Problems1

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Problems1

Section 13.2 Mutual Inductance

13.1 For the three coupled coils in Fig. 13.72, calculate the total inductance.

Figure 13.72

For Prob. 13.1.

13.2 Using Fig. 13.73, design a problem to help other students better understand mutual inductance.

For Prob. 13.2.

• 13.3 Two coils connected in series-aiding fashion have a total inductance of 500 mH. When connected in a series-opposing configuration, the coils have a total inductance of 300 mH. If the inductance of one coil (L1) is three times the other, find L1, L2, and M. What is the coupling coefficient?
• 13.4 (a) For the coupled coils in Fig. 13.74(a), show that

$$L_{\text{eq}} = L_1 + L_2 + 2M$$

(b) For the coupled coils in Fig. 13.74(b), show that

eq: Cous in Fig. 13.74(

$$L_{eq} = \frac{L_1 L_2 - M^2}{L_1 + L_2 - 2M}$$

13.5 Two coils are mutually coupled, with L1 = 50 mH, L2 = 120 mH, and k = 0.5. Calculate the maximum possible equivalent inductance if:

(a) the two coils are connected in series (b) the coils are connected in parallel

13.6 Given the circuit shown in Fig. 13.75, determine the value of V1 and I2.

Figure 13.76

For Prob. 13.7.

Figure 13.77

For Prob. 13.8.

13.9 Find Vx in the network shown in Fig. 13.78.

¹Remember, unless otherwise specified, assume all values of currents and voltages are rms.

For Prob. 13.10.

13.11 Use mesh analysis to find ix in Fig. 13.80, where is = 4 cos(600t) A and v^s = 110 cos(600t + 30°)

Figure 13.80 For Prob. 13.11.

13.12 Determine the equivalent Leq in the circuit of Fig. 13.81.

For Prob. 13.13.

in Fig. 13.83 at terminals a-b.

Figure 13.83

For Prob. 13.14.

13.15 Find the Norton equivalent for the circuit in Fig. 13.84 at terminals a-b.

Figure 13.84

For Prob. 13.15.

13.16 Obtain the Norton equivalent at terminals a-b of the circuit in Fig. 13.85.

Figure 13.85 For Prob. 13.16.

13.17 In the circuit of Fig. 13.86, ZL is a 15-mH inductor having an impedance of j40 Ω. Determine Zin when k = 0.6.

Figure 13.86 For Prob. 13.17.

13.18 Find the Thevenin equivalent to the left of the load Z in the circuit of Fig. 13.87.

Figure 13.87

For Prob. 13.18.

13.19 Determine an equivalent T-section that can be used to replace the transformer in Fig. 13.88.

Figure 13.88

For Prob. 13.19.

Section 13.3 Energy in a Coupled Circuit

13.20 Determine currents I1, I2, and I3 in the circuit of Fig. 13.89. Find the energy stored in the coupled coils at t = 2 ms. Take ω = 1,000 rad/s.

Figure 13.89 For Prob. 13.20.

13.21 Using Fig. 13.90, design a problem to help other students better understand energy in a coupled circuit.

Figure 13.90 For Prob. 13.21.

Figure 13.91 For Prob. 13.22.

13.23 Let is = 5 cos (100t) A. Calculate the voltage across the capacitor, vc. Also calculate the value of the energy stored in the coupled coils at t = 2.5π ms.

Figure 13.92

For Prob. 13.23.

13.24 In the circuit of Fig. 13.93,

• (a) find the coupling coefficient, (b) calculate vo,
• (c) determine the energy stored in the coupled inductors at t = 2 s.

* An asterisk indicates a challenging problem.

For Prob. 13.25.

13.26 Find Io in the circuit of Fig. 13.95. Switch the dot on the winding on the right and calculate Io again.

For Prob. 13.26.

Figure 13.96

For Prob. 13.27.

For Prob. 13.28.

Section 13.4 Linear Transformers

13.29 In the circuit of Fig. 13.98, find the value of the coupling coefficient k that will make the 10-Ω resistor dissipate 1.28 kW. For this value of k, find the energy stored in the coupled coils at t = 1.5 s.

Figure 13.98

For Prob. 13.29.

• 13.30 (a) Find the input impedance of the circuit in Fig. 13.99 using the concept of reflected impedance.
  • (b) Obtain the input impedance by replacing the linear transformer by its T equivalent.

Figure 13.100 For Prob. 13.31.

*13.32 Two linear transformers are cascaded as shown in Fig. 13.101. Show that

$$\omega^{2}R(L_{a}^{2} + L_{a}L_{b} - M_{a}^{2})$$ $$Z_{in} = \frac{+j\omega^{3}(L_{a}^{2}L_{b} + L_{a}L_{b}^{2} - L_{a}M_{b}^{2} - L_{b}M_{a}^{2})}{\omega^{2}(L_{a}L_{b} + L_{b}^{2} - M_{b}^{2}) - j\omega R(L_{a} + L_{b})}$$

For Prob. 13.32.

13.33 Determine the input impedance of the air-core transformer circuit of Fig. 13.102.

Figure 13.102 For Prob. 13.33.

