðtÞ + 20 de−10t dt u t ð Þ
9–15 Use MATLAB to find the Laplace transform of the following waveform f t ð Þ= 50+2e−10t uðtÞ + ½2:5 cos 100ðt−0:05Þ uðt−0:05Þ
9–14 Find the Laplace transforms of the following waveforms.Use MATLAB to verify your results.(a) f1ðtÞ = 2δðt−1Þ(b) f2ðtÞ = 5e−100ðt−2Þuðt−2Þ(c) f3ðtÞ = 200e−50 t−10−3 ð Þu t−10−3
9–13 Find the Laplace transforms of the following waveforms.Locate the poles and zeros of F s ð Þ. Use MATLAB to verify your results.(a) f1ðtÞ = 2δðtÞ + 144t e−12t u t ð Þ(b) f2ðtÞ = 100 + 50e−10tðcos100t + sin100tÞ u t ð Þ
9–12 Find the Laplace transforms of the following waveforms and plot their pole-zero diagrams.(a) f1ðtÞ = δðtÞ + 100e−100t + 200e−200t u t ð Þ(b) f2ðtÞ = 15e−2000t + 15 cos 5000t u t ð Þ
9–11 Find the Laplace transforms of the following waveforms and plot their pole-zero diagrams. Then use MATLAB to validate your results.(a) f1ðtÞ = 15e−5t −20e−10t u t ð Þ(b) f2ðtÞ = 10 ½cos 1000t + cos 2000t uðtÞ
9–10 Find the Laplace transform of f t ð Þ = 2− 5t − 2e−25t u t ð Þ. Locate the poles and zeros of F s ð Þ.
9–9 Find the Laplace transformof f ðtÞ = δ0ðtÞ + δðtÞ − e−t uðtÞ.Locate the poles and zeros of F s ð Þ.
9–8 Find the Laplace transform of f ðtÞ = 5½4cosð10tÞ − 3sinð10tÞ uðtÞ. Locate the poles and zeros of FðsÞ.
9–7 Find the Laplace transform of f ðtÞ = ½5 − 5 cosð500tÞ uðtÞ.Locate the poles and zeros of F s ð Þ.
9–6 Find the Laplace transform of f ðtÞ = A ðB + αtÞ e−αt ½ u t ð Þ.Locate the poles and zeros of F s ð Þ.
9–5 Find the Laplace transform of f t ð Þ = e−2t − 2et u t ð Þ.Locate the poles and zeros of F s ð Þ.
9–4 Find the Laplace transform of f t ð Þ = 10 e−2000t −2e−1000 t u t ð Þ. Locate the poles and zeros of F s ð Þ.
9–3 Find the Laplace transform of f ðtÞ = – 5 δðtÞ + 50 uðtÞ.Locate the poles and zeros of F s ð Þ.
9–2 Find the Laplace transform of f t ð Þ = 20×103 sinð60πtÞ u t ð Þ. Locate the poles and zeros of F s ð Þ.
9–1 Find the Laplace transform of f ðtÞ = 3 1 − e−1000 t u t ð Þ.Locate the poles and zeros of F s ð Þ.
Find the initial and final values of the waveforms corresponding to the following transforms: (a) Fi(s) 100- (b) F2(s) 80- $+3 s(s+5)(s+20) s(s+5) (s+4)(s+20)
Use the initial and final value properties to find the initial and final values of the waveform whose transform is F(s)=2- (s+3) s(s+1)(s+2)
The RL circuit of Figure 9–16 is in the zero state when the input iSðtÞ = ½2 cos 1000tuðtÞA is applied. Find iLðtÞ for t ≥ 0. 4.00) 100 200 mH is(t)=2 cos1000r u(t) A FIGURE 9-16
Find υCðtÞ when the input to the RC circuit in Figure 9–15 is υSðtÞ = ½VAcos βtuðtÞ V R www vs(t)= [VACOSr] u(t) V FIGURE 9-15 + C vc(1)
The RC circuit in Figure 9–14 has R=10 kΩ,C =0:2 μF, and V0 = −5 V. The input isυSðtÞ = 10e−1000tuðtÞ V. Find υCðtÞ for t ≥ 0. R ww Cvc(t) vs(t)=Veu(t) V FIGURE 9-14
Find υCðtÞ in the RC circuit in Figure 9–14 when the input is the waveformυSðtÞ = VAe−αt ½ uðtÞ V. R ww vs(1) = Ve u(t) V FIGURE 9-14 + C=vc(t)
Find the transforms that satisfy the following equations and the given initial conditions. (a) v(t)dt+10v(t)=10u(t) V (b) dv(t)+4()+3(1)=5e-2 V, (0-)=2V/s, (0-)=-2V di dt
Find the transforms that satisfy the following equations and the given initial conditions. (a) dv(t) (b) 4- dt dv(t) dt + 6v(t)=4u(t) V, v(0-)=-3 V + 12v(t) 16 cos31 V, v(0-)=2V
The switch in Figure 9–13 has been open a long time. At t = 0 the switch is closed.(a) Find the differential equation for iLðtÞ and initial condition iLð0Þ.(b) Transform the equation into the s domain.(c) Solve the equation for ILðsÞ.(d) Take the inverse transform of ILðsÞ to find
The switch in Figure 9–12 has been open for a long time. At t = 0 the switch is closed.Find iðtÞ for t ≥ 0. R R ww + L 000 VR(1) VL(1) + i(1) vc(f) C 1=09 R 400 L= 1H FIGURE 9-12 C=5F V = 10 V
The inductor in Figure 9–11 is replaced by a capacitor C. The switch has been in position A for a long time. At t = 0 it is moved to position B. Find VCðsÞ and υCðtÞ for t ≥ 0. 0=1 R iL(t) B VR(f) + VL(D) L FIGURE 9-11
The switch in Figure 9–11 has been in positionAfor a long time. At t = 0 it is moved to position B. Find iLðtÞ for t ≥ 0. 1=0 R -ma iL(t) B VR(t) + VL(7) FIGURE 9-11
Find the transform FðsÞ from the pole-zero diagram of Figure 9–9. K is 3 × 104.
Find the transform FðsÞ from the pole-zero diagram of Figure 9–8. K is 106.
Use MATLAB to find the inverse transform f ðtÞ of the following function F(s)= (s+100)2 (s+50)(s+200)
Find the inverse transforms of the following functions (a) F(s)= (b) F(s)= (c) F(s)=. S (s+1)(s+2) 16 2(s+4) 800s(s+1) (s+2)(s+10)
Find the inverse transform of F(s) 400- (s+100) s(s+200)
Find the inverse transform of F(s)=- 4(s+3) s(s+2) 2
Find the inverse transforms of the following functions: S+10 (a) F(s)= S+100 (b) F(s)= (c) F(s)= 28+35+5 S $3 +28+8+3 s+2
Find the inverse transforms of the following functions: s+4s+5 (a) F(s)=+4+3 (b) F(s)= -4 $2 +4

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