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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

7–18 Repeat Problem 7–17. However, after the switch is moved to position B at t = 0, the switch is moved back to positionAat t = 50 μs. Find vCðtÞ for t ≥ 0.
7–19 Find the function that satisfies the following differential equation and the initial condition for an input vSðtÞ = 10 cosð250tÞ V: dv(t) +50v(t)=vs(t), v(0)=0V dt
7–20 Repeat Problem 7–19 for vSðtÞ = 1000e−200tuðtÞV. Plot your result using MATLAB.
7–21 The switch in Figure P7–21 has been open long enough for iLð0Þ to reach 0A and is closed at t = 0.(a) If vSðtÞ = 100 uðtÞ V, find vLðtÞ for t ≥ 0.(b) If vSðtÞ = 100 cosð100tÞ V, find vLðtÞ for t ≥ 0.(c) If vSðtÞ = 100 e – 100t V, find vLðtÞ for t ≥ 0. vs (f) + 330
7–22 Repeat Problem 7–21 using Multisim.
7–23 Repeat Problem 7–21 using MATLAB to plot the waveforms.
7–24 The switch in Figure P7–21 has been closed a long time after being excited by vSðtÞ = 100 uðtÞ V. After the circuit reached equilibrium, a new t = 0 is established and the switch is suddenly opened. Find vLðtÞ for t ≥ 0.
7–25 The switch in FigureP7–25 has been in positionAfor a long time and is moved to position B at t = 0. Find iLðtÞ for t ≥ 0. 100 B 1=0 i(t) w ww 10 A 100 mH 15 V 000 FIGURE P7-25
7–26 The switch in Figure P7–25 has been in position B for a long time and is moved to position A at t = 0. Find iLðtÞ for t ≥ 0.
7–27 The switch in Figure P7–25 has been in position A for a long time and is moved to position B at t =0. At t = 100 μs the switch returns to position A. Find iLðtÞ for t ≥ 0.
7–28 The follower circuit in Figure P7–28 is in the zero state and is driven by a step input vSðtÞ = 2 uðtÞ. If R1 = 50 kΩ, R2 = 2:2 kΩ, and C = 0:1 μF, find v2ðtÞ for t ≥ 0. R + W Vs(1) R V(1) ww FIGURE P7-28 C
7–29 The inverting OP AMP in Figure P7–29 is driven by a step input vSðtÞ = 2 uðtÞ. Let R1 = 10 kΩ, R2 = 20 kΩ, and C =0:1 μF.(a) If vCð0Þ = 4 V find v2ðtÞ for t ≥ 0.(b) What is the value of v2ðtÞ when t = 1:386ms?(c) Validate your analysis using Multisim. vc(0) + C www R R + o w
7–30 The switch in Figure P7–30 has been in position A for a long time and is moved to position B at t = 0. The switch suddenly returns to positionAafter 10ms. Find vCðtÞ for t ≥ 0 and sketch its waveform. 24 V 1+ 20 www 1=0 A B + 1 F vc(f) 80 20 FIGURE P7-30
7–31 Switches 1 and 2 in Figure P7–31 have both been in position A for a long time. Switch 1 is moved to position B at t =0 and Switch 2 is moved to position B at t = 20 ms. Find the voltage across the 0:22-μF capacitor for t ≥ 0 and sketch its waveform. SW#1 150 Bo 1=0 120 V SW#2 50 W +
7–32 The switch in Figure P7–32 has been open for a long time and is closed at t = 0. The switch is reopened at t = 3 ms. Find vCðtÞ for t ≥ 0. 2.2 ww t = 3 ms + 1=0 +36 V 15 0.68 F= vc(t) FIGURE P7-32
7–33 Find the sinusoidal steady-state response of vCðtÞ in Figure P7–33 when R= 100 kΩ, C =0:02 μF, and the input voltage is vSðtÞ = 15 cosð50tÞ uðtÞ V. Repeat for an input voltage of vSðtÞ = 15 cosð500tÞ uðtÞ V, and one more time for an input voltage of vSðtÞ = 15 cosð5ktÞ
7–34 On the circuit of Figure P7–33 the input is vSðtÞ = 5e – 1000t uðtÞ V. Find the output vCðtÞ when R= 100 kΩ, C = 0:01 μF, and vCð0Þ = 0 V.
