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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

16–9 A load draws 15AðrmsÞ, 6 kW, and 4:5 kVARS (lagging)from a 60-Hz source. Find the load power factor and impedance.
16–8 A load draws 8 kWat a leading power factor of 0:8 from an 880-V ðrmsÞ source. Find the load current and the load impedance.
16–7 An inductive load draws an apparent power of 30 kVA at a power factor of 0:6 from a 2400-V ðrmsÞ source. Find the complex power S and the load impedance Z.
16–6 Find the load impedance Z for the following complex powers when jIj = 100 A.(a) S = 1000 + j250 kVA,(b) jSj = 15 kVA, P = 12 kW, Q < 0.
16–5 Given the load circuit in Figure P16–3, find the complex power delivered to the load impedance Z.(a) V = 150ff45 V ðrmsÞ, Z = 50ff15 Ω(b) Z = 30 − j40 Ω, I = 10ff−25 AðrmsÞ
16–4 The following sets of V and I apply to the circuit in Figure P16–3. Calculate the complex power and the power factor. State whether the power factor is lagging or leading.(a) V = 120ff30 V ðrmsÞ, I = 3:3ff−15 AðrmsÞ (b) V = 440ff45 V ðrmsÞ, I = 8ff95 AðrmsÞ
16–2 The following sets of vðtÞ and iðtÞ apply to the load circuit in Figure P16–1. Calculate the average power and the reactive power.(a) vðtÞ = 135 sinðωtÞ V, iðtÞ = 1:5 cos ðωt + 30ÞA(b) vðtÞ = 370 cos ðωtÞ V, iðtÞ = 10 sinðωt + 20ÞA
An average power of 20 kW is delivered to a balanced Δ-connected load with ZΔ =30+j45 Ω=phase. Find the line voltage VL at the load and the complex power delivered to the load
In a balanced three-phase circuit, the line voltage at the load is 4160 VðrmsÞ and the apparent power delivered to the load is 60 kVA. Find the line current
The phase B line current in a Δ-connected load with ZΔ =14+j9Ω per phase is IB = 26ff−165 AðrmsÞ. Find IAB and VAB for a positive phase sequence.
A balanced three-phase source with VL = 200 VðrmsÞ feeds a Δ-connected load with ZΔ =12+j6 Ω per phase through a three-wire line with ZW =0:1+j0:55 Ω per phase.Find the line current and phase current phasors using ffIA =0 as the phase reference
The line voltage at a Δ-connected load with ZΔ = 520 + j400 Ω per phase is VL = 1300 VðrmsÞ. Find IA and IAB using ffVA = 0 as the phase reference. Verify your results using Multisim. Assume 60 Hz.
The line voltage at a Δ-connected load (see Figure 16–22(a)) with ZΔ =40+j30 Ω per phase is VL = 2:4 kVðrmsÞ. Find the line and phase current phasors using ffVAN =0 as the phase reference. Verify your results using Multisim. Assume 60 Hz
In a balanced Y-Y circuit, the load and line impedances are ZY =16+j12 Ω per phase and ZW = 0:25 + j1:5 Ω per phase. The line current is IL = 14:2AðrmsÞ. Find the line voltage phasors at the source using ffV~AN = 0 as the phase reference. Verify your reults using Multisim.Assume 60 Hz.
In a balanced Y-Y circuit, the line voltage is VL = 480 VðrmsÞ and the phase impedance is ZY =24+j9 Ω per phase. Using ffVAN =0 as the phase reference, find the line current and line voltage phasors for a positive phase sequence.
Figure 16–17 shows a balanced Δ-connected load in parallel with a balanced Y-connected load. The two-phase impedances are ZΔ = 120 + j40 Ω and ZY =50+j30 Ω. Find the phase impedance of an equivalent Y-connected load.
In a balanced three-phase circuit VBC = 480ff−120 VðrmsÞ. Find the phase voltages for a positive phase sequence.
In a balanced three-phase circuit, the rms line voltage is VL = 7:2 kVðrmsÞ. Find all of the phase and line voltages for a positive phase sequence using ffVAN =0 as the phase reference.
