Fault Calculation  Symmetrical Components
Symbol 
 
Definition 

 
system voltage
, ,  unbalanced voltages
, ,  symmetrical voltages 

 
circuit impedance
 earth fault impedance
 positive sequence impedance
 negative sequence impedance
 zero sequence impedance 
For unbalanced conditions, the calculation of fault currents is more complex. One method of dealing with this is symmetrical components. Using symmetrical components, the unbalanced system is broken down into three separate symmetrical systems:
 Positive sequence – where the three fields rotate clockwise
 Negative sequence – where the three fields rotate anticlockwise
 Zero sequence – a single field which does not rotate
The positive sequence network rotates clockwise, with a phase of 120° between phases as per any standard a.c. system.
The negative sequence network rotates anticlockwise, and the zero sequence network with each phase together (0° apart).
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:
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:
By using the a operator, any unbalanced any unbalance threephase system V_{a}, V_{b}, V_{c} can be broken down into three balanced (positive, negative and zero sequence) networks V_{1}, V_{2}, V_{0}.
Unbalanced Network 

Symmetrical Network 









The operator a, is the unit 120° vector: a = 1120°. Note: a^{3}=1 and a^{1} = a^{2}
Fault Solutions
Once the sequence networks are known, the determination of the magnitude of the fault is relatively straightforward.
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 U_{n}, be the nominal voltage across the branch, the use of symmetrical components gives the following solutions (excluding fault impedance):
Type of Fault 
Initial Fault Current 
Comments 
three phase 


phase to phase 


phase to phase 


phase to phase to earth 

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 nonrotating 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