13.34 Using Fig. 13.103, design a problem to help other students better understand how to find the input impedance of circuits with transformers.

Figure 13.103 For Prob. 13.34.

Section 13.5 Ideal Transformers

13.36 As done in Fig. 13.32, obtain the relationships between terminal voltages and currents for each of the ideal transformers in Fig. 13.105.

For Prob. 13.36.

• 13.37 A 240∕2,400-V rms step-up ideal transformer delivers 50 kW to a resistive load. Calculate:
  • (a) the turns ratio
  • (b) the primary current
  • (c) the secondary current

13.38 Design a problem to help other students better understand ideal transformers.

13.39 A 1,200∕240-V rms transformer has impedance 60⧸−30° Ω on the high-voltage side. If the transformer is connected to a 0.8⧸10°-Ω load on the low-voltage side, determine the primary and secondary currents when the transformer is

13.40 The primary of an ideal transformer with a turns ratio of 5 is connected to a voltage source with Thevenin parameters vTh = 10 cos 2000t V and RTh = 100 Ω. Determine the average power delivered to a 200-Ω load connected across the secondary winding.

For Prob. 13.41.

13.42 For the circuit in Fig. 13.107, determine the power absorbed by the 2-Ω resistor. Assume the 120 V is an rms value.

Figure 13.107 For Prob. 13.42.

For Prob. 13.43.

13.45 For the circuit shown in Fig. 13.110, find the value of the average power absorbed by the 8-Ω resistor.

Figure 13.110

For Prob. 13.45.

13.46 (a) Find I1 and I2 in the circuit of Fig. 13.111 below. (b) Switch the dot on one of the windings. Find I¹ and I2 again.

For Prob. 13.47.

For Prob. 13.46.

Figure 13.113 For Prob. 13.48.

13.49 Find current ix in the ideal transformer circuit shown in Fig. 13.114.

For Prob. 13.49.

13.50 Calculate the input impedance for the network in Fig. 13.115.

13.53 Refer to the network in Fig. 13.118.

• (a) Find n for maximum power supplied to the 200-Ω load.
• (b) Determine the power in the 200-Ω load if n = 10.

Figure 13.118 For Prob. 13.53.

• 13.51 Use the concept of reflected impedance to find
• the input impedance and current I1 in

Fig. 13.116.

13.54 A transformer is used to match an amplifier with

an 8-Ω load as shown in Fig. 13.119. The Thevenin equivalent of the amplifier is: VTh = 10 V, ZTh = 128 Ω.

• (a) Find the required turns ratio for maximum energy power transfer.
• (b) Determine the primary and secondary currents.
• (c) Calculate the primary and secondary voltages.

Figure 13.119

For Prob. 13.54.

13.56 Find the power absorbed by the 100-Ω resistor in the ideal transformer circuit of Fig. 13.121.

Figure 13.121 For Prob. 13.56.

13.57 For the ideal transformer circuit of Fig. 13.122 below, find:

• (b) V1, V2, and Vo,
• (c) the complex power supplied by the source.

13.58 Determine the average power absorbed by each resistor in the circuit of Fig. 13.123.

Figure 13.123

For Prob. 13.58.

Figure 13.124

For Prob. 13.59.

13.60 Refer to the circuit in Fig. 13.125 on the following page.

(a) Find currents I1, I2, and I3.

Figure 13.125 For Prob. 13.60.

*13.61 For the circuit in Fig. 13.126, find I1, I2, and Vo.

13.62 For the network in Fig. 13.127, find: (a) the complex power supplied by the source, (b) the average power delivered to the 18-Ω resistor.

Figure 13.128

For Prob. 13.63.

13.64 For the circuit in Fig. 13.129, find the turns ratio so that the maximum power is delivered to the 30-kΩ resistor.

*13.65 Calculate the average power dissipated by the 20-Ω resistor in Fig. 13.130.

Section 13.6 Ideal Autotransformers

13.66 Design a problem to help other students better understand how the ideal autotransformer works.

13.67 An autotransformer with a 40 percent tap is supplied by an 880-V, 60-Hz source and is used for stepdown operation. A 5-kVA load operating at unity power factor is connected to the secondary terminals. Find:

(a) the secondary voltage,

• (b) the secondary current,
• (c) the primary current.
• 13.68 In the ideal autotransformer of Fig. 13.131, calculate I1, I2, and Io. Find the average power delivered to the load.

For Prob. 13.68.

*13.69 In the circuit of Fig. 13.131, N¹ = 190 turns and N² = 10 turns. Determine the Thevenin equivalent circuit looking into terminals a and b. What would be the value of ZL that would absorb maximum power from the circuit?

For Prob. 13.69.

13.70 In the ideal transformer circuit shown in Fig. 13.133, determine the average power delivered to the load.

Figure 13.133

For Prob. 13.70.

13.71 When individuals travel, their electrical appliances need to have converters to match the voltages required by their appliances to the local voltage available to power their appliances. Today these converters use power electronics to convert voltages. In the past these converters were autotransformers. The autotransformer shown in Fig. 13.134 is used to convert 115 to 220 V. What is the value of the turns? If the maximum current available from the 115 V source is 15 A, what will be the maximum current available for the 220-V appliance?