In a balanced three-phase circuit, the line voltages have an rms value of VL =480 VðrmsÞ.Using ffVAB as the phase reference, find all of the line and phase voltages for a positive phase sequence
For the load conditions in Example 16–11, find the capacitance needed to raise the power factor to unity.
A single-phase source supplies a load through a two-wire line with an impedance of ZW =2+j10 Ω per wire. The rms load voltage is 2:4 kV and the load receives an apparent power of 25 kVA at a lagging power factor of 0.85. Find the required source power and rms voltage.
Repeat Example 16–8 when the load power factor is increased to 0.95 and all other conditions remain the same.
A load consisting of a 50-Ω resistor in parallel with an inductor whose reactance is 75 Ω is connected across a 500-VðrmsÞ source. Find the complex power delivered to the load and the load power factor. State whether the power factor is leading or lagging.
A load consisting of a 2:5-kΩ resistor in parallel with a 2-μF capacitor is connected across a 440-VðrmsÞ, 60-Hz voltage source. Find the complex power delivered to the load and the load power factor. State whether the power factor is leading or lagging.Validate your answers using Multism.
Find the impedance of a two-terminal load under the following conditions.(a) V= 120ff30 V ðrmsÞ and I = 20ff75 AðrmsÞ (b) jSj = 3:3 kVA, Q= −1:8 kVAR, and Irms = 7:5A
At 440 VðrmsÞ a two-terminal load draws 3 kVA of apparent power at a lagging power factor of 0.9. Find(a) Irms(b) P(c) Q(d) the load impedance Draw the power triangle for the load.
Determine the average power, reactive power, and apparent power for the following voltage and current phasors. State whether the power factor is lagging or leading.(a) V= 208ff−90 VðrmsÞ, I = 1:75ff−75 AðrmsÞ (b) V= 277ff + 90 VðrmsÞ, I = 11:3ff0 AðrmsÞ
Find the average power, reactive power, and apparent power for the following voltage and current phasors. Find the power factor and state whether it is lagging or leading.V= 350ff45 VðrmsÞ, I = 6ff65 AðrmsÞ
16-6 Three-Phase Power Flow (Sect. 16–7)Given a single-line diagram of a balanced three-phase system, find the source outputs and bus voltages that produce a prescribed load power flow.
16-5 Three-Phase Circuit Analysis (Sect. 16–6)For a given balanced three-phase circuit:(a) Find the line and phase current phasors for a specified phase reference.(b) Find the source or load power using the scalars VL and IL.
16-4 Balanced Three-Phase Circuits (Sect. 16–5)For a balanced three-phase circuit:(a) Find all of the phase and line voltage phasors for a given phase reference.(b) Find equivalent Y- or Δ-connected sources and loads.
16-3 Single-Phase Power Flow (Sect. 16–4)Given a specified load power in a single-phase circuit:(a) Find the required source outputs.(b) Find the parallel capacitance needed to produce a specified power factor.
16-2 Single-Phase Circuit Analysis (Sect. 16–3)Given a single-phase circuit operating in the ac steady state, find the power produced by sources or delivered to specified loads.
16-1 Complex Power (Sects. 16–1 and 16–2)Given a linear circuit in the sinusoidal steady state:(a) Find the average, reactive, and instantaneous power for a specified voltage and current.(b) Find the load impedance for a specified load power flow.
