EE 30372 Homeworks, Spring 2012

No. 1, Due 27 January Chapman: 1-19
Supplemental:
(a) You need to supply a 5 horsepower motor, having 80% efficiency, power factor of 0.75 lagging, and operating at 240V. Find the gauge of wire you'll need to use for single-phase and three-phase motors having these characteristics. The AWG capacity ratings (very conservative) for power transmission for several wire gauges are: 18ga: 2.3A, 16ga: 3.7A, 14ga: 5.9A, 12ga: 9.3A, 10ga: 15A, 8ga: 24A, 6ga: 37A
(b) The type of system commonly known as "two-phase" had the configuration we discussed in class for household power, but with the two sources 90 degrees apart. Show that if such a system drives any balanced two-phase load, the total power delivered does not vary with time.
(c) A single-phase system includes two voltage sources, the first at 120/_0 degrees V and the second at 110/_45 degrees V with opposite orientation, and the two are separated by a line impedance of 1.5 +j6 Ohms. Find the real and reactive power produced/absorbed by each of the two sources. Determine the real and reactive power produced/consumed by the line impedance.

No. 2, Due 3 February Chapman: 2-1, 2-2, 2-5
Supplemental:
(a) Suppose that the following is true of the motor we ran at 208V, 3-phase on Wednesday morning: with zero power produced, the line current was 1.9A, and at 1 HP, the current rose to 3.2A (remember these are RMS currents). If we are to model each phase of the Y-connected motor as a passive circuit in each of these two states, specify parameters in a diagram which matches a phase in each state.
(b) We can produce constant power in a balanced system of two or three (or more) phases, with two-phase of the type in HW1 (b). Each system can be implemented using three wires to carry current. Using each of the two systems, both having line-to-neutral voltage of 250V, find the RMS current magnitude in each of the three wires when you are supplying a 10 HP motor with power factor 0.8 lagging. You may assume 100% efficiency.

No. 3, Due 10 February Chapman: 2-6
Supplemental:
(a) We have a 600V 3-phase source running in open delta configuration, with the CA source removed. It is driving a Y-connected load having impedance 10 + j5 Ohms in each phase. Find the line current I_B, the real and reactive power being consumed by the load, and the real and reactive power being output by each of the two source phases.
(b) Suppose the transformer we've been using in class shows an impedance of j200 Ohms at 60 Hz between the terminals of a single coil on the X (low voltage) side. What will be the impedance between the outside ends of the two X coils if we connect their inner terminals? What are the impedances of the two possible parallel configurations of the two coils?

No. 4, Due 17 February Chapman: 3-1, 3-2, 3-3
Supplemental:
(a) Show a wiring diagram for the transformer we've been using in class as an autotransformer to raise the voltage from the wall outlet from 120V to 150V.

No. 5, Due 24 February Chapman: 3-11, 3-12, 3-18, 3-22a,c, 4-4
Supplemental:
(a) Under our standard circuit model for a single-phase transformer operating at 60 Hz, with all impedances referenced to the primary side, we have equivalent values of X_eq = 1.0 Ohms, R_eq = 1.0 Ohms, X_M = 99 Ohms. The turns ratio is a = 0.5 and V_p is 100V. Find the no-load secondary voltage. Next, suppose we place in series on the primary side a capacitor of C = 26.53 microfarads. What is now the no-load secondary voltage? This sort of configuration is known as a resonant transformer.
(b) Show that the two-phase system in homework supplemental problem 1(b) can be used to create a uniform-magnitude, rotating magnetic field in a stator, similarly to the three-phase stator.

No. 6, Due 2 March Chapman: 5-1, 5-15, 6-2, 6-5, 7-2
Supplemental:
(a) Under our simple model for rotor and stator field interaction, sketch the relationship of the two in a two-pole, three-phase generator rotating clockwise, generating terminal voltage 480V and supplying a 12 HP motor at full load plus capacitor bank with total power factor 0.85 leading. Each phase of the Y-connected generator has output impedance 1 + j3.
(b) Draw an accurate phasor diagram of the generator in the state described in (a).

No. 7, Due 6 March (practice) Chapman: 7-7, 7-13, 7-21

No. 8, Due 23 March Chapman: 9-3, 9-5, 9-6 (in 9-4, resistance is 4.004e-5 Ohms/meter; "charging current" is current input to line when receiving end is open-circuited. Treat as medium-length line for this problem.), 9-7(treat as short line for this problem), 9-8, 9-9(again, as medium-length for this problem)
Supplemental: In class we found the inductance and capacitance per meter for a two-line, single-phase system. If we have a specific type of cable with the two conductors distance D apart and have inductance of L/meter and C/meter in single-phase, what will be the capacitance and inductance per meter in the per-phase circuit of a three-phase system using the same cable, and having the three conductors uniformly distance D apart?

No. 9, Due 30 March Chapman: 9-15(treat as short line), 9-17, 10-3(set base values according to generator 1; put per-phase circuit in pu), 10-4
Schaum's: 2.21, 2.25, 2.29 (correct the given answer)
Supplemental:
(a) For the transmission line connecting the two busses in Chapman's Problem 9-17, find the impedance matrix which relates the two bus voltages to the two bus currents.
(b) Find the admittance matrix for the line of 9-17.

