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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

8–77 For the circuit of Figure P8–77 find the Thévenin equivalent circuit seen at the output 1/0 V + 2 ww 12 ee Vx FIGURE P8-77 1+ 10VX VTZT
8–78 The two competingOPAMP circuits in Figure P8–78 are operating in the sinusoidal steady state with ω = 100 krad=s.The two manufacturers both claim that their circuit meets the following specifications:(a) Find the magnitude of the ratio of the output phasor V2 to the input phasor V1 at
8–79 Find the phasor input VS in Figure P8–79 when the phasor output is VO = 300 + j200 V. 102 /2002 ww -m 100-200 = Vo FIGURE P8-79
8–80 The dependent source circuit in Figure P8–80 is operating in the sinusoidal steady state with ω=1 krad=s and μ=104. Find the phasor gain K =VO=VS and the input impedance ZIN seen by VS. Validate your answer using Multisim 10 www -j10 kQ 10 ww - -j10 kQ= +1 +1 Vo Vx ZIN FIGURE P8-80
8–81 Find the phasor gainK =VO=VS and input impedance ZIN of the circuit in Figure P8–81. ZIN 10 w -25 -25 (+1) Vs 47 FIGURE P8-81 Vo
8–82 Find the phasor gain K = VO=VS, input impedance ZIN of the circuit, output impedance ZOUT of the circuit, and the capacitor current IX in Figure P8–82. Vs + I 100 w j100 2 000 -100 100 Vo ZIN ZOUT FIGURE P8-82
8–83 Given the circuit in Figure P8–83:(a) Use node-voltage or mesh-current analysis to develop a set of matrix equations for the circuit.(b) Use MATLAB to solve the matrix equations and then find the phasor gain K ¼VO=VS and input impedance ZIN of the circuit.(c) Without using the matrix
8–84 A load consisting of a 3:3-kΩ resistor in series with a 1:5-μF capacitor is connected across a voltage source vSðtÞ = 169:7 cos ð377tÞ V. Find the phasor voltage, current, and average power delivered to the load.
8–85 A load consisting of a 1-kΩ resistor in parallel with a 0:33-μF capacitor is connected across a current source delivering iSðtÞ = 15 cos ð3000tÞmA. Find the average power delivered to the load.
8–86 The circuit in Figure P8–86 is operating in the sinusoidal steady state at a frequency of 10 krad=s. Use Multisim to find the average power delivered to the 100-Ω resistor 120 V(+ -j402 HH 180 18-90 V + PAVG 100 FIGURE P8-86
8–87 You have a task of designing a load that ensure maximum power is delivered to it. The load needs to be connected to a source circuit that is not readily observable, but that you can make measurements at its output terminals.You measure the open-circuit voltage and read 120ff0 V.You then
8–88 The load in Figure P8–88 needs to be designed for maximum power transfer.(a) Find the maximum available average power at the interface shown in the figure.(b) Specify the load required to extract the maximum average power. 1 mH m 500 pF 1 ww +1 + V PMAX Load 30 cos 10% V FIGURE P8-88
8–89 The RC load in Figure P8–89 needs to be designed for maximum power transfer.(a) Find the maximum average power available at the interface in the figure.(b) Specify the values of R and C that will extract the maximum power from the source circuit R ww Load i(f) 500 mH m 500 ww + + 10 cos
8–90 AC Voltage Measurement An ac voltmeter measurement indicates the amplitude of a sinusoid and not its phase angle. The magnitude and phase can be inferred by making several measurements and using KVL. For example, Figure P8–90 shows a relay coil of unknown resistance and inductance. The
8–91 Home Power Distribution The circuit of Figure P8–91 emulates a typical 60-Hz residential power system. There are three wires entering the house, two are called “hot” and the remaining one is called the return or “neutral.”Each hot line is protected by a circuit breaker—but not
8–92 OP AMP Bandpass Filter Use the analysis methods discussed in Example 8–30 to find the input–output relationshipVO=VS for the active bandpass filter of Figure P8–92. Treat each stage separately and then multiply the input–output relationships from each stage to obtain the overall
8–93 Power Transmission Efficiency A power transmission circuit with a source voltage of VS = 880 + j0 V can be modeled as shown in Figure P8–93.(a) Find the average power produced by the source, lost in the wires, and delivered to the load.(b) Whatis the transmission efficiency defined as η =