For the coupled inductors in Figure 15–4:(a) Write the i−υ characteristics using the reference marks shown in the figure.(b) For υSðtÞ = 200 sin 400t V, find υ2ðtÞ when the output terminals are open-circuitedði2 =0Þ. i(1) M Vs(f) (+ V(f) Li i2(t) V2(1) L = L = 10 mH M = 2 mH FIGURE 15-4
In Figure 15–4, i1ðtÞ = −20 cos 8000t mAand i2ðtÞ = 0. Find υ1ðtÞ and υ2ðtÞ. vs(f) (+ +1. i (1) V(f) Li ee M i2(t) p+ V2(1) L = L = 10 mH M = 2 mH FIGURE 15-4
For the coupled inductors in Figure 15–5:(a) Write the i− υ characteristics using the reference marks shown in the figure.(b) For iSðtÞ = 2cos500t A, find υ1ðtÞ and i2ðtÞ when the output terminals are shortcircuitedðυ2 =0Þ. in(t) M i2(t) (1)A is vt) LL V(1) L =50 mH, L2 75 mH, M = 60
Find υ1ðtÞ and υ2ðtÞ for the circuit in Figure 15–6. + M i(t) i2(t) V(t) v(1) 5 cos 10,000r A LI | 12 2 sin 50001 A FIGURE 15-6 L =0.2 mH, L2 = 0.5 mH M = 0.3 mH
Since a transformer changes impedance levels, it can be used to match a source and load to achieve maximum power transfer. Figure 15–15 shows a circuit model of an audio amplifier with an output impedance of 600 Ω feeding an 8-Ω speaker. Find the transformer turns ratio needed to achieve
In a transformer, the primary and secondary windings are magnetically coupled but are usually electrically isolated. Transformer performance in some applications can be improved by electrically connecting the two magnetically coupled windings in a configuration called an autotransformer. Figure
The source circuit in Figure 15–17 has ZS =0+j20 Ω and VS = 2500 ff0 Vatω = 377 rad=s. The transformer has L1 =2H,L2 =0:2H,andM=0:6H. The load impedance is ZL =25+j15 Ω. Find IA, IB,V1,V2,ZIN, and the average power delivered by the source at input interface Vs +1 Zs ZIN + joM 12 m IA joL
Find IA, IB,V1,V2, and the impedance seen by the voltage source in Figure 15–18. www I 12 50 ww j2002 V 100 100 2 V 30100 100/0 V IA IB - 175 600
Figure 15–19 shows an ideal transformer connected as an autotransformer. Find the voltage and average power delivered to the load ZL for VS = 500 ff0 V, ZS = j10 Ω,ZL =50+j0 Ω,N1 = 200,and N2 = 280. Zs Vs + V2 ee CH Ideal FIGURE 15-19
Using the results in Example 15–11, find the input impedance seen by the source in Figure 15–19. Zs Vs + V 12 N ee C Ideal FIGURE 15-19
The linear transformer in Figure 15–20 is in the sinusoidal steady state with reactances of X1 =25 Ω,X2 =16 Ω,XM =18 Ω, and a load impedance of ZL =25−j10 Ω. Find V2 and I2 when VS = 100 ff0 V. 12 +1 + jXM VsV V X1 X2 V2 ZL FIGURE 15-20
Using the results found in Example 15–12, find the input impedance seen by the source in Figure 15–20. 12 jXM + V X1 X2 V2 ZL Vs (V +1 FIGURE 15-20
A transformer transfers signals or power from one circuit to another strictly by magnetic induction without conductive paths between the circuits. Such circuits are said to be electrically isolated since there is no electric current flowing directly from one to the other. Any transformer provides
15–1 In Figure P15–1 L1 = 10 mH, L2 = 5 mH, M = 7 mH, and vSðtÞ = 200 sin 100t V.(a) Write the i – v relationships for the coupled inductors using the reference marks in the figure.(b) Solve for v2ðtÞ when the output terminals are open circuited (i2 = 0). Vs(f) +1 (1) M (1) V(1) V(1) LI
15–4 In Figure P15–4 L1 = L2 = 3 mH, M = 2 mH, and iSðtÞ = 50 sin 100t A. Solve for v1ðtÞ and v2ðtÞ when the output terminals are open circuited (i2 = 0).15–5 In Figure P15–4 L1 =L2 = 3 mH, M= 2 mH, and i2ðtÞ= 0:5 sin 1000t A when the output terminals are shortcircuited.Solve for
15–7 In Figure P15–7 L1 =2H, L2 =8H, M= 4 H, and i1ðtÞ = 5 sinð1000tÞmA. Find the input voltage vXðtÞ. + vx(t) i(t) Vi(t) M ell 000 FIGURE P15-7 i2(t) V2(f)
