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

Find the inverse transform of the following function 53 +352 +1 F(s) = $3+6s+11s+6
Use the sum of residues to find the unknown residue in the following expansions: 21(s+5) 6 k (a) + 58s (b) (s+3)(s+10) s+3 s+10 k 2+j5 2-j5 + (s+2)(s2+25) s+2's+j5 s-j5
Use the sum of residues to find the unknown residue in the following expansion: F(s)= 660(s+10) 60 k + (s+1)(s+100) s+1 5+100
Find the inverse transforms of the following rational functions: (a) F(s)= (b) F(s)=. 8 s(s2+4s+1 +8) 4s +4s+8
Find the inverse transforms of the following rational functions: (a) F(s)= 16 (s+2)(s+4) 2(s+2) (b) F(s)=s(s+4)
Find the inverse transform of F(s)= 20(s+3) (s+1)(s+2s+5)
Find the waveforms corresponding to the following transforms: (a) F(s)= (b) F(s)= 6(s+2) s(s+1)(s+4) 4(s+1) s(s+1)(s+4)
Find the waveforms corresponding to the following transforms: (a) Fi(s)= 4 (s+1)(s+3) (b) F(s)=e5s (c) F3(s)= 2s [(s+1)(s+3)] 4(s+2) (s+1)(s+3)
Find the waveform corresponding to the transform F(s)=2- (s+3) s(s+1)(s+2)
Find the poles and zeros of the transform of the following waveform and plot the results on a pole-zero diagram.f ðtÞ = ½e−10tcos 200t +0:05e−10tsin 200tuðtÞ
Find the poles and zeros of the transform of the waveform f ðtÞ = ½4−3 cos 500tuðtÞ and plot the results.
Find the poles and zeros of the transform of the following waveform and plot the results on a pole-zero diagram.f ðtÞ = ½−2e−t −t +2uðtÞ
Find the poles and zeros of the waveform f ðtÞ = ½e−2t + cos2t−sin2tuðtÞ
Find the Laplace transforms of the following waveforms for T >0:(a) f ðtÞ =AuðtÞ−2Auðt−TÞ +Auðt−2TÞ(b) f ðtÞ =Ae−αðt−TÞuðt−TÞ
Find the Laplace transforms of the following waveforms:(a) f ðtÞ =A½cos ðβt−ϕÞuðtÞ(b) f ðtÞ =A½e−αtcos ðβt−ϕÞuðtÞ
Find the Laplace transforms of the following waveforms (a) f(t)= [e-"]u(t) +5 sin4x dx (b) f(t)=5e-40 (1)+5e-4(1) dt
Find the Laplace transforms of the following waveforms:(a) f ðtÞ = e−2t uðtÞ + 4tuðtÞ−uðtÞ(b) f ðtÞ = ½2 + 2 sin2t−2 cos2tuðtÞ
Find the Laplace transform of the waveform f(t)=2u(t)-5[e]u(t)+3[cos2t]u(t) +3[sin2t]u(t)
If iðtÞ=30 e−1200tuðtÞ mA, find the Laplace transform of div. v(t)=0.1- dt
Let υ1ðtÞ =VArðtÞ V. Show that the Laplace transform of υ2ðtÞ = dVArðtÞ=dtV is equal to sV1ðsÞ−υ1ð0−Þ.
Show that the Laplace transform of f ðtÞ = ½cos βtuðtÞ is FðsÞ = s=ðs2 + β2Þ.
If iðtÞ=6 e−1000tuðtÞ mA, find the Laplace transform of = v(t) == 1 10- 6 fi(x) dxV.
Let υ1ðtÞ =VAe−αtuðtÞ V. Show that the Laplace transform ofis equal to V1ðsÞ=s. 02(t) = Vedx V J
Show that the Laplace transform of the ramp function rðtÞ = tuðtÞ is 1=s2.
Use the linearity property to find the Laplace transform of f ðtÞ =A½cosðβtÞ.
Transform the sinusoid iðtÞ = 100½sinð200tÞuðtÞmA into the Laplace domain.
Show that the Laplace transform of the sinusoid f ðtÞ =A½sinðβtÞuðtÞ is FðsÞ = Aβ=ðs2 + β2Þ.