8–94 60-Hz Filter A 1-kΩ resistor models an important and sensitive laboratory instrument. The instrument measures a desired signal that varies from 1Hz to 500 Hz. However, interference from power lines in the laboratory causes the instrument to saturate. A vendor has designed a device that he
8–95 AC Circuit Design Select values ofL and C in Figure P8–95 so that the input impedance seen by the voltage source is 50 + j0 Ω when the frequency is ω = 106 rad=s. For these values of L and C, find the output Thévenin impedance seen by the 300-Ω load resistor. Vs +1 50 joL m 1 300 2
8–96 AC Circuit Analysis Ten years after graduating with a BSEE, you decide to go to graduate schools for a master’s degree. In desperate need of income, you agree to sign on as a grader in the basic circuit analysis course. One of the problems asks the students to find vðtÞ in Figure P8–96
7-1 First-order Circuit Analysis (Sects. 7–1 to 7–4)Given a first-order RC or RL circuit:(a) Find the circuit differential equation, the circuit time constant, and the initial conditions (if not given).(b) Find the zero-input response.(c) Find the complete response for step function,
7-2 First-order Circuit Design (Sects. 7–1 to 7–4)Given responses in a first-order RC or RL circuit:(a) Find the circuit parameters or other responses.(b) Design a circuit to produce the given responses.
7-3 Second-order Circuit Analysis (Sects. 7–5 to 7–7)Given a second-order circuit:(a) Find the circuit differential equation.(b) Find the circuit natural frequencies and the initial conditions (if not given).(c) Find the zero-input response.(d) Find the complete response for a step function
7-4 Second-order Circuit Design (Sects. 7–5 to 7–7)Given responses in a second-order RLC circuit:(a) Find the circuit parameters or other responses.(b) Design a circuit to produce the given responses
Find the time constants for circuits C1 and C2 in Figure 7–4. W R3 www L m R C1 C C HA R C2 R ww 48 FIGURE 7-4 C3 25 R R www
Find the time constant TC for circuit C3 in Figure 7–4 R3 www L R C C C1 R C2 R w ee 48 FIGURE 7-4 C3 R www 25 R
The switch in Figure 7–5(a) closes at t = 0, connecting a 1-μF capacitor with 10 V initially stored across it to two resistors in series. Find the responses vCðtÞ and iðtÞ for t ≥ 0. Write an equation for the power pRðtÞ absorbed by the equivalent resistance.Validate your answers using
The switch in Figure 7–6 closes at t = 0. For t ≥ 0 the current through the resistor is iRðtÞ = e−100t mA.(a) What is the capacitor voltage at t =0?(b) Write an equation for υðtÞ for t ≥ 0.(c) Write an equation for the power absorbed by the resistor for t ≥ 0.(d) How much energy does
Find the response of the state variable of the RL circuit in Figure 7–7 using L1 = 10 mH,L2 = 30 mH,R1 =2 kΩ,R2 =6 kΩ, and iLð0Þ = 100 mA. RR ww Given circuit RR 12 i(0) L m R + R v(t) L+L Equivalent circuit FIGURE 7-7
Find the current through R2 and the power dissipated in R1 in Example 7–3.
The switch in Figure 7–8 is closed at t = 0, connecting a capacitor with an initial voltage of 30 V to the resistances shown. Find the responsesυCðtÞ, iðtÞ, i1ðtÞ, and i2ðtÞ for t ≥ 0. 0 10 www i(t) i(t) 12(1) + vc (f). 30 V - 20 20 0.5 F FIGURE 7-8 REQ
The switch in the RL circuit of Figure 7–9(a) moves instantly from position A to position B at t = 0. If the current flowing through the inductor at t = 0 is 1 mA, how long after the switch moves to position B does it take for the voltage across the resistor to reach−5 V? Validate your answer
UseMultisim to analyze the zero-input transient behavior of the first-order OPAMP RC circuit shown in Figure 7–10, when C = 1 μF, R1 =R2 = 200 kΩ, and vCð0Þ = −10 V.Find the responses vOðtÞ, iR1 ðtÞ, iR2 ðtÞ, and iCðtÞ for t ≥ 0. Display your results on one Grapher View plot +1
A100-mH inductor and two resistors are all connected in parallel. One resistor is 100 Ω and the second is 470 Ω. At time t = 0, the inductor has 100 mA flowing through it. Use Multisim to calculate a transient plot of the current through the inductor and through each resistor for t ≥ 0.
