The line-to-neutral voltage at the terminals of the balanced three-phase load in the

What is the phase sequence of each of the following sets of voltages?

For each set of voltages, state whether or not the voltages form a balanced three-phase set. If the set is balanced, state whether the phase sequence is positive or negative. If the set is not balanced, explain why. a) b) c) d) vc = 1121 cos (2000t + 100) V. vb = 1121 sin (2000t - 50) V, va = 1121 cos (2000t - 20 ) V, vc = 426 cos(100t - 120) V. vb = 462 cos(100t + 120) V, va = 426 cos 100t V, vc = 188 cos (250t - 60 ) V. vb = -188 cos 250t V, va = 188 cos(250t + 60) V, vc = 48 cos (314t + 75 ) V. vb = 48 cos (314t - 165 ) V, va = 48 cos(314t - 45) V, vc

Verify that Eq. 11.3 is true for either Eq. 11.1 or Eq. 11.2.

Refer to the circuit in Fig. 11.5(b).Assume that there are no external connections to the terminals a, b, c. Assume further that the three windings are from a three-phase generator whose voltages are those described in Problem 11.2(b). Determine the current circulating in the -connected generator.

Repeat Problem 11.4 but assume that the three-phase voltages are those described in Problem 11.2(c).

a) Is the circuit in Fig. P11.6 a balanced or unbalanced three-phase system? Explain. b) Find Figure P11.6 5  5  5  35  27  1  A B n N C a b 110 0 V 110 120 V 110 120 V c j8  j8  I0 j22  j32  j8  Io. vc =

a) Find in the circuit in Fig. P11.7. b) Find c) Find d) Is the circuit a balanced or unbalanced threephase system?

Find the rms value of in the unbalanced threephase circuit seen in Fig. P11.8.

The time-domain expressions for three line-to-neutral voltages at the terminals of a Y-connected load are What are the time-domain expressions for the three line-to-line voltages and ?

A balanced three-phase circuit has the following characteristics: Y-Y connected; The line voltage at the source, is The phase sequence is positive; The line impedance is The load impedance is a) Draw the single phase equivalent circuit for the a-phase. b) Calculated the line current in the a-phase. c) Calculated the line voltage at the load in the a-phase.

The magnitude of the line voltage at the terminals of a balanced Y-connected load is 6600 V. The load impedance is \(240-j 70 \Omega / \phi\). The load is fed from a line that has an impedance of \(0.5+j 4 \Omega / \phi\). a) What is the magnitude of the line current? b) What is the magnitude of the line voltage at the source?

The magnitude of the phase voltage of an ideal balanced three-phase Y-connected source is 125 V. The source is connected to a balanced Y-connected load by a distribution line that has an impedance of \(0.1+j 0.8 \Omega / \phi\). The load impedance is \(19.9+j 14.2 \Omega / \phi\). The phase sequence of the source is acb. Use the a-phase voltage of the source as the reference. Specify the magnitude and phase angle of the following quantities: (a) the three line currents, (b) the three line voltages at the source, (c) the three phase voltages at the load, and (d) the three line voltages at the load.

A balanced \(\Delta \text {-connected }\) load has an impedance of \(216-j 288 \Omega / \phi\). The load is fed through a line having an impedance of \(3+j 5 \Omega / \phi\). The phase voltage at the terminals of the load is 7.2 kV. The phase sequence is negative. Use \(\mathbf{V}_{\mathrm{AB}}\) as the reference. a) Calculate the three phase currents of the load. b) Calculate the three line currents. c) Calculate the three line voltages at the sending end of the line.

A balanced, three-phase circuit is characterized as follows: • \(\mathrm{Y}-\Delta\) connected; • Source voltage in the b-phase is \(150 \angle 135^{\circ} \mathrm{V}\); • Source phase sequence is acb; • Line impedance is \(2+j 3 \ \Omega / \phi\); • Load impedance is \(129+j 171 \ \Omega / \phi\). a) Draw the single phase equivalent for the a-phase. b) Calculate the a-phase line current. c) Calculate the a-phase line voltage for the three-phase load.

An acb sequence balanced three-phase Y-connected source supplies power to a balanced, three-phase \(\Delta\)-connected load with an impedance of \(12+j 9 \Omega / \phi\). The source voltage in the b-phase is \(240 \angle-50^{\circ} \mathrm{V}\). The line impedance is \(1+j 1 \Omega / \phi\). Draw the single phase equivalent circuit for the a-phase and use it to find the current in the a-phase of the load.

