Two -connected loads are connected in parallel and powered by a balanced Y-connected

Let \(\mathbf{V}_{ab}=100 \angle 0^{\circ}\mathrm{\ V}\), \(\mathbf{V}_{bd}=40 \angle 80^{\circ}\mathrm{\ V}\), and \(\mathbf{V}_{ca}=70 \angle 200^{\circ}\mathrm{\ V}\). Find (a) \(\mathbf{V}_{ad}\) (b) \(\mathbf{V}_{bc}\); (c) \(\mathbf{V}_{cd}\).

Refer to the circuit of Fig. 12.6 and let \(\mathbf{I}_{fj}=3\mathrm{\ A}\), \(\mathbf{I}_{de}=2\mathrm{\ A}\), and \(\mathbf{I}_{hd}=-6\mathrm{\ A}\). Find (a) \(\mathbf{I}_{cd}\); (b) \(\mathbf{I}_{ef}\); (c) \(\mathbf{I}_{ij}\).

Modify Fig. 12.9 by adding a \(1.5\ \Omega\) resistance to each of the two outer lines, and a \(2.5\ \Omega\) resistance to the neutral wire. Find the average power delivered to each of the three loads.

A balanced three-phase three-wire system has a Y-connected load. Each phase contains three loads in parallel: \(?j100\ \Omega\), \(100\ \Omega\), and \(50 + j50\ \Omega\). Assume positive phase sequence with \(\mathbf{V}_{ab}=400\angle0^{\circ}\mathrm{\ V}\). Find (a) \(\mathbf{V}_{an}\); (b) \(\mathbf{I}_{aA}\); (c) the total power drawn by the load.

A balanced three-phase three-wire system has a line voltage of 500 V. Two balanced Y-connected loads are present. One is a capacitive load with \(7 ? j2\ \Omega\) per phase, and the other is an inductive load with \(4 + j2\ \Omega\) per phase. Find (a) the phase voltage; (b) the line current; (c) the total power drawn by the load; (d) the power factor at which the source is operating.

Three balanced Y-connected loads are installed on a balanced three-phase four-wire system. Load 1 draws a total power of 6 kW at unity PF, load 2 pulls 10 kVA at PF = 0.96 lagging, and load 3 demands 7 kW at 0.85 lagging. If the phase voltage at the loads is 135 V, if each line has a resistance of \(0.1\ \Omega\), and if the neutral has a resistance of \(1\ \Omega\), find (a) the total power drawn by the loads; (b) the combined PF of the loads; (c) the total power lost in the four lines; (d) the phase voltage at the source; (e) the power factor at which the source is operating.

Each phase of a balanced three-phase \(\Delta\)-connected load consists of a 200 mH inductor in series with the parallel combination of a \(5\ \mu F\) capacitor and a \(200\ \Omega\) resistance. Assume zero line resistance and a phase voltage of 200 V at \(\omega = 400\ rad/s\). Find (a) the phase current; (b) the line current; (c) the total power absorbed by the load.

A balanced three-phase three-wire system is terminated with two \(\Delta\)-connected loads in parallel. Load 1 draws 40 kVA at a lagging PF of 0.8, while load 2 absorbs 24kW at a leading PF of 0.9. Assume no line resistance, and let \(\mathbf{V}_{ab}=440\angle30^{\circ}\mathrm{\ V}\). Find (a) the total power drawn by the loads; (b) the phase current \(\mathbf{I}_{A B 1}\) for the lagging load; (c) \(\mathbf{I}_{A B 2}\); (d) \(\mathbf{I}_{a A}\).

Determine the watt meter reading in Fig. 12.24, state whether or not the potential coil had to be reversed in order to obtain an upscale reading, and identify the device or devices absorbing or generating this power. The (+) terminal of the wattmeter is connected to (a) x; (b) y; (c) z.

For the circuit of Fig. 12.26, let the loads be \(\mathbf{Z}_A=25 \angle 60^{\circ}\ \Omega\), \(\mathbf{Z}_B=50 \angle -60^{\circ}\ \Omega\), \(\mathbf{Z}_C=50 \angle 60^{\circ}\ \Omega\), \(\mathrm{V}_{AB}=600 \angle 0^{\circ}\mathrm{\ V}\mathrm{\ rms}\) with (+) phase sequence, and locate point x at C. Find (a) \(P_A\); (b) \(P_B\); (c) \(P_C\).