15–8 In Figure P15–8 show that LEQ = L1 1 − k2 , where k is the coupling coefficient. LEQ FIGURE P15-8 M 5
15–9 In Figure P15–9 show that the indicated open-circuit voltage is Voc = (kL2/L)vi where k is the coupling coefficient. i(t) Mi2(t) + + vs(1) () v(t) V2(t) voc(t) L FIGURE P15-9
15–10 In Figure P15–10 show that the indicated short-circuit current iswhere k is the coupling coefficient. Assume that i1 has no dc component isc=(kL1/L2i
15–12 In Figure P15–12RS = 40Ω, RL = 1280 Ω, the turns ratio is n = 8, and the source voltage is vSðtÞ = 240 cos 377t V.Find expressions for i1ðtÞ and i2ðtÞ. Validate your answer using Multisim. V(1) Ideal FIGURE P15-12 V(1) RL (1)?! + (1) Sy (1)SA
15–14 The turns ratio of the ideal transformer in Figure P15–14 is n = 5. The source and load impedances are ZS =40 + j45 Ωand ZL = 500 −j350 Ω. Find I1, I2, and VO when VS =200ff0 V. Vs + 1:n Zs + V2 Z Vo Ideal FIGURE P15-14
15–16 In Figure P15–16 the turns ratio is n = 4, X = 45Ω, and RL = 720 Ω. Find IIN and VO when VS = 100 ff0 V. Vs JIN +1 jX 1:n Ideal FIGURE P15-16 www RL Vo
15–21 In Figure P15–21 the impedances are Z1 = 20 −j45 Ω, Z2 = 45 + j30 Ω, and Z3 = 300 + j250 Ω. Find I1, I2, and I3 13 1:3 22 Z3 Ideal 4. 20020 v 1:2 Ideal 12 FIGURE P15-21 (+-
15–26 The input voltage to the transformer in Figure P15–26 is a sinusoid vSðtÞ = 220 cos 400t V.With the circuit operating in the sinusoidal steady state, transform the circuit into the phasor domain and write mesh-current equations. Solve for the mesh currents and find the output voltage V2
15–30 The linear transformer in Figure P15–30 is sinusoidal steady state with VS = 500 V and ZL = 20 + j10 Ω. Use mesh-current analysis to find the input impedance seen by the source and the average power delivered by the source +1 j7002 50 w 10 00 + 100 250 ZVO FIGURE P15-30
15–32 Find the phasor current I and the input impedance seen by the source in Figure P15–32. 200/0 V I 12 100 200 $2 + j8 V j j16 V2 + +1 ZIN FIGURE P15-32
15–33 The circuit in Figure P15–33 is in the sinusoidal steady state with VS = 200ff0 V and RL = 50Ω. Use mesh-current analysis to find VO and the input impedance seen by the source j6002 10 m m 00 j40 | 1100 Vs -130 FIGURE P15-33 RL Vo
15–34 Find IA and IB in Figure P15–34 and the input impedance seen by the voltage source. 120L0 +1 IA n=5 00 ZIN FIGURE P15-34 -j50 IB
15–38 The linear transformer in Figure P15–38 is in the sinusoidal steady state with reactances of X1 = 32Ω, X2 = 50Ω, XM = 40Ω, and a load impedance of ZL = 150 – j50 Ω. Find the input impedance seen by the voltage source. Vs +1 Vi jXM 12 V2 Z jX jX2 ZIN FIGURE P15-38
15–42 Transformer Thévenin Equivalent In the time domain, the i–v equations for a linear transformer areAssuming zero initial conditions, transform these equations into the s-domain and show that the s-domain parameters of the Thévenin equivalent at the outputwhere k is the coupling
15–43 Perfectly Coupled Transformer FigureP15–43isanequivalent circuit ofaperfectlycoupled transformer.Thismodel is the basis for the transformer equivalent circuits used in the analysis of power systems. The inductanceLm is called the magnetizing inductance. The current through this inductance
15–44 Equivalent T-Circuit Transformer Model Thetransformermodelshownin Figure 15–17 can also bemodeled using an Equivalent T-Circuit as shown in Figure P15–44.The three inductors are related to the transformer inductances as follows:LA = L1M LB = L2M LC = M where the upper sign applies for
15–41 Equivalent Capacitance A capacitor C is connected across the secondary of an ideal transformer whose turns ratio is 1: n. Derive an expression for the equivalent capacitance CEQ seen looking into the primary winding.
