PROBLEMS

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

2.1 A transformer is made up of a 1200-turn primary coil and an open-circuited 75-turn secondary coil wound around a closed core of cross-sectional area 42 cm2cm2 size 12{ ital "cm" rSup { size 8{2} } } {}. The core material can be considered to saturate when the rms applied flux density reaches 1.45T. What maximum 60-Hz rms primary voltage is possible without reaching this saturation level? What is the corresponding secondary voltage? How are these values modified if the applied frequency is lowered to 50 Hz?

2.2 A magnetic circuit with a cross-sectional area of 15 cm2cm2 size 12{ ital "cm" rSup { size 8{2} } } {} is to be operated at 60 Hz from a 120-V rms supply. Calculate the number of turns required to achieve a peak magnetic flux density of 1.8 T in the core.

2.3 A transformer is to be used to transform the impedance of a 8 ΩΩ size 12{ %OMEGA } {} resistor to an impedance of 75 ΩΩ size 12{ %OMEGA } {}. Calculate the required turns ratio, assuming the transformer to be ideal.

2.4 A 100 ΩΩ size 12{ %OMEGA } {} resistor is connected to the secondary of an idea transformer with a turns ratio of 1:4 (primary to secondary). A 10-V rms, 1-kHz voltage source is connected to the primary. Calculate the primary current and the voltage across the 100 ΩΩ size 12{ %OMEGA } {} resistor.

2.5 A source which can be represented by a voltage source of 8 V rms in series with an internal resistance of 2 kΩkΩ size 12{k %OMEGA } {} is connected to a 50 ΩΩ size 12{ %OMEGA } {} load resistance through an ideal transformer. Calculate the value of turns ratio for which maximum power is supplied to the load and the corresponding load power? Using MATLAB, plot the the power in milliwatts supplied to the load as a function of the transformer ratio, covering ratios from 1.0 to 10.0.

2.6 Repeat Problem 2.5 with the source resistance replaced by a 2- kΩkΩ size 12{k %OMEGA } {} reactance.

2.7 A single-phase 60-Hz transformer has a nameplate voltage rating of 7.97 kV:266 V, which is based on its winding turns ratio. The manufacturer calculates that the primary (7.97-kV) leakage inductance is 165 mH and the primary magnetizing inductance is 135 H. For an applied primary voltage of 7970 V at 60 Hz, calculate the resultant open-circuit secondary voltage.

2.8 The manufacturer calculates that the transformer of Problem 2.7 has a secondary leakage inductance of 0.225 mH.

a. Calculate the magnetizing inductance as referred to the secondary side.

b. A voltage of 266 V, 60 Hz is applied to the secondary. Calculate (i) the resultant open-circuit primary voltage and (ii) the secondary current which would result if the primary were short-circuited.

2.9 A 120-V:2400-V, 60-Hz, 50-kVA transformer has a magnetizing reactance (as measured from the 120-V terminals) of 34.6 ΩΩ size 12{ %OMEGA } {}. The 120-V winding has a leakage reactance of 27.4 mΩmΩ size 12{m %OMEGA } {} and the 2400-V winding has a leakage reactance of 11.2 ΩΩ size 12{ %OMEGA } {}.

a. With the secondary open-circuited and 120 V applied to the primary (120-V) winding, calculate the primary current and the secondary voltage.

b. With the secondary short-circuited, calculate the primary voltage which will result in rated current in the primary winding. Calculate the corresponding current in the secondary winding.

2.10 A 460-V:2400-V transformer has a series leakage reactance of 37.2 ΩΩ size 12{ %OMEGA } {} as referred to the high-voltage side. A load connected to the low-voltage side is observed to be absorbing 25 kW, unity power factor, and the voltage is measured to be 450 V. Calculate the corresponding voltage and power factor as measured at the high-voltage terminals.