Transform the responseV of a particular RC circuit into the Laplace domain. |p(t)= [10e-1000-5]u(t)
Show that the Laplace transform of f ðtÞ =Að1−e−αtÞuðtÞ is F(s)= Aa s(s+)
Find the Laplace transform of iðtÞ=0:5 δðtÞ A.
Show that the Laplace transform of the impulse function f ðtÞ = δðtÞ is FðsÞ = 1.
Find the Laplace transform of v(t)=8e5tu(t) V.
Show that the Laplace transform f(t) = [e]u(t) is F(s) = 1/(s+a).
Find the Laplace transform of υðtÞ = −7 uðtÞ V.
Show that the Laplace transform of the unit step function f ðtÞ = uðtÞ is FðsÞ = 1=s.
9-4 Initial and Final Value Properties (Sect. 9–6)Given the Laplace transform of a signal, find the initial and final values of the signal waveform.
9-3 Circuit Response Using Laplace Transforms(Sect. 9–5)Given a first- or second-order circuit:(a) Determine the circuit differential equation and the initial conditions (if not given).(b) Transform the differential equation into the s domain and solve for the response transform.(c) Use the
9-2 Inverse Transforms (Sects. 9–4)(a) Find the inverse transform of a given Laplace transform using partial fraction expansion, basic transform properties and pairs, or using software tools.(b) Given a pole-zero diagram, find the respective transform.
9-1 Laplace Transform (Sects. 9–1–9–3)Find the Laplace transform of a given signal waveform using transform properties and pairs, using the integral definition of the Laplace transformation, or using software applications.Locate the poles and zeros of the transform and construct a pole-zero
8-1 Sinusoids and Phasors (Sect. 8–1)Use the additive and derivative properties of phasors to convert sinusoids into phasors and vice versa.
8-2 Impedance (Sects. 8–2, and 8–3)Given a linear circuit in the sinusoidal steady state:(a) Convert R, L, and C elements into impedances in the phasor domain.(b) Use series and parallel equivalence to find the equivalent impedance at a specified pair of terminals.
8-3 Basic Phasor Circuit Analysis and Design (Sects. 8–3, and 8–4)(a) Given a linear circuit in the sinusoidal steady state, find phasor responses using equivalent circuits, circuit reduction, Thévenin or Norton equivalent circuits, and proportionality, or superposition.(b) Given a desired
8-4 General Circuit Analysis (Sect. 8–5)Given a linear circuit in the sinusoidal steady state, find equivalent impedances and phasor responses using node-voltage or mesh-current analysis.
8-5 Average Power and Maximum Power Transfer(Sect. 8–6)Given a linear circuit in the sinusoidal steady state:(a) Find the average power delivered at a specified interface.(b) Find the maximum average power available at a specified interface.(c) Find the load impedance required to draw the maximum
Convert the following sinusoids to phasors in polar and rectangular form: (a) v(t)=20 cos(1501-60) V (b) v(t)=10 cos (1000 +180) V (c) i(t)=-4 cos 31 +3 cos(31-90) A
Convert the following phasors to sinusoids: (a) V-169-45 V at f = 60 Hz (b) V=10/90 +66-j10 V at o= 10 krad/s (c) 1=15+j5+10/180 mA at = 1000 rad/s
(a) Construct the phasors for the following signals:(b) Use the additive property of phasors and the phasors found in(a) to find υðtÞ = υ1ðtÞ + υ2ðtÞ. vi(t) = 10 cos(1000t-45)V 12(t) = 5 cos(1000t+30) V
(a) Construct the phasors for the following signals:(b) Use the additive property to find iðtÞ = i1ðtÞ + i2ðtÞ and check the results using MATLAB. i(t) = 100 cos (2000r) mA i2(t) = 50 cos(20001-60) mA
(a) Construct the phasors representing the following signals:(b) Use the additive property of phasors and the phasors found in(a) to find the sum of these waveforms. iA (t) = 5 cos(377t+50) A iB(t) = 5 cos(377t+170) A ic(t) = 5 cos(3771-70) A
Show that the phasors IA, IB, and IC would still sum to zero if they were all rotated 90counterclockwise
Use the derivative property of phasors to find the time derivative of υðtÞ = 15 cosð200t−30Þ V.