An unknown three terminal device has leads named b, c, and e. When installed in one particular circuit, measurements indicated that \(V_{ec}=-9\mathrm{\ V}\) and \(V_{eb}=-0.65\mathrm{\ V}\). (a) Calculate \(V_{c b}\). (b) Determine the power dissipated in the b-e junction if the current \(I_b\) flowing into the terminal marked b is equal to \(1\ \mu A\).

A common type of transistor is known as the MESFET, which is an acronym for metal-semiconductor field effect transistor. It has three terminals, named the gate (g), the source (s), and the drain (d). As an example, consider one particular MESFET operating in a circuit such that \(V_{sg} = 0.2\ V\) and \(V_{ds} = 3\ V\). (a) Calculate \(V_{gs}\) and \(V_{dg}\). (b) If a gate current \(I_g = 100\ pA\) is found to be flowing into the gate terminal, compute the power lost at the gate-source junction.

For a certain Y-connected three-phase source, \(\mathbf{V}_{an}=400 \angle 33^{\circ}\mathrm{\ V}\), \(\mathbf{V}_{bn}=400 \angle 153^{\circ}\mathrm{\ V}\), and \(\mathbf{V}_{cx}=160 \angle 208^{\circ}\mathrm{\ V}\). Determine (a) \(\mathbf{V}_{cn}\); (b) \(\mathbf{V}_{a n}-\mathbf{V}_{b n}\); (c) \(\mathbf{V}_{ax}\); (d) \(\mathbf{V}_{bx}\).

Describe what is meant by a “polyphase” source, state one possible advantage of such sources that might outweigh their additional complexity over single phase sources of power, and explain the difference between “balanced” and “unbalanced” sources.

Several of the voltages associated with a certain circuit are given by \(\mathbf{V}_{12}=9\angle30^{\circ}\mathrm{\ V}\), \(\mathbf{V}_{32}=3\angle130^{\circ}\mathrm{\ V}\), and \(\mathbf{V}_{14}=2\angle10^{\circ}\mathrm{\ V}\). Determine \(\mathbf{V}_{21}\), \(\mathbf{V}_{13}\), \(\mathbf{V}_{34}\), and \(\mathbf{V}_{24}\).

The nodal voltages which describe a particular circuit can be expressed as \(\mathbf{V}_{14}=9-j\mathrm{\ V}\), \(\mathbf{V}_{24}=3+j3\mathrm{\ V}\), and \(\mathbf{V}_{34}=8\mathrm{\ V}\). Calculate \(\mathbf{V}_{12}\), \(\mathbf{V}_{32}\), and \(\mathbf{V}_{13}\). Express your answers in phasor form.

In the circuit of Fig. 12.29, the resistor markings unfortunately have been omitted, but several of the currents are known. Specifically, \(I_{ad}=1\mathrm{\ A}\). (a) Compute \(I_{ab}\), \(I_{cd}\), \(I_{de}\), \(I_{fe}\), and \(I_{be}\). (b) If \(V_{ba}=125\mathrm{\ V}\), determine the value of the resistor linking nodes a and b.

Additional resistors are added in parallel to the resistors between terminals d and e, and terminals f and j, respectively, of the circuit in Fig. 12.30. (a) Which voltages may still be described using double-subscript notation? (b) Which line currents may still be described by double-subscript notation?

Most consumer electronics are powered by 110 V outlets, but several types of appliances (such as clothes dryers) are powered from 220 V outlets. Lower voltages are generally safer. What, then, motivates manufactures of some pieces of equipment to design them to run on 220 V?

The single-phase three-wire system of Fig. 12.31 has three separate load impedances. If the source is balanced and \(\mathbf{V}_{an}=110+j0\mathrm{\ V\ }\mathrm{rms}\), (a) express \(\mathbf{V}_{an}\) and \(\mathbf{V}_{bn}\) in phasor notation. (b) Determine the phasor voltage which appears across the impedance \(\mathbf{Z}_{3}\). (c) Determine the average power delivered by the two sources if \(\mathbf{Z}_1=50+j0\ \Omega\), \(\mathbf{Z}_2=100+j45\ \Omega\), and \(\mathbf{Z}_3=100-j90\ \Omega\). (d) Represent load \(\mathbf{Z}_{3}\) by a series connection of two elements, and state their respective values if the sources operate at 60 Hz.

