Study Guide for Chapter 11

Transcription

1 Study Guide for Chapter 11 Objectives: 1. Know how to analyze a balanced, three-phase Y-Y connected circuit. 2. Know how to analyze a balanced, three-phase Y-Δ connected circuit. 3. Be able to calculate power (average, reactive, and complex) in any three-phase circuit. Study guide questions: 1. Read Section a) Use the definition of a phasor to show that Eq. (11.3) holds for the positive phase sequence (Eq. 11.1) and also for the negative phase sequence (Eq. 11.2). b) Use basic trigonometry to show that both sets of the three vectors in Fig sum to zero, again confirming Eq. (11.3). c) Summarize your understanding of the material in this section by answering the i. What are the two requirements of a balanced three-phase source? ii. If you are given the phasor voltage in the a-phase of a balanced threephase source, what additional information must you know in order to determine the phasor voltages in the other two phases? iii. Describe how to calculate the a-phase and c-phase voltage phasors from the b-phase voltage phasor in a balanced, three-phase, abc source. iv. Describe how to calculate the a-phase and b-phase voltage phasors from the c-phase voltage phasor in a balanced, three-phase, acb source. v. What is the sum of the three phasor voltages in a balanced three-phase source? a) Solve the following problems: Problems Read Section a) Re-label the source voltages in Fig to distinguish the Y-connected sources from the Δ-connected sources by calling the Y-connected source V ya, V yb, and V yc and calling the Δ-connected sources V Δa, V Δb, and V Δc. Then express each Δ- connected voltage source in terms of the Y-connected voltage sources by equating the voltage drops between nodes ab, bc, and ca in both circuits. Assume the Y- connected sources use the positive phase sequence. b) Summarize your understanding of the material in this section by answering the i. What are the four configurations for three-phase circuits? ii. Which configuration is the key to solving all balanced three-phase circuits? iii. What node is available in a Y-connected source that is not available in a Δ-connected source? iv. What three letters are used to label the nodes of a three-phase source? a) Use the result in 2(a) to solve Problem 11.8.

2 3. Read Section a) Show that Eqs. (11.18) (11.20) match the equations you derived in 2(a). b) Derive the equations that describe the line-to-line voltages (V AB, V BC, and V CA ) in terms of the line-to-neutral voltages (V AN, V BN, and V CN ), assuming the line-toneutral voltages have a negative phase sequence. c) Draw the entire three-phase circuit described in Example 11.1, labeling all line voltages, line currents, phase voltages, and phase currents. d) The following questions refer to the solution for Example 11.1: i. What circuit analysis technique is used to calculate the a-phase line current in part (b)? ii. Use voltage division to calculate the a-phase voltage of the load and show that you get the same answer as in part (c). iii. Refer to the three-phase circuit you drew in 3(c). Write a KVL equation for V AB in terms of V AN and V BN. Then substitute the values of V AN and V BN into this equation and confirm the answer in part (d) of the solution. Use the same technique to check the answers for V BC and V CA. iv. Again refer to the three-phase circuit you drew in 3(c). Write a KVL equation for V ab in terms of V an and V bn. Then plug the values of V an and V bn into this equation and confirm the answers in part (f) of the solution. Use the same technique to check the answers for V bc and V ca. v. Example 11.1 asked you to calculate both the line voltages and the phase voltages at the terminals of the load. But this example asked to you to calculate only the line currents for the load, and not the phase currents for the load. Why weren t you asked to calculate the phase currents for the load? a) Summarize your understanding of the material in this section by answering the i. What is the easiest circuit analysis technique to use when analyzing a general (not necessarily balanced) three-phase Y-Y circuit? ii. What are the four conditions that must be satisfied if a three-phase circuit is balanced? iii. When a three-phase Y-Y circuit is balanced, what is the current in the neutral line? iv. When a three-phase Y-Y circuit is balanced, what is the relationship between the three line currents? v. Define the following quantities for a Y-Y circuit and give an example phasor name for each: line voltage, phase voltage, line current, phase current. vi. In a balanced Y-Y circuit, what is the relationship between the line current and the phase current? What is the relationship between the line voltage and the phase voltage? vii. What circuit analysis technique would you use to calculate the line current I aa in the single phase equivalent circuit? viii. What circuit analysis technique would you use to calculate the phase voltage V AN in the single phase equivalent circuit? ix. What is the phasor symbol for the phase current in a Y-connected load? How do you calculate its value from the line current phasor whose value you determined from the single phase equivalent circuit? x. What is the phasor symbol for the line voltage in a Y-connected load?