7–35 For t ≥ 0 the zero-input response of the circuit in Figure P7–35 is vCðtÞ = 20e−10ktV.(a) Find C and iCðtÞ when R= 10 kΩ.(b) Find the energy stored in the capacitor at t = 2 ms.(c) Suppose the only capacitor you had available has a value of 0:022 μF. Could you achieve the same
7–36 For t ≥ 0 the zero-input response of the circuit in Figure P7–36 is iLðtÞ = 150e−500t mA.(a) Find L and vLðtÞ when R= 500 Ω.(b) Find the energy stored in the inductor at t =0:5 ms.(c) Suppose the only inductor you had available has a value of 500 mH. Could you achieve the same
7–37 Design a series RC circuit using a dc voltage source that delivers the following voltage across the capacitor for t >0. vc(t)=2e-00V 120
7–38 Design a parallel RL circuit using a dc current source that delivers the following voltage across the resistor for t >0. VR(t)=50e-2000 V 120
7–39 Design a series RC circuit using a dc voltage source that delivers a voltage across the capacitor for t > 0 that fits entirely within the nonshaded region of Figure P7–39. vc(f) 10 9 6. 2 0 -t (ms) 01 5 10 12 20 FIGURE P7-39
7–40 Design a series RC circuit using dc voltage sources that delivers a voltage across the capacitor for t > 0 that fits entirely within the nonshaded region of Figure P7–40. vc(1)(V) 10 9 6 0 t(s) 2 12 20 24 40 -4 FIGURE P7-40
7–41 For t ≥ 0 the step response of the voltage across the capacitor in Figure P7–41 is vCðtÞ = 10−20e−10,000t V. Find the IV, FV, TC, R, and iCðtÞ when C = 0:15 μF. R ic(t) vs(t) (+ ww FIGURE P7-41 + vc(1) -
7–42 Design a first-order RC circuit using standard parts(see inside rear cover) that will produce the following voltage across the capacitor: vCðtÞ = 10−20e−2000t V.
7–43 For t ≥ 0 the step responses of the current through and voltage across the inductor in Figure P7–43 are iLðtÞ = 5−10e−2000t mA and vLðtÞ = e−2000tV. Find IV, FV, TC, R, and L. vs(t) +1 R iL (1) ww FIGURE P7-43 + VL(t)
7–44 Design a first-order RL circuit that will produce the following current through the inductor: iLðtÞ =25 + 50e−50,000t mA for t ≥ 0.
7–45 The switch in Figure P7–45 has been in position B for a long time and is moved to position A at t =0.Design the first-order RC interface circuit such that vOðtÞ = 10−10e−2500t V. 20 V (+ B t=0 Interface circuit vo(t) FIGURE P7-45 www : 300
7–46 The switch in Figure P7–45 has been in position A for a long time and is moved to position B at t =0.Design the first-order RC interface circuit such that vOðtÞ = 5e−5000tV.
7–47 A timing circuit is required that feeds into an OP AMP’s noninverting terminal (i.e., draws no current.) The circuit’s output response vOðtÞ must beFigure P7–47 shows two commercial products and the vendors claim each will meet the requirement. Which will you select and why?
