The momentary current is used when specifying the
closing current of switchgear. Typically, the AC and DC components
decay to 90% of their initial values after the first half cycle.
From this, the value of the r.m.s. current would then be:
Usually, a factor of 1.6 is used by manufacturers and in international standards so that, in general, this value should be used when carrying out similar calculations.
The peak value is
obtained by arithmetically
adding together the AC and
DC components. It should be noted that, in this case, the AC
component is multiplied by a factor of Thus:
Replacing the DC component by its exponential
expression gives: The expression ( ) has been drawn for different
Values of
X/R,
and for different switchgear contact-separation times, in ANSI
Standard
C37.5–1979.
The multiplying factor graphs are reproduced in Figure 6
NOTE: Fed predominantly through two or more transformations or with external reactance in series equal to or above 1.5 times generator subtransient reactance As an illustration of the validity of the curves for any situation, Consider a circuit breaker with a total contact-separation time of two cycles one cycle due to the relay and one related to the operation of the breaker mechanism. If the frequency, f is 60 Hz and the ratio X/R
With this arrangement,
voltage values of any three-phase system, V_{b} =V_{bo} + V_{b1} + V_{b2} V_{c} =V_{co} + V_{c1} + V_{c2}
It can be demonstrated that: ^{ } V_{b}= V_{ao}+a^{2}V_{a1}+aV_{a2} V_{c}= V_{ao}+aV_{a1}+ a^{2}V_{a2}
where a is a so called operator which gives a phase
shift of 120° clockwise and a multiplication of unit
magnitude, i.e.
a=1°,
Inverting the matrix of coefficients:
From the above
matrix it
can be deduced that:
The foregoing procedure can also be
applied directly to currents, and gives:
Therefore: In three-phase systems, the neutral current is equal to In = (I_{a} + Ib + I_{c)} and, therefore, l_{n}=3I0
By way of
illustration, a three-phase unbalanced system
is
shown in
Figure 8 together with the associated symmetrical components. 2.1 Importance and construction of sequence networks The impedance of a circuit in which only positive-sequence currents are circulating is called the positive-sequence impedance and, similarly, those in which only negative and zero-sequence currents flow are called the negative and zero-sequence impedances. These sequence impedances are designated Z_{1}, Z2 and Z0, respectively, and are used in calculations involving symmetrical components. Since generators are designed to supply balanced voltages, the generated voltages are of positive sequence only. Therefore, the positive-sequence network is composed of an e.m.f source in series with the positive-sequence impedance. The negative and zero-sequence net-works do not contain e.m.f but only include impedances to the flow of negative and zero-sequence currents, respectively. The positive- and negative-sequence impedances of overhead-line circuits are identical, as are those of cables, being independent of the phase if the applied voltages are balanced. The zero-sequence impedances of lines different from the positive and negative-sequence impedances since the magnetic field creating the positive and negative-sequence currents is different from that for the zero-sequence currents. The following ratios may be used in the absence of detailed information. For a single-circuit line, Zo/Z1 = 2 when no earth wire is present and 3.5 with an earth wire. For a double-circuit line Zo/Z1 = 5.5. For underground cables Zo/Z1 can be taken as 1 to 1.25 for single core, and 3 to 5 for three-core cables:
However, for most studies only the reactance's of synchronous machines are used. Three values of positive reactance are normally quoted-subtransient, transient and synchronous reactance, denoted by X", X_{d}' and X_{d}. In fault studies the subtransient and transient reactance of generators grid motors must be included as appropriate, depending on^{ } the machine characteristics and fault clearance time. |
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