15–40 The self and mutual inductances of a transformer can be calculated from measurements of the steady-state ac voltages and currents with the secondary winding open-circuited and short-circuited. Suppose the measurements are |V1 | = 60 V,| I1 | = 120 mA, and |V2 | = 180 V when the secondary
15–39 The linear transformer in Figure P15–38 is in the sinusoidal steady state with reactances of X1 = 15Ω, X2 = 60Ω, and XM = 30Ω. Find the transformer secondary response V2 and I2 when ZL = 200 −j100 Ω and VS = 250 ff0 V.
15–37 A transformer operating in the sinusoidal steady state has inductances L1 = 510 mH, L2 = 2 H, and M= 1 H. The load connected across the secondary is ZL = 200 + j150 Ω.The 60-Hz voltage source connected to the primary side has a peak amplitude of 2:5 kV. Find the amplitudes of the secondary
15–36 A transformer operating in the sinusoidal steady state has inductances L1 = 800 mH, L2 = 320 mH, and M= 500 mH.A load ZL =6 + j0 Ω is connected across the secondary.The 60-Hz voltage source connected to the primary side has a peak amplitude of 250 V. Find the impedance seen by the voltage
15–35 If f = 60 Hz, find V1 and V2 in Figure P15–34 using Multisim.
15–31 Repeat Problem 15–30 with ZL = 20 −j10 Ω.
15–29 Repeat Problem 15–28 when a 50-mH inductive load is connected across the secondary winding.
15–28 A transformer has self-inductances L1 = 200 mH, L2 =200 mH, and a coupling coefficient of k = 0:5. The transformer is operating in the sinusoidal steady state withω = 500 rad=s and a 50-Ω resistive load connected across the secondary winding. Find the transformer input impedance.Assume
15–27 Repeat Problem 15–26 with vSðtÞ = 100 cos 2000t V.
15–25 A transformer that can be treated as ideal has 480 turns in the primary winding and 240 turns in the secondary winding. The primary winding is connected to a 60-Hz source with a peak amplitude of 400 V. The secondary winding delivers 5 kWto a resistive load. Find the primary and secondary
15–24 The primary winding of an ideal transformer with N1 = 100 and N2 = 250 is connected to a 480-V source.A 100-Ω load is connected across the secondary windings.Find amplitudes of the primary and secondary currents.
15–23 An ideal transformer has a turns ratio of n = 5. The secondary winding is connected to a load ZL = 300 + j100 Ω. The primary winding is connected to a voltage source with a peak amplitude of 300 V and an internal impedance of ZS = j2 Ω.Find the average power delivered to the load.
15–22 In Figure P15–21 the impedances are Z1= 35 + j20 Ω, Z2 =70 + j8 Ω, and Z3 = 270 −j0 Ω. Find the input impedance seen by the source.
15–20 The number of turns in the primary and secondary of an ideal transformer are N1 = 50 and N2 = 400. The primary winding is connected to a 120-V, 60-Hz source, with a source resistance of 50 Ω. The secondary winding is connected to a 4800-Ω load. Find the primary and secondary currents.
15–19 The primary voltage of an ideal transformer is a 120-V, 60-Hz sinusoid. The secondary voltage is a 24-V, 60-Hz sinusoid.The secondary winding is connected to an 800-Ω resistive load.(a) Find the transformer turns ratio.(b) Write expressions for the primary current and voltage.
15–18 A 440-V source with a source resistance of 25 Ω is connected to the primary of an ideal transformer. Design the turns ratio needed to deliver the maximum average power to a 400-Ω load connected across the secondary.