Find the phasor corresponding to the time derivative of the waveform: v(t)=100 cos(1000r)V
(a) Convert the following phasors into sinusoidal waveforms:(b) Use phasor addition to find the sinusoidal waveform υ3ðtÞ = υ1ðtÞ + υ2ðtÞ. V = 20+120 V, w=500 rad/s V = 102e-145 V, co= 500 rad/s
Find the phasor corresponding to the waveform υðtÞ =VA cosðωtÞ + 2VA sinðωtÞ
The circuit in Figure 8–11(a) is operating in the sinusoidal steady state with iðtÞ = 4 cosð5000tÞA. Find the steady-state voltage vðtÞ by hand and by using Multisim
A series circuit is composed of a 1-kΩ resistor, a 1-μF capacitor, and a 100-mH inductor.(a) At what frequency will the magnitude of the impedance of the inductor equal that of the resistor?(b) At what frequency will the magnitude of the impedance of the capacitor equal that of the resistor?(c)
An element in a circuit operating in the sinusoidal steady state has a voltage across it and a current through it as followsFind the impedance of the element. v(t)=50 cos(5001) V and i(t) =4 cos(5001-60) A
A series connection consists of a 12-mH inductor and a 20-pF capacitor. The current flowing through the circuit is iLðtÞ = 20 cos 106t mA.(a) Find the impedance of each element.(b) Find the phasor voltage across each element.(c) Using both hand calculations and a Multisim simulation, find the
The circuit in Figure 8–15(a) is operating in the sinusoidal steady state with υSðtÞ = 35 cos 1000t V.(a) Transform the circuit into the phasor domain.(b) Solve for the phasor current I.(c) Solve for the phasor voltage across each element.(d) Find the waveforms corresponding to the phasors
The circuit in Figure 8–15(a) is operating in the sinusoidal steady state with υSðtÞ = 100 cosð2000t−45Þ V.(a) Transform the circuit into the phasor domain.(b) Solve for the phasor current I.(c) Solve for the phasor voltage across each element.(d) Find the waveforms corresponding to the
Design the voltage divider in Figure 8–16(a) so that an input υs = 15 cos 2000t V produces a steady-state output υOðtÞ = 2 sin 2000t V. Vs + Z FIGURE 8-16 (a) N Z2 Vo
Design the voltage divider in Figure 8–16(a) so that an input υSðtÞ = 50 cosð2000tÞ V produces an output υOðtÞ = 25 cosð2000t−30Þ V. Vs +1 Z + Z2 Vo FIGURE 8-16 (a)
The circuit shown in Figure 8–17(a) is operating in the sinusoidal steady state at a frequency of 100 krad=s. It requires a load of ZL = 1500ff−57:5 Ω to operate properly. Design the load using standard parts to within 5% of the desired values. Rest of the circuit (a) ZL
The purpose of the impedance bridge in Figure 8–18 is to measure the unknown impedance ZX by adjusting known impedances Z1, Z2,and Z3 until the detector voltage VDET is zero. The circuit consists of a sinusoidal source VS driving two voltage dividers connected in parallel. Using the voltage
Equating the real and imaginary parts on each side of this equation yields the parameters of the unknown impedance in terms of the known impedances:Note that adjusting R1 affects only RX. The Maxwell bridge measures inductance by balancing the positive reactance of an unknown inductive device with
Consider the Maxwell bridge shown in Figure 8–18. Suppose we know that the unknown impedance is an unknown capacitor CX in parallel with an unknown resistance RX. Let Z1 be a resistance R1 in series with an inductance L1. Let Z2 be a resistance R2 and Z3 be a resistance R3. Find the relationships
The circuit in Figure 8–21(a) is operating in the sinusoidal steady state with iSðtÞ = 50 cos 2000t mA.(a) Transform the circuit into the phasor domain.(b) Solve for the phasor voltage V.(c) Solve for the phasor current through each element.(d) Find the waveforms corresponding to the phasors