Find the response of the RC circuit in Figure 7–15. iR, (1) R1 w +1 + Vu(t) R C1 v(t) ++ Vol V02 VA = 100 V at f=0 Vol 5 V C = 0.1 F C = 0.5 F V02 = 10 V R = 30 k R = 10 k FIGURE 7-15
Use the results from Example 7–6 and find the current through R1 in Figure 7–15 for t ≥ 0. IR, (1) R ww + VAM(1) Vu(t) + R v(1) C Vol C2V02 VA = 100 V at=0 Vol 5 V C = 0.1 F C = 0.5 F R = 30 k V02 = 10 V R = 10 kQ FIGURE 7-15
Find the step response of the RL circuit in Figure 7–16(a). The initial condition is i(0) = I0. Iu(t) R ww i(t) R v(t) L w FIGURE 7-16 (a)
Use the results from Example 7–7 and find the voltage across the current source in Figure 7–16(a) for t ≥ 0. Iu(t) R ww i(t) R v(t) L w FIGURE 7-16 (a)
The state variable response of a first-order RC circuit for a step function input is(a) What is the circuit time constant?(b) What is the initial voltage across the capacitor?(c) What is the amplitude of the forced response?(d) At what time is υCðtÞ = 0?(e) Use MATLAB to display the state
Given the first-order circuit step response(a) What is the amplitude of the step input?(b) What is the circuit time constant?(c) What is the initial value of the state variable?(d) What is the circuit differential equation? vc(1) 20-20e-1000r V 10
Find the solutions of the following first-order differential equations (a) 10-4duc(1) +vc(t)=-5u(t) vc(0)=5V di (b) 5 x 10-dir()+2000iL (t)=10u(t) i (0)=-5mA dt
The operation of a digital system is controlled by a clock waveform that provides a standard timing reference. At its source a clock waveform can be described by a rectangular pulse of the formIn this example the pulse amplitude is VA = 5 V and the pulse duration is T = 10 ns.This clock pulse
The element in Figure 7–19 is a 1-μF capacitor. The switch closes at t = 0. Find the zero-state response of the capacitor voltage vCðtÞ for t ≥ 0. 10 V 10 m 4(1) Element + v(t). ww 10 + 1=0 10 5kQ FIGURE 7-19
The element in Figure 7–19 is a 1-mH inductor. The switch closes at t = 0. Find the zero-state response of the inductor current iLðtÞ for t ≥ 0. 4(1) Element + vc(1) 10 W ww 10 1=0 10 V 10 5 FIGURE 7-19
The switch in Figure 7–22(a) has been open for a long time and is closed at t = 0. Find the inductor current for t ≥ 0. 1=00 R ww i(t) VA R v(t) L (a)
The switch in Figure 7–22(a) has been closed for a long time. The switch opens at t = 0. Find the inductor current for t ≥ 0. 10=1 R ww VA R2 i(t) v(t) L (a)
The switch in the circuit of Figure 7–23 has been open for a long time. It closes at t =0.Find the current iRðtÞ for t ≥ 0. 20 w 10 kQ=0 50 V 20 ks2iR(!) FIGURE 7-23 + 0.1 F (t)
The switch in the circuit of Figure 7–23 has been closed for a long time. It opens at t = 0. Find the voltage υCðtÞ and the current iRðtÞ for t ≥ 0 20 w 10kQ=0 =50 V + 0.1uFvc() 20 ksIR(1) FIGURE 7-23
Design a first-order RC circuit using standard parts (see inside rear cover) that will produce the following voltage across the capacitor: υC(t) = 50−100e−2000t V.
Design a first-order RL circuit that will produce the following current through the inductor:iLðtÞ = 5−5e−500t mA for t ≥ 0. Use standard values for the components.
