Calculati** of short circuit current
The current that flows through an element of a power system is a parameter which can be used to detect faults, given the large increase in current flow when a short circuit occurs.
For this reas** a review of the c**cepts and procedures for calculating fault currents will be made in this chapter, together with some calculati**s illustrating the methods used.
Although the use of these short-circuit calculati**s in relati** to protecti** settings will be-c**sidered in detail, it is important to bear in mind that these calculati**s are also required for other applicati**s, for example calculating the substati** Earthing grid, the selecti** of c**ductor sizes and for the specificati**s of equipment such as power-circuit breakers.
1 Mathematical derivati** of fault currents
The treatment of electrical faults should be carried out as a functi** of time,
from the start of the event at time  until stable c**diti**s are reached, and therefore it is necessary to use differential equati**s when calculating these currents. In order to illustrate the transient nature of the current,
c**sider an RL circuit as a simplified equivalent of the circuits in electricity-distributi** networks. This simplificati** is important because all the system equipment must be modeled in some way in order to quantify the transient values which can occur during the fault c**diti**.
For the circuit shown in Figure 1, the mathematical expressi** which defines the behaviour of the current is:
e(t) = L di + Ri(t) 2.1
Figure 1 RL, circuit for transient analysis studyThis is a differential equati** with c**stant coefficients, of which the soluti** is in two parts:
Where:
ih(t) Is the soluti** of the homogeneous equati** corresp**d ing to the transient period and ip(t) is the soluti** to the particular equati** corresp**ding to the steady-state period.
By the use of differential equati** theory, which will not be discussed in detail here, the complete soluti** can be determined and expressed iii the following form:
Where:
α = the closing angle which defines the point ** the source sinusoidal voltage when the fault occurs and
It can be seen that, in eqn. 2.2, the first term varies sinusoidally, while the sec**d term decreases exp**entially with a time c**stant of L/R. The latter term can be recognised as the DC comp**ent of the current, and has an initial maximum
value when , and zero value when Φ=α, see Figure 2.
It is impossible to predict at what point the fault will be applied ** the sinusoidal cycle and therefore what magnitude the DC comp**ent will reach. If the tripping of the circuit, owing to a fault, takes place when the sinusoidal comp**ent is at its negative peak, the DC comp**ent reaches its theoretical maximum value half a cycle later.
Figure 2 Variati** of fault current with time
a (α–Φ) =0
b (α–Φ)=π/2
An approximate formula for calculating the effective value of the total asymmetric current,
including the AC and DC comp**ents, with acceptable accuracy can be obtained from the following expressi**:
The fault current which results when an alternator is short circuited can easily be analysed since this is similar to the case which has alreadybeen analysed, i.e. when voltage is, applied to an RL circuit. The reducti** in current from its value at the **set, owing to the gradual decrease in the magnetic flux caused by the reducti** of the e.m.f. of the inducti** current, can be seen in Figure 3. This effect is known as armature reacti**.
The physical situati** that is presented to a generator, and which makes the calculati**s quite difficult, can be interpreted as a reactancewhich varies with time. Notwithstanding this, in the majority of practical applicati**s it is possible to take account of the variati** of reactance in **ly three stages without producing significant errors. In Figure 4 it will be noted that the variati** of current with time, 1(t), comes close to the three discrete levels of current, I", 1' and I, the subtransient, transient and steady-state currents, respectively. The corresp**ding values of direct axis reactance are denoted by  and Xd, 
Figure 3 Transient short-circuit currents in a synchr**ous generator
Figure 4 Variati** of current with time during a fault
Figure 5 Variati** of generator reactance with time during a fault And the typical variati** with, time for each of these is illustrated in
Figure 5.
To sum up, when calculating short-circuit currents it is necessary to take into account two factors which could result in the currents varying with time:
· the presence of the DC comp**ent;
· the behaviour of the generator under short circuit c**diti**s.
In studies of electrical protecti** some adjustment has to be made to the values of instantaneous short circuit current calculated using subtransient reactance's which result in higher values of current.
Time delay units can be set using the same values but, in some cases, short-circuit values based ** the transient reactance are used, depending ** the operating speed of the protecti** relays. Transient reactance values are generally used in stability studies.
Of necessity, switchgear specificati**s require reliable calculati**s of the short-circuit levels which can be present ** the electrical network. Taking into account the rapid drop of the short-circuit current due to the armature reacti** of the synchr**ous machines, and the fact that extincti** of an electrical arc is never achieved instantaneously, ANSI Standards C37.010 and C37.5 recommend using different values of subtransient reactance when calculating the so-called momentary and interrupting duties of switchgear.
Asymmetrical or symmetrical r.m.s. values can be defined depending ** whether or not the DC comp**ent is included. The peak values are obtained by multiplying the R.M.S. values by  .
The asymmetrical values are calculated as the square root of the sum of the squares of the DC comp**ent and the r.m.s. value of the AC current, i.e.:
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