Fault Calculation - Symmetrical Components 

By on

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 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 anti-clockwise
  • 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 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  
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 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 Un, 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 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

myElectrical Engineering

comments powered by Disqus



International System of Units (SI System)

The International System of Units (abbreviated SI) is the world's most widely used system of units.  The system consists of a set of units and prefixes...

Lightning Protection and Earth Electrode Resistance

Most installations involve some form of lightning protection system which is connected to an earth electrode.  The function of the earth electrode is to...

IEC 61439 Verification Methods

The (relatively new) switchgear and control gear standard, IEC 61439 'Low-voltage switchgear and controlgear assemblies' has three methods which can be...

Our internet address and Vanity URLs

Visitors who like to type web address rather then click menus may be interested in how our URL structure works.

IEEE Winds of Change

IEEE TV has a part series of videos on wind power and it's implication. For a really good overview to the technologies and issues around wind power, these...

Low Voltage Circuit Breakers

Circuit breakers are switching devices whose primary function is to isolate parts of an electrical distribution system in the even of abnormal conditions...

Windows Live Writer and myElectrical

When making adding a Note to our site we have a great online WYSIWYG editor and things are pretty simple.  However, if you prefer you can write, manage...

MIT OpenCourseWare

MIT OpenCourseWare, makes the materials used in teaching all MIT subjects available on the Web, free of charge, to any user in the world.

Introduction to Traction Substations

Following on from my post on railway electrification voltages, I thought an introduction to traction substations would be a good idea. Traction substations...

GE's Shingijutsu Factory

GE's latest thinking on product manufacturing is he Shingijutsu philosophy or Lean production system. They have started applying this at the Louisville...

Have some knowledge to share

If you have some expert knowledge or experience, why not consider sharing this with our community.  

By writing an electrical note, you will be educating our users and at the same time promoting your expertise within the engineering community.

To get started and understand our policy, you can read our How to Write an Electrical Note