For t ≥ 0 the state variable response of the RL circuit in Figure 7–25(a) is observed to be(a) Identify the forced and natural components of the response.(b) Find the circuit time constant.(c) Find the Thévenin equivalent circuit seen by the inductor. iL(t)=50+100e-5000 mA
Use Multisim to to find the inductor current iLðtÞ and voltage υLðtÞ for t ≥ 0 for the circuit in Figure 7–25. (Hint: Make certain you have the direction of the inital condition correct.)\ Voltage (V) Transient Analysis 0 -10 -20 -30 150 m 140 m 130 m Inductor voltage v (f) 120 m 110 m -40
The switch in Figure 7–26 moves from positionAto position B at t = 0. The first-order RC circuit in the figure must be designed to produce an output ofEvaluate the two proposed circuit designs shown in the figure using the following criteria.(a) A design must produce the required output.(b) If
There is a need to design an interface circuit in Figure 7–27(a) so that the output voltage vOðtÞ across the 100-Ω load equals 10 1−e−100t V for t ≥ 0.Use the fewest number of components possible 7-0 B + Interface 15 V Vo(t) 100 circuit FIGURE 7-27 (a) www 50 300 F (b)
In the circuit in Figure 7–28 the switch has been in position A for a long time and is moved to position B at t = 0. For t ≥ 0 find the output voltage υOðtÞ 10 V t=0 50 50 A w is(f) B 25 2volt) 1+ 10 V 150 mH +1 FIGURE 7-28
Find the response of the RC circuit in Figure 7–29 to an exponential forcing function.The initial capacitor voltage is υ(0) =V0.
The capacitor in the circuit of Figure 7–30 is in the zero state. Find the voltage across and the current through the capacitor for t ≥ 0. 300 w Vs()+200 k2 0.001 F Vs(t)=100 e 1000 (1)V FIGURE 7-30 ic(1) + vc(f)
The switch in Figure 7–32(a) has been open for a long time and is closed at t = 0. Find the voltage υðtÞ for t ≥ 0 when υSðtÞ = ½20 sin 1000tuðtÞ V. 4 i(t) 1=0 ww (vs (1) 4 kv(t)=1 F FIGURE 7-32 (a)
Find the sinusoidal steady-state response of the output voltage υOðtÞ in Figure 7–33 when the input current is iSðtÞ = ½IA cos ωtuðtÞA. i(f) + 1=0 + R LVO(1) is (f) FIGURE 7-33
Find the forced component solution of the differential equationfor the following frequencies:(a) ω = 500 rad=s (b) ω = 1000 rad=s (c) ω = 2000 rad=s 10-3 du(1) dt +v(t)=10 cos oor V
The RC circuit in Figure 7–29 is driven by an input vSðtÞ = 10 sinð2π100 tÞ uðtÞ and has an RT of 47 kΩ and a C of 0:1 μF. The capacitor has an initial voltage of −20 V.(a) Use Multisim to plot the transient output voltage vðtÞ across the capacitor. In your plot include the input
Aseries RLC circuit has a C =0:25 μF and L= 1 H. Find the roots of the characteristic equation for RT =8:5 kΩ, 4kΩ, and 1 kΩ.
For a series RLC circuit:(a) Find the roots of the characteristic equation when RT =2 kΩ, L= 100 mH,and C =0:4 μF.(b) For L= 100 mH, select the values of RT and C so the roots of the characteristic equation are s1, s2 = −1000j2000.(c) Select the values of RT,L, and C so s1 = s2 = −104.
The circuit of Figure 7–36 has C =0:25 μF and L= 1 H. The switch has been open for a long time and is closed at t = 0. Find the capacitor voltage for t ≥ 0 for ðaÞ R= 8:5 kΩ,ðbÞ R= 4 kΩ, and ðcÞ R= 1 kΩ. The initial conditions are I0 = 0 and V0 = 15 V. t=0 Li(0)=0 C + + R vc(t): 15 V
The circuit in Figure 7–36 has C =0:02 μF and L= 100 mH. Select a value for R that will produce the critically damped case. 1=0 R FIGURE 7-36 L (0)=0 m + C + vc(t) 15 V
In a series RLC circuit the zero-input voltage across the 1-μF capacitor isυCðtÞ = 10e−1000tsin 2000t V t ≥ 0(a) Find the circuit characteristic equation.(b) Find R and L.(c) Find iLðtÞ for t ≥ 0.(d) Find the initial values of the state variables.
