Voltage, Current, Ohm’s Law, and Resistance

12-15 Direction of current S N 12-16 S N N S conductor that ultimately moves the electrons in a flow. The symbol for emf is the capital letter “E.” Across the terminals of the typical aircraft battery, voltage can be measured as the potential difference of 12 volts or 24 volts. That is to say that between the two terminal posts of the battery, there is an emf of 12 or 24 volts available to push current through a circuit. Relatively free electrons in the negative terminal move toward the excessive number of positive charges in the positive terminal. Recall from the discussion on static electricity that like charges repel each other but opposite charges attract each other. The net result is a flow or current through a conductor. There cannot be a flow in a conductor unless there is an applied voltage from a battery, generator, or ground power unit. The potential difference, or the voltage across any two points in an electrical system, can be determined by: Where E = E Q E = Potential difference in volts E = Energy expanded or absorbed in joules (J) Q = Charge measured in coulombs current. Two interconnected water tanks demonstrate that when a difference of pressure exists between the two tanks, water flows until the two tanks are equalized. The illustration shows the level of water in tank A to be at a higher level, reading 10 psi (higher potential energy) than the water level in tank B, reading 2 psi (lower potential energy). Between the two tanks, there is 8-psi potential difference. If the valve in the interconnecting line between the tanks is opened, water flows from tank A into tank B until the level of water (potential energy) of both tanks is equalized. It is important to note that it was not the pressure in tank A that caused the water to flow; rather, it was the difference in pressure between tank A and tank B that caused the flow.

This comparison illustrates the principle that electrons move, when a path is available, from a point of excess electrons (higher potential energy) to a point deficient in electrons (lower potential energy). The force that causes this movement is the potential difference in electrical energy between the two points. This force is called the electrical pressure or the potential difference or the electromotive force (electron moving force).

Current

Electrons in motion make up an electric current. This electric current is usually referred to as “current” or “current flow,” no matter how many electrons are moving. Current is a measurement of a rate at which a charge flows through some region of space or a conductor. The moving charges are the free electrons found in conductors, such as copper, silver, aluminum, and gold. The term “free electron” describes a condition in some atoms where the outer electrons are loosely bound to their parent atom. These loosely bound electrons can be easily motivated to move in a given direction when an external source, such as a battery, is applied to the circuit.

These electrons are attracted to the positive terminal of the battery, while the negative terminal is the source of the electrons. The greater amount of charge moving through the conductor in a given amount of time translates into a current. Current = Charge Time or I = Q t Where: I = current in amperes (A) Q = charge in coulombs (C) T = time 12-17 A B The System International (SI) unit for current is the ampere (A), where 1A = 1 C s One ampere (A) of current is equivalent to 1 coulomb (C) of charge passing through a conductor in 1 second. One coulomb of charge equals 6.28 billion electrons. The symbol used to indicate current in formulas or on schematics is the capital letter “I.” When current flow is one direction, it is called direct current (DC). Later in the handbook, the form of current that periodically oscillates back and forth within the circuit is discussed. The present discussion is only concerned with the use of DC.

The velocity of the charge is actually an average velocity and is called drift velocity. To understand the idea of drift velocity, think of a conductor in which the charge carriers are free electrons. These electrons are always in a state of random motion similar to that of gas molecules. When a voltage is applied across the conductor, an emf creates an electric field within the conductor and a current is established. The electrons do not move in a straight direction but undergo repeated collisions with other nearby atoms. These collisions usually knock other free electrons from their atoms, and these electrons move on toward the positive end of the conductor with an average velocity called the drift velocity, which is relatively a slow speed. To understand the nearly instantaneous speed of the effect of the current, it is helpful to visualize a long tube filled with steel balls as shown in of the tube, which represents the conductor, will immediately cause a ball to be emitted at the opposite end of the tube.

Thus, electric current can be viewed as instantaneous, even though it is the result of a relatively slow drift of electrons.

Ohm’s Law (Resistance)

The two fundamental properties of current and voltage are related by a third property known as resistance. In any electrical circuit, when voltage is applied to it, a current results. The resistance of the conductor determines the amount of current that flows under the given voltage. In most cases, the greater the circuit resistance, the less the current. If the resistance is reduced, then the current increases. This relation is linear in nature and is known as Ohm’s Law. By having a linearly proportional characteristic, it is meant that if one unit in the relationship increases or decreases by a certain percentage, the other variables in the relationship increase or decrease by the same percentage. An example would be if the voltage across a resistor is doubled, then the current through the resistor doubles. It should be added that this relationship is true only if the resistance in the circuit remains constant. If the resistance changes, current also changes. A graph of this relationship is shown in relationship between voltage and current in this example shows voltage plotted horizontally along the X axis in values from 0 to 120 volts, and the corresponding values of current are plotted vertically in values from 0 to 6.0 amperes along the Y axis. A straight line drawn through all the points where the voltage and current lines meet represents the equation I = E⁄20 and is called a linear relationship.