In a balanced three-phase system, the source is a balanced Y with an abc phase sequence and a line voltage \(\mathbf{V}_{\mathrm{ab}}=208 \angle 50^{\circ} \mathrm{V}\). The load is a balanced Y in parallel with a balanced \(\Delta\).The phase impedance of the Y is \(4+j 3 \Omega / \phi\) and the phase impedance of the \(\Delta\) is \(3-j 9 \Omega / \phi\). The line impedance is \(1.4+j 0.8 \Omega / \phi\). Draw the single phase equivalent circuit and use it to calculate the line voltage at the load in the a-phase.

A balanced Y-connected load having an impedance of \(60-j 45 \Omega / \phi\) is connected in parallel with a balanced \(\Delta\)-connected load having an impedance of \(90 \sqrt{2} \angle 45^{\circ} \Omega / \phi\). The paralleled loads are fed from a line having an impedance of \(2+j 2 \Omega / \phi\). The magnitude of the line-to-line voltage of the \(\Delta\)-load is \(300 \sqrt{3} \mathrm{~V}\). a) Calculate the magnitude of the phase current in the Y-connected load. b) Calculate the magnitude of the phase current in the \(\Delta\)-connected load. c) Calculate the magnitude of the current in the line feeding the loads. d) Calculate the magnitude of the line voltage at the sending end of the line.

A three-phase \(\Delta\)-connected generator has an internal impedance of \(9+j 90 \mathrm{~m} \Omega / \phi\). When the load is removed from the generator, the magnitude of the terminal voltage is 13,800 V. The generator feeds a \(\Delta\)-connected load through a transmission line with an impedance of \(20+j 180 \mathrm{~m} \Omega / \phi\). The per-phase impedance of the load is \(7.056+j 3.417 \Omega\). a) Construct a single-phase equivalent circuit. b) Calculate the magnitude of the line current. c) Calculate the magnitude of the line voltage at the terminals of the load. d) Calculate the magnitude of the line voltage at the terminals of the source. e) Calculate the magnitude of the phase current in the load. f) Calculate the magnitude of the phase current in the source.

The impedance Z in the balanced three-phase circuit in Fig. P11.19 is \(100-j 75 \Omega\). Find a) \(\mathbf{I}_{\mathrm{AB}}, \mathbf{I}_{\mathrm{BC}} \text {, and } \mathbf{I}_{\mathrm{CA}}\), b) \(\mathbf{I}_{\mathrm{aA}}, \mathbf{I}_{\mathrm{bB}}, \text { and } \mathbf{I}_{\mathrm{cC}}\), c) \(\mathbf{I}_{\mathrm{ba}}, \mathbf{I}_{\mathrm{cb}}, \text { and } \mathbf{I}_{\mathrm{ac}}\).

For the circuit shown in Fig. P11.20, find a) the phase currents and b) the line currents and when and Z3 = 20 + j 0 .

A balanced three-phase \(\Delta\)-connected source is shown in Fig. P11.21. a) Find the Y-connected equivalent circuit. b) Show that the Y-connected equivalent circuit delivers the same open-circuit voltage as the original \(\Delta\)-connected source. c) Apply an external short circuit to the terminals A, B, and C. Use the \(\Delta\)-connected source to find the three line currents \(\mathbf{I}_{\mathrm{aA}}, \mathbf{I}_{\mathrm{bB}}\), and \(\mathbf{I}_{\mathrm{cC}}\). d) Repeat (c) but use the Y-equivalent source to find the three line currents.

The \(\Delta\)-connected source of Problem 11.21 is connected to a Y-connected load by means of a balanced three-phase distribution line. The load impedance is \(1192+j 1584 \Omega / \phi\). and the line impedance is \(6.5+j 15 \Omega / \phi\). a) Construct a single-phase equivalent circuit of the system. b) Determine the magnitude of the line voltage at the terminals of the load. c) Determine the magnitude of the phase current in the \(\Delta\)-source. d) Determine the magnitude of the line voltage at the terminals of the source.

a) Find the rms magnitude and the phase angle of \(\mathbf{I}_{\mathrm{CA}}\) in the circuit shown in Fig. P11.23. b) What percent of the average power delivered by the three-phase source is dissipated in the three-phase load?

A balanced three-phase source is supplying 60 kVA at 0.6 lagging to two balanced Y-connected parallel loads. The distribution line connecting the source to the load has negligible impedance. Load 1 is purely resistive and absorbs 30 kW. Find the per-phase impedance of Load 2 if the line voltage is \(120 \sqrt{3} \mathrm{~V}\) and the impedance components are in series.