In a series RLC circuit, R= 250 Ω, L= 10 mH, C =1 μF, V0 = 0, and I0 = 30 mA. Find the capacitor voltage and inductor current for t ≥ 0.
In a series RLC circuit the zero-input responses are(a) Find the circuit characteristic equation.(b) Find the initial values of the state variables.(c) Find R, L, and C. vc(t) 2000te-500 V iL(t)=3.2e-500-1600te-500 mA
In a parallel RLC circuit RT = 500 Ω, C =1 μF, L=0:2H. The initial conditions are I0 = 50mA and V0 = 0. Find the zero-input response of inductor current, resistor current, and capacitor voltage
A parallel RLC circuit has R=1 kΩ,C =1 μF,and L= 100 mH. The initial conditions are I0 = 100mA and V0 =0V.(a) Use Multisim to plot the zero-input response of the inductor, resistor, and capacitor currents on one y-axis.(b) From your current plots, show that Kirchhoff’s current law (KCL) holds
The switch in Figure 7–40 has been open for a long time and is closed at t =0.(a) Find the initial conditions at t =0.(b) Find the inductor current for t ≥ 0.(c) Find the capacitor voltage and current through the switch for t ≥ 0. 9V +1 1H4uF: + 250 50 ww t=0 isw(t) iL(t) FIGURE 7-40 vc(t)
The zero-input responses of a parallel RLC circuit are observed to be(a) What is the circuit characteristic equation?(b) What are the initial values of the state variables?(c) What are the values of R, L, and C?(d) Write an expression for the current through the resistor.(e) Use MATLAB to plot your
The series RLC circuit in Figure 7–42(a) is driven by a step function and is in the zero state at t = 0. Find the capacitor voltage for t ≥ 0, when VA =10V, R=1 kΩ, C =0:5 μF, and L= 2H. +1 R w VAu(t) (a) FIGURE 7-42 L C + vc(f)
The series RLC circuit of Figure 7–42 is excited by a 10-V step source and has the initial conditions iLð0Þ = 0A and vCð0Þ = −5 V. The parameters are L= 2H and C =0:5 μF. Use Multisim to vary R from underdamped to overdamped to view the effects of the damping ratio ζ on the voltage across
In Example 7–23 we plotted the effect of ζ on the circuit response by varying the series resistor R. In this exercise, we would like to look at the effect on the circuit response by varying the series capacitor C. Let R= 1kΩ, L= 2H, VA = 10 V, iLð0Þ = 0A, and vCð0Þ = −5 V. Set C at a
The circuit in Figure 7–45 is in the zero state. Find the current through the resistor for t ≥ 0. iR(f) Fillo 0.1F 25 u(t) mA 47002 100 mH FIGURE 7-45
Use Multisim to plot the currents through the three elements in circuit of Figure 7–45. Show that sum of all three element currents equals the source current for all time. iR(1) iL(1) 0.1F 25 u(t) mA 470 100 mH FIGURE 7-45
(a) Select a value for R that will cause the RLC circuit of Figure 7–47 to produce a critically damped response. All parameters except R remain the same.(b) UseMultisimto determine themaximumvalue of υOðtÞ and the time at which it reaches that value.(c) What is the value of the maximum power
Design a series RLC circuit whose zero-state step response iswhere VA is the amplitude of the step function input Dc(t)=VA-VA-400 -2000r
The step response of a series RLC circuit is observed to be(a) What is the circuit characteristic equation?(b) What are the initial values of the state variables?(c) What is the amplitude of the step input?(d) What are the values of R, L, and C?(e) What is the voltage across the resistor? vc(1)
What range of source resistance will produce an underdamped natural response in a parallel RLC circuit with L= 200mH and C =0:032 μF?
(a) What range of source resistance will produce an underdamped natural response in a series RLC circuit with L= 200mH and C =0:032 μF?(b) Compare your answer with the parallel circuit solution in Example 7–26.
Design a series RLC circuit with ζ=1:5 and ω0 = 50 krad=s. You must use a 0:1-μF capacitor.