If E = 10 V Then 10 V 20 Ω = 0.5 A If E = 60 V Then 60 V 20 Ω = 3 A If E = 120 V Then 120V 20 Ω = 6 A Ohm’s Law may be expressed as an equation, as follows: Equation 1 I = E R I = current in amperes (A) E = voltage (V) R = resistance (Ω) Where I is current in amperes, E is the potential difference measured in volts, and R is the resistance measured in ohms. 12-18 If any two of these circuit quantities are known, the third may be found by simple algebraic transposition. With this equation, we can calculate current in a circuit if the voltage and resistance are known. This same formula can be used to calculate voltage. By multiplying both sides of the equation 1 by R, we get an equivalent form of Ohm’s Law, which is: Equation 2 E = I (R) Finally, if we divide equation 2 by I, we solve for resistance, This relationship is true only if the resistance in the circuit remains constant. If the resistance changes, current also changes. A graph of this relationship is shown in relationship between voltage and current in this example shows voltage plotted horizontally along the X axis in values from 0 to 120 volts. The corresponding values of current are plotted vertically in values from 0 to 6.0 amperes along the Y axis. A straight line drawn through all the points where the voltage and current lines meet represents the equation I = E⁄20 and is called a linear relationship.

Equation 3 R = E I All three formulas presented in this section are equivalent to each other and are simply different ways of expressing Ohm’s Law. The various equations, which may be derived by transposing the basic law, can be easily obtained by using the triangles in Figure 12-40. The triangles containing E, R, and I are divided into two parts, with E above the line and I × R below it. To determine an unknown circuit quantity when the other two are known, cover the unknown quantity with a thumb. The location of the remaining uncovered letters in the triangle indicate the mathematical operation to be performed. For example, to find I, refer to Figure 12-40A, and cover I with the thumb. The uncovered letters indicate that E is to be divided by R, or I = E⁄R. To find R, refer to Figure 12-40B, and cover R with the thumb. The result indicates that E is to be divided by I, or R = E⁄I. To find E, refer to Figure 12-40C, and cover E with the thumb. The result indicates I is to be multiplied by R, or E = I × R. This chart is useful when learning to use Ohm’s Law.

It should be used to supplement the beginner’s knowledge of the algebraic method. Resistance of a Conductor While wire of any size or resistance value may be used, the word “conductor” usually refers to materials that offer low resistance to current flow, and the word “insulator” describes materials that offer high resistance to current. There is no distinct dividing line between conductors and insulators; under the proper conditions, all types of material conduct some current. Materials offering a resistance to current flow midway between the best conductors and the poorest conductors (insulators) are sometimes referred to as “semiconductors,” and find their greatest application in the field of transistors.

The best conductors are materials, chiefly metals, which possess a large number of free electrons; conversely, insulators are materials having few free electrons. The best conductors are silver, copper, gold, and aluminum; but some nonmetals, such as carbon and water, can be used as conductors. Materials such as rubber, glass, ceramics, and plastics are such poor conductors that they are usually used as insulators. The current flow in some of these materials is so low that it is usually considered zero. The unit used to measure resistance is called the ohm. The symbol for the ohm is the Greek letter omega (Ω). In mathematical formulas, the capital letter “R” refers to resistance. The resistance of a conductor and the voltage applied to it determine the number of amperes of current flowing through the conductor. Thus, 1 ohm of resistance limits the current flow to 1 ampere in a conductor to which a voltage of 1 volt is applied.

Factors Affecting Resistance 1. The resistance of a metallic conductor is dependent on the type of conductor material. It has been pointed out that certain metals are commonly used as conductors because of the large number of free electrons in their outer orbits. Copper is usually considered the best available conductor material, since a copper wire of a particular diameter offers a lower resistance to current flow than an aluminum wire of the same diameter. However, aluminum is much lighter than copper, and for this reason, as well as cost considerations, aluminum is often used when the weight factor is important.

2. The resistance of a metallic conductor is directly proportional to its length. The longer the length of a given size of wire, the greater the resistance. lengths. If 1 volt of electrical pressure is applied across 12-19 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 10 20 30 40 50 60 70 80 90 100 110 120 Volts (E) Amperes (I) R = 20 Ohms (Constant) E I X R E I X R E I X R A B C To find I (amperes), place thumb over I and divide E by R as indicated. To find R (ohms), place thumb over R and divide as indicated. To find E (volts), place thumb over E and multiply as indicated. the two ends of the conductor that is 1 foot in length, and the resistance to the movement of free electrons is assumed to be 1 ohm, the current flow is limited to 1 ampere. If the same size conductor is doubled in length, the same electrons set in motion by the 1 volt applied now find twice the resistance; consequently, the current flow is reduced by one-half.