For the system represented in Fig. 12.32, the ohmic losses in the neutral wire are so small they can be neglected and it can be adequately modeled as a short circuit. (a) Calculate the power lost in the two lines as a result of their non zero resistance. (b) Compute the average power delivered to the load. (c) Determine the power factor of the total load.

Referring to the balanced load represented in Fig. 12.33, if it is connected to a three-wire balanced source operating at 50 Hz such that \(V_{AN}=115\mathrm{\ V}\), (a) determine the power factor of the load if the capacitor is omitted; (b) determine the value of capacitance C that will achieve a unity power factor for the total load.

(a) Show that if \(V_{an}=400\angle33^{\circ}\mathrm{\ V}\), \(V_{bn}=400\angle-87^{\circ}\mathrm{\ V}\), and \(V_{bn}=400\angle-207^{\circ}\mathrm{\ V}\), that \(\mathbf{V}_{a n}+\mathbf{V}_{b n}+\mathbf{V}_{c n}=0\). (b) Do the voltages in part (a) represent positive or negative phase sequence? Explain.

Consider a simple positive phase sequence, three-phase, three-wire system operated at 50 Hz and with a balance load. Each phase voltage of 240 V is connected across a load composed of a series-connected \(50\ \Omega\) and 500 mH combination. Calculate (a) each line current; (b) the power factor of the load; (c) the total power supplied by the three-phase source.

Assume the system shown in Fig. 12.34 is balanced, \(R_{w} = 0\), \(\mathbf{V}_{an}=208 \angle 0^{\circ}\mathrm{\ V}\), and a positive phase sequence applies. Calculate all phase and line currents, and all phase and line voltages, if \(\mathbf{Z}_{p}\) is equal to (a) \(1\ k \Omega\); (b) \(100 + j48\ \Omega\) ; (c) \(100 ? j48\ \Omega\).

Repeat Exercise 17 with \(R_w = 10\ \Omega\), and verify your answers with appropriate PSpice simulations if the operating frequency is 60 Hz.

Each impedance \(\mathbf{Z}_{p}\) in the balanced three-phase system of Fig. 12.34 is constructed using the parallel combination of a 1 mF capacitance, a 100 mH inductance, and a \(10\ \Omega\) resistance. The sources have positive phase sequence and operate at 50 Hz. If \(\mathbf{V}_{ab}=208\angle0^{\circ}\mathrm{\ V}\), and \(R_w = 0\), calculate (a) all phase voltages; (b) all line voltages; (c) all three line currents; (d) the total power drawn by the load.

With the assumption that the three-phase system pictured in Fig. 12.34 is balanced with a line voltage of 100 V, calculate the line current and per-phase load impedance if \(R_w = 0\) and the load draws (a) 1 kW at a PF of 0.85 lagging; (b) 300 W per phase at a PF of 0.92 leading.

The balanced three-phase system of Fig. 12.34 is characterized by a positive phase sequence and a line voltage of 300 V. And \(\mathbf{Z}_{p}\) is given by the parallel combination of a \(5 ? j3\ \Omega\) capacitive load and a \(9 + j2\ \Omega\) inductive load. If \(R_w = 0\), calculate (a) the power factor of the source; (b) the total power supplied by the source. (c) Repeat parts (a) and (b) if \(R_w = 1\ \Omega\).

A balanced Y-connected load of \(100 + j50\ \Omega\) is connected to a balanced three-phase source. If the line current is 42 A and the source supplies 12 kW, determine (a) the line voltage; (b) the phase voltage.

A three-phase system is constructed from a balanced Y-connected source operating at 50 Hz and having a line voltage of 210 V, and each phase of the balanced load draws 130 W at a leading power factor of 0.75. (a) Calculate the line current and the total power supplied to the load. (b) If a purely resistive load of \(1\ \Omega\) is connected in parallel with each existing load, calculate the new line current and total power supplied to the load. (c) Verify your answers using appropriate PSpice simulations.

Returning to the balanced three-phase system described in Exercise 21, determine the complex power delivered to the load for both \(R_w = 0\) and \(R_w = 1\ \Omega\).

Each load in the circuit of Fig. 12.34 is composed of a 1.5 H inductor in parallel with a \(100\ \mu F\) capacitor and a \(1\ k \Omega\) resistor. The resistance is labeled \(R_w = 0\ \Omega\). Using positive phase sequence with \(\mathbf{V}_{ab}=115 \angle 0^{\circ}\mathrm{\ V}\) at f = 60 Hz, determine the rms line current and the total power delivered to the load. Verify your answers with an appropriate PSpice simulation.