The circuit in Figure 8–21(a) is operating in the sinusoidal steady state with iSðtÞ = 100 cosð1000t−45ÞmA.(a) Transform the circuit into the phasor domain.(b) Solve for the phasor voltage V.(c) Solve for the phasor current through each element.(d) Find the waveforms corresponding to the
Find the steady-state currents iðtÞ, iCðtÞ, and iRðtÞ in the circuit of Figure 8–22 for υS = 100 cos 2000tV, L= 250 mH,C =0:5 μF, and R=3 kΩ. (1)SA FIGURE 8-22 + (1)! L m ic(0) C R(1) R
Using the values in Example 8–10, find the voltage υLðtÞ across the inductor in the circuit shown in Figure 8–22. Validate your answer using Multisim. vs(1) i(t) L m ic(0) FIGURE 8-22 IR(1) C R
In general, the equivalent impedance seen at any pair of terminals can be written in rectangular form asIn a passive circuit the equivalent resistance REQ must always be nonnegative, that is, REQ ≥ 0. However, the equivalent reactance XEQ can be either positive (inductive)or negative
The impedance ZRC is connected in series with the inductor. Therefore, the overall equivalent impedance ZEQ isNote that the equivalent resistance REQ is positive for all ω. However, the equivalent reactance XEQ can be positive or negative. The resonant frequency is found by setting the reactance
The circuit in Figure 8–25 is operating in the sinusoidal steady state. If R=1 kΩ,L= 200 mH, and C =1 μF:(a) Find the value of ω that will cause the circuit to be in resonance.(b) What will the value of ZEQ be under those conditions? ZRC ZEQ m 00 jool. FIGURE 8-25 joc www R
The circuit in Figure 8–26 is operating in the sinusoidal steady state with ω = 5 krad=s.(a) Find the value of capacitance C that causes the input impedance Z to be purely resistive.(b) Find the real part of the input impedance for this value of C. 10 mH 00! 100 Z FIGURE 8-26 EC
The circuit in Figure 8–25 is operating in the sinusoidal steady state at ω = 1 krad=s. If R=1 kΩ,L= 200 mH, and C =1 μF:(a) Find the value of ZEQ classically under those conditions.(b) Repeat the problem using Multisim. (Hint: Drive the circuit with a 1-V ac signal source at 1 krad=s. The
A circuit is operating in the sinusoidal steady state with ω = 377 rad=s. A load ZL of 327ff63:4 Ω is required to be connected to the circuit. It is desired that the load be made purely resistive by the addition of an appropriate reactance. Select the appropriate reactance using standard parts
The circuit in Figure 8–27 is operating at ω = 10 krad=s.(a) Find the equivalent impedance Z.(b) What element should be connected in series with Z to make the total reactance zero? Z 500 -/125 FIGURE 8-27 000 100 j400
In Figure 8–28 υSðtÞ = 12:5 cos 1000t V and iSðtÞ = 0:2 cosð1000t−36:9ÞA. What is the impedance seen by the voltage source and what element is in the box? is(t) Vs(f) FIGURE 8-28 ? 50
Consider the RC circuit in Figure 8–29(a). Find a relationship for the ratio of the output voltage phasor to the input voltage phasor. Plot the magnitude of this ratio as a function of frequency and comment on the result. +1 R www 1/(joC) Vc FIGURE 8-29 (a)
For the circuit of Figure 8–29(a), design a low-pass filter using standard parts so that the cutoff frequency is 1000 rad=s. 1+ FIGURE 8-29 (a) R www 1/(joC) Vc
For the circuit of Figure 8–29(a), replace the capacitor with an inductor.(a) Find the ratio jVL=VSj, and(b) comment on its behavior as the frequency changes from 0 to ∞. 1+ FIGURE 8-29 (a) R www 1/(joC) Vc
Use the unit output method to find the input impedance, current I1, output voltage VC, and current I3 of the circuit in Figure 8–30 for VS =10ff0 V.