3. The resistance of a metallic conductor is inversely proportional to the cross-sectional area. This area may be triangular or even square, but is usually circular. If the cross-sectional area of a conductor is doubled, the resistance to current flow is reduced in half. This is true because of the increased area in which an electron can move without collision or capture by an atom. Thus, the resistance varies inversely with the cross-sectional area of a conductor. 4. The fourth major factor influencing the resistance of a conductor is temperature. Although some substances, such as carbon, show a decrease in resistance as the ambient (surrounding) temperature increases, most materials used as conductors increase in resistance as temperature increases. The resistance of a few alloys, such as constantan and Manganin™, change very little as the temperature changes. The amount of increase in the resistance of a 1 ohm sample of a conductor, per degree rise in temperature above 0° Centigrade (C), the assumed standard, is called the temperature coefficient of resistance. For each metal, this is a different value.

For example, for copper the value is approximately 0.00427 ohm. Thus, a copper wire having a resistance of 50 ohms at a temperature of 0 °C has an increase in resistance of 50 × 0.00427, or 0.214 ohm, for each degree rise in temperature above 0 °C. The temperature coefficient of resistance must be considered where there is an appreciable change in temperature of a conductor during operation. Charts listing the temperature coefficient of resistance for different materials are available. Figure 12-42 shows a table for “resistivity” of some common electric conductors. The resistance of a material is determined by four properties: material, length, area, and temperature. The first three properties are related by the following equation at T = 20 °C (room temperature): (ρ × 1) AR = Where R = resistance in ohms ρ = resistivity of the material in circular mil-ohms per foot 12-20 + 2 feet 0.5 amp ( 2 ohms) 1 foot 1 amp (1 ohm) Conductor Material Resistivity (ohm meters @ 20 °C) Silver 1.64 × 10-8 Copper 1.72 × 10-8 Aluminum 2.83 × 10-8 Tungsten 5.50 × 10-8 Nickel 7.80 × 10-8 Iron 12.0 × 10-8 Constantan 49.0 × 10-8 Nichrome II 110 × 10-8 l = length of the sample in feet A = area in circular mils Resistance and Relation to Wire Sizing Circular Conductors (Wires/Cables) Because it is known that the resistance of a conductor is directly proportional to its length, and if we are given the resistance of the unit length of wire, we can readily calculate the resistance of any length of wire of that particular material having the same diameter. Also, because it is known that the resistance of a conductor is inversely proportional to its cross-sectional area, and if we are given the resistance of a length of wire with unit cross-sectional area, we can calculate the resistance of a similar length of wire of the same material with any cross-sectional area. Therefore, if we know the resistance of a given conductor, we can calculate the resistance for any conductor of the same material at the same temperature. From the relationship: (ρ × 1) AR = It can also be written: R1 R2 = 11 12 = A1 A2 If we have a conductor that is 1 meter (m) long with a cross- sectional area of 1 (millimeter) mm2 and has a resistance of 0.017 ohm, what is the resistance of 50 m of wire from the same material but with a cross-sectional area of 0.25 mm2?

R1 R2 = 11 12 = A1 A2 R 2 = 0.017 Ω × 50 m 1 m × 1 mm2 0.25 mm2 = 3.4 Ω While the SI units are commonly used in the analysis of electric circuits, electrical conductors in North America are still being manufactured using the foot as the unit length and the mil (one thousandth of an inch) as the unit of diameter. Before using the equation R = (ρ × l)⁄A to calculate the resistance of a conductor of a given American wire gauge (AWG) size, the cross-sectional area in square meters must be determined using the conversion factor 1 mil = 0.0254 mm. The most convenient unit of wire length is the foot. Using these standards, the unit of size is the mil-foot. Thus, a wire has unit size if it has a diameter of 1 mil and length of 1 foot.

In the case of using copper conductors, we are spared the task of tedious calculations by using a table as shown in the table are such that each decrease of one gauge number equals a 25 percent increase in the cross-sectional area. Because of this, a decrease of three gauge numbers represents an increase in cross-sectional area of approximately 2:1. Likewise, change of ten wire gauge numbers represents a 10:1 change in cross-sectional area—also, by doubling the cross- sectional area of the conductor, the resistance is cut in half. A decrease of three wire gauge numbers cuts the resistance of the conductor of a given length in half.

Rectangular Conductors (Bus Bars) To compute the cross-sectional area of a conductor in square mils, the length in mils of one side is squared. In the case of a rectangular conductor, the length of one side is multiplied by the length of the other. For example, a common rectangular bus bar (large, special conductor) is 3⁄8 inch thick and 4 inches wide. The 3⁄8-inch thickness may be expressed as 0.375 inch. Since 1,000 mils equal 1 inch, the width in inches can be converted to 4,000 mils. The cross-sectional area of the rectangular conductor is found by converting 0.375 to mils (375 mils × 4,000 mils = 1,500,000 square mils).

Power and Energy

Power in an Electrical Circuit This section covers power in the DC circuit and energy consumption. Whether referring to mechanical or electrical systems, power is defined as the rate of energy consumption