Figure 13.134

For Prob. 13.71.

Section 13.7 Three-Phase Transformers

13.72 In order to meet an emergency, three single-phase transformers with 12,470∕7,200 V rms are connected in Δ-Y to form a three-phase transformer which is fed by a 12,470-V transmission line. If the transformer supplies 60 MVA to a load, find:

(a) the turns ratio for each transformer,

• (b) the currents in the primary and secondary windings of the transformer,
• (c) the incoming and outgoing transmission line currents.

13.73 Figure 13.135 on the next page shows a three-phase transformer that supplies a Y-connected load.

• (a) Identify the transformer connection.
• (b) Calculate currents I2 and Ic.
• (c) Find the average power absorbed by the load.

Figure 13.135 For Prob. 13.73.

• 13.74 Consider the three-phase transformer shown in Fig. 13.136. The primary is fed by a three-phase source with line voltage of 2.4 kV rms, while the secondary supplies a three-phase 120-kW balanced load at pf of 0.8. Determine:
  • (a) the type of transformer connections,

(b) the values of ILS and IPS,

(c) the values of ILP and IPP,

• (d) the kVA rating of each phase of the transformer.
• 13.75 A balanced three-phase transformer bank with the Δ-Y connection depicted in Fig. 13.137 is used to step down line voltages from 4,500 V rms to 900 V rms. If the transformer feeds a 120-kVA load, find:
  • (a) the turns ratio for the transformer,
  • (b) the line currents at the primary and secondary sides.

Figure 13.137 For Prob. 13.75.

13.76 Using Fig. 13.138, design a problem to help other students better understand a Y-Δ, three-phase transformer and how they work.

Figure 13.138 For Prob. 13.76.

• 13.77 The three-phase system of a town distributes power with a line voltage of 13.2 kV. A pole transformer connected to single wire and ground steps down the high-voltage wire to 120 V rms and serves a house as shown in Fig. 13.139.
  • (a) Calculate the turns ratio of the pole transformer to get 120 V.
  • (b) Determine how much current a 100-W lamp connected to the 120-V hot line draws from the high-voltage line.

Figure 13.139 For Prob. 13.77.

Section 13.8 PSpice Analysis of Magnetically Coupled Circuits

13.78 Use PSpice or MultiSim to determine the mesh currents in the circuit of Fig. 13.140. Take ω = 1 rad/s. Use k = 0.5 when solving this problem.

For Prob. 13.78.

13.79 Use PSpice or MultiSim to find I1, I2, and I3 in the circuit of Fig. 13.141.

Figure 13.141 For Prob. 13.79.

• 13.80 Rework Prob. 13.22 using PSpice or Multisim.
• 13.81 Use PSpice or MultiSim to find I1, I2, and I3 in the circuit of Fig. 13.142.

Figure 13.142

For Prob. 13.81.

13.82 Use PSpice or MultiSim to find V1, V2, and Io in the circuit of Fig. 13.143.

Figure 13.143

For Prob. 13.82.

13.83 Find Ix and Vx in the circuit of Fig. 13.144 using PSpice or MultiSim.

13.84 Determine I1, I2, and I3 in the ideal transformer circuit of Fig. 13.145 using PSpice or MultiSim.

Section 13.9 Applications

• 13.85 A stereo amplifier circuit with an output impedance of 7.2 kΩ is to be matched to a speaker with an input impedance of 8 Ω by a transformer whose primary side has 3,000 turns. Calculate the number of turns required on the secondary side.
• 13.86 A transformer having 2,400 turns on the primary and 48 turns on the secondary is used as an impedancematching device. What is the reflected value of a 3-Ω load connected to the secondary?
• 13.87 A radio receiver has an input resistance of 300 Ω. When it is connected directly to an antenna system with a characteristic impedance of 75 Ω, an

impedance mismatch occurs. By inserting an impedance-matching transformer ahead of the receiver, maximum power can be realized. Calculate the required turns ratio.

• 13.88 A step-down power transformer with a turns ratio of n = 0.1 supplies 12.6 V rms to a resistive load. If the primary current is 2.5 A rms, how much power is delivered to the load?
• 13.89 A 240∕120-V rms power transformer is rated at 10 kVA. Determine the turns ratio, the primary current, and the secondary current.
• 13.90 A 4-kVA, 2,400∕240-V rms transformer has 250 turns on the primary side. Calculate:
  • (a) the turns ratio,
  • (b) the number of turns on the secondary side,
  • (c) the primary and secondary currents.
• 13.91 A 25,000∕240-V rms distribution transformer has a primary current rating of 75 A.
  • (a) Find the transformer kVA rating.
  • (b) Calculate the secondary current.
• 13.92 A 4,800-V rms transmission line feeds a distribution transformer with 1,200 turns on the primary and 28 turns on the secondary. When a 10-Ω load is connected across the secondary, find:
  • (a) the secondary voltage,
  • (b) the primary and secondary currents,
  • (c) the power supplied to the load.