2.11 The resistances and leakage reactances of a 30-kVA, 60-Hz, 2400-V:240-V distribution transformer are

R1 = 0.68 ΩΩ size 12{ %OMEGA } {} R2 = 0.0068 ΩΩ size 12{ %OMEGA } {}

X11X11 size 12{X rSub { size 8{1 rSub { size 6{1} } } } } {} = 7.8 ΩΩ size 12{ %OMEGA } {}{}X12X12 size 12{X rSub { size 8{1 rSub { size 6{2} } } } } {} = 0.0780 ΩΩ size 12{ %OMEGA } {}

where subscript 1 denotes the 2400-V winding and subscript 2 denotes the 240-V winding. Each quantity is referred to its own side of the transformer.

a. Draw the equivalent circuit referred to (i) the high- and (ii) the low-voltage sides. Label the impedances numerically.

b. Consider the transformer to deliver its rated kVA to a load on the low-voltage side with 230 V across the load. (i) Find the high-side terminal voltage for a load power factor of 0.85 power factor lagging. (ii) Find the high-side terminal voltage for a load power factor of 0.85 power factor leading.

c. Consider a rated-kVA load connected at the low-voltage terminals operating at 240V. Use MATLAB to plot the high-side terminal voltage as a function of the power-factor angle as the load power factor varies from 0.6 leading through unity power factor to 0.6 pf lagging.

2.12 Repeat Problem 2.11 for a 75-kVA, 60-Hz, 4600-V:240-V distribution transformer whose resistances and leakage reactances are

R1 = 0.846 ΩΩ size 12{ %OMEGA } {} R2 = 0.00261 ΩΩ size 12{ %OMEGA } {}

X11X11 size 12{X rSub { size 8{1 rSub { size 6{1} } } } } {} = 26.8 ΩΩ size 12{ %OMEGA } {}X12X12 size 12{X rSub { size 8{1 rSub { size 6{2} } } } } {}= 0.0745 ΩΩ size 12{ %OMEGA } {}

where subscript 1 denotes the 4600-V winding and subscript 2 denotes the 240-V winding. Each quantity is referred to its own side of the transformer.

2.13 A single-phase load is supplied through a 35-kV feeder whose impedance is 95+j360 ΩΩ size 12{ %OMEGA } {} and a 35-kV:2400-V transformer whose equivalent impedance is 0.23 + j1.27 ΩΩ size 12{ %OMEGA } {} referred to its low-voltage side. The load is 160 kW at 0.89 leading power factor and 2340 V.

a. Compute the voltage at the high-voltage terminals of the transformer.

b. Compute the voltage at the sending end of the feeder.

c. Compute the power and reactive power input at the sending end of the feeder.

2.14 The nameplate on a 50-MVA, 60-Hz single-phase transformer indicates that it has a voltage rating of 8.0-kV:78-kV. An open-circuit test is conducted from the low-voltage side, and the corresponding instrument readings are 8.0 kV, 62.1 A, and 206 kW. Similarly, a short-circuit test from the low-voltage side gives readings of 674 V, 6.25 kA, and 187 kW.

a. Calculate the equivalent series impedance, resistance, and reactance of the transformer as referred to the low-voltage terminals.

b. Calculate the equivalent series impedance of the transformer as referred to the high-voltage terminals.

c. Making appropriate approximations, draw a T equivalent circuit for the transformer.

d. Determine the efficiency and voltage regulation if the transformer is operating at the rated voltage and load (unity power factor).

e. Repeat part (d), assuming the load to be at 0.9 power factor leading.

2.15 A 550-kVA, 60-Hz transformer with a 13.8-kV primary winding draws 4.93 A and 3420 W at no load, rated voltage and frequency. Another transformer has a core with all its linear dimensions 22 size 12{ sqrt {2} } {} times as large as the corresponding dimensions of the first transformer. The core material and lamination thickness are the same in both transformers. If the primary windings of both transformers have the same number of turns, what no-load current and power will the second transformer draw with 27.6 kV at 60 Hz impressed on its primary?

