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 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).

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 three phase system V_a, V_b, V_c can be broken down into three balanced (positive, negative and zero sequence) networks V₁, V₂, V₀.

Unbalanced Network		Symmetrical Network

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 U_n, 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
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 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:

Approximate Zero Sequence Data (source: Schneider)

Related Notes

• Fault Calculations - Introduction