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By : Ramakrishna Perumal, Sr. Engineer, Technicas Reunidas
Industry : Power
Activity:  1 comments  3875 views  last activity : 07 06 2010 20:18:04 +0000
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SHORT CIRCUIT CALCULATION

A short circuit (sometimes abbreviated to short or s/c) allows a charge to flow along a different path from the one intended. The electrical opposite of a short circuit is an open circuit, which is infinite resistance between two nodes. It is common to misuse "short circuit" to describe any electrical malfunction, regardless of the actual problem.

Explanation

A short circuit is an accidental low-resistance connection between two nodes of an electrical circuit that are meant to be at different voltages. This results in an excessive electric current limited only by the Thevenin equivalent resistance of the rest of the network and potentially causes circuit damage, overheating, fire or explosion. Although usually the result of a fault, there are cases where short circuits are caused intentionally, for example, for the purpose of voltage-sensing crowbar circuit protectors.

In circuit analysis, the term short circuit is used by analogy to designate a zero-impedence connection between two nodes. This forces the two nodes to be at the same voltage. In an ideal short circuit, this means there is no resistance and no voltage drop across the short. In simple circuit analysis, wires are considered to be shorts. In real circuits, the result is a connection of nearly zero impedance, and almost no resistance. In such a case, the current drawn is limited by the rest of the circuit.

Example

A short circuit is to connect the positive and negative terminals of a battery together with a low-resistance conductor, like a wire. With low resistance in the connection, a high current flows, causing the cell to deliver a large amount of energy in a short time. (See also: Ohm's law, power).

In electrical devices, unintentional short circuits are usually caused when a wire's insulation breaks down, or when another conducting material is introduced, allowing charge to flow along a different path than the one intended.

A large current through a battery (also called a cell) can cause the rapid buildup of heat, potentially resulting in an explosion or the release of hydrogen gas and electrolyte, which can burn tissue and may be either an acid or a base. Overloaded wires can also overheat, sometimes causing damage to the wire's insulation, or a fire. High current conditions may also occur with electric motor loads under stalled conditions, such as when the impeller of an electrically driven pump is jammed by debris.

Damage from short circuits can be reduced or prevented by employing fuses, circuit breakers, or other overload protection, which disconnect the power in reaction to excessive current. Overload protection must be chosen according to the maximum prospective short circuit current in a circuit. For example, large home appliances (such as clothes dryers) typically draw 10 to 20 amperes, so it is common for them to be protected by 20 - 30 ampere circuit breakers, whereas lighting circuits typically draw less than 10 amperes and are protected by 10 - 15 ampere breakers. Wire sizes are specified in building and electrical codes, and must be carefully chosen for their specific application to ensure safe operation in conjunction with the overload protection.

In mains circuits, short circuits are most likely to occur between two phases, between a phase and neutral or between a phase and earth (ground). Such short circuits are likely to result in a very high current flowing and therefore quickly trigger an overcurrent protection device. However, it is possible for short circuits to arise between neutral and earth conductors, and between two conductors of the same phase. Such short circuits can be dangerous, particularly as they may not immediately result in a large current flowing and are therefore less likely to be detected. Possible effects include unexpected energisation of a circuit presumed to be isolated. To help reduce the negative effects of short circuits, power distribution transformers are deliberately designed to have a certain amount of leakage reactance. The leakage reactance (usually about 5 to 10% of the full load impedance) helps limit both the magnitude and rate of rise of the fault current.

Short Circuit Analysis

Determining available short circuit fault currents. International standards (IEEE, IEC, and DIN VDE) define calculation procedure for determining the fault currents.

Types of short circuits

1) Symmetrical short circuit 2) Asymmetrical short circuit

3 Phase / 3 phase to ground fault is a symmetrical short circuit. All three conductors are equally involved and carry the same short circuit current. Single phase to ground fault,phase to phase fault are some examples of asymmetrical short circuit. The magnitude of current varies in all the phases during an asymmetrical short circuit.

For more information on calculating short circuit fault currents refer to IEEE Std 242, "Recommended Practice for Protection and Coordination of Industrial and Commercial Power Systems" Chapter 2 (Buff Book).

 

 Symmetrical components

These were developed to ease the calculations for unbalanced 3-phase systems and as a help to numerical solution using network analyzers. Even with present day digital computation, the symmetrical components help in solution of unbalanced systems, besides explaining many phenomena such as rotor heating in machines, neutral current etc.

The basic concept is to convert a set of three phasors into another set of three phasors with certain desirable properties. The symmetrical components (introduced by Fortescue) is only one such set, the other set is the Kimbark/Clarke components.

