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:
 


When considering the specification for the switchgear-opening cur-rent, the so-called r.m.s. value of interrupting current is used in which, again, the AC and DC components are taken into account, and therefore:

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
 


Figure 6 Multiplying factors for three-phase and line-to-earth faults (total current rating basis) (from. IEEE Standard C37.5-1979; reproduced by permis­sion of the IEEE)

 

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,
Va Vb and Vc can be represented thus:
                                  Va =Vao + Va1 + Va2

                                  Vb =Vbo + Vb1 + Vb2

                                  Vc =Vco + Vc1 + Vc2

 

It can be demonstrated that:

 

Vb= Vao+a2Va1+aVa2

 Vc= Vao+aVa1+ a2Va2

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°,
and a 2 similarly gives a phase shift
of 240°, i.e. a 2=1 Therefore,
 the following matrix relationship can be established:
 

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 = (Ia + Ib + Ic) and, therefore, ln=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 Z1, 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:


For transformers, the positive and negative-sequence impedances
are equal because in static circuits these impedances are independent of the phase order, provided that the applied voltages are balanced. The zero-sequence impedance is either the same as the other two impedances, or infinite, depending on the transformer connections. The resistance of the windings is much smaller and can generally be neglected in short-circuit calculations. When modelling small generators and motors it may be necessary to take resistance into account.

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", Xd' and Xd. 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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