2.1 Problems - chapter 2

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 ${\text{cm}}^2$ . 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 ${\text{cm}}^2$ 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 $\Omega$ resistor to an impedance of 75 $\Omega$ . Calculate the required turns ratio, assuming the transformer to be ideal.

2.4 A 100 $\Omega$ 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 $\Omega$ 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 $k\Omega$ is connected to a 50 $\Omega$ 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- $k\Omega$ 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 $\Omega$ . The 120-V winding has a leakage reactance of 27.4 $m\Omega$ and the 2400-V winding has a leakage reactance of 11.2 $\Omega$ .

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.