Introduction - Line to line faults are a common type of short circuit event on a power system. During line to line faults, one phase conductor comes into direct, electrical contact with another phase conductor. On industrial systems, these kinds of faults are often caused by a breakdown in insulation between conductors or improper operation and maintenance of equipment during or following energization.
Figure 1: Line to Line Fault Circuit Diagram
Why Study Line to Line Faults? - The interrupting rating of overcurrent protection devices and the withstand duty of equipment is generally referenced to symmetrical faults (bolted three phase faults). However, proper analysis of a power system requires a detailed look at all types of fault current. If you need a refresher on bolted three phase faults, check out my article here.
Line to line faults are generally a lesser concern on power systems than bolted three phase faults or line to ground faults. Line to line faults do not produce a ground current that can lead to shock hazards, and in most cases the magnitude of a line to line fault is lower than a three phase fault.
Line to line faults produce negative sequence currents, unlike three phase faults. Depending on the system configuration, the use of negative sequence protection for line to line fault detection can be a real asset in protection and controls design. In many cases, using negative sequence detection allows us to identify the type of fault that has occurred and respond with better specificity than simply tripping the whole system offline.
Symmetrical Components - The line to line fault current is typically calculated using symmetrical components, an alternative representation of power systems for unbalanced fault analysis. A detailed explanation of symmetrical components won't be provided here, but for a quick refresher:
Positive Sequence Impedance is the impedance of a power system in response to normal phasor rotation voltage sources. Positive sequence impedance is the "normal" impedance of a system that is used for things like voltage drop calculations and symmetrical fault analysis. When we say "impedance" without any modifiers, we're almost always talking about positive sequence impedance.
Negative Sequence Impedance is the impedance of a power system in response to reverse phasor rotation voltage sources. For non-rotating equipment, like transformers and cables, the positive sequence impedance is the same as the negative sequence impedance.
Zero Sequence Impedance is the impedance of a power system in response to the application of a single phase voltage source. Zero sequence impedance is unique because there is no concept of phase rotation.
Figure 2: Symmetrical Components Visualized
Calculating the Line to Line Fault Current - Performing detailed analysis of symmetrical components by hand can be cumbersome. Typically, manual analysis is only concerned with simplified systems, usually involving a transformer, like the case in Figure 3. For more information on transformer impedances in the sequence domain, see my article here.
Figure 3: The Typical Infinite Bus Condition for Line to Line Faults
Figure 4 is a representation of a line to line fault as viewed in the sequence domain. For this type of fault, the positive sequence and negative sequence networks are connected in parallel. In other words, the positive sequence current is equal to the negative of the negative sequence current. For typical power systems, the only source voltage will be the positive-sequence (normal operating) voltage. Note that the zero sequence network is not involved in this calculation. During a line to line fault, zero sequence current should not be detected.
Figure 4: A Line to Line Fault in the Sequence Domain
Considering the system voltage source to be positive phase rotation and the only source in the system, we can obtain a relationship for our fault current magnitude:
Where:
Z1 is the positive sequence impedance in Ohms
Z2 is the negative sequence impedance in Ohms
VLL is the line to line voltage in Volts
ILL is the line to line fault current magnitude in Amps
In the case of a line to line fault through a transformer like in Figure 3 (assuming no source impedance), the positive and negative sequence impedance are equal. Then, the impedance is simplified to:
Both of these equations neglect any additional "fault impedance" that could be in place between the faulting phases. This fault impedance would lower the magnitude of the current.
Example: Compute the symmetrical fault current and the line to line fault current for the power system below. Assume an infinite grid (no source impedance).
Solution: This is an easy one! Start by computing the symmetrical fault current (the bolted three phase fault current). We do that by using the positive sequence impedance:
Where:
S is the transformer apparent power rating in VA
VLL is the line to line voltage on the faulted winding in Volts
Z1 is the positive sequence impedance of the transformer in %
Ibf is the bolted three-phase fault current magnitude in Amps
Plugging in our values:
Ibf = 100 / (√3 * 34.5 * .1) = 16.7 kA
The line to line fault current is computed simply by using our equation above, remembering that the positive sequence impedance and negative sequence impedance of a transformer are equal. To begin solving this problem, we need to convert transformer percent impedances into Ohms. This starts with identifying the base value of the impedance at the 34.5 kV side of the transformer:
Zbase = 34.5 ^ 2 / 100 = 11.9 Ohms
Then, convert the impedances to Ohms from percent:
Z1 = Z2 = .1 * 11.9 = 1.19 Ohms
Lastly, we can apply our equation for line to line fault current:
ILL = 34.5 / (1.19 + 1.19) = 14.5 kA
These are the final results of this calculation. Notice that line to line fault current is lower than the transformer three phase fault current. For devices like transformers and conductors, where the positive sequence and negative sequence impedance are identical, this is always the result. In these cases, the line to line fault current will be equal to 86.6% of the 3-phase fault current.
For devices that behave differently with altered phase rotation, like motors or generators, negative sequence impedance may be different and produce alternative results.