A threephase system is said to be symmetrical when the system viewed from any phase is similar. Thus, in a threephase symmetrical system the selfimpedance of all the three phases are equal and the mutual impedances, if any between the three phases are the same any three phase system having unbalanced phasor quantities can be represented in terms of three phase balanced phasor components as a combination of positive, negative and zero sequence component, which are as follows
It represents a set of balance phasors V_{a1}, V_{b1} & V_{c1}. These components have three phasors equal in magnitude displaced by 120^{o} and having same phase sequence as original phasors.
Negative Phase Sequence Components
These components have three phasors V_{a2}, V_{b2} & V_{c2} equal in magnitude displaced by 120^{o} but having phase sequence opposite to original phasors. It represents a set of balanced phasors
Zero Sequence Components
These phasors V_{ao}, V_{bo} & V_{co} are equal in magnitude and having zero phase displacement.
α operator = e^{+/120°}
= 0.5 + j 0.866
α^{2} = e^{240º}= 0.5  j 0.86
α^{3}=1
α^{4}=α
1 + α +α^{2} = 0
and
In matrix form
Three Phase Power in Terms of Symmetrical Components
= sum of symmetrical components power
The positive sequence impedance of an equipment is the impedance offered by the equipment to the flow of positive sequence current. similarly, the negative sequence or zero sequence impedance of the equipment is the impedance offered by the equipment to the flow of corresponding sequence current.
Sequence Impedance of a Transmission Line
Positive sequence impedance Z_{1} = Z_{s}  Z_{m}
Negative sequence impedance Z_{2} = Z_{s}  Z_{m}
Zero sequence impedance Z_{0} = Z_{s} – 2Z_{m} + 3Z_{n}
where, Z_{s} = Selfimpedance per phase
Z_{m} = Mutual impedances between phases
Sequence Impedance of Synchronous Machine
Sequence Networks Equations
Sequence Network of Unloaded Alternator
Sequence Impedance of Transformers
where Z0 = Zero sequence impedance of transformer.
Examples
Fault may occur at different points in power system.
Faults that occur on a power system are broadly classified as follows
Transient on a Transmission Line
When a sudden change in voltage and current distribution takes place, a power energy distribution also changes. But energy distribution is called transient phenomenon in power system.
The current after short circuit I(t) has components i.e.,
I = I_{s} + I_{t}
Where, Is = Steady state value of current
I_{t }= Transient current
3ϕ Sudden Short Circuit of an Unloaded Alternator
Whenever a threephase short circuit occurs at the terminals of an alternator, the armature current suddenly increases to a large value and voltage across its terminals drops.
Where V_{t} = RMS voltage from one terminal to neutral at no load
= Direct axis subtransient reactance
= Direct axis transient reactance
X_{d} = Direct axis synchronous reactance
X_{d} = X_{L} + X_{m}
Where X_{L} = Leakage reactance
X_{m} = Main winding reactance
X_{t} = Field winding reactance
X_{d} = Damper winding reactance
Note:
Symmetrical Fault Analysis
Three Phase Fault
Symmetrical short circuit on Synchronous Machine The selection of a circuit breaker for a power system depends not only upon the current the breaker is to carry under normal operating conditions but also upon the maximum current it may have to carry momentarily and the current it may have to interrupt at the voltage of the line in which it is placed.
In order to approach the problem of calculating the initial current, we need to study the behavior of a synchronous generator when it is shortcircuited. When an ac voltage is applied suddenly across a series RL circuit, the current which flows has two components 1.
a steady state sinusoidally varying component of constant amplitude and 2. a nonperiodic and exponentially decaying with a time constant of L/R. (which is also referred as the dc component current). The initial value of the dc component of current depends on the magnitude of the ac voltage when the circuit is closed.
In such types of faults, all the three phases are shortcircuited to each other and often to earth also. Such faults are balanced and symmetrical in the sense that the system remains balanced even after the fault.
Fault Level Calculations
In a power system, the maximum the fault current (or fault MVA) that can flow into a zero impedance fault is necessary to be known for switchgear solution. This can either be the balanced three phase value or the value at an asymmetrical condition.
The Fault Level defines the value for the symmetrical condition. The fault level is usually expressed in MVA (or corresponding perunit value), with the maximum fault current value being converted using the nominal voltage rating.
The Short circuit capacity (SCC) of a busbar is the fault level of the busbar. The strength of a busbar (or the ability to maintain its voltage) is directly proportional to its SCC. An infinitely strong bus (or Infinite bus bar) has an infinite SCC, with a zero equivalent impedance and will maintain its voltage under all conditions.
MVA_{base }= √3. Nominal Voltage (kV). I_{base} (kA)
MVA_{Fault}_{ }= √3. Nominal Voltage (kV). I_{sc} (kA)
where
MVA_{Fault}  Fault Level at a given point in MVA
I_{base}  Rated or base line current
I_{sc}  Short circuit line current flowing in to a fault
The per unit value of the fault Level may thus be written as
The per unit value for nominal value of unity, so that
27 videos350 docs

1. What is symmetrical component analysis in electrical engineering? 
2. How does symmetrical component analysis help in fault analysis? 
3. What are positive, negative, and zero sequence components in symmetrical component analysis? 
4. How is symmetrical component analysis applied in power system protection? 
5. What are the advantages of using symmetrical component analysis in electrical engineering? 
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