# The Art of Protecting Electrical Systems, Part 10: Assigning Impedance Values

10/17/2006

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 10 continues the series with a discussion of calculating the effects of fault current by assigning resistance and reactance values to an electrical system.

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.

In Part 9 of this series, we introduced methods of diagramming electrical systems and reducing a diagram to its Thevenin Equivalent. Here, we continue the study by considering a simple system, assigning impedance values to its components and determining the fault current at representative locations.

Figure 1 illustrates a portion of a radial electrical distribution system, with all components in a series arrangement. To simplify procedures for this example, it is assumed no rotating machinery is installed in the system, hence, the single source of fault current is the utility connection.

In the figure, the one-line diagram on the left shows all significant items of equipment in the system. The impedance diagram of the system is to the right of the one-line diagram.

Impedance identified
Each item of equipment is represented by an impedance, each numbered so the calculations presented here may be indexed. (In the typical short-circuit current calculation, the connections between the impedances are numbered, rather than the impedances themselves. This node-notation facilitates identification of faults. In this example, the first fault is assumed to be at the node between impedances 6 and 7.) Each impedance is identified by a phrase defining the type and size of the equipment for ease in following the calculations.

Alongside the impedance diagram is a resistance and reactance chart.This shows the operating voltage of the various components and the reactance and resistance (in ohms) for each item of equipment. These are the values developed in the calculations we will describe later.

If the resistance and reactance values were at the same operating voltage, the total resistance and reactance to the point of fault could be simply added as an impedance:

Z =
R2 + X2

and then solved for fault current by:

I = E / Z

However, the matter is complicated by the various voltages employed on different projects so that proper multiplying factors must be applied to the resistance and reactance at each voltage level. Step-by-step calculations for both the first and second fault points are illustrated in Formula Box A and Formula Box B. In each case, the source of the equation is given in the calculations. If no source preference is given, the equation is being mentioned for the first time.

Per-unit vs. ohmic method
Calculations for the fault current at the second fault point—at 208 volts—can be made and compared with the values determined by the simpler per-unit method. (The preferred method of calculating short-circuit currents is the per-unit method, described in Part 8 .) It is preferred because the ohmic method involves conversion of values between one voltage base and another and the very small numbers may introduce inaccuracies. With the former method, all resistances and reactances are reduced to per-unit at a standard base kVA and base voltage.

These bases and calculated per-unit values of resistance and reactance are shown in the block on the right side of Figure 1. Also given here are the total values at the two fault points. This article illustrates the procedure used to assign resistance and reactance values to an electrical system where no rotating machinery is employed. It describes how fault currents may be calculated using both the ohmic and per-unit methods for this type of system. The next article in our series will develop the procedures used when rotating machinery is installed in the system.

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