7–48 Aproduct line needs an RC circuit that will meet the following response specifications ±5%:Design a circuit to meet the specifications and validate your results using Multisim. IV FV T R C -5 V +5 V 150 s 21 0.1 F
7–49 There is a need for a timing circuit to trip a house alarm. Design an RC circuit that reaches 5 V across a capacitor in exactly 5 s. Your source is 12 dc and you have a 10- μF capacitor available
7–50 Find the v t ð Þ that satisfies the following differential equation and initial conditions: d'v(t) dr +4 dv(t) dt +36v(t)=0, v(0)=0 V, dv(0)-24V/s dt
7–51 Find the v t ð Þ that satisfies the following differential equation and initial conditions: dv(t) dv(t) di +10- dt +100v(t)=0, v(0)=5 V, dv(0) =0V/s di
7–52 Find the v t ð Þ that satisfies the following differential equation and initial conditions: dv(1) +10dv(t) +125v(t)=250u(t), dt di v(0)=5V, dv(0)-25V/s dt
7–53 Find the iðtÞ that satisfies the following differential equation and initial conditions di(t) dr +4di(1) di(t) dt +4i(t)=16u(t), i(0)=0, di(0) =0 dt
7–54 The switch in Figure P7–54 has been open for a long time and is closed at t = 0. The circuit parameters are L=1H, C = 0:5 μF, R= 100 Ω, and vCð0Þ = 5 V.(a) Find vCðtÞ and iLðtÞ for t ≥ 0.(b) Is the circuit overdamped, critically damped, or underdamped?(c) Use Multisim to simulate
7–55 The switch in Figure P7–55 has been open for a long time and is closed at t = 0. The circuit parameters are L=1H, C = 1 μF, R= 500 Ω, and vCð0Þ = 20 V.(a) Find vCðtÞ and iLðtÞ for t ≥ 0.(b) Is the circuit overdamped, critically damped, or underdamped?(c) Use Multisim to simulate
7–56 Use Multisim to study how the voltage across the circuit in Figure P7–55 changes as the value of the resistor is varied. Let L=1H, C =1 μF, R= 500 Ω, and vCð0Þ = – 10 V. Under “Analyses” perform a “Parameter sweep.” The parameter we wish to sweep is the resistor R1.We select
7–57 The switch in Figure P7–57 has been open for a long time and is closed at t = 0. The circuit parameters are L=4H, C =1 μF, R1 =2:2 kΩ, R2 =3:3 kΩ, and VA =12V.(a) Find vCðtÞ and iLðtÞ for t ≥ 0.(b) Is the circuit overdamped, critically damped, or underdamped?(c) Use Multisim to
7–58 The switch in Figure P7–58 has been open for a long time and is closed at t = 0. The circuit parameters are L=1:25 H, C =0:05 μF, R1 =33 kΩ, R2 =33 kΩ, and VA =20V.(a) Find vCðtÞ and iLðtÞ for t ≥ 0.(b) Is the circuit overdamped, critically damped, or underdamped?(c) Use Multisim
7–59 Repeat Problem 7–58 with R1 =2 kΩ, R2 =2 kΩ.
7–60 The switch in Figure P7–60 has been in position A for a long time. At t = 0 it is moved to position B. The circuit parameters are R1 =20 kΩ, R2 =4 kΩ, L=1:6H, C =1:25 μF, and VA =24V.(a) Find vCðtÞ and iLðtÞ for t ≥ 0.(b) Is the circuit overdamped, critically damped, or
7–61 You have a need for an interface circuit that will connect your source to a load with a very high input as shown in Figure P7–61(a) . Your interface must have a response that fits within the boundaries shown in Figure P7–61(b).A vendor offers a suitable circuit shown in Figure P7–61(a)
7–62 The switch in Figure P7–62 has been in position A for a long time and is moved to position B at t = 0. The circuit parameters are R1 =1 kΩ, R2 = 100 Ω, L= 250 mH, C= 3:3 μF, and VA =24V.(a) Find vCðtÞ and iLðtÞ for t ≥ 0.(b) Is the circuit overdamped, critically damped, or
7–63 The switch in Figure P7–62 has been in position B for a long time and is moved to positionAat t = 0. The circuit parameters are R1 =1 kΩ, R2 = 100 Ω, L= 250 mH, C= 3:3 μF, and VA =24V.(a) Find vCðtÞ and iLðtÞ for t ≥ 0.(b) Is the circuit overdamped, critically damped, or
7–64 The circuit in Figure P7–64 is in the zero state when the step function input is applied. The circuit parameters are L= 250mH, C = 1 μF, R= 4:7 kΩ, and VA = 5 V. Find vOðtÞ for t ≥ 0. (Hint: Find the capacitor voltage first.) R C wh VAu(t) L vo(f) FIGURE P7-64
7–65 The circuit in Figure P7–65 is in the zero state when the step function input is applied.(a) If VA =15V, R=1:5 kΩ, L= 250 mH, and C= 0:25 μF, derive an expression for the voltage vOðtÞ for t ≥ 0.(b) Validate your solution by plotting it using MATLAB and comparing it to a Multisim
7–66 Derive expressions for the damping ratio and undampednatural frequency of the circuit inFigure P7–66 in terms of the circuit parameters R, L, and C. Which parameter(s)affect the damping ratio? Can you change the damping ratio without affecting the undamped natural frequency? 1 + R w vs(f)
7–67 Derive expressions for the damping ratio and undamped natural frequency of the circuit in Figure P7–67 in terms of the circuit parameters R, L, and C. Which parameter(s) affect the damping ratio? Can you change the damping ratio without affecting the undamped natural frequency? is(t) 2R ww
7–68 The circuit of Figure P7–68 is a two-stage, first order cascade circuit that will be studied extensively in Chapter 14.It is a second-order circuit whose zero-state, step response can be solved by recognizing that since R and C are the same, the solution fits Case B for the natural
7–69 The circuit in Figure P7–69 is in the zero state when the step function input is applied. If the input source is VA = 10 V and L=0:5H, select values of R and C so that the circuit’s output vOðtÞ for t ≥ 0 is critically damped. Use MATLAB or Multisim to show your result for vOðtÞ.
