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Rethinking X/R as Time Constants

X/R ratios are a critical parameter in power system design. In short, the X/R ratio is the ratio of the system reactance (X) to the system resistance (R), when viewed from a particular point in the network.


X/R ratios are crucial for short circuit studies. Traditional breaker ratings are based on symmetrical three-phase fault currents with an assumed maximum X/R ratio. When the system's X/R ratio exceeds the tested value of the overcurrent device, the circuit breaker will need to have its interrupting rating derated, normally to match the design total asymmetrical fault current (RMS).


Why is this the case? Circuit breakers normally rely on zero crossings during the short circuit waveform to be able to interrupt the circuit and clear the fault. When the X/R ratio is higher, there is more inductance (reactance) in the system. This prolongs the fault duration and will delay the occurrence of the first zero crossing.


The worst-case DC fault current can be written as:


DC Fault Current

Where:

  • IDC is the DC Fault Current

  • IAC  is the AC Symmetrical Fault Current

  • f is the system frequency

  • X/R is the X/R ratio

  • t is the time after the fault


Essentially, the DC fault current decays exponentially. When X/R is higher, the factor in the exponential gets lower. This means bigger X/R equals longer DC offsets.



Asymmetrical Fault Current

Figure 1: A Short Circuit has Two Components: The AC (Symmetrical Fault Current) and the DC Offset. Higher X/R ratios lead to longer decay times for the DC offset.



Thinking directly in terms of X/R ratios can be difficult. With enough experience, engineers will become comfortable identifying values that are "high" or "low". However, the meanings behind these X/R ratios will likely remain nebulous.


I recommend rethinking X/R in terms of time constants. Short circuits in power systems are just like an RL circuit-that's why you see the exponential decay in the DC fault current above. We can define a time constant:


RL Circuit Time Constant

Where L is the system inductance. This time constant describes the time it take for the DC component to decay from its maximum value to 1/e (approximately 36.7%). After 3 time constant, less than 5% of the DC offset will remain. After 5 time constants, less than 1% of the DC offset will remain. For this reason, the time constant is a very intuitive and natural way of taking inductance and resistance and converting it to something meaningful for decay of the DC offset.


By recognizing that inductance L is directly proportional to reactance, we can write the time constant directly in terms of X/R.


X/R Time Constant

This is a convenient way of viewing things. Now, for typical X/R in the range of 5-60 with a 60 Hertz frequency, we can relate it directly to a time constant:

X/R Ratio

Time Constant (Cycles)

5

.8

10

1.6

20

3.2

30

4.8

40

6.4

50

8.0

60

9.5


The table above demonstrates how DC offset become critical to consider as we move upstream in the power system. An X/R ratio of 5-10 is normal for low voltage power systems, where resistive effects are significant in transformers and cables. In these networks, we can expect that the DC fault current will have a time constant of no more than 2 cycles. That's not bad at at all for clearing the fault and maintaining equipment ratings.


Large power transformers, on the other hand, can have X/R ratios on the order of 50-60. In these cases, the time constant could be up to 10 cycles. Failing to account for DC offset here could lead to major problems with clearing faults.


Understanding X/R as time constants will give you the power to put real, physical behavior to your networks. Instead of just knowing that the X/R is "high" or "low", we can say exactly how big this impact will be.


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