Use the unit output method to find the output current IO in the circuit of Figure 8–31. 100 20 ww 50 190 50/15 mA 3 50 240 FIGURE 8-31 www 50 20
Use superposition to find the steady-state voltage υRðtÞ in Figure 8–32 for R=20 Ω, L1 =2mH,L2 =6mH, C =20 μF, υS1 > = 100 cos 5000t V, and υS2 =120 cos ð5000t + 30Þ V. Validate your answer using Multisim. +1 LI m 12 + R C VS2(1) VSI() VR(f) FIGURE 8-32
The two sources in Figure 8–34 have the same frequency. Use superposition to find the phasor current IX. 00 175 w 10 100 --100 0.1-90 A FIGURE 8-34 20/45 V
Use superposition to find the steady-state current iðtÞ in Figure 8–35 for R= 10 kΩ,L= 200mH,υS1 = 24 cos 20,000t V, and υS2 = 8 cosð60,000t + 30Þ V. Comment on using Multisim to solve this problem VSI(t) VS2(f) + i(t) L R FIGURE 8-35
Use superposition to find the output voltage υOðtÞ in the circuit of Figure 8–36 if iSðtÞ = 100 cosð10,000tÞmAand υSðtÞ = 20 cosð20,000t − 45Þ V. 100 w is(t) 1 uF vo(f) Vs(f) FIGURE 8-36
The voltage source in Figure 8–37 produces a 60-Hz sinusoid with a peak amplitude of 200 V plus a 180-Hz third harmonic with a peak amplitude of 10 V, that is, υSðtÞ = 200 cos ð2π 60tÞ + 10 cos ð2π 180tÞ V. The purpose of the LC T-circuit is to reduce the relative size of the
Analyze the sinusoidal steady-state behavior of the circuit shown in Figure 8–37 in more detail. To do so, find the magnitude of the ratio of the output voltage to the input voltage for the range of frequencies from 1Hz to 1 kHz. For simplicity, assume that the input signal always has a magnitude
The Thévenin circuit in Figure 8–41(a) is operating in the phasor domain. If VT = 120ff30 V and ZT = 100 − j50 Ω, perform a source transformation resulting in the Norton circuit of Figure 8–41(b). VT + ZT ww (a) IN w FIGURE 8-41 (b) + V ZN V
The Norton circuit in Figure 8–41(b) has a current source IN = 300 j400mA and a Norton impedance ZN of 100 þ j100 Ω. Find the equivalent Thévenin circuit. IN ZN V FIGURE 8-41 (b)
The circuit of Figure 8–42 is in the phasor domain. Find the Thévenin equivalent circuit that the load ZL sees.Then design a load ZL so that 10ff−90 V are delivered across it. ww 50/0 V -jl kQ FIGURE 8-42 jl kQ + ZL VO= 10/90 V
Convert the Thévenin circuit found in Example 8–20 into its Norton equivalent. Then repeat the design task in that example.
Both sources in Figure 8–43(a) operate at a frequency of ω = 5000 rad=s. Find the steady-state voltage υRðtÞ using source transformations. 1000 V j102 VR 2002 m j30 2 2002-j100 (a) + 120/30 V
Repeat Example 8–21 using Multisim if both sources are operating at a frequency of ω = 20 krad=s. (Hint: Find L1, L2, and C from the data in Example 8–21 first.)
Use Thévenin’s theorem to find the current IX in the bridge circuit shown in Figure 8–44(a). 50 50 1200 (A) B 750 V -j1200 60 (a) D
(a) Find the Thévenin equivalent circuit seen by the inductor in Figure 8–45.(b) Use the Thévenin equivalent to calculate the current IX. ww Ix 500 500 2 50/120 V + j1002 FIGURE 8-45
By inspection, determine the Thévenin equivalent circuit seen by the capacitor in Figure 8–30 for VS =10ff0 V. 500 VA 1000 VB 1000 VC w m Vs +1 I 12 13 -j500 50 FIGURE 8-30 VG=0 ww
In the steady state, the open-circuit voltage at an interface is observed to beWhen a 50-mH inductor is connected across the interface, the interface voltage is observed to beFind the Thévenin equivalent circuit at the interface. Doc(t) =12 cos 2000r V
In the steady state the short-circuit current at an interface is observed to beWhen a 150-Ω resistor is connected across the interface, the interface current is observed to be Find the Norton equivalent phasor circuit at the interface. isc (t)=0.75 sin of A

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