# Audel electrical course for apprentices and journeymen - Rosenberg P.

ISBN: 0-764-54200-1

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Three-phase windings may be connected either wye or delta. Figure 23-10 shows the actual connections in an alternator for a delta-connected winding. It should be noted that phase-winding connections to A are reversed as opposed to phases B and C.

Polyphase Circuits 261

The internal connections of an alternator must be visually checked to ascertain if the winding is connected in wye or delta.

Sometimes a wye-connected alternator may be connected four-wire, that is, the three-phase wires and the common internal connection are all brought out externally.

Questions

1. Sketch a three-wire, two-phase connection.

2. Sketch a four-wire, two-phase connection.

3. Give voltage relations for both of the above.

4. Windings of a three-phase alternator are connected so that the voltages are apart. (True or false?)

5. Which is the most economical: a single-phase or a three-phase alternator?

6. Draw a delta connection.

7. Draw a wye connection.

8. Sketch three-phase sine waves, with voltages 120° apart.

9. Sketch three-phase sine waves with voltages 60° apart.

Chapter 24

Power in Polyphase Circuits

In a three-phase machine-connected wye (Y), the emf between any two lines is equal to the voltage of one phase times the square root of 3.

In Figure 24-1, lines 1 and 2 are supplied by phases A and B of the alternator winding. As described in Chapter 14, phases A and B are 120° apart, so the voltage of the two windings can’t be added by arithmetic. It may be noted from this illustration that the emf of any one phase winding is 277 volts and, as stated, the voltage of one phase times 23 is the voltage between two lines. Thus, 277 X 1.73 = 479 volts, line to line.

LINE 1

Figure 24-1 Voltages in a wye system.

This may be arrived at mathematically through the use of trigonometry. In Figure 24-2, it may be observed that the voltage, E, due to two phases in series 120° apart, is to the sine of the angle opposite it, which is 120°, as the voltage of one phase, A, is to the sine of the angle opposite it, which is 30°. The sine of 120° is the same as the sine of 60°. Thus

E sin120° sin60° 0.866 rr

— =-----------=---------=---------= 1 732 = V3

A sin30° sin30° 0.500

263

264 Chapter 24

Figure 24-2 Calculating voltage delivered by two-phase windings 120° apart when connected in series.

Three-Phase Wye Connection

The relation of voltages, currents, and power in separate phases to the total power of the system in both wye and delta connections may be considered now:

For a Wye Connection at 100% PF

e E= (1)

E

e = 23 (2)

i = I (3)

p = eI (4)

p =1 = 23 (2) & (4)

IE

P = 3p = 3— 23 (5)

p = ie 23 (6)

where

p = volt-amperes per phase P = total power in system in volt-amperes E = voltage between any two line wires e = emf of one phase i = current per phase I = current in each line wire

Power in Polyphase Circuits 265

In analyzing these formulas, it will be seen that in a three-phase system, the total power P is not the current I in each line wire multiplied by each of the voltages between adjacent line wires and then totaled. Due to the angles of displacement, the total power P is the current I in a single-line wire multiplied by the voltage E between any two lines times the square root of 3, which is formula (6) above.

The power in a three-phase system is always equal to the sum of the power in the three separate phases (5). To verify, use Figure 24-3, in which the current per phase is 10 A and the voltage per phase is 1000 V; so, p = ei = 1000 X 10 = 10,000 VA (formula (4)). Now

10.000 X 3 = 30,000 VA total. To check this, note that E = e 23 = 1000 X 1.73 = 1732 V (1), and P = IE 23 (6), and I is the same as i in a wye connection, so P = 10 X 1732 X 23 =

30.000 VA.

If the load has 100% PF, then the watts and volt-amperes will be equal. If the PF is less than 100%, the formula for P in watts will be

P = 23 X IE cos f

10 A

Figure 24-3 Relationships of voltage and currents delivered by a wye-connected alternator.

Lest there be some confusion, apparent power is rated in VA or kVA and true power is rated in watts (W) or kilowatts (kW). A thorough understanding of kVA and kW is essential to a thorough understanding of power in AC circuits.

266 Chapter 24

Three-Phase Delta Connection

Current and voltage relationships in a delta-connected (A) alternator or other AC delta connections are different than in a wye-connected alternator or other wye-connected circuits.

i = i 23 (1)

I ‘ = 23 (2)

e = E (3)

p = ei (4)

p = Ei (5)

P = 3p = IE r-3P = IE 23 23 (6)

where the letter symbols have the same meaning as before.

The formula for total power in VA in a delta connection is the same as for a wye connection. The square root of 3 is a factor used in both wye and delta connections. With wye connections, 23 is a factor of the line voltage, and with delta connections the 23 is a factor with line current. Thus for Figure 24-4,

I = i23 = 10 X 1.732 — 17.32 amperes (1)

E = e or 1000 volts = 1000 volts (3)

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