power system

300771 POWER SYSTEMS

TUTORIAL 1-2-3: THREE PHASE, POWER FACTOR

Q1.

Q2.

Q3.

Q4.

Q5.

Question 1 (GSO Q2.28 p. 82) Three loads are connected in parallel across a single source voltage of 240 V r.m.s., 50 Hz. Load 1 absorbs 12 kW and 6.667 kvar. Load 2 absorbs 4 kVA at 0.96 p.f. lagging. Load 3 absorbs 15 kW at unity power factor. Calculate the equivalent impedance, Z, for the three parallel loads, for the two cases: (i) series combination of R and X (ii) parallel combination of R and X.

Q6. Consider the high voltage example, from Lecture 2. Calculate the line current, voltage drop, and line losses:

Now consider the case of using HV transmission, in conjunction with a step-up and step-down transformer:

Re-calculate the line current, voltage drop, and line losses. Comment on the results. Q7. Consider the “Dyn11” 3-phase transformer winding arrangement, as shown in your lecture notes (Sect 2).

Example: Dy11 Transformer • Note: The A-ph HV winding A1-A2 is magnetically coupled to the A-

ph LV winding a1-a2. Hence the winding voltage are in phase.

A2

B2

C2

A1

B1

C1

a1

b1

c1

a2

b2

c2

Draw all the voltage vectors, for HV to LV, for all 3 phases. For a line voltage ratio of 1:1, what are the vector values (magnitude and phase) for all HV and LV windings, in magnitude/angle format? Q8. A “Dyn11” 3-phase transformer and a “Dy1” transformer, with the same LV voltage output, are accidentally connected in parallel. What voltage difference will there be, across the LV terminals? Q9. A 3-phase power system operates at angular frequency ω and has a peak voltage of V1a, V1b, V1c, on its fundamental frequency component, a, b and c phases. (a) From first principles, write down the equations for the a, b and c phases, for

the – Fundamental frequency, The second harmonic, The third harmonic. (b) Show that the second harmonic has a negative sequence of phase rotation.

What are the implications for rotating machines? (c) Show that the third harmonic has no phase rotation (i.e., a, b and c phases are all in phase at the 3rd harmonic frequency). Show that this will create a “third harmonic short circuit” in delta connected windings. Q10. From the Fourier Series definitions given in the lecture notes, determine the harmonic components of the simple square wave shown below:

Take a frequency of f and a magnitude of +1/-1. Why are the an terms zero?