The unique property of symmetrical components is that they retain the concept of 3-phase system associated with each component phasor. Thus the positive sequence retains the concept of 3 balanced phasors having the same phase sequence as the original phasors whereas the negative sequence component retains the concept of 3 balanced phasors but rotating in opposite direction of rotation. The zero sequence component is balanced set of 3- coincident phasors but rotating in the same direction as the original unbalanced phasors.

In electrical engineering, the method of Symmetrical components is used to simplify analysis of unbalanced three phase power systems.

Charles Legeyt Fortescue in a paper presented in 1918 (Method of Symmetrical Co-Ordinates Applied to the Solution of Polyphase Networks) demonstrated that any set of N unbalanced polyphase quantities could be expressed as the sum of N symmetrical sets of balanced phasors. Only a single frequency component is represented by the phasors.

In a three-phase system, one set of phasors has the same phase sequence as the system under study (positive sequence - say ABC), the second set has the reverse phase sequence (negative sequence - BAC), and in the third set the phasors A, B and C are in phase with each other (zero sequence).

By expanding a one-line diagram to show the positive sequence, negative sequence and zero sequence impedances of generators and transformers and other devices, analysis of such unbalanced conditions as a single line to ground short-circuit fault is greatly simplified. The technique can also be extended to higher phase order systems.

Physically, in a three phase winding a positive sequence set of currents produces a normal rotating field, a negative sequence set produces a field with the opposite rotation, and the zero sequence set produces a field that oscillates but does not rotate. Since these effects can be detected physically, the mathematical tool became the the basis for the design of protection relays, which used negative-sequence voltages and currents as a reliable indicator of fault conditions. Such relays may be used to trip circuit breakers or take other steps to protect electrical systems.

The analytical technique was adopted and advanced by engineers at General Electric and Westinghouse and after World War II it was an accepted method for asymmetric fault analysis.

The Three-Phase Case

Symmetrical components are most commonly used for analysis of three-phase electrical power systems. If the phase quantities are expressed in phasor notation using complex numbers, a vector can be formed for the three phase quantities. For example, a vector for three phase voltages could be written as

and the three symmetrical components phasors arranged into a vector as

where the subscripts 0, 1, and 2 refer respectively to the zero, positive, and negative sequence components.

A phase rotation operator is defined to rotate a phasor vector forward by 120 degrees or radians. A matrix can be defined using this operator to transform the phase vector into symmetrical components.

 

The basic definition for a set of unbalanced three-phase system in terms of the sequence components is

Ia = I0a + I1a +I2a

Ib = I0b + I1b + I2b + I0 a + a2 I1 a +a I2 a

Ic = I0 c +I1 c +I2 c = I0 a + a I1 a + a 2 I2 a

These equations may be written in matrix form as

I a,b,c = [Ts] I 0,1,2

Where Ts is called the symmetrical component transformation matrix.

I 0,1,2 = [Ts]-1 I a,b,c

Similar transformation may be applied for unbalanced voltages also.

Illustrate these transformations by phasor additions.

Symmetrical components in Fault Calculation

Any unbalance three phase system Va, Vb, Vc can be broken down into three balanced (positive, negative and zero sequence) networks V1, V2, V0.

Where:

 


and

 


Equivalent voltage for each network are given by:


,    ,   

During a fault and letting V, be the voltage across the branch prior to the fault, use of systematical components give the following solutions (excluding fault impedance):

Single Phase → Earth Fault

 

 


Phase → Phase Fault

 

 

 

Phase → Phase → Earth Fault

 

 


Where:


Three Phase Fault

 

 

 

V is the normal voltage across the branch before the fault:

Three phase & phase to phase faults:

Phase to phase & phase to phase to earth:

Where there is a fault impedance Zf this needs to be taken into account.

 

 Power invariance

If the above transformation Ts is used simultaneously on the voltage and current values of three phase network elements, then

Spq abc = P pq + j Qpq =[( I pq abc )*]t e pq abc

And

Spq012 = [( I pq012)*]t e pq012

Note that Spqabc is not equal to Spq012.

Spqabc = 3Spq012 since Spq012 refers to only phase "a" power , and similar amount of power is in phases "b" and "c" also. Thus all three phases of symmetrical components must be used. Often for computer work, the symmetrical transformation given by

Tsi  3) TÖ= (1/s

is use where Tsi*t . Tsi = unity matrix. This is a property of orthogonal matrices. Further since Tsi* = Tsi-1 we can show that Spqabc = Spq012. However, in earlier works, and even now, the originally defined non-power invariant transformation is being used

Impedance transformation

To solve the network in terms of the sequence components, sequence components of impedance are required. These are obtained from their corresponding three phase values.

Eabc = Zabc I abc

Or,

Ts E012 = Z abc Ts I 012

Or,

E012 = Tsi-1 Z abc Ts I 012

= Z012 I 012

Thus Z012 = Ts-1 Z abc Ts

0r,

Zabc = Ts Z 012 Ts-1

 

Forms of Z012 for balanced stationary and rotating elements should be known. Decoupling of sequence components and its limitations are to be stressed.

