Page 89 - Power Electronic Control in Electrical Systems

//SYS21/F:/PEC/REVISES_10-11-01/075065126-CH002.3D ± 78 ± [31±81/51] 17.11.2001 9:49AM

78 Power systems engineering ± fundamental concepts

lowest-order harmonics appearing in the line current are of order 1 12 1 ˆ 11
and 13.
The harmonics of orders kq 1 are called characteristic harmonics. In practice
the commutation is imperfect (it takes time to transfer current from one SCR to
the next), and the DC load current may not be perfectly smooth. Also, the supply
voltages may not be perfectly balanced. These factors give rise to the appearance
of non-characteristic harmonics. For example, a 5th or 7th harmonic in a 12-pulse
rectifier is a non-characteristic harmonic: it would not appear under ideal condi-
tions. Also, if the SCR firing is phase-shifted, additional harmonics are generated.

2.13 Per-unit quantities

Per-unit quantities are quantities that have been normalized to a base quantity. For
example, consider a 5 kW motor operating at 3 kW. The actual power is 3 kW. The
base power is 5 kW. The per-unit power is therefore 3/5 ˆ 0:6p:u. As another
example, a cable might be rated at 150 A. If it is carrying 89 A, and if we take the
base current to be the same as the rated current, then the per-unit current is
89/150 ˆ 0:593 p:u.
In general, 17
X
X ˆ p:u: (2:70)
X b
where x is a per-unit value, X is an actual value, and X b is the base value. X and X b
are expressed in ordinary units, such as volts, amps, watts, Nm, etc. Per-unit quan-
tities have no dimensions but it is normal practice to express them `in p.u.', for
example, 1.05 p.u.
Per-unit quantities express relative values ± that is, relative to the base value. The
choice of base value is important. In the simple example quoted above, the most
natural choice of base was the rated value, which is the value associated with `normal
full-load operation'. However, base values may be freely chosen, as the next example
shows.
Imagine two motors in parallel, one of 5 kW rating, the other of 50 kW rating,
connected to a supply that is rated at 150 A. Suppose that the smaller motor draws its
rated current of 7.5 A from the supply, while the larger one draws 75 A. If we
consider each motor individually we could say that each one is working at 1 p.u. or
100% of its own rating. But if we choose 150 A as the common base current, then the
first motor is drawing 7:5/150 ˆ 0:05 p:u: while the second one draws 75/150 ˆ
0:5p:u: The total current is 0.55 p.u., or 82.5 A. This clearly expresses the fact that
the smaller motor is taking 5% of the supply current while the larger one is taking
50%. The supply is working at only 55% of its capacity. Evidently we could add
motors and almost double the load.
Per-unit systems are especially useful when we have a more complicated network
with transformers, in which there may be several different voltage and current levels.

17
Boldface lowercase letters denote per-unit values. Italic letters denote values in ordinary units.