Question 5.4: The following data are taken from the open- and short-circui......
The following data are taken from the open- and short-circuit characteristics of a 45-kVA, three- phase, Y-connected, 220-V (line-to-line), six-pole, 60-Hz synchronous machine. From the open-circuit characteristic:
Line-to-line voltage = 220 V Field current = 2.84 A
From the short-circuit characteristic:
From the air-gap line:
Field current = 2.20 A Line-to-line voltage = 202 V
Compute the unsaturated value of the synchronous reactance, its saturated value at rated voltage in accordance with Eq. 5.29, and the short-circuit ratio. Express the synchronous reactance in ohms per phase and in per unit on the machine rating as a base.
X_s=\frac{V_{a,rated}}{I^′_a} (5.29)
| Armature current, A | 118 | 152 |
| Field current, A | 2.20 | 284 |
At a field current of 2.20 A the line-to-neutral voltage on the air-gap line is
V_{a,ag}= \frac{202}{\sqrt{3}} = 116.7 V
and for the same field current the armature current on short circuit is
I_{a,sc} = 118 A
From Eq. 5.28
X_{s,u}=\frac{V_{a,ag}}{I_{a,sc}} (5.28)
X_{s,u}= \frac{116.7}{118}=0.987 Ω/\text{phase}
Note that rated armature current is
I_{a,rated}= \frac{45,000}{\sqrt{3}×220}=118 A
Therefore, I_{a,sc}= 1.00 per unit. The corresponding air-gap-line voltage is
V_{a,ag}= \frac{202}{220} = 0.92 per unit
From Eq. 5.28 in per unit
X_{s,u}=\frac{0.92}{1.00}= 0.92 per unit
The saturated synchronous reactance can be found from the open- and short-circuit characteristics and Eq. 5.29
X_s=\frac{V_{a,rated}}{I^′_a}=\frac{(220/\sqrt{3})}{152} = 0.836 Ω/phase
In per unit I^′_a= \frac{152}{118}= 1.29, and from Eq. 5.29
X_s =\frac{1.00}{1.29}= 0.775 per unit
Finally, from the open- and short-circuit characteristics and Eq. 5.30, the short-circuit ratio is given by
SCR=\frac{\mathrm{O}f^′}{Of^{″}} (5.30)
SCR = \frac{2.84}{2.20}=1.29
Note that as was indicated following Eq. 5.30, the inverse of the short-circuit ratio is equal to the per-unit saturated synchronous reactance
X_s=\frac{1}{ SCR} =\frac{1}{ 1.29} = 0.775 per unit