Inductive reactance

Electrical resistance is not the only property of materials that resists the flow of current. Let us consider an experiment. Let’s purchase a 12,000-foot spool of insulated 20 AWG copper wire commonly used as communications wire and pull the wire off the spool and lay it out on the ground. If we take the two ends and connect them to a male electrical plug and plug it into a 120 V AC source, we would expect the current magnitude that would flow in the wire to be equal to the source voltage divided by the resistance of the wire. Since the resistance of 20 AWG copper wire is 10.1W/ 1000 ft at a temperature of 20°C, then 12,000 feet would have a resistance 12 x 10.1 or 121.2W. From Ohm’s law, the current we would expect to see would be the voltage divided by the resistance, 120 / 121.2 = 0.9900 amp. If we take the same wire and wrap it tightly around a 4-inch diameter solid rod of iron and then energize it from the 120-volt source, as before, we would find that the current is considerably less. If the current is less, then the resistance must have increased. If the current is now 0.7682 amp, the resistance must now be equal to the voltage divided by the current or 120 / 0.7682 = 156.21W. How did the resistance increase from 121.2 to 156.21? It didn’t. By wrapping the wire around the iron rod, we introduced another form of resistance to current flow into the circuit. This second form is called inductive reactance. Inductive reactance is created any time we wrap wire in a coil. The current in the coiled wire creates a magnetic field around the coil. The inductive reactance is particularly high when the space inside the coil is filled with iron. Electrical resistance in a conductor resists the flow of current by generating heat. Inductive reactance resists the flow of current by generating a magnetic field. The symbol we use in calculations to represent inductive reactance is XL. The unit of measure is the same as for electrical resistance, the ohm symbolized byW.

Impedance

In our voltage-drop calculations we performed as part of this series of articles, to calculate the current in the circuit, we divided the source voltage by the total resistance in the circuit. The total resistance in the circuit was the sum of the resistance of each element of the circuit, the service wire, the wire within the house, the extension cord, and the load. Resistance elements that are in series can be added to determine the total resistance. When we deal with resistance and inductive reactance in the same circuit, the current in the circuit is the source voltage divided by the total impedance. Impedance is used in place of total resistance because what impedes the flow of current is a combination of resistance and inductive reactance. The term impedance is also used because resistance and inductive reactance cannot be added together. For calculation purposes, the symbol for impedance is Z. To calculate the total impedance, we have to refer to what we call the impedance triangle. Resistance forms one side of the triangle, inductive reactance forms the other side of the triangle, and the total impedance is the hypotenuse. The Pythagorean theorem covers the relationship between resistance, inductive reactance, and impedance; the square of the hypotenuse is equal to the sum of the squares of the other two sides. Z^{2}= R^{2}+ X_{L}^{2} If we solve for Z we get:

Z = square root (R^{2}+ X_{L}^{2})

In our experiment, we found the impedance of the 12,000 feet of wire wrapped around the iron rod is equal to the source voltage divided by the current, 120 / 0.7682 = 156.21W. Since the resistance of the wire is 121.2W, and the total impedance of the wire is 156.21W, the inductive reactance must be:

XL = square root (Z^{2}– R^{2})

XL = 98.54W

Electric Motors

In general, the amount of power consumed by an electric motor is a function of the rated horsepower of the motor and how much mechanical load is placed on the motor. A motor with 100 percent efficiency would draw 746 watts of power for every horsepower of mechanical load placed on the motor. A motor with a more realistic efficiency of 80 percent would draw almost 900 watts of power for every horsepower of mechanical load placed on it. When motors run without load, they draw anywhere from 30% to 70% of their full load power. When motors are first energized, some draw several times their full load power in the form of a current surge until the motor is up to speed. For example, my hand-held circular saw draws 22 amps for about a second when it is first turned on. After the initial surge of current, it draws 9 amps. When I’m cutting through a 4 by 4 of treated lumber, it draws 13 amps. In general, electric motors have high levels of inductive reactance because the magnetic field is a very important part of motors. When my saw is cutting through the 4 by 4, the resistance of the saw is 7.384 W and the inductive reactance 5.538 W. The total impedance is:

Z = square root (R^{2}+ X_{L}^{2})

Z = 9.23W

Power

When we looked at conductors and loads with little or no inductive reactance, we only considered the resistance, and we calculated the power consumed by the resistance by multiplying the voltage across the resistance times the current flowing through the resistance. Now that we are looking at a load that has both resistance and inductive reactance, we have to consider other forms of power. The power associated with the heat generated by current flowing through a resistance is called real power. It is calculated as we did before by multiplying the voltage across the resistance times the current flowing through the resistance. The symbol for real power is P. The unit of measure is watt. The abbreviation for watt is W. The power associated with inductive reactance, which maintains the magnetic field within the motor, is called reactive power. The symbol for reactive power is Q. The unit of measure is var. There is no abbreviation. The combination of real power and reactive power is called apparent power. The symbol for apparent power is S. The unit of measure is volt-amperes. The abbreviation is VA. To calculate the apparent power, we have to refer to what we call the power triangle. Real power forms one side of the triangle, reactive power forms the other side of the triangle, and the apparent power is the hypotenuse. The Pythagorean theorem covers the relationship between real power, reactive power, and apparent power; the square of the hypotenuse is equal to the sum of the squares of the other two sides. S^{2}= P^{2}+ Q^{2}. If we solve for S we get:

S = square root (P^{2}+ Q^{2})

The apparent power can also be calculated by multiplying the voltage times the current. For my saw, the apparent power is 120 x 13 = 1560 VA. The real power is I^{2}x R or 13 x 13 x 7.384 = 1248W. The reactive power is I^{2}x X_{L}or 13 x 13 x 5.538 = 935.9 var. The term power factor is commonly used to measure how much inductive reactance is in a load. Power factor is abbreviated PF and is calculated by dividing the real power by the apparent power. For my saw, the PF is 1248 / 1560 = 0.8, commonly referred to as 80% power factor. Another way of calculating PF is to divide the load resistance by the load impedance. For my saw, the PF is 7.384 / 9.23 = 0.80 or 80%.

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