2.16 The following data were obtained for a 20-kVA, 60-Hz, 2400:240-V distribution transformer tested at 60 Hz:

a. Compute the efficiency at full-load current and the rated terminal voltage at 0.8 power factor.

b. Assume that the load power factor is varied while the load current and secondary terminal voltage are held constant. Use a phasor diagram to determine the load power factor for which the regulation is greatest. What is this regulation?

2.17 A 75-kVa, 240-V:7970-V, 60-Hz single-phase distribution transformer has the following parameters referred to the high-voltage side:

R1 = 5.93 ΩΩ size 12{ %OMEGA } {} X1 = 43.2 ΩΩ size 12{ %OMEGA } {}

R2 = 3.39 ΩΩ size 12{ %OMEGA } {} X2 = 40.6 ΩΩ size 12{ %OMEGA } {}

Rc = 244 kΩkΩ size 12{k %OMEGA } {} X m = 114 kΩkΩ size 12{k %OMEGA } {}

Assume that the transformer is supplying its rated kVA at its low-voltage terminals. Write a MATLAB script to determine the efficiency and regulation of the transformer for any specified load power factor (leading or lagging). You may use reasonable engineering approximations to simplify your analysis. Use your MATLAB script to determine the efficiency and regulation for a load power factor of 0.87 leading.

2.18 The transformer of Problem 2.11 is to be connected as an autotransformer. Determine (a) the voltage ratings of the high- and low-voltage windings for this connection and (b) the kVA rating of the autotransformer connection.

2.19 A 120:480-V, 10-kVA transformer is to be used as an autotransformer to supply a 480-V circuit from a 600-V source. When it is tested as a two-winding transformer at rated load, unity power factor, its efficiency is 0.979.

a. Make a diagram of connections as an autotransformer.

b. Determine its kVA rating as an autotransformer.

c. Find its efficiency as an autotransformer at full load, with 0.85 power factor lagging.

2.20 Consider the 8-kV:78-kV, 50-MVA transformer of Problem 2.14 connected as an autotransformer.

a. Determine the voltage ratings of the high- and low-voltage windings for this connection and the kVA rating of the autotransformer connection.

b. Calculate the efficiency of the transformer in this connection when it is supplying its rated load at unity power factor.

2.21 Write a MATLAB script whose inputs are the rating (voltage and kVA) and rated-load, unity-power-factor efficiency of a single-transformer and whose output is the transformer rating and rated-load, unity-power-factor efficiency when connected as an autotransformer.

2.22 The high-voltage terminals of a three-phase bank of three single-phase transformers are supplied from a three-wire, three-phase 13.8-kV (line-to-line) system. The low-voltage terminals are to be connected to a three-wire, three-phase substation load drawing up to 4500kVA at 2300 V line-to-line. Specify the required voltage, current, and kVA ratings of each transformer (both high- and low-voltage windings) for the following connections:

2.23 Three 100-MVA single-phase transformers, rated at 13.8 kV:66.4 kV, are to be connected in a three-phase bank. Each transformer has a series impedance of 0.0045 + j0.19 ΩΩ size 12{ %OMEGA } {} referred to its 13.8-kV winding.

a. If the transformers are connected Y-Y, calculate (i) the voltage and power rating of the three-phase connection, (ii) the equivalent impedance as referred to its low-voltage terminals, and (iii) the equivalent impedance as referred to its high-voltage terminals.

b. Repeat part (a) if the transformer is connected Y on its low-voltage side and ΔΔ size 12{Δ} {} on its high-voltage side.

2.24 A three-phase Υ−ΩΥ−Ω size 12{Υ - %OMEGA } {} transformer is rated 225-kV:24-kV, 400 MVA and has a series reactance of 11.7 ΩΩ size 12{ %OMEGA } {} as referred to its high-voltage terminals. The transformer is supplying a load of 325 MVA, with 0.93 power factor lagging at a voltage of 24 kV (line-to-line) on its low-voltage side. It is supplied from a feeder whose impedance is 0.11 + j 2.2 ΩΩ size 12{ %OMEGA } {} connected to its high-voltage terminals. For these conditions, calculate (a) the line-to-line voltage at the high-voltage terminals of the transformer and (b) the line-to-line voltage at the sending end of the feeder.