In a balanced three-phase system, the source has an abc sequence, is Y-connected, and \(\mathbf{V}_{\mathrm{an}}=120 \angle 20^{\circ} \mathrm{V}\). The source feeds two loads, both of which are Y-connected. The impedance of load 1 is \(8+j 6 \Omega / \phi\). The complex power for the a-phase of load 2 is \(600 \angle 36^{\circ} \mathrm{VA}\). Find the total complex power supplied by the source.

The line-to-neutral voltage at the terminals of the balanced three-phase load in the circuit shown in Fig. P11.26 is 1600 V. At this voltage, the load is absorbing 480 kVA at 0.8 pf lag. a) Use \(\mathbf{V}_{\mathrm{AN}}\) as the reference and express \(\mathbf{I}_{\mathrm{na}}\) in polar form. b) Calculate the complex power associated with the ideal three-phase source. c) Check that the total average power delivered equals the total average power absorbed. d) Check that the total magnetizing reactive power delivered equals the total magnetizing reactive power absorbed.

A three-phase positive sequence Y-connected source supplies 14 kVA with a power factor of 0.75 lagging to a parallel combination of a Y-connected load and a \(\Delta\)-connected load. The Y-connected load uses 9 kVA at a power factor of 0.6 lagging and has an a-phase current of \(10 \angle-30^{\circ} \mathrm{A}\). a) Find the complex power per phase of the \(\Delta\)-connected load. b) Find the magnitude of the line voltage.

A balanced three-phase distribution line has an impedance of \(5+j 10 \Omega / \phi\). This line is used to supply three balanced three-phase loads that are connected in parallel. The three loads are \(\mathrm{L}_{1}=180 \mathrm{kVA}\) at 0.866 pf lag, \(\mathrm{L}_{2}=150 \mathrm{kVA}\) at 0.28 pf lead, and \(\mathrm{L}_{3}=72.12 \mathrm{~kW}\) at unity pf. The magnitude of the line voltage at the terminals of the loads is \(1800 \sqrt{3} \mathrm{~V}\). a) What is the magnitude of the line voltage at the sending end of the line? b) What is the percent efficiency of the distribution line with respect to average power?

The three tools described below are part of a university’s machine shop. Each piece of equipment is a balanced three-phase load rated at 220 V(rms). Calculate (a) the magnitude of the line current supplying these three tools and (b) the power factor of the combined load. • Drill press: 10.2 kVA at 0.87 pf lag. • Lathe: 4.2 kW at 0.91 pf lag. • Band saw: line current 36.8 A(rms), 7.25 kVAR.

Calculate the complex power in each phase of the unbalanced load in Problem 11.20.

Show that the total instantaneous power in a balanced three-phase circuit is constant and equal to \(1.5 V_{m} I_{m} \cos \theta_{\phi}\), where \(V_{m}\) and \(I_{m}\) represent the maximum amplitudes of the phase voltage and phase current, respectively.

The total apparent power supplied in a balanced, three-phase \(\text { Y- } \Delta\) system is 4800 VA. The line voltage is 240 V. If the line impedance is negligible and the power factor angle of the load is \(-50^{\circ}\), determine the impedance of the load.

A balanced three-phase load absorbs 150 kVA at a leading power factor of 0.96 when the line voltage at the terminals of the load is 600 V. Find four equivalent circuits that can be used to model this load.

At full load, a commercially available 100 hp, three-phase induction motor operates at an efficiency of 97% and a power factor of 0.88 lag.The motor is supplied from a three-phase outlet with a line-voltage rating of 208 V. a) What is the magnitude of the line current drawn from the 208 V outlet? (1 hp = 746 W. ) b) Calculate the reactive power supplied to the motor.

A three-phase line has an impedance of \(0.1+j 0.8 \Omega / \phi\). The line feeds two balanced three-phase loads connected in parallel. The first load is absorbing a total of 630 kW and absorbing 840 kVAR magnetizing vars. The second load is Y-connected and has an impedance of \(15.36-j 4.48 \Omega / \phi\). The line-to-neutral voltage at the load end of the line is 4000 V. What is the magnitude of the line voltage at the source end of the line?

