# The Art of Protecting Electrical Systems: Short-Circuit Calculation Methods

09/19/2006

Editor’s Note: From 1965 through 1970, Consulting-Specifying Engineer ’s predecessor, Actual Specifying Engineer, ran a series of articles on overcurrent protection. Due to the immense popularity of the 31 installments in the series, the authors, George Farrell and Frank Valvoda, P.E., reprised the series in an updated version beginning in the Feb. 1989 issue of CSE. Over the years since the last installment ran in the late ’90s, we have received many requests to re- run this series. Mr. Valvoda passed away in Dec. 2001, and his long-time friend and editorial partner, George Farrell, passed away earlier this year.

Part nine of this ongoing series continues the discussion of short-circuit calculations and details the use of two manual per-unit steps: diagramming the system and finding the Thevenin Equivalent.

We also offer readers a “Talk Back” option at the end of the story, to comment on or update any of the technical information in this article.

By GEORGE W. FARRELL and FRANK R. VALVODA, P.E.

This article continues the discussion of short-circuit calculations introduced in Part eight of this series. Basically, determining short-circuit currents by the manual per-unit method involves four steps:

1. Diagramming the system;

2. Assigning per-unit values;

3. Reducing the system diagram to its Thevenin Equivalent, a single equivalent reactance;

4. calculating the symmetrical and asymmetrical fault currents.

Examples given here detail only the first and third steps; the other two will be covered in subsequent articles in our series.
Defining the system
The first step in calculating short-circuit currents is the preparation of a simplified one-line diagram of the system. Determining what to include or omit from this diagram requires judgment based on experience. Because all current-carrying system components have some impedance, calculated fault current will increase when any are omitted. However, the impedance of many components is so small that, for the average system, omitting them will change the calculated current by an insignificant amount.
To ensure an accurate calculation, it is necessary to include all transformers, cables more than a few feet in length, busway, generators and motors in the system. In larger systems that have very high fault currents (very low total impedance), even small additions can reduce available current significantly. In such systems, impedance of circuit breakers, fused switches and other circuit-protective devices should be included.
Describing systems via diagrams
Figure 9.1 illustrates a simple radial system, from its single-line representation to a Thevenin Equivalent of the system. To simplify the interpretation, the circuit-protective devices have been omitted. The single-line diagram in 9.1(a) shows the system includes a generator supplying two motors and a transformer. The fault is assumed to be on the secondary side of the transformer.
In 9.1(b), the impedance diagram, the generator and motors are represented by reactances and a voltage source; the cable and transformer are represented by impedances. The values of reactances and impedances used in the calculations are in per-unit ohms, according to the methods developed in

of this series. Actual values to be used will be discussed in the next several articles.
The modified impedance diagram in 9.1(c) has all sources connected to the infinite bus (sometimes called the zero reactance bus) which is the reference point for all source reactances and impedances. Note that the fault point is at the farthest distance from the infinite bus. That is, fault current flows from all sources of fault current, through all reactances and impedances (dividing into as many paths as are available), finally concentrating in the fault path. The current through the fault path is, therefore, the fault current of the system at the point of fault.
A simplified impedance diagram is shown in 9.1(d). The parallel combination of generator reactance, motor 1 reactance and motor 2 reactance has been reduced to a single reactance by utilizing the procedure of equation 9 in

.
The final impedance diagram is given in 9.1(e). The Thevenin Equivalent has been obtained by adding the series combination of the remaining reactances and impedances of the system using equation 7 from

. The per-unit fault current is then simply the per-unit voltage divided by the per-unit equivalent impedance.

Calculating impedance
The reactances and impedances have been lumped loosely together in this discussion. In many cases, especially in computer programs, the impedance (resistance and reactance) will be fully calculated. This will ensure that more accurate values will be realized, and an X/R ratio may be calculated to permit determination of asymmetrical fault current.
In some situations, and at certain voltages, the resistance is often neglected, especially when approximations to the fault current are desired. Rule-of-thumb methods addressing these approximations will be presented later in the series.
Figure 9.2 illustrates how determining fault current at multiple locations in a system requires different modified impedance diagrams; a separate diagram must be made for each point being studied. For hand calculations, this can be a very time-consuming and error-prone task; with the digital computer such calculations are made quick and accurately, and an entire system may be expeditiously investigated.

Figure 9.3 shows how a simple loop system may be evaluated. The single-line diagram appears in 9.3(a). Again, protective devices have not been shown in order to simplify the discussion.
A modified impedance diagram with the infinite bus is shown in 9.3(b). The elements have been rearranged in 9.3(c), with the centermost group of reactances (impedances) circled.

The conversion from a delta to a wye for the encircled portion of the diagram is shown in 9.3(d). This conversion This conversion is made using formula 10 in

.
The diagram in 9.3(e) is the same as 9.3(c), but with the modifications of 9.3(d) for the delta-to-wye conversion. In 9.3(f), the series elements have been combined. And, further combination of parallel and series elements leads to the Thevenin Equivalent in 9.3(g).
Actual numbers will be applied to these examples in future articles. The next article will begin the data book of typical reactances and impedances to be used in short-circuit calculations.
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