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Units
This manual uses the International System of Units (SI).
The manual also uses the per unit (p.u.) system on occasion to define the model parameters.
What Is the Per Unit System?
The per unit system is widely used in the power system industry to express values of voltages, currents, powers, and impedances of various power equipment. It is mainly used for transformers and AC machines.
For a given quantity (voltage, current, power, impedance, torque, etc.) the per unit value is the value related to a base quantity.
Generally the following two base values are chosen:
All other base quantities are derived from these two base quantities. Once the base power and the base voltage are chosen, the base current and the base impedance are determined by the natural laws of electrical circuits.
For a transformer with multiple windings, each having a different nominal voltage, the same base power is used for all windings (nominal power of the transformer). However, according to the above definitions, there are as many base values as windings for voltages, currents, and impedances.
For AC machines, the torque and speed can be also expressed in p.u. The following base quantities are chosen:
Instead of specifying the rotor inertia in kg*m2, you would generally give the inertia constant H defined as
The inertia constant is expressed in seconds. For large machines, this constant is around 3 to 5 seconds. An inertia constant of 3 seconds means that the energy stored in the rotating part could supply the nominal load during 3 seconds. For small machines, H is lower. For example, for a 3 HP motor, it can be between 0.5 and 0.7 seconds.
Example 1: Three-Phase Transformer
Consider, for example, a three-phase two-winding transformer. The following typical parameters could be provided by the manufacturer:
The base values for each single phase transformer are first calculated:
Base power |
300 kVA/3 = 100e3 VA |
Base voltage |
600 V rms |
Base current |
100e3/600 = 166.7 A rms |
Base impedance |
600/166.7 = 3.60 ![]() |
Base resistance |
600/166.7 = 3.60 ![]() |
Base inductance |
3.60/(2![]() |
The values of the winding resistances and leakage inductances expressed in SI units are therefore
For the magnetizing branch, magnetizing losses of 1% resistive and 1% inductive mean a magnetizing resistance Rm of 100 p.u. and a magnetizing inductance Lm of 100 p.u. Therefore, the values expressed in SI units referred to winding 1 are
Example 2: Asynchronous Machine
Now consider the three-phase four-pole Asynchronous Machine in SI units provided in the Machines library of powerlib. It is rated 3 HP, 220 V rms line-to-line, 60 Hz.
The stator and rotor resistance and inductance referred to stator are
The mutual inductance is Lm = 69.31 mH. The rotor inertia is J = 0.089 kg.m2.
The base quantities for one phase are calculated as follows:
Using the above base values, you can compute the values in per units.
Rs= 0.435 / 21.62 = 0.0201 p.u. Ls= 2 / 57.35 = 0.0349 p.u.
Rr= 0.816 / 21.62 = 0.0377 p.u. Lr= 2 / 57.35 = 0.0349 p.u.
Lm = 69.31/57.35 = 1.208 p.u.
The inertia is calculated from inertia J, synchronous speed, and nominal power.
If you open the dialog box of the Asynchronous Machine block in p.u. units provided in the Machines library of powerlib, you find that the parameters in p.u. are the ones calculated above.
Base Values for Instantaneous Voltage and Current Waveforms
When displaying instantaneous voltage and current waveforms on graphs or oscilloscopes, you normally consider the peak value of the nominal sinusoidal voltage as 1 p.u. In other words, the base values used for voltage and currents are the rms values given above multiplied by .
Why Use the Per Unit System Instead of the Standard SI Units?
Here are the main reasons for using the per unit system:
Similarly, for salient pole synchronous machines, the synchronous reactance Xd is generally between 0.60 and 1.50 p.u., whereas the sub transient reactance X'd is generally between 0.20 and 0.50 p.u.
It means that if you do not know the parameters for a 10 kVA transformer, you will not make a big mistake by assuming an average value of 0.02 p.u. for leakage reactances and 0.0075 p.u. for winding resistances.
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