7–70 In a series RLC circuit the step response across the 1-μF capacitor is vCðtÞ = 15−e−200t ½15 cosð1000tÞ + 3 sinð1000tÞ V t ≥ 0(a) Find R and L. (b) Find iLðtÞ for t ≥ 0.
7–71 In a parallel RLC circuit the zero-input response in the 220-mH inductor is iLðtÞ = 50e−6000t −40e−3000t mA t ≥ 0(a) Find R and C. (b) Find vCðtÞ for t ≥ 0.
7–72 In a parallel RLC circuit the state variable responses are vCðtÞ = e−100t ½5 cosð300tÞ + 15 sinð300tÞ V t ≥ 0 iLðtÞ = 20−25e−100tcosð300tÞmA t ≥ 0 Find R,L, and C.
7–73 The zero-input response of a series RLC circuit with R=50 Ω is vCðtÞ = 2e−1000t cosð500tÞ−4e−1000t sinð500tÞ V t ≥ 0 If the initial conditions remain the same, what is the zero-input response when R= 100 Ω?
7–74 In a parallel RLC circuit the inductor current is observed to be iLðtÞ = 20e−20tsinð20tÞmA t ≥ 0 Find vCðtÞ when vCð0Þ = 0:6 V.
7–75 Design a parallel RLC circuit whose natural response has the form vLðtÞ =K1e−20,000t +K2te−20,000t V t ≥ 0
7–76 Design a series RLC circuit with ζ=0:5 andω0 = 100 krad=s.(a) What is the form of the natural response of vCðtÞ for your design?(b) Simulate your circuit in Multisim.
7–77 Design a series RLC circuit with ζ = 1 andω0 = 10 krad=s.(a) What is the form of the natural response of vCðtÞ for your design?(b) Simulate your circuit in Multisim.
7–78 Design a series RLC circuit whose output voltage resides entirely within the nonshaded region of Figure P7–78. Validate your design using MATLAB or Multisim. Vo(1)(V) 7 6 5 15.25 4.5 14.75 4 3 2 1 0 0 2 FIGURE P7-78 t (ms) 6
7–79 A circuit is needed to produce the following step response:A vendor has proposed using the circuit shown in Figure P7–79 to produce the desired response. The vendor realizes that the proposed circuit does not exactly meet the desired response and is willing to make a single change for no
7–80 What range of damping ratios is available in the circuit in Figure P7–80? 200 w 10 w 0.01 F= 2.5 mH -000 FIGURE P7-80
7–81 A variable capacitor is used in the circuit of P7–81 to vary the damping ratio. What range of damping ratios is available in the circuit? 47 ww 1-1000 pF 1 mH m FIGURE P7-81
7–82 A particular parallel RLC circuit has the step response observed on an oscilloscope and shown in Figure P7–82. Four points on the waveform were measured and are shown.Determine the circuit’s initial value, final value, the dominant exponential’s time constant, and the likely case (A,
7–83 First-Order OP AMP Circuit Step Response Find the zero-state response of the OP AMP output voltage in Figure P7–83 when the input is vSðtÞ =VA uðtÞ V. Validate your solution using Multisim when R1 =1 kΩ, R2 =10 kΩ, C2 =0:01 μF, and VA =1V. C R + vs(f) R ww w + FIGURE P7-83 Vo(1)
7–84 Intermittent Timing Circuit for Windshield Wipers Acar maker needs an RCtiming circuit to trigger the windshield wiper relay. The circuit should be driver selectable to trigger at 1, 2, 5, and 10 s ± 5%. The source circuit is the car voltage of 12 V with a series resistance of 10 Ω. You
7–85 RC Circuit Design Design the first-order RC circuit in Figure P7–85 so an input vSðtÞ = 20 uðtÞ V produces a zero-state response vOðtÞ = 20−5e−1000t V. Validate your design using MATLAB or Multisim. vs(t) +1 First-order RC circuit + vo(t) FIGURE P7-85