15–17 Design the turns ratio in Figure P15–16 if VS = 200 ff0 V, VO = 100 ff0 V, X = 10Ω, and RL = 5Ω.Then find IIN.
15–15 Design the turns ratio of the ideal transformer in Figure P15–14 so that VO = 70:8 ff7:973 V when VS = 440 ff0 V. The source and load impedances are ZS = 50+ j0 Ω and ZL = 5 + j2 Ω. Validate your design using Multisim.
15–13 In Figure P15–12 RS = 50Ω, RL = 2Ω, the turns ratio is n = 1=5, and the source voltage is vSðtÞ = 440 cos 400t V.Find expressions for v1ðtÞ and v2ðtÞ. Validate your answer using Multisim.
15–11 A perfectly coupled transformer has a turns ratio of n = 5. The voltage across the primary winding is v1ðtÞ = 120 cos 377t V. Find the secondary voltage and the current delivered to an 800- Ω resistive load. Assume additive coupling. Validate your answer using Multisim.
15–6 A pair of coupled inductors have L1 =3:6H, L2 =2:5H, and k=0:5 When the output terminals are open-circuited(i2 = 0), the output voltage is observed to be v2ðtÞ = 30 sin 1000t V. Find the input voltage v1ðtÞ for additive coupling.
15–3 In Figure P15–1 L1 = 10 mH, L2 = 5 mH,M = 7 mH, and the outputs are v2ðtÞ = 0 and i2ðtÞ = 35 sin 1000t A.(a) Write the i–v relationships for the coupled inductors using the reference marks given.(b) Solve for the source voltage vSðtÞ.
15–2 In Figure P15–1 L1 = 10 mH, L2 = 5 mH, M = 7 mH, and vSðtÞ = 100 sin 1000t V.(a) Write the i – v relationships for the coupled inductors using the reference marks in the figure.(b) Solve for i1ðtÞ and i2ðtÞ when the output terminals are short-circuited (v2 = 0).
Using the values of the mesh currents found in Example 15–9, find the average power delivered to the load ZL.
The input impedance on the primary side of an ideal transformer is 1250 Ω when the load connected to the secondary is 50 Ω. What is the transformer turns ratio?
Electric power is generated by a power plant and transmitted to a distribution station using very high voltage power lines to significantly reduce power line i2R losses. From the distribution station, the electricity is reduced to lower, but still high, voltages for final distribution to commercial
The turns ratio of the ideal transformer in Figure 15–12 is n=1=4 and the source and load impedances are ZS =50+j0 Ω and ZL =50−j2 Ω. Find the impedance seen by the voltage source
The turns ratio of the ideal transformer in Figure 15–12 is n = 5. The source and load impedance are ZS =2:5+j1:5 Ω and ZL =75+j10 Ω. Find I1, V1, I2, and V2 when the input is VS = 220 ff 0 V.
For a transformer with perfect coupling, find the secondary voltage υ2ðtÞ when the primary voltage is υ1ðtÞ = 50 cos 3000t V, n = 50, and the mutual inductance is subtractive. Validate your answer using Multisim.
A transformer with perfect coupling has N1 = 250 turns and N2 = 25 turns. The voltage across the primary winding is υ1ðtÞ = 120 sin 377t V, and a resistive load RL = 50 Ωis connected across the secondary winding. Find the current, voltage, and average power delivered to the load. Assume
A pair of coupled inductors have self-inductances L1 =4:5mH and L2 = 8 mH. What is the maximum possible mutual inductance?
A pair of coupled inductors have self-inductances L1 =2:5H, L2 =1:6H, and a coupling coefficient of k = 0:8 . When the terminals of L2 are short-circuited ðυ2ðtÞ = 0Þ the short-circuit current is observed to be i2ðtÞ = −50 sinð2000tÞmA. Find the input voltage υ1ðtÞ for additive
15-3 Linear Transformers (Sect. 15–5)Given a linear circuit with a transformer operating in the sinusoidal steady-state, find phasor voltages and currents, average powers, and equivalent impedances.

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