Fault Calculation - Symmetrical Components 

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Symbol - Definition
image23 - system voltage

image24, image25, image26 - unbalanced voltages
image27, image28, image29 - symmetrical voltages
image21 - circuit impedance

image25 - earth fault impedance

image16 - positive sequence impedance
image17 - negative sequence impedance
image18 - zero sequence impedance

For unbalance conditions the calculation of fault currents is more complex. One method of dealing with this is symmetrical components. Using symmetrical components, the unbalance system is broken down in to three separate symmetrical systems:

  • Positive sequence – where the three fields rotate clockwise
  • Negative sequence – where the three fields rotate anti-clockwise
  • Zero sequence – a single fields which does not rotate

The positive sequence network rotates clockwise, with a phase and of 120° between phase as per any standard a.c. system. 

Negative sequence network, rotates anti-clockwise and the zero sequence network with each phase together (0° apart).

 

Image(29)

Basic Symmetrical Component Theory

Mathematically, the relationship between the symmetrical networks and the actual electrical systems, make use of a rotational operator, denoted by a and given formally by:

Image(27)

Perhaps more simply, the a operator can be looked at as a 120° shift operator. It can also be shown that the following conditions hold true:

Image(28)

By using the a operator, any unbalanced any unbalance three phase system Va, Vb, Vc can be broken down into three balanced (positive, negative and zero sequence) networks V1, V2, V0.

Unbalanced Network Symmetrical Network
image36   image39
image37   image40
image38

image41

The operator a, is the unit 120° vector: a = 1|120°. Note: a3=1 and a-1 = a2

Fault Solutions

Once the sequence networks are know, determination of the magnitude of the fault is relatively straight forward.

The a.c. system is broken down into it's symmetrical components as shown above.  Each symmetrical system is then individually solved and the final solution obtained by superposition of these (as shown above).

For the more common fault conditions,  once the sequence networks are known we can jump directly to the fault current.  During a fault and letting Un, be the nominal voltage across the branch, use of symmetrical components give the following solutions (excluding fault impedance):

 

Type of Fault Initial Fault Current Comments
three phase image34  
phase to phase image35  
phase to phase image36  
phase to phase to earth image37

image39

image38

Note: the example given assumes a phase to phase fault between L2 and L3 (then shorted to earth)

 

Sequence Impedance Data

Positive, negative and zero sequence impedance data is often available from manufacturers. 

A common assumption is that for non rotating equipment the negative sequence values are taken to be the same as the positive. 

Zero sequence impedance values are closely tied to the type of earthing arrangements and do vary with equipment type.  While it is always better to use actual data, if it is not available (or at preliminary stages), the following approximations can be used:

Element Z(0)
Transformer  
No neutral connection ∞ (infinity)
Yyn or Zyn 10 to 15 x X(1)
Dyn or YNyn X(1)
Dzn or Yzn 0.1 to 0.2 x X(1)
Rotating Machine
Synchronous 0.5 x Z(1)
Asynchronous zero
Transmission Line 3 x Z(1)

Approximate Zero Sequence Data (source: Schneider)

Related Notes



Steven McFadyen's avatar Steven McFadyen

Steven has over twenty five years experience working on some of the largest construction projects. He has a deep technical understanding of electrical engineering and is keen to share this knowledge. About the author

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