Sequence generated voltages in a balanced network are:

E1 = Ea, E2 = 0 , E0 =0

Sequence networks have the advantage since for balanced network there is no mutual coupling between sequence component elements unlike what happens in balanced 3-phase components. Thus balanced 3-phase network can be assembled component for component in three separate sequence networks. Sequence networks of generator, transmission line, transformer, and loads should be known. Zero sequence networks for different transformer connections should also be known. The ideal earth and the neutral point should be distinguished. Relative magnitudes of sequence impedances of generator and transmission lines should be known.

  • Find Z012 for rotating and non-rotating elements in terms of corresponding phase quantities
  • Study the measurements of zero sequence impedances of transformer for different connections
  •  transformationU DFind phase shifts in sequence components in  
  • If the positive, negative and zero sequence impedances of a transmission line are 0.3,0.3, and 0.5 respectively, find the self and mutual impedances between the phases.

Fault analysis

An essential part of the design of a power system is the calculation of currents which flow in the components and the resulting voltages , when the faults of various type occur. Common faults on a transmission system are;

  1. LG
  2. LL
  3. LLG
  4. LLL or LLLG
  5. Open conductor
  6. Simultaneous faults that may be any combination of the above five.

Faults may also occur in switchgear, transformers and machines but their nature may be different than those in transmission lines.

Causes of system faults

Faults are mostly due to lightning and switching. Most of them are temporary. LG fault is most common and LLL fault is least common. Relays provided on the system detect these faults and produce a trip signal for the circuit breaker to isolate the faulty portion form the system. Thus to determine the circuit breaker and relay operating times, fault currents & voltages have to be calculated and for many other applications.

Except for the three phase faults ,all other types of faults cause unbalanced operation, an fault currents & voltages under such conditions are required to be obtained using symmetrical components, or phase components ( the latter analysis is much more difficult even with digital computers.

The calculation of 3-phase balanced faults is relatively simple but forms the basis of determining the circuit breaker ratings.

Symmetrical faults

When a sudden short circuit occurs on the electric supply system, the currents & voltages are of transient nature before they settle down to steady state values.

The fault current at any time consists of

  1. Dc transient , also called the DC offset, arising out of the terms of the type

Ae-Rt/L

  1. Ac transients consisting of the terms of the type

Ae-Rt/L  twSin( +  = supply frequencyw ) where f

  1.  twSteady state value of the type A Sin( +  ).f

For a single machine system, the maximum value of the Dc transient could be equal to the peak value of the total Ac component. Fig.5 shows the symmetrical current in one phase with no DC offset.( Other phases will have DC offset). The cause of subtransient/ transient current could be explained on the basis of the theorem of constant flux linkages associated with the field winding. The damper winding affects the initial value of the fault current. The initial value decays faster(region PQ in Fig.5) because of the very low time constant of this circuit. Later, the closed field circuit effects dominate (region QR). The region QR may be extended up to the origin as the dotted line bQ.

Definition of reactances

Xd"  2 .E. /OaÖ=

Xd 2 .E. /ObÖ'=

Xd 2 .E. /OcÖ=

What are the typical values of these reactances for turbo and hydro generators?

For the interruption current rating of a circuit breaker (time involved being 2-5 cycles), the subtransient current is important. The transient current is important for transient stability studies lasting from 1-2 seconds to 10 seconds.

 3. (short-circuit current). [Nominal Voltage (line value)]ÖShort circuit volt-amperes =

 

System representation during short circuit

Simplifying assumptions

  1. Each machine is represented by a constant voltage behind the machine reactance, subtransient or transient depending on the situation.
  2. Shunt connections i.e.; loads , line charging etc. are neglected.
  3. All transformers are assumed to be at nominal tap settings.
  4. The above three assumptions do not imply that the system is unloaded before the fault.. However , it is assumed that the network is unloaded and that all the generator voltages are at 1 pu. This assumption is justified because fault currents in the network are much larger than the load currents.

If assumptions 2-4 are not taken into account, pre-fault currents are added to the fault currents obtained by using the Thevenin's equivalent of the network.

Thevenin's equivalent

The Thevenin's equivalent of the network at the point of the fault location (either in phase or sequence components) is obtained for calculating the fault currents. If the network is balanced before the fault, then at the fault point , Thevenin's sequence voltages become

Ea1= Ea

Ea2= 0

Ea0= 0.