No. 10, Due 11 April Chapman: 10-7, 10-8, 10-9 (Use Matlab for any significant matrix/vector operations in these problems and attach your code to your homework.)
Supplemental:
(a) In the text's Example 10-3, we solved for voltages at all buses. Suppose that while the system is in the given state, a load is attached to bus 4 which demands 0.2 Amps pu, at fixed angle -5.0 degrees. What will be the new voltages at all 4 buses?
(b) Again suppose the 4-bus system to be in the state computed for Example 10-3. Now a source is attached to bus 2 which raises its voltage to 1.1 V pu, at 0 degrees, held regardless of current and power flow. Other bus voltages are unchanged. What are the currents now flowing into or out of buses 1 and 3?
(c) Read/work through all the text and examples in the Schaum's book, chapter 7, except the matrix partitioning stuff on pages 98 and 99. These would be nice examples to reproduce on a quiz and/or exam.

Binary-valued project, Due 13 April
Using the schematic from ND Utilities and the wire parameter sheets handed out in class, find the admittance matrix for Feeder 33, of the type used for power flow. Consider any line shorter than 100 feet to be impedance free, and place buses at the entrances to buildings, which keeps the transformers out of the admittance matrix. Assume all wires are identical (500kcmil, three 1/C cables). Minimize the number of buses you use.

No. 11, Due 18 April For all Matlab problems, turn in your input and ouput files by email to xzou@nd.edu and sauer@nd.edu with answers to any questions appended to output files.
Chapman: 11-4, 11-5, 11-6, 11-7 (use power_flow.m for these)
Supplemental:
(a) Using the line values for Feeder 33 from the April 13 project, converted to pu under line Vbase of 4160V and Sbase of 3 MVA, use power_flow.m to analyze the state of the feeder system under the following conditions: The Bus labeled "Bus 3" on the schematic is the slack bus. All loads (buildings) are consuming 60% of the total rated capacity of their transformers (note that McKenna has a 3x100kVA=300kVA transformer) at power factor 0.8 lagging. Leave the 400kVA capacitor at Hanks disconnected. How's everything look? Compute the complex power flowing out of the Hanks bus going "downstream."
(b) Repeat (a) with the Hanks capacitor connected to the bus. Comment on the difference from the state of (a).
(c) Find the Gauss-Seidel acceleration factor for which the system converges most rapidly for the configuration of (b). Turn in output file including your value for alpha.
(d) For the 4-bus example from class, compute the first Gauss-Seidel updates of load buses 2 and 3, using a flat start.

No. 12, Due 27 April Chapman: 13-2a,b, 13-4c,d, 13-6a,b(subtransient for both parts)
Supplemental:
(a) A three-phase distribution transformer with output voltage 13.8kV is rated at 30 MVA, and is Y/Y-connected with solidly grounded neutrals. We know that the currents on the three phases can vary +/- 15% of rated maximum in magnitude, and phase may vary +/- 15 degrees around standard positive sequence values. How much current should the grounding wire be equipped to carry under this range of normal operating conditions?
(b) Under the worst-case condition you considered in (a), what is the symmetric component decomposition of the three-phase current?

Binary-valued project, Due 30 April
Columbus, Indiana, is noted as the home of Cummins Diesel and an unusually large collection of noteworthy architecture for a town its size. To enhance its image yet more, it decides to purchase its entire electric power supply from a new wind farm sited on Clint Manning's ranch near Macy, IN. (When there's no wind, the city will get its power from a generator driven by the world's largest Cummins Diesel engine.) Columbus needs 200 MW, which it consumes with power factor 0.95 lagging. You need to build a dedicated transmission line to get the power to Columbus, and also have to pass through or past Indianapolis. Indy will not cooperate with siting the lines unless they can connect to the line and draw at least 100MW for themselves, also at power factor 0.95 lagging. The distance from Macy to Indianapolis is 85 miles, and from Indianapolis to Columbus is 50 miles.
(a) You must choose whether to use 138kV or 345kV for transmission. Use Table 3.1, handed out in class, for the parameters of each type of line (medium-length model) and compare the voltage options in terms of voltage regulation, power angle and line wattage losses between Macy and Columbus with Indianapolis connected as well. You will supply the rated voltage at the Columbus bus.
(b) Now consider using a 400kV DC line, with each of the two cables having the same resistance per unit length as the 345kV AC line. Again compute voltage regulation and line losses. Here you may assume the reactive power is generated locally by capacitor banks.
Write a Matlab program to do your calculations, or modify power_flow.m to solve this problem. Email your program to Dr. Sauer and Xinzhi.

No. 13, Due 2 May Chapman: 13-6d, 13-7(Use faults.m or write your own Matlab for all these problems plus Supplemental (a).)
Supplemental:
(a) Find the A,B,C fault currents at bus 2 and the A,B,C voltages at all buses in the system of Problem 13-6 if there is a line-to-line fault at bus 2. You may assume the the system is holding voltage of 1.0 at all buses before the fault. You may use your own Matlab program or faults.m
(b) (Extra Credit) In the text, line-to-line and both single and double line-to-ground faults on an unloaded generator are analyzed. The author forgot to include the deadly line-to-line(B&C phases)-plus-line-to-ground(A phase) fault. Create the circuit for analysis of this fault similarly to what was derived for the other types.