A particular balanced three-phase system is supplying a \(\Delta\)-connected load with 10 kW at a leading power factor of 0.7. If the phase voltage is 208 V and the source operates at 50 V, (a) compute the line current; (b) determine the phase impedance; (c) calculate the new power factor and new total power delivered to the load if a 2.5 H inductor is connected in parallel with each phase of the load.

If each of the three phases in a balanced \(\Delta\)-connected load is composed of a 10 mF capacitor in parallel with a series-connected \(470\ \Omega\) resistor and 4 mH inductor combination, assume a phase voltage of 400 V at 50 Hz. (a) Calculate the phase current; (b) the line current; (c) the line voltage; (d) the power factor at which the source operates; (e) the total power delivered to the load.

A three-phase load is to be powered by a three-wire three-phase Y-connected source having phase voltage of 400 V and operating at 50 Hz. Each phase of the load consists of a parallel combination of a \(500\ \Omega\) resistor, 10 mH inductor, and 1 mF capacitor. (a) Compute the line current, line voltage, phase current, and power factor of the load if the load is also Y-connected. (b) Rewire the load so that it is \(\Delta\)-connected and find the same quantities requested in part (a).

For the two situations described in Exercise 28, compute the total power delivered to each of the two loads.

Two \(\Delta\)-connected loads are connected in parallel and powered by a balanced Y-connected system. The smaller of the two loads draws 10 kVA at a lagging PF of 0.75, and the larger draws 25 kVA at a leading PF of 0.80. The line voltage is 400 V. Calculate (a) the power factor at which the source is operating; (b) the total power drawn by the two loads; (c) the phase current of each load.

For the balanced three-phase system shown in Fig. 12.35, it is determined that 100 W is lost in each wire. If the phase voltage of the source is 400 V, and the load draws 12 kW at a lagging PF of 0.83, determine the wire resistance \(R_w\).

Repeat Exercise 32 if \(R_w = 1\ \Omega\). Verify your solution using an appropriate PSpice simulation.

Compute \(\mathbf{I}_{a A}\), \(\mathbf{I}_{A B}\), and \(\mathbf{V}_{a n}\) if the \(\Delta\)-connected load of Fig. 12.35 draws a total complex power of \(1800 + j700\ W\), \(R_w = 1.2\ \Omega\), and the source generates a complex power of \(1850 + j700\ W\).

A balanced three-phase system having line voltage of 240 V rms contains a \(\Delta\)-connected load of \(12 + j\ k \Omega\) per phase and also a Y-connected load of \(5 + j3\ k \Omega\) per phase. Find the line current, the power taken by the combined load, and the power factor of the load.

Determine the wattmeter reading (stating whether or not the leads had to be reversed to obtain it) in the circuit of Fig. 12.36 if terminals A and B, respectively, are connected to (a) x and y; (b) x and z; (c) y and z.

Find the reading of the wattmeter connected in the circuit of Fig. 12.38.

(a) Find both wattmeter readings in Fig. 12.39 if \(\mathbf{V}_A=100 \angle 0^{\circ}\mathrm{\ V\ }\mathrm{rms}\), \(\mathbf{V}_B=50 \angle 90^{\circ}\mathrm{\ V\ }\mathrm{rms}\), \(\mathbf{Z}_A=10-j10\ \Omega\), \(\mathbf{Z}_B=8+j6\ \Omega\), and \(\mathbf{Z}_C=30+j10\ \Omega\). (b) Is the sum of these readings equal to the total power taken by the three loads? Verify your answer with an appropriate PSpice simulation.

Circuit values for Fig. 12.40 are \(\mathbf{V}_{a b}=200 \angle 0^{\circ}\), \(\mathbf{V}_{b c}=200 \angle 120^{\circ}\), \(\mathbf{V}_{ca}=200\angle240^{\circ}\ \mathrm{V} \ \mathrm{rms}\), \(\mathbf{Z}_4=\mathbf{Z}_5=\mathbf{Z}_6=25 \angle 30^{\circ}\ \Omega\), \(\mathbf{Z}_1=\mathbf{Z}_2=\mathbf{Z}_3=50 \angle -60^{\circ}\ \Omega\). Find the reading for each wattmeter.

Explain under what circumstances a \(\Delta\)-connected load might be preferred over a Y-connected load which draws the same average and complex powers.