2.25 Assume the total load in the system of Problem 2.24 to remain constant at 325 MVA. Write a MATLAB script to plot the line-to-line voltage which must be applied to the sending end of the feeder to maintain the load voltage at 24 kV line-to-line for load power factors in range from 0.75 lagging to unity to 0.75 leading. Plot the sending-end voltage as a function of power factor angle.

2.26 A Δ-Y-connected bank of three identical 100-kVA, 2400-V:120-V, 60-Hz transformers is supplied with power through a feeder whose impedance is 0.065 + j0.87 ΩΩ size 12{ %OMEGA } {} per phase. The voltage at the sending end of the feeder is held constant at 2400 V line-to-line. The results of a single-phase short-circuit test on one of the transformers with its low-voltage terminals short-circuited are

VH=53.4V f=60Hz IH=41.7A P=832W

a. Determine the line-to-line voltage on the low-voltage side of the transformer when the bank delivers rated current to a balanced three-phase unity power factor load.

b. Compute the currents in the transformer's high- and low-voltage windings and in the feeder wires if a solid three-phase short circuit occurs at the secondary line terminals.

2.27 A 7970-V: 120-V, 60-Hz potential transformer has the following parameters as seen from the high-voltage (primary) winding:

X1 = 1721 ΩΩ size 12{ %OMEGA } {}X2'X2' size 12{ { {X}} sup { ' } rSub { size 8{2} } } {} = 1897 ΩΩ size 12{ %OMEGA } {} Xm = 782 kΩkΩ size 12{k %OMEGA } {}

R1 = 1378 ΩΩ size 12{ %OMEGA } {}R2'R2' size 12{ { {R}} sup { ' } rSub { size 8{2} } } {} = 1602 ΩΩ size 12{ %OMEGA } {}

a. Assuming that the secondary is open-circuited and that the primary is connected to a 7.97-kV source, calculate the magnitude and phase angle (with respect to the high-voltage source) of the voltage at the secondary terminals.

b. Calculate the magnitude and phase angle of the secondary voltage if a 1 kΩkΩ size 12{k %OMEGA } {} resistive load is connected to the secondary terminals.

c. Repeat part (b) if the burden is changed to a 1 ***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.*** reactance.

2.28For the potential transformer of Problem 2.27, find the maximum reactive burden (mimimum reactance) which can be applied at the secondary terminals such that the voltage magnitude error does not exceed 0.5 percent.

2.29Consider the potential transformer of Problem 2.27.

a. Use MATLAB to plot the percentage error in voltage magnitude as a function of the magnitude of the burden impedance (i) for a resistive burden of 100 Ω≤Rb≤Ω≤Rb≤ size 12{ %OMEGA <= R rSub { size 8{b} } <= {}} {} 3000 ΩΩ size 12{ %OMEGA } {} and (ii) for a reactive burden of 100 Ω≤Xb≤Ω≤Xb≤ size 12{ %OMEGA <= X rSub { size 8{b} } <= {}} {} 3000 ΩΩ size 12{ %OMEGA } {}. Plot these curves on the same axis.

b. Next plot the phase error in degrees as a function of the magnitude of the burden impedance (i) for a resistive burden of 100 Ω≤Rb≤Ω≤Rb≤ size 12{ %OMEGA <= R rSub { size 8{b <= {}} } } {} ≤ 3000 ΩΩ size 12{ %OMEGA } {} and (ii) for a reactive burden of 100 Ω≤Xb≤Ω≤Xb≤ size 12{ %OMEGA <= X rSub { size 8{b} } <= {}} {} 3000 ΩΩ size 12{ %OMEGA } {}. Again, plot these curves on the same axis.