7–1 Find the function iðtÞ that satisfies the following differential equation and the initial condition di(t) 500- +25ki(t)=0, i(0)=25 mA dt
7–2 Find the function vðtÞ that satisfies the following differential equation and initial condition: 10-dv(1) dt + v(t)=0, v(0) 100 V
7–3 Find the time constants of the circuits in Figure P7–3. 250 mH 250 2 150 mH ww - 150 C1 3.3 mH 220 22202 10 mH C2 FIGURE P7-3 20 mH
7–4 Find the time constants of the circuits in Figure P7–4. 0.33 F= 1.5 ww 1 w ww 3 C1 w 1 0.1 F 47 47 0.1 F C2 FIGURE P7-4
7–5 Each of the two circuits in Figure P7–5 has a switch that affects their time constants. For circuit C1, find the time constant when the switch is in position A and repeat for position B. For circuit C2, find the time constant when the switch is closed and repeat when it is open. B R R C R w
7–6 The switch in Figure P7–6 is closed at t = 0. The initial voltage on the capacitor is vCð0Þ = 100 V.(a) Find vCðtÞ and iOðtÞ for t ≥ 0.(b) UseMATLAB to plot the waveforms for vCðtÞ and iOðtÞ.(c) Simulate the problem using Multisim and compare the results to the plots in part
7–7 In Figure P7–7 the initial current through the inductor is iLð0Þ = 5mA.(a) Find iLðtÞ and vOðtÞ for t ≥ 0.(b) UseMATLAB to plot the waveforms for iLðtÞ and vOðtÞ.(c) Simulate the problem using Multisim and compare the results to the plots in part (b). 100 k + 150 kovo(1) 0.5 H
7–8 The switch in Figure P7–8 has been in position A for a long time and is moved to position B at t = 0. Find iLðtÞ for t ≥ 0. 1-0 B 100. 401 25 mH 1.50 2 10 mA FIGURE P7-8
7–9 The circuit in Figure P7–9 has 10 V stored across the two capacitors, plus on top, at t =0. If C1 = 10,000 pF, C2 = 15,000 pF, R1 =R2 =R3 = 330 kΩ, find vOðtÞ for t ≥ 0. vo(t) R w C FIGURE P7-9
7–10 The switch in Figure P7–10 has been in position A for a long time and is moved to position B at t = 0. Find vCðtÞ for t ≥ 0. www A B 10 811=0. vc(f)= 0.05 F 100 + 15 V FIGURE P7-10
7–11 The switch in Figure P7–11 has been open for a long time and is closed at t = 0. Find iLðtÞ for t ≥ 0. 100 mA 1502 w FIGURE P7-11 100 ww 1=0 i(1) 15 mH
7–12 The switch in the circuit in Figure P7–12 has been in position A for a long time. At t = 0 it switches to position B; find vRðtÞ for t ≥ 0. Verify your result using Multisim + VR(t) 1 1=0 B A 10 ks 100 mH 10 mA k2 FIGURE P7-12 100 mH
7–13 The circuit in Figure P7–13 is in the zero state. Find the voltage vOðtÞ for t ≥ 0 when an input of iSðtÞ = IAuðtÞ is applied. Identify the forced and natural components in the output. W is(t) R C FIGURE P7-13 vo(t)
7–14 The circuit in Figure P7–14 is in the zero state when the input vSðtÞ =VAuðtÞ is applied. Find vOðtÞ for t ≥ 0. Identify the forced and natural components in the output. L eee + +- +vs(t) R vo(t) FIGURE P7-14
7–15 The circuit in Figure P7–15 is in the zero state when the input vSðtÞ = 150uðtÞ is applied. If C = 0:022 μF and R= 82 kΩ, find vOðtÞ for t ≥ 0. Identify the forced and natural components in the output. C R ww +1 + vs(t) R vo(t) FIGURE P7-15
7–16 The circuit in Figure P7–16 is in the zero state when the input vSðtÞ = 24 uðtÞ is applied. If L= 150mH and R= 1:5 kΩ, find vOðtÞ for t ≥ 0. Identify the forced and natural components in the output. On a single set of axes, use MATLAB to plot the forced response, the natural
7–17 The switch in Figure P7–17 has been in position A for a long time and is moved to position B at t = 0. Find vCðtÞ for t ≥ 0. Identify the forced and natural components in the response. 1=0 A w 10 B 12 V 10 ww + 10 k2 0.01 F: vc(t) FIGURE P7-17

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