```
In [1]:
```%pylab inline
%precision 1

```
Out[1]:
```

A 25-MVA, 12.2-kV, 0.9-PF-lagging, three-phase, two-pole, Y-connected, 60-Hz synchronous generator was tested by the open-circuit test, and its air-gap voltage was extrapolated with the following results:

- Open-citcuit test

Field current [A] | Line voltage [kV] | Extrapolated air-gap voltage [kV] |
---|---|---|

275 | 12.2 | 13.3 |

320 | 13.0 | 15.4 |

365 | 13.8 | 17.5 |

380 | 14.1 | 18.3 |

475 | 15.2 | 22.8 |

570 | 16.0 | 27.4 |

- Short-circuit test

Field current [A] | Armature current [A] |
---|---|

275 | 890 |

320 | 1040 |

365 | 1190 |

380 | 1240 |

475 | 1550 |

570 | 1885 |

The armature resistance is $0.6\,\Omega$ per phase.

- Find the unsaturated synchronous reactance of this generator in ohms per phase and in per-unit.

- Find the approximate saturated synchronous reactance $X_S$ at a field current of 380 A. Express the answer both in ohms per phase and in per-unit.

- Find the approximate saturated synchronous reactance at a field current of 475 A. Express the answer both in ohms per phase and in per-unit.

- Find the short-circuit ratio for this generator.

- What is the internal generated voltage of this generator at rated conditions?

- What field current is required to achieve rated voltage at rated load?

```
In [2]:
```Sbase = 25e6 # [VA]
Vbase = 12.2e3 # [V]
PF = 0.9
Ra = 0.6 # [Ohm]

```
In [3]:
```if_a = 380.0 # [A]

The extrapolated air-gap voltage at this point is 18.3 kV, and the short-circuit current is 1240 A

```
In [4]:
```Vag_a = 18.3e3 # [V]
isc_a = 1240.0 # [A]

Since this generator is Y-connected, the phase voltage is:

```
In [5]:
```Vphi_a = Vag_a / sqrt(3)
print('Vphi_a = {:.0f} V'.format(Vphi_a))

```
```

and the armature current is:

```
In [6]:
```Ia_a = isc_a
print('Ia_a = {:.0f} A'.format(Ia_a))

```
```

Therefore, the unsaturated synchronous impedance $Z_{s} = \sqrt{R_a^2 + X_s^2}$ is:

```
In [7]:
```Zsu_a = Vphi_a / Ia_a
print('Zsu_a = {:.2f} Ω'.format(Zsu_a))

```
```

Which leads to the unsaturated syncronous *reactance* $X_{s} = \sqrt{Z_s^2 - R_a^2}$:

```
In [8]:
```Xsu_a = sqrt(Zsu_a**2 - Ra**2)
print('''
Xsu_a = {:.2f} Ω
==============
'''.format(Xsu_a))

```
```

*As you can see the impact of the armature resistance is negligible small. This is also the reason why $R_a$ is often simply ignored in calculations of the synchronous reactance. Especially for larger machines.*

The base impedance of this generator is:

$$Z_\text{base} = \frac{3V^2_{\phi,\text{base}}}{S_\text{base}}$$```
In [9]:
```Vphi_base = Vbase/sqrt(3)
Zbase = 3*Vphi_base**2 / Sbase
print('Zbase = {:.2f} Ω'.format(Zbase))

```
```

Therefore, the per-unit unsaturated synchronous reactance is:

```
In [10]:
```xsu_a = Xsu_a / Zbase
print('''
xsu_a = {:.2f}
============
'''.format(xsu_a))

```
```

```
In [11]:
```If_b = 380.0 # [A]
Vocc_b = 14.1e3 # [V]
isc_b = 1240.0 # [A]

Since this generator is Y-connected, the corresponding phase voltage is:

```
In [12]:
```Vphi_b = Vocc_b / sqrt(3)
print('Vphi_b = {:.0f} V'.format(Vphi_b))

```
```

and the armature current is:

```
In [13]:
```Ia_b = isc_b
print('Ia_b = {:.0f} A'.format(Ia_b))

```
```

Therefore, the saturated synchronous reactance is:

```
In [14]:
```Zs_b = Vphi_b / Ia_b
Xs_b = sqrt(Zs_b**2 - Ra**2)
print('''
Xs_b = {:.2f} Ω
=============
'''.format(Xs_b))

```
```

and the per-unit unsaturated synchronous reactance is:

```
In [15]:
```xs_b = Xs_b / Zbase
print('''
xs_b = {:.2f}
===========
'''.format(xs_b))

```
```

```
In [16]:
```If_c = 475.0 # [A]
Vocc_c = 15.2e3 # [V]
isc_c = 1550.0 # [A]

Since this generator is Y-connected, the corresponding phase voltage is:

```
In [17]:
```Vphi_c = Vocc_c / sqrt(3)
print('Vphi_c = {:.0f} V'.format(Vphi_c))

```
```

and the armature current is:

```
In [18]:
```Ia_c = isc_c
print('Ia_c = {:.0f} A'.format(Ia_c))

```
```

Therefore, the saturated synchronous reactance is:

```
In [19]:
```Zs_c = Vphi_c / Ia_c
Xs_c = sqrt(Zs_c**2 - Ra**2)
print('''
Xs_c = {:.2f} Ω
=============
'''.format(Xs_c))

```
```

and the per-unit unsaturated synchronous reactance is:

```
In [20]:
```xs_c = Xs_c / Zbase
print('''
xs_c = {:.3f}
============
'''.format(xs_c))

```
```

```
In [21]:
```If_d = 275.0 # [A]

The rated line and armature current of this generator is:

```
In [22]:
```Il = Sbase / (sqrt(3) * Vbase)
print('Il = {:.0f} A'.format(Il))

```
```

```
In [23]:
```If_d_2 = 365.0 # [A]
SCR = If_d / If_d_2
print('''
SCR = {:.2f}
==========
'''.format(SCR))

```
```

```
In [24]:
```Xs_e = Xs_b
If_e = If_b
Ia_e = Il # rated current as calculated in part d

Since the power factor is 0.9 lagging, the armature current is:

```
In [25]:
```IA_e_angle = -arccos(PF)
IA_e = Ia_e * (cos(IA_e_angle) + sin(IA_e_angle)*1j)
print('IA_e = {:.0f} Ω ∠{:.2f}°'.format(*(abs(IA_e), IA_e_angle/ pi*180)))

```
```

Therefore, $$\vec{E}_A = \vec{V}_\phi + R_A\vec{I}_A + jX_S\vec{I}_A$$

```
In [26]:
```EA = Vphi_base + Ra*IA_e + Xs_e*IA_e*1j
EA_angle = arctan(EA.imag / EA.real)
print('''
EA = {:.0f} V ∠{:.1f}°
===================
'''.format(*(abs(EA), EA_angle/pi*180)))

```
```

```
In [27]:
```abs(EA)

```
Out[27]:
```

Volts per phase, the corresponding line value would be:

```
In [28]:
```Vline_f = abs(EA)* sqrt(3)
print('Vline_f = {:.0f} V'.format(Vline_f))

```
```

```
In [29]:
```If_f=(475-380)/(22.8e3-18.3e3)*(abs(EA)*sqrt(3)-18.3e3)+380
print('''
If_f = {:.0f} A
============
'''.format(If_f))

```
```