3 How do you calculate its value from the phase voltage phasor whose value you determined from the single phase equivalent circuit? (Hint there are two answers to this question, one for each phase sequence!) a) Solve the following problems: Assessment Problems ; Chapter Problems Read Section a) Show that Eq. (11.21 follows from Eqs. (9.51) (9.53) on p b) Derive the line current in terms of the phase currents for a negative phase sequence for a Y-Δ circuit. Follow the example for the positive phase sequence from Eqs. (11.22) (11.27). c) Draw the entire three-phase circuit described in Example 11.2, labeling the line currents, phase currents, line voltages and phase voltages in each phase. Use this circuit to show that the line voltage V ab at the terminals of the source can be calculated by summing the voltage drop across the a-phase line (V aa ), the voltage drop across the a-phase load (V AB ), and the voltage drop across the b-phase line (V Bb ). What circuit analysis technique did you use? Now perform this calculation to confirm the value computed in part (e) of the solution. d) Summarize your understanding of the material in this section by answering the i. Define the following quantities for a Y-Δ circuit and give an example phasor name for each: line voltage, phase voltage, line current, phase current. ii. In a balanced Y-Δ circuit, what is the relationship between the line current and the phase current? What is the relationship between the line voltage and the phase voltage? iii. What simple calculation do you need to make to create a single phase equivalent circuit from a balanced Y-Δ circuit? iv. What is the phasor symbol for the phase current in a Δ-connected load? How do you calculate its value from the line current phasor whose value you determined from the single phase equivalent circuit? (Hint there are two answers to this question, one for each phase sequence!) v. What is the phasor symbol for the phase voltage in a Δ-connected load? How do you calculate its value from the voltage phasor whose value you determined from the single phase equivalent circuit? (Hint there are two answers to this question, one for each phase sequence!) a) Complete the following problems: Assessment Problems Read Section a) What is the other name for the quantity cos(θ va - A θ ) in Eq. (11.28)? Rewrite Eqs. ia (11.35) and (11.36) using this alternate name. b) What is the other name for the quantity sin θ φ in Eq. (11.37)? Rewrite Eqs. (11.37) and (11.38) using this alternate name. c) Look at the solution to Example What equation for average power is used in parts (c) and (d)? What equation for per-phase complex power is used in part (f)? Show that the total complex power in the circuit for this example balances by calculating the total complex power of the source, the source impedance, the line, and the load and summing these complex power values. d) Look at the solution to Example Show that the total complex power in the circuit for this example balances by calculating the total complex power of the

4 source, the source impedance, the line, and the load and summing these complex power values. e) Look at the solution to Example How was the voltage V AN (in Fig ) calculated from the information supplied in the problem statement? Where did the first equation in part (b) come from? Note the alternative solution for the line current magnitude at the end of part (b) this is an important solution technique to master, but it is only useful in calculating the line current magnitude, not its phase angle. What circuit analysis technique is used to construct the first equation in part (c)? Show that the complex power balances for the single line equivalent circuit in Fig by calculating the complex power at the sending end of the line, the complex power in the line, and the complex power of the load and summing these complex power values. Make sure you follow the passive sign convention! f) Summarize your understanding of the material in this section by answering the i. If you use phase quantities to calculate the real, reactive, and complex power in the a-phase of a Y-connected load (that is, you use the phase voltage and phase current of a single phase of the load), what are the ii. If you use line quantities to calculate the real, reactive, and complex power in the a-phase of a Y-connected load (that is, you use the line voltage and line current of a single phase of the load), what are the iii. If you use phase quantities to calculate the real, reactive, and complex power in the a-phase of a Δ-connected load (that is, you use the phase voltage and phase current of a single phase of the load), what are the iv. If you use line quantities to calculate the real, reactive, and complex power in the a-phase of a Δ-connected load (that is, you use the line voltage and line current of a single phase of the load), what are the v. What is the advantage of using the formulas for P, Q, and S that contain the line voltage and line current when calculating the power in a single phase of a three-phase circuit? What phase angle should be used in these formulas? vi. If you are given the total P, Q, or S in a balanced three-phase circuit, how can you calculate the P, Q, or S per phase? a) Complete the following problems: Problems 11.19, 11.24, 11.29, and

5 Answers to Assigned Problems: 11.1 a) acb; b) abc 11.2 a) balanced, positive; b) balanced, negative; c) unbalanced (phase angle); d) unbalanced (amplitude); e) unbalanced (phase angle); e) unbalanced (frequency) 11.8 v AB (t) = cos(ωt + 56 ) V(rms); v BC (t) = cos(ωt - 64 ) V(rms); v AB (t) = cos(ωt ) V(rms) 11.9 a) A(rms); b) V(rms) a) I aa = A(rms), I bb = A(rms), I cc = A(rms); b) V ab = V(rms), V bc = V(rms), V ca = V(rms); c) V an = V(rms), V bn = V(rms), V cn = V(rms); d) V AB = V(rms), V BC = V(rms), V CA = V(rms); a) I AB = A(rms), I BC = A(rms), I AB = A(rms); b) I aa = A(rms), I bb = A(rms) I cc = A(rms); c) V ab = V(rms), V bc = V(rms), V ca = V(rms) a) j3510 kva; b) 99.29% V(rms) a) V(rms); b) 30, j28, VA a) V(rms); b) V(rms); c) 95.05%; d) 96.77%; e) μf