If the network is not balanced , the above voltages may have finite values and the sequence component approach may not be useful. The performance equation for the balanced network in terms of sequence components is :

[Vs] = [Eph]- [Zs][Is],

where

[Vs]=[V0a,V1a, V2a]

[Eph] = [0,E1a,0]

[Zs] = diagonal matrix of [Z0,Z1,Z2]

[Is] = [I0a,I1a,I2a]

The network at the fault point F appears as shown in Fig.6 where Z1,Z2 and Z0 are the lumped values of the Thevenin's impedances between the fault point and the neutral.

Depending upon the type of fault, the sequence components of currents and voltages are constrained leading to particular connections of sequence network. After the fault currents coming out of the network at the fault point are determined, the current & voltage distributions inside the network are found out . If originally, there are load currents , these are restricted to the positive sequence network , and they are superposed on the fault currents for accurate results, but this is often not done.

 

Three-phase fault

Here ,

Ia1 = Ea/(Z1 +ZF)

Ia2 = 0

Ia0= 0

Single-line to ground (SLG)fault

Fault is assumed in phase "a". If it is in any other phase, symmetrical shifting will be required.

The connections at the fault point are given in FIG.7 for various types of faults. The connections of the sequence networks for SLG fault are given in FIG.8 .

Study the network connections for L-L and DLG faults.

  • How do you distinguish switching transient currents from sub-transient currents?
  • Sketch the subtransient/ transient current wave-shapes in phase b and c respectively for a synchronous generator when the current in phase a is symmetrical.
  • Under what conditions can you expect a negative or zero sequence Thevenin's voltage at the fault point?
  • Determine and draw the sequence network connection diagrams for
  1. Single conductor opening at a point
  2. Two conductors opening at a point.

SHORT-CIRCUIT ANALYSIS -OBJECTIVE TYPE QUESTIONS

  1. If a positive sequence current passes through a transformer and its phase shift is 30 degrees, the negative sequence current flowing through the transformer will have a phase shift of
  1. 30 deg.
  2. -30deg
  3. 120 deg.

Ans: (b)

  1. The zero sequence impedances of an ideal star-delta connected transformer (star-grounded)
  1. looking from star side is zero and looking from delta side is infinite
  2. looking from star side is zero and looking from delta side is also zero
  3. looking from star side is infinite and looking from delta side is zero

Ans: (a)

  1. The positive, negative and zero sequence impedances of a transmission line are 0.5,0.5 and 1.1 pu respectively. The self (Zs) and mutual (Zm) impedances of the line will be given by

(a) Zs = 0.7 pu, Zm =0.2 pu

(b) Zs = 0.5 pu, Zm =0.6 pu

(c)Zs = 1 pu, Zm =0.6 pu

Ans: (a)

  1. Symmetrical component method of analysis is more useful when
  1. system has unsymmetrical fault and the network is otherwise balanced
  2. system has symmetrical fault and the network is otherwise unbalanced
  3. system has unsymmetrical fault and the network is unbalanced.

Ans: (a)

  1. For a line to line fault analysis using symmetrical components,

(a) the positive and negative sequence networks at the fault point are

connected in series

(b) the positive and negative sequence networks at the fault point are

connected in parallel

(c) the positive, negative, and zero sequence networks at the fault point are

connected in parallel

Ans: (b)

  1. The machine reactances used for computation of short circuit current ratings of a circuit breaker are
  1. synchronous reactance
  2. transient reactance
  3. sub-transient reactance

Ans: (c)

  1. The load currents in short-circuit calculations are neglected because
  1. short-circuit currents are much lager than load currents
  2. short -circuit currents are greatly out of phase with load currents

The correct alternative is

  1. both (i) and (ii) are wrong
  2. (i) is wrong and (ii) is correct
  3. both (i) and (ii) are correct.

Ans: (c)

  1. The positive sequence network of a sample power system is shown in Fig.2 and the primitive reactance of each element is marked in ohms. The element Zcc of the bus impedance matrix Zbus will be
  1. 1.0
  2. 0.1
  3. 0.21

Ans: (c)

  1. Which one of the following statements is not true?
  1. Fault levels in an all a. c system are less than in an a. c system with a D.C. link operating
  2. Large systems may be interconnected with d. c link of small capacity
  3. Limitation on the critical length of underground cables for use in A. C no longer exists if D.C transmission by cables is used.
  4. Corona loss and radio and TV interference with D.C. transmission is less

Ans. a

  1. Advantage of power system interconnection is
  1. Large size circuit breakers are required because of large short-circuit currents
  2. Machines of one system remain in step with machines of another system
  3. Fewer machines are required as reserve for operation at peak loads

Ans. c

  1. The interenal voltages of a 3-phase synchronous generator correspond to
  1. Positive sequence
  2. Negative sequence
  3. Zero sequence
 
1 comments on "Short Circuit Calculations"
  Commented by  Vineet Prakash, Sales/BD Manager, Tata Power    | 11 04 2009 05:22:26 +0000
Nice article Ramakrishna, really very informative. Thanks for sharing...
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