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Chapter 9—Wind Power Plants 9–41 The last column in Table C.1 shows the temperature coefficient of resistance, α, whichis the rate at which the resistance changes with temperature. In the case of metals, the change in resistance is positive; that is, the resistance increases with the rise in temperature. This change is very small for special alloys such as Manganin (composed of copper, manganese, and nickel). There are other materials, such as carbon, electrolytes, gaseous arcs, and ceramic materials, that possess a negative temperature coefficient, which means that the resistance decreases with the rise in temperature. The usual explanation for this phenomenon is that the electron current carriers in a metal experience more collisions in a thermally excited lattice, thus increasing the resistance. On the other hand, a rise in temperature in electrolytes and conducting gases results in the presence of more ions serving as carriers, thus tending to increase the current for a given potential difference, which is a decrease in resistance. The temperature coefficients change somewhat with temperature, but these values in Table C.1 can be considered reasonably accurate over the temperature range of −50 o C to 100 o C, which includes most cases of interest. If the resistance of a wire is R 1 at temperature T 1 , the resistance R 2 at temperature T 2 is given by R 2 = R 1 [1 + α(T 2 − T 1 )] (C.2) Example If the resistance of a coil of copper wire is 10 Ω at 20 o C, what is the resistance at 50 o C? R 2 = 10[1 + 0.00393(50 − 20)] = 11.18 Ω The resistance has increased almost 12% with a 30 o C increase in temperature. This shows that if precise values of electrical losses are required, one must use the actual temperature of the wire in calculating the resistance. Only two metals are widely used as electrical conductors, copper and aluminum, so we will focus our attention on them. Conductors are made in many different sizes to meet needs for a given current carrying ability. If the diameter is 460 mils (0.46 inch) or smaller, the size is specified by a number called the American Wire Gauge (AWG). If the diameter is 500 mils or larger, the size is specified by the wire area expressed in thousands of circular mils, kcmil. A partial list of wire sizes is given in Table C.2, along with the diameter, area, and resistance per 1000 ft at a temperature of 20 o C. The odd sizes are omitted for wires of 3 gauge through 29 gauge, since they are rarely used. The very small sizes are also omitted since their fineness makes them difficult to handle. If intermediate values are needed, one can use the fact that the areas of two adjacent wire gauges are always in a constant ratio with each other, for the AWG portion of the table. For example, the ratio of areas of 4/0 and 3/0 wires is 211.6/167.8 = 1.261, and likewise for all Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001
Chapter 9—Wind Power Plants 9–42 the other adjacent sizes. For wires that are two gauge sizes apart, like 6 gauge and 4 gauge, the ratio is 41.74/26.25 = 1.590 = 1.261 2 . A span of 10 gauge numbers yields a change of area of (1.261) 10 = 10.166 or approximately a factor of 10. The resistance is inversely proportional to area, so if the resistance of a 10 gauge copper wire is about 1 Ω per 1000 ft, then the resistance of a 20 gauge wire will be about 10 Ω per 1000 ft. The selection of a value of 1.261 for the ratio between areas of adjacent gauge sizes means that the potential benefit of having diameters expressed in integers has been lost, except for the 250 and 1000 kcmil conductors. It could have been done in a manner similar to resistor sizes, where each standard value is about 10 % larger than the previous size, but rounded off to two digits. But it is too late to make such a choice now, so we must learn to live with the values given in the table. There are two different constraints which must be considered in selecting a wire size. First, the I 2 R loss in the wire must not result in a temperature rise in the wire that will damage the insulation. Second, the voltage drop in the wire must not excessively lower the load voltage. We will illustrate the second constraint with a short example. Example The new well at your country home is 400 ft from your house. The 120-V single-phase induction motor driving the pump draws 30 A of starting current and 10 A of running current. The starting current flows only for a few seconds at most, so is not a factor in determining the wire temperature. You are told that 12 gauge copper wire is rated for 20 A service in residential wiring. This is twice the running current for this situation, which seems like an ample safely margin for thermal considerations. But you want to check the voltage drop also. The current has to flow to the well and back to the house, so the total length of conductor is 2(400) = 800 ft. The resistance from Table C.2 is R = 800 1.588 = 1.270 Ω 1000 The voltage drop while the starting current is flowing is V drop = IR = 30(1.27) = 38.1 V The voltage available to the pump motor is only 120 - 38.1 = 81.9 V during the starting interval. Single-phase motors do not have very good starting characteristics, and may not start under load if the terminal voltage is less than about 90% of rated, or 108 V for the 120 V system. The proposed wire size is too small. We see that selecting wire sizes based on thermal considerations alone may easily lead to situations where performance is poor due to voltage drop. In general, we have two choices in such situations. We can increase the voltage rating or we can increase the wire size (or both). A 240 V pump motor costs about the same as a 120 V motor, but draws only half the current for the same power. Insulation on most 12 gauge wire is rated at 600 V, so that is not Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001
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