Three balanced three-phase loads are connected in parallel. Load 1 is Y-connected with an impedance of \(400+j 300 \Omega / \phi\); load 2 is \(\Delta\)-connected with an impedance of \(2400-j 1800 \Omega / \phi\) and load 3 is 172.8 + j2203.2 kVA. The loads are fed from a distribution line with an impedance of \(2+j 16 \Omega / \phi\). The magnitude of the line-to-neutral voltage at the load end of the line is \(24 \sqrt{3} \mathrm{kV}\). a) Calculate the total complex power at the sending end of the line. b) What percentage of the average power at the sending end of the line is delivered to the loads?

The output of the balanced positive-sequence three-phase source in Fig. P11.37 is 41.6 kVA at a lagging power factor of 0.707. The line voltage at the source is 240 V. a) Find the magnitude of the line voltage at the load. b) Find the total complex power at the terminals of the load.

A balanced three-phase source is supplying 540 kVA at 0.96 pf lag to two balanced \(\Delta\)-connected parallel loads. The distribution line connecting the source to the load has negligible impedance. The power associated with load 1 is 38.4 - j208.8 kVA. a) Determine the types of components and their impedances in each phase of load 2 if the line voltage is \(1600 \sqrt{3} \mathrm{~V}\) and the impedance components are in series. b) Repeat (a) with the impedance components in parallel.

The total power delivered to a balanced three-phase load when operating at a line voltage of \(2500 \sqrt{3} \mathrm{~V}\) is 900 kW at a lagging power factor of 0.6. The impedance of the distribution line supplying the load is \(1+j 3 \Omega / \phi\). Under these operating conditions, the drop in the magnitude of the line voltage between the sending end and the load end of the line is excessive. To compensate, a bank of Y-connected capacitors is placed in parallel with the load. The capacitor bank is designed to furnish 1125 kVAR of magnetizing reactive power when operated at a line voltage of \(2500 \sqrt{3} \mathrm{~V}\). a) What is the magnitude of the voltage at the sending end of the line when the load is operating at a line voltage of \(2500 \sqrt{3} \mathrm{~V}\) and the capacitor bank is disconnected? b) Repeat (a) with the capacitor bank connected. c) What is the average power efficiency of the line in (a)? d) What is the average power efficiency in (b)? e) If the system is operating at a frequency of 60 Hz, what is the size of each capacitor in microfarads?

A balanced bank of delta-connected capacitors is connected in parallel with the load described in Assessment Problem 11.9. The effect is to place a capacitor in parallel with the load in each phase. The line voltage at the terminals of the load thus remains at 2450 V. The circuit is operating at a frequency of 60 Hz.The capacitors are adjusted so that the magnitude of the line current feeding the parallel combination of the load and capacitor bank is at its minimum. a) What is the size of each capacitor in microfarads? b) Repeat (a) for wye-connected capacitors. c) What is the magnitude of the line current?

The two-wattmeter method is used to measure the power at the load end of the line in Example 11.1. Calculate the reading of each wattmeter.

The wattmeters in the circuit in Fig. 11.20 read as follows: \(W_{1}=40,823.09 \mathrm{~W}\), and \(W_{2}=103,176.91 \mathrm{~W}\). The magnitude of the line voltage is \(2400 \sqrt{3} \mathrm{~V}\). The phase sequence is positive. Find \(Z_{\phi}\).

In the balanced three-phase circuit shown in Fig. P11.43, the current coil of the wattmeter is connected in line aA, and the potential coil of the wattmeter is connected across lines b and c. Show that the wattmeter reading multiplied by \(\sqrt{3}\) equals the total reactive power associated with the load. The phase sequence is positive.

The line-to-neutral voltage in the circuit in Fig. P11.43 is 680 V, the phase sequence is positive, and the load impedance is \(16-j 12 \Omega / \phi\). a) Calculate the wattmeter reading. b) Calculate the total reactive power associated with the load.

The two wattmeters in Fig. 11.20 can be used to compute the total reactive power of the load. a) Prove this statement by showing that \(\sqrt{3}\left(W_{2}-W_{1}\right)=\sqrt{3} V_{\mathrm{L}} I_{\mathrm{L}} \sin \theta_{\phi}\). b) Compute the total reactive power from the wattmeter readings for each of the loads in Example 11.6. Check your computations by calculating the total reactive power directly from the given voltage and impedance.

Derive Eqs. 11.56 and 11.57.

a) Calculate the complex power associated with each phase of the balanced load in Problem 11.19. b) If the two-wattmeter method is used to measure the average power delivered to the load, specify the reading of each meter.

