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customary to express the comparative conductivity of samples of copper in terms of Matthiessen's standard. Unfortunately, Matthiessen gave several values which do not agree exactly. To overcome this difficulty, a committee of the American Institute of Electrical Engineers carefully investigated this question, and recommended the general acceptance of a definite value for Matthiessen's standard. It is possible to obtain copper of greater purity and higher density than that used by Matthiessen, so that a somewhat better conductivity of 102 or even 105 per cent of the standard may be reached. But practically all commercial copper, is below Matthiessen's standard.
A complete table was prepared by the committee of the American Institute of Electrical Engineers,* giving the resistance, weight, etc., of the various sizes of wire, for both the American (B. and S.) and Birmingham (Stubs) gauges. This table, for the A. W. G. and covering wires, from Nos. 0000 to 18 inclusive, is given on page 8. The resistance of any copper wire at 20 degrees C. or 68 degrees Fahr., according to Matthiessen's standard,
may be calculated by the following simple formula: R = —
R being the resistance in international ohms, / the length of the wire in feet, and d its diameter in mils. The latter is easily determined with accuracy by means of the ordinary screw micrometer.
A very simple and convenient rule to remember is the fact that 1,000 feet of No. 10 A. W. G. wire, which is practically onetenth of an inch in diameter (.1019), has 1 ohm resistance at 20 degrees C. (68 degrees Fahr.), and weighs 31.4 pounds. A wire three sizes larger, that is, No. 7, has almost exactly twice the cross-section and weight per thousand feet, and one-half the resistance. A wire three sizes smaller, that is, No. 13, has one-half the cross-section and weight, and twice the resistance per thousand feet. Three sizes smaller than No. 13, that is, No. 16, has onefourth the cross-section and weight and four times the resistance of No. 10; and similarly a No. 4 wire has four times the crosssection and weight and one-fourth the resistance of No. 10. This may be carried to the extreme limits of the gauge in either direction, the cross-section doubling with each three numbers. Intermediate sizes may be approximated by interpolation; for example,
* Transactions, vol. x., 1893.
one size larger has about li (1.261) times, and two sizes larger about lfis (1.59) times the cross-section. The cross-section of the next size smaller is always found by multiplying by .793, and for two sizes smaller by .629.
Temperature Coefficient of Copper. — The effect of variations in temperature upon the conductivity of copper was given by Matthiessen * in the following formula : —
C, = C0 (1 - .0038701/ + .000009009/2), (1)
in which C, is the conductivity at any temperature / in Centigrade degrees, and C0 is the conductivity at zero degrees C. This expression is sometimes converted into one for resistance by merely changing signs; but this is incorrect algebraically, since it is necessary to take the reciprocals of both members of the equation. If this is done the following formula is obtained : —
R, = R, (1 + .0038701/ + .000005969/2). . (2)
This is usually simplified by reducing the number of decimal places, giving the form : —
R, = Ra (1 + .00387/ + .00000597/2). (3)
In these expressions for resistance, terms containing /8 and other higher powers of / are neglected, hence, they do not give results agreeing exactly with Matthiessen's original formula (Equa. 1). The correct method is to find the value of the temperature coefficient for conductivity (1 ~ .0038701/ + .000009009/2), take its reciprocal, which gives the temperature coefficient for resistance, then multiply the resistance at 0° C. by this amount in order to find the resistance at the given temperature /. This is the process by which the figures in the table on page 8 were obtained. But for moderate ranges of temperature the error resulting from the use of Equation 3"is slight, being about \ of 1 per cent too high at 50° C, and about A of 1 per cent too high at 80° C.
Indeed, it is doubtful if any of these somewhat complicated formulae are actually more correct than the very simply expression, — f
R, = R0 (1 + .004/), (4)
* Philosophical Transactions, 1862.
in which, as before, Rt is the resistance of a copper conductor at the given temperature t, and R0 is its resistance at 0° C.
Matthiessen found nearly all pure metals to have substantially the same temperature coefficient as copper, the only important exceptions being iron and liquid mercury. The values given for these are somewhat variable, but are about .0045 for the former, and about .0009 for the latter. The temperature coefficients for alloys are less than those of pure metals, being only about onetenth as great for German silver as for copper. ,
The resistance in ohms of a soft copper conductor at a given temperature / in Centigrade degrees may be obtained from the following expression :—
This assumes the ordinary form of Matthiessen's formula (Equa. 3), and gives slightly different results from those set forth in the table on page 8, as already explained.
The very simple expression, —
gives values agreeing exactly with Matthiessen's original formula at 23° C, and not differing by more than two-tenths of one per cent between 0° and 35° C, which covers the ordinary temperature range of conductors used in electrical distribution. At 60° C, which is the usual heating limit allowed in electrical machinery, the results obtained from Equation 6 are three-quarters of one per cent less than those of Matthiessen. This expression is therefore sufficiently accurate for almost any practical calculation.
In fact, the variation in resistance of copper is so great with ordinary changes in temperature that it is rarely possible to predetermine it with great accuracy. The temperature of an overhead line may vary from natural causes enough to alter the resistance about 25 per cent. A further increase due to the heating effect of the current would make a total change of about 40 per cent in resistance. Underground and interior conductors are not subject to such extreme variations in the temperature of their environment, but they often amount to many degrees, particularly for the latter; and the heating effect of the current may be equally
great. Fortunately, however, in electrical transmission or distribution the current does not ordinarily heat wires more than 5 or 10 degrees, the loss of voltage being usually the controlling factor. Under the head of current capacity the rise in temperature of the various kinds of conductors will be discussed later.
In any given case the probable temperature around a conductor and the rise due to the current can be at least approximately determined, so that resistance calculations can be made accordingly.
It is customary to specify in plans and contracts that copper for electrical purposes shall have a conductivity not less than 98 per cent of Matthiessen's standard. In some cases only 96 per cent is required; but it should not be allowed to fall below this limit, as it is perfectly practicable to obtain copper of such quality, and inferior grades would not be enough cheaper to make up for their lower conductivity. Allowance should be made for this fact in calculating the resistance of conductors.
The Drop, or Lost Pressure in Volts. —This is the first effect of resistance in electrical distribution, and is very easily and definitely determined from Ohm's law by changing its ordinary form / = EjR into E = IR. This is not only true of the whole circuit, but also applies to any portion or branch of the circuit; and ordinarily it is far simpler and more likely to avoid errors if each part of the circuit is considered separately. In the case of a very complicated electrical system, it would be practically out of the question to treat the circuit as a whole; but it is always possible to divide the system of conductors into separate lengths, in each of which we can determine the current, the resistance, and therefore the fall of potential which takes place. In most practical work the current in amperes is given, since it is usually known how many lamps or how much power are to be supplied. It then becomes necessary to calculate the value of the resistance in order to have the proper value for the drop, the latter being assumed or fixed by the conditions in each case. The common idea that a short conductor of very large diameter has no appreciable resistence is quite fallacious. Tor example, a bar of copper one foot long and one inch in diameter has about one hundred-thousandth of an ohm resistance. While this may be a negligible amount in most cases, it is always perfectly definite, and is often quite appreciable. Such a rod would carry one thousand amperes with a drop of one hun