7–86 Sample-Hold Circuit Figure P7–86 is a simplified diagram of a sample-hold circuit.When the switch is in position A, the circuit is in the sample mode and the capacitor voltage must charge to at least 99%of the source voltage VA in less than 1 μs. When the switch is moved to position B,
7–87 Super Capacitor Super capacitors have very large capacitance (typically from 0.1 to 3000F), small sizes, and very long charge-holding times, making them useful in nonbattery backup power applications. The charge-holding quality of a super capacitor is measured using the circuit in Figure
7–88 Cost-Conscious RLC Circuit Design You are assigned a task to design a series, passive RLC circuit with a characteristic equation of s2 + 2000s+5×106 = 0. To save money, your supervisor wants you to use a previously purchased 150-mH inductor with a 10-Ω parasitic resistance. The RLC circuit
7–89 Combined First- and Second-Order Response The switch in Figure P7–89 has been in position A for a long time and is moved to position B at t = 0 and then to position C when t = 10 ms. For 0 V.For t > 10 ms, the capacitor voltage is a sinusoid vCðtÞ = 6:321 cos½1000ðt−0:01Þ V.(a)
7–90 Undesired Ringing A digital clock has become corrupted by a ringing (undesired oscillations) as shown in Figure P7–90(a). The unwanted oscillations can cause false triggers and must be reduced. The clock can be modeled as an RLC series circuit as shown in Figure P7–55(b) with the voltage
7–91 Triangular Wave Design There is a need to generate a 12-V ± 10%, 1-kHz triangular wave. You have a ± 5-V, 1-kHz square wave. You recall from your first Circuits course that you can easily design an OP AMP integrator that should be able to produce a triangular wave from a square wave. But
7–92 Optimum Fusing A sensitive instrument that can be modeled by the series RLC circuit shown in Figure P7–92 is to be protected by a fuse. The voltage across the capacitor isThe peak current was found to occur at about 13 ms after t =0.Engineer A suggests a 10-mA fuse; Engineer B suggests a
7–93 Lightning Pulser Design The circuit in Figure P7–93 is a simplified diagram of a pulser that delivers simulated lightning transients to the test article at the output interface. Closing the switch must produce a short-circuit current of the form iSCðtÞ = IAe−αtcosðβtÞ, withα = 100
7–94 RLC Circuit Design Losses in real inductors can be modeled by a series resistor as shown in Figure P7–94. In this problem, we include the effect of this resistor on the design of the series RLC circuit shown in the figure. The design requirements include a source resistance of 50 Ω, an
7–95 Competing Circuit Designs Figure P7–88 shows the step responses vCðtÞ of two competing series RLC circuits from two different vendors. The circuits are designed to switch from 0 to 10 V and to meet a specification for a desired circuit with the following characteristic equation:s2 +10 s
7–96 Solving Differential Equations with MATLAB MATLAB has a built-in function for solving ordinary differential equations called dsolve. We can use this function to quickly explore the solution to a second-order differential equation when the forcing function is a sinusoidal or exponential
6–70 Supercapacitor Supercapacitors have very large capacitances (typically from 0.1 to 3000 F), very long charge-holding times, and small sizes, making them useful in nonbattery backup power applications.To measure its capacitance, a supercapacitor is charged to an initial voltage vOð0Þ = 6 V.