2.30 A 200-A:5-A, 60-Hz current transformer has the following parameters as seen from the 200-A (primary) winding:

X1 = 745 μΩμΩ size 12{μ %OMEGA } {}X2'X2' size 12{ { {X}} sup { ' } rSub { size 8{2} } } {}= 813 μΩμΩ size 12{μ %OMEGA } {} Xm = 307 mΩmΩ size 12{m %OMEGA } {}

R1 = 136 μΩμΩ size 12{μ %OMEGA } {}{}R2'R2' size 12{ { {R}} sup { ' } rSub { size 8{2} } } {}= 128 μΩμΩ size 12{μ %OMEGA } {}

a. Assuming a current of 200 A in the primary and that the secondary is short-circuited, find the magnitude and phase angle of the secondary current.

b. Repeat the calculation of part (a) if the CT is shorted through a 250 μΩμΩ size 12{μ %OMEGA } {} burden.

2.31 Consider the current transformer of Problem 2.30.

a. Use MATLAB to plot the percentage error in current magnitude as a function of the magnitude of the burden impedance (i) for a resistive burden of 100 Ω≤Rb≤Ω≤Rb≤ size 12{ %OMEGA <= R rSub { size 8{b <= {}} } } {} 1000 ΩΩ size 12{ %OMEGA } {} and (ii) for a reactive burden of 100 Ω≤Xb≤Ω≤Xb≤ size 12{ %OMEGA <= X rSub { size 8{b} } <= {}} {} 1000 ΩΩ size 12{ %OMEGA } {}. Plot these curves on the same axis.

b. Next plot the phase error in degrees as a function of the magnitude of the burden impedance (i) for a resistive burden of 100Ω≤Rb≤1000Ω100Ω≤Rb≤1000Ω size 12{"100" %OMEGA <= R rSub { size 8{b} } <= "1000" %OMEGA } {}and (ii) for a reactive burden of 100Ω≤Xb≤1000Ω100Ω≤Xb≤1000Ω size 12{"100" %OMEGA <= X rSub { size 8{b} } <= "1000" %OMEGA } {}. Again, plot these curves on the same axis.

2.32 A 15-kV: 175-kV, 125-MVA, 60-Hz single-phase transformer has primary and secondary impedances of 0.0095 + j0.063 per unit each. The magnetizing impedance is j148 per unit. All quantities are in per unit on the transformer base. Calculate the primary and secondary resistances and reactances and the magnetizing inductance (referred to the low-voltage side) in ohms and henrys.

2.33 The nameplate on a 7.97-kV:460-V, 75-kVA, single-phase transformer indicates that it has a series reactance of 12 percent (0.12 per unit).

a. Calculate the series reactance in ohms as referred to (i) the low-voltage terminal and (ii) the high-voltage terminal.

b. If three of these transformers are connected in a three-phase Y-Y connection, calculate (i) the three-phase voltage and power rating, (ii) the per unit impedance of the transformer bank, (iii) the series reactance in ohms as referred to the high-voltage terminal, and (iv) the series reactance in ohms as referred to the low-voltage terminal.

c. Repeat part (b) if the three transformers are connected in Y on their HV

side and ΔΔ size 12{Δ} {} on their low-voltage side.

2.34 a. Consider the Y-Y transformer connection of Problem 2.33, part (b). If the rated voltage is applied to the high-voltage terminals and the three low-voltage terminals are short-circuited, calculate the magnitude of the phase current in per unit and in amperes on (i) the high-voltage side and (ii) the low-voltage side.

b. Repeat this calculation for the Y- ΔΔ size 12{Δ} {} connection of Problem 2.33, part (c).

2.35 A three-phase generator step-up transformer is rated 26-kV:345-kV, 850 MVA and has a series impedance of 0.0035 + j0.087 per unit on this base. It is connected to a 26-kV, 800-MVA generator, which can be represented as a voltage source in series with a reactance of j1.57 per unit on the generator base.

a. Convert the per unit generator reactance to the step-up transformer base.

b. The unit is supplying 700 MW at 345 kV and 0.95 power factor lagging to the system at the transformer high-voltage terminals. (i) Calculate the transformer low-side voltage and the generator internal voltage behind its reactance in kV. (ii) Find the generator output power in MW and the power factor.