(a) Is the load represented in Fig. 12.41 considered a three-phase load? Explain. (b) If \(\mathbf{Z}_{AN}=1-j7\ \Omega\), \(\mathbf{Z}_{BN}=3\ \angle22^{\circ}\ \Omega\) and \(\mathbf{Z}_{AB}=2+j\ \Omega\), calculate all phase (and line) currents and voltages assuming a phase to neutral voltage of 120 VAC (the two phases are \(180^{\circ}\) out of phase). (c) Under what circumstances does current flow in the neutral wire?

The computer equipment in a small manufacturing plant all runs on standard 120 VAC, but only 208 VAC three-phase power is available. Explain how the computer equipment can be connected to the existing power wiring.

For the circuit of Fig. 12.26, let the loads be \(Z_A = 25 \ang {60^{\circ}}\ \Omega\), \(Z_B = 50 \ang {-60^{\circ}}\ \Omega\), \(Z_C = 50 \ang {60^{\circ}}\ \Omega\), \(V_{AB} = 600 \ang {0^{\circ}}\ V\ rms\) with (+) phase sequence, and locate point x at C. Find (a) \(P_A\); (b) \(P_B\); (c) \(P_C\).

For the circuit shown in Fig. 12.30, (a) determine \(I_{gh}\), \(I_{cd}\) , and \(I_{dh}\). (b) Calculate \(I_{ed}\), \(I_{ei}\) , and \(I_{jf}\). (c) If all resistors in the circuit each have a value of \(1\ \Omega\), determine the three clockwise-flowing mesh currents.

In the three-wire system of Fig. 12.32, (a) replace the \(50\ \Omega\) resistor with a \(200\ \Omega\) resistor, and calculate the current flowing through the neutral wire. (b) Determine a new value for the \(50\ \Omega\) resistor such that the neutral wire current magnitude is 25% that of line current \(I_{aA}\).

The balanced \(\Delta\)-connected load in Fig. 12.35 is demanding 10 kVA at a lagging PF of 0.91. If line losses are negligible, calculate \(I_{bB}\) and \(V_{an}\) if \(V_{ca} = 160 \ang {30^{\circ}}\ V\) and the source voltages are described using positive phase sequence.

A wattmeter is connected into the circuit of Fig. 12.37 so that \(I_1\) enters the (+) terminal of the current coil, while \(V_2\) is the voltage across the potential coil. Find the wattmeter reading, and verify your solution with an appropriate PSpice simulation.

A certain 208 V, 60 Hz, three-phase source is connected in a Y configuration and exhibits positive phase sequence. Each phase of the balanced load consists of a coil best modeled as a \(0.2\ \Omega\) resistance in series with a 580 mH inductance. (a) Determine the line voltages and the phase currents if the load is \(\Delta\)-connected. (b) Repeat part (a) if the load is Y-connected instead.

For the circuit shown in Fig. 12.30, (a) determine \(I_{gh}\), \(I_{cd}\) , and \(I_{dh}\). (b) Calculate \(I_{ed}\), \(I_{ei}\) , and \(I_{jf}\). (c) If all resistors in the circuit each have a value of \(1\ \Omega\), determine the three clockwise-flowing mesh currents.

In the three-wire system of Fig. 12.32, (a) replace the \(50\ \Omega\) resistor with a \(200\ \Omega\) resistor, and calculate the current flowing through the neutral wire. (b) Determine a new value for the \(50\ \Omega\) resistor such that the neutral wire current magnitude is 25% that of line current \(I_{aA}\).

The balanced \(\Delta\)-connected load in Fig. 12.35 is demanding 10 kVA at a lagging PF of 0.91. If line losses are negligible, calculate \(I_{bB}\) and \(V_{an}\) if \(V_{ca} = 160 \ang {30^{\circ}}\ V\) and the source voltages are described using positive phase sequence.

A wattmeter is connected into the circuit of Fig. 12.37 so that \(I_1\) enters the (+) terminal of the current coil, while \(V_2\) is the voltage across the potential coil. Find the wattmeter reading, and verify your solution with an appropriate PSpice simulation.

A certain 208 V, 60 Hz, three-phase source is connected in a Y configuration and exhibits positive phase sequence. Each phase of the balanced load consists of a coil best modeled as a \(0.2\ \Omega\) resistance in series with a 580 mH inductance. (a) Determine the line voltages and the phase currents if the load is \(\Delta\)-connected. (b) Repeat part (a) if the load is Y-connected instead.