6–61 A capacitor bank is required that can be charged to 5 kVand store at least 250 J of energy. Design a series–parallel combination that meets the voltage and energy requirements using 33-μF capacitors each rated at 1.5 kV max.
6–56 What is the equivalent capacitance and initial voltage of a series connection of a 100-μF capacitor with 100 V stored and a 47-μF capacitor with 200 V stored?
6–55 Verify Eqs. (6–30) and (6–31) .
6–54 You need to have an equivalent inductance of 235 mH for a particular application. However, you only have 100-mH inductors available. How might you connect these to get within 5% of the desired value?
6–52 A 2-H inductor is connected in series with a 2-mH inductor and the combination connected in parallel with a 1-mH inductor. All are 5%. Find the equivalent inductance of the connection. Which inductor played no effective role in this combination and could have been ignored?
6–45 Repeat Problem 6–44 but use an RL OP AMP circuit.
6–40 The OP AMP differentiator in Figure P6–35 with R = 5 kΩ and C = 220 pF has the input vSðtÞ = 2:5½sin ðωtÞ u t ð Þ V. Determine the frequency at which the OP AMP saturates at 15 V. Validate your answer using Multisim. [Hint:Use an AC source and an AC analysis with a linear sweep
6–39 The input to the OP AMP differentiator in Figure P6–35 is vSðtÞ = 5 sin 2π × 106t uðtÞmV. Select R and C so that the output sinusoid has extreme values of at least 14 V but does not saturate the OP AMP at 15 V.
6–38 The OP AMP differentiator shown in Figure P6–35 has R = 100 kΩ and C = 0:1 μF and an output vOðtÞ =1½sin ð100tÞ uðtÞ V. What is its input vS(t)?
6–36 Redesign the circuit of Figure P6–35 using an RL circuit rather than the RC approach shown. Refer to Problem 6–29.
6–34 The OP AMP integrator in Figure P6–27 has R = 22kΩ, C = 0:001 μF, and vOð0Þ = 0 V. The input is vSðtÞ =2 sin ðωtÞ uðtÞ V. Derive an expression for vOðtÞ and find the smallest allowable value of ω for linear operation of the OP AMP. Assume VCC = 15 V.
6–33 The OP AMP integrator in Figure P6–27 has R = 50kΩ, C = 120 μF, and vOð0Þ = – 2 V. The input is vSðtÞ =10 u t ð Þ V. Use Multisim to determine how long it takes for the OP AMP to saturate when VCC = 12 V.
6–32 Design appropriate OP AMP circuits that will realize each of the functions in problem 6–31.
6–28 Build the OP AMP circuit of Figure P6–27 in Multisim.Let R = 33 kΩ, C = 0:056μF, and vOð0Þ = 15 V. The input is 10 1 − e−500t u t ð Þ V. The OP AMP has a VCC = 15V.Plot the output vOðtÞ for t > 0. Over what period of time is the OP AMP in the linear range? [Hints: Use the
6–26 For t > 0 the voltage across an energy storage element is vðtÞ = 5 − 20 e−500t V and the current through the element is iðtÞ = 2000t + 16 e−500t mA. What are the element, the element value, and its initial condition?
6–25 For t > 0 the voltage across a circuit element is vðtÞ = 5 te−100t cosð1000tÞ V and the current through the element is iðtÞ = 2:5 te−100t cosð1000tÞ μA.What are the element, the element value, and its initial condition?
6–24 For t > 0 the voltage across an energy storage element is vðtÞ = 5 e−100t V and the current through the element is iðtÞ = 10 – 5 e−100t A. What are the element, the element value, and its initial condition?
6–23 A 500-μH inductor is connected in parallel with a 330-kΩresistor. The current through the inductor is iLðtÞ =200 e−1000t μA. What is the current through the resistor?
6–22 A 0:033-μF capacitor is connected in series with a 10-kΩ resistor. The voltage across the capacitor is vCðtÞ =10 cosð5000tÞ V. What is the voltage across the resistor?
6–21 The inductor in Figure P6–20 carries an initial current of iLð0Þ = 20mA. At t = 0, the switch opens, and thereafter the voltage across the inductor is vLðtÞ = −6 e−1000t mV. Derive expressions for iLðtÞand pLðtÞ for t > 0. Is the inductor absorbing or delivering power?

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