The two-wattmeter method is used to measure the power delivered to the unbalanced load in Problem 11.20. The current coil of wattmeter 1 is placed in line aA and that of wattmeter 2 is placed in line bB. a) Calculate the reading of wattmeter 1. b) Calculate the reading of wattmeter 2. c) Show that the sum of the two wattmeter readings equals the total power delivered to the unbalanced load.

The balanced three-phase load shown in Fig. P11.49 is fed from a balanced, positive-sequence, three-phase Y-connected source. The impedance of the line connecting the source to the load is negligible. The line-to-neutral voltage of the source is 7200 V. a) Find the reading of the wattmeter in watts. b) Explain how you would connect a second wattmeter in the circuit so that the two wattmeters would measure the total power. c) Calculate the reading of the second wattmeter. d) Verify that the sum of the two wattmeter readings equals the total average power delivered to the load.

a) Calculate the reading of each wattmeter in the circuit shown in Fig. P11.50. The value of is b) Verify that the sum of the wattmeter readings equals the total average power delivered to the -connected load.

a) Calculate the reading of each wattmeter in the circuit shown in Fig. P11.51 when \(Z=13.44+j 46.08 \Omega\). b) Check that the sum of the two wattmeter readings equals the total power delivered to the load. c) Check that \(\sqrt{3}\left(W_{1}-W_{2}\right)\) equals the total magnetizing vars delivered to the load.

a) Find the reading of each wattmeter in the circuit shown in Fig. P11.52 if \(Z_{\mathrm{A}}=20 \angle 30^{\circ} \ \Omega\), \(Z_{\mathrm{B}}=60 \angle 0^{\circ} \ \Omega\), and \(Z_{\mathrm{C}}=40 \angle-30^{\circ} \ \Omega\). b) Show that the sum of the wattmeter readings equals the total average power delivered to the unbalanced three-phase load.

Refer to the Practical Perspective example: a) Construct a power triangle for the substation load before the capacitors are connected to the bus. b) Repeat (a) after the capacitors are connected to the bus. c) Using the line-to-neutral voltage at the substation as a reference, construct a phasor diagram that depicts the relationship between \(\mathbf{V}_{\mathrm{AN}}\) and \(\mathbf{V}_{\mathrm{an}}\) before the capacitors are added. d) Assume a positive phase sequence and construct a phasor diagram that depicts the relationship between \(\mathbf{V}_{\mathrm{AB}}\) and \(\mathbf{V}_{\mathrm{ab}}\).

Refer to the Practical Perspective example.Assume the frequency of the utility is 60 Hz. a) What is the \(\mu \mathrm{F}\) rating of each capacitor if the capacitors are delta-connected? b) What is the \(\mu \mathrm{F}\) rating of each capacitor if the capacitors are wye-connected?

Choose a single capacitor from Appendix H that is closest to the \(\mu \mathrm{F}\) rating of the wye-connected capacitor from Problem 11.54(b). a) How much reactive power will a capacitor bank using this new value supply? b) What line-to-line voltage at the generating plant will be required when this new capacitor bank is connected to the substation bus?

Choose a single capacitor from Appendix H that is closest to the \(\mu \mathrm{F}\) rating of the delta-connected capacitor from Problem 11.54(a). a) How much reactive power will a capacitor bank using this new value supply? b) What line-to-line voltage at the generating plant will be required when this new capacitor bank is connected to the substation bus?

In the Practical Perspective example, what happens to the voltage level at the generating plant if the substation is maintained at 13.8 kV, the substation load is removed, and the added capacitor bank remains connected?

In the Practical Perspective example, calculate the total line loss in kW before and after the capacitors are connected to the substation bus.

Assume the load on the substation bus in the Practical Perspective example drops to 180 kW and 480 magnetizing kVAR. Also assume the capacitors remain connected to the substation. a) What is the magnitude of the line-to-line voltage at the generating plant that is required to maintain a line-to-line voltage of 13.8 kV at the substation? b) Will this power plant voltage level cause problems for other customers?

Assume in Problem 11.59 that when the load drops to 180 kW and 480 magnetizing kVAR the capacitor bank at the substation is disconnected. Also assume that the line-to-line voltage at the substation is maintained at 13.8 kV. a) What is the magnitude of the line-to-line voltage at the generating plant? b) Is the voltage level found in (a) within the acceptable range of variation? c) What is the total line loss in kW when the capacitors stay on line after the load drops to ? d) What is the total line loss in kW when the capacitors are removed after the load drops to ? e) Based on your calculations, would you recommend disconnecting the capacitors after the load drops to ? Explain.