A 1000 kW, 3-phase, star-connected, 3.3 kV, 24-pole, 50 Hz synchronous motor has a synchronous reactance of 3.24 Ω per phase; the resistance being negligible. (a) The motor is fed from infinite bus-bars at 3.3 kV. Its field excitation is adjusted to result in unity pf operation at rated load.

Question

A 1000 kW, 3-phase, star-connected, 3.3 kV, 24-pole, 50 Hz synchronous motor has a synchronous reactance of [latex] 3.24 \Omega [/latex] per phase; the resistance being negligible.

(a) The motor is fed from infinite bus-bars at 3.3 kV. Its field excitation is adjusted to result in unity pf operation at rated load. Compute the maximum power and torque that the motor can deliver with its excitation remaining constant at this value.

(b) The motor is now fed from a 1200 kVA, 3-phase star-connected, 3.3 kV, 2-pole, 50 Hz synchronous generator with a synchronous reactance of [latex] 4.55 \Omega [/latex] per phase, the resistance being negligible. Compute the field excitations of motor and generator when the set is operating at rated terminal voltage at unity pf and the motor is delivering full-load power. The field excitations of both the machines remaining constant, the motor load is gradually raised. Compute the maximum power and torque that the motor can deliver. Also compute the terminal voltage when the motor is delivering maximum power.

Accepted Answer

(a) The operation of motor at infinite bus-bars is shown in Fig. 8.45

[latex] V_{t}=3300 / \sqrt{3}=[/latex] 1905 V

[latex] I_{a}=\frac{1000 imes 1000}{\sqrt{3} imes 3300 imes 1}=[/latex] 175 A

[latex] \cos \phi=1, \quad \phi=0^{\circ}[/latex]

Taking the terminal voltage as reference,

[latex] \bar{V}_{t}=1905 \angle 0^{\circ}[/latex] V

[latex] \bar{I}_{a}=175 \angle 0^{\circ} A[/latex]

Then the excitation emf is computed as

[latex] E_{f n}=1905 \angle 0^{\circ}-j 175 \angle 0^{\circ} imes 3.24=1905-j 567[/latex]

which gives [latex] E_{f m}=1987 V[/latex]

Excitation remaining fixed, the maximum power delivered by the motor is

[latex] p_{e, \max }=P_{m, \max }( gross )=3 imes \frac{V_{t} E_{f}}{X_{s m}}=3 imes \frac{1905 imes 1987}{3.24 imes 1000}=3505 kW[/latex] (3-phase)

[latex] \omega_{s m}=\frac{120 imes 50 imes 2 \pi}{24 imes 60}=26.18[/latex] rad / s

[latex] herefore[/latex] [latex] T_{\max }=\frac{3505 imes 1000}{26.18}=133.9 imes 10^{3} N m[/latex]

(b) Figure 8.46(a) shows the generator feeding the motor. At rated terminal voltage, unity pf, full-load operation

[latex] \bar{V}_{t}=1905 \angle 0^{\circ}[/latex]

[latex] \bar{I}_{a}=175 \angle 0^{\circ}[/latex]

As calculated before [latex] E_{f m}=1905 V[/latex]

Now [latex] \bar{E}_{f g}=1905 \angle 0^{\circ}+j 175 \angle 0^{\circ} imes 4.55[/latex]

= 1905 + j 796

Or [latex] E_{f g}=2065 V[/latex]

The total series reactance [latex] X=X_{s m}+X_{s g}=3.24+4.55=7.79 \Omega[/latex]

The maximum power output delivered by the motor is

[latex] P_{e, \max }=P_{m, \max }( ext { gross })=3 imes \frac{E_{f g} imes E_{f m}}{X}=3 imes \frac{2065 imes 1987}{7.79 imes 1000}=1580 kW[/latex] (3-phase)

[latex] T_{\max }=\frac{1580 imes 1000}{26.18}=60.35 imes 10^{3} Nm[/latex]

The phasor diagram under condition of maximum power output is drawn in Fig. 8.46(b). For convenience, choosing the motor excitation as reference

At maximum power

Output [latex] \delta=90^{\circ}[/latex]

[latex] \bar{E}_{f m}=1987 \angle 0^{\circ} [/latex] V

[latex] \bar{E}_{f g}=2065 \angle 90^{\circ} V[/latex]

Then [latex] \bar{I}_{a}=\frac{\bar{E}_{f g}-\bar{E}_{f m}}{j X}=\frac{2065 \angle 90^{\circ}-1987 \angle 0^{\circ}}{j 7.79}[/latex]

[latex] j \bar{I}_{a} X_{s m}=\frac{-1987+j 2065}{j 7.79} imes j 3.24_{=}=-826.4+j 858.9[/latex]

Now [latex] \bar{V}_{t}=\bar{E}_{f m}+j \bar{I}_{a} X_{s m}=1987-826.4+j 858.9=1160.6+j 858.9[/latex]

Or [latex] V_{t}=[/latex] 1443.85 or 2500 V (line)

Remark The reduction in [latex] P_{e,max}[/latex] in case (b) compared to case (a) is explained by the fact that [latex] V_{t}[/latex] in this case is only 2500 V (line) compared to 3300 V in case (a).

Question 8.8: A 1000 kW, 3-phase, star-connected, 3.3 kV, 24-pole, 50 Hz s...

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