Thinking Like an Engineer An Active Learning Approach

Transcription

1 GlOBAl EDITION Thinking Like an Engineer An Active Learning Approach THIRD EDITION Stephan Bowman Park Sill Ohland

2 Thinking Like an engineer

3 242 Chapter 8 UNiVErSal UNiTS 8.11 electrical concepts learn to: describe the relationship between electric charge and electric current Use relationships between common electrical properties and their units to facilitate problem solution The basic concepts of electricity and electrical devices are perhaps less familiar to most students than are many of the other physical phenomena covered previously in this text. This is due partly to lack of practical experience, and partly to the fact that in general these phenomena are themselves invisible, only their effects being perceptible by people. These effects range from receiving an electric shock to the almost magical performance of touch-screen devices. Table 8-14 summarizes the concepts discussed in this section. Warning: Electrical parameters can get confusing at times since V is used as both a variable name for voltage and for the unit volt, and C is used as a variable name for capacitance as well as the unit coulomb. Table 8-14 summary of electrical properties Property symbol related equations typical units equivalent units Q Charge Q 1 Q 2 Coulomb s Law: = k e coulomb [C] C = A s r 2 Current I Q = I t ampere [A] A = C>s Voltage V volt [V] V = J>C Resistance R Ohm s Law: V = I R ohm [Ω] = V>A Conductance G G = 1>R siemens [S] S = A>V Electric power P P = V I = I 2 R = V 2 >R watt [W] W = V A Capacitance C Q = C V E C = 1 2 CV 2 farad [] = C>V Inductance L E L = 1 2 LI2 henry [H] H = V s>a How big is 6.24 * 10 18? This is estimated to be the number of stars found in 10 million galaxies the size of the Milky Way. electric Charge Electrons and protons, as well as some subatomic particles, have a property known as electric charge. On a small scale, charge can be measured in terms of the elementary charge (e). The magnitude of the elementary charge on either an electron or a proton is 1, and by convention, the charge of a proton is called positive (e = +1) and that of an electron negative (e = -1). A force acts on a charged particle when in the vicinity of another charged particle. If the charges are alike, both positive or both negative, the force is a repulsive force, and the charges tend to accelerate away from each other. If the charges are unlike, one positive and one negative, the force is attractive with the particles tending to accelerate toward each other. The value of the elementary charge (e) is inconveniently small, so charge (Q) is generally quantified using the derived unit coulomb [C]. A charge of one coulomb represents the total charge on approximately 6.24 * protons. Another way to say this: the elementary charge of a single electron is 1>6.24 * = 1.6 * C.

4 8.11 ElECTriCal CONCEpTS 243 note This is actually a specific case for two charges only. In general, there are more than two charges in threedimensional space and a three-dimensional vector representation is required. Such mathematics, however, is beyond the scope of this introductory course. You will study vectors in both your calculus courses and in your physics courses. These concepts will then be used in applications specific to your discipline. V I R The actual force exerted on a charged object varies with both the amount of charge on each object (Q 1 and Q 2 ) and the distance (r) between the charges. This relationship is defined by Coulomb s Law, named after the rench physicist Charles-Augustin de Coulomb ( ) who first described and quantified the attractive and repulsive electrostatic force. = k e Q 1 Q 2 r 2 In this equation, k e is Coulomb s constant, and is approximately equal to 9 * 10 9 N m 2 >C 2. electric Current Electric current is superficially analogous to a current of water or other fluid. Just as a current of water is a movement of water molecules in a pipe or channel, electric current is a movement of electric charge in a wire or other solid material. Electric current (I) is measured in amperes [A], one of the base units in the metric system, and is named for Andre-Marie Ampere, ( ), the rench physicist who is credited with discovering electromagnetism. The derived unit coulomb is defined in terms of the ampere as one ampere second: 1 C = 1 A s. This may be easier to understand on an intuitive level by rephrasing this as one ampere equals one coulomb per second: 1 A = 1 C>s. In other words, a current of one ampere represents a movement of one coulomb of charge past any given point in the wire every second. To put the magnitude of the ampere in context: for those who have received an electric shock by sticking your finger in a light socket, for instance, you realize the sensation is rather unpleasant. In general, in countries that use 120 volts in domestic appliances, such a shock is typically about 5 milliamperes [ma] or one two-hundredth of an ampere. This level of current is small enough that although unpleasant, your muscles will still respond to the commands from your brain, and you can release the live wire or pull your hand away. However, in circuits powering large appliances such as stoves or air conditioners that use 240 volts, the current from a typical shock is roughly twice the above value, or 10 ma. This is very close to the current level that will overload your nervous system so your muscles will no longer obey and you become unable to let go. This is AR more dangerous. When denoting a current on a circuit diagram, an arrow is used to indicate the assumed direction. If the current actually flows the other way, the numeric value will be negative. A couple of centuries ago when people were just beginning to seriously experiment with electricity, they hypothesized that there was a flow of some substance from one terminal of their primitive devices to the other. Perhaps needless to say, they knew nothing about electrons, since the structure of the atom was completely unknown at that time. They had a chance of correctly guessing the direction this substance flowed, but luck was against them and they got it wrong. or many decades, scientists assumed that this substance, called charge, flowed from the terminal they called positive (an excess of charge) to the other terminal they called negative (a deficiency of charge). Eventually, the structure of the atom was deciphered and scientists realized that for many years they had been working with the opposite assumption to the correct one, since in most situations it is electrons flowing from the negative terminal to the positive terminal. As a consequence, even to this day, most engineers solve problems using conventional current, that assumes charge flows from positive to negative. If you want the actual direction of the flow of electrons, you just multiply the current by minus one.

5 244 Chapter 8 UNiVErSal UNiTS voltage To really understand voltage requires knowledge of the concept of the electric field. This is unfortunately a bit too complicated for the limited time and space we have here, so we will merely attempt to help you develop a feel for how voltage affects other electrical parameters. You will study electric fields in some depth in physics, typically the second physics course, and may learn even more in other courses, particularly if you choose to study electrical or computer engineering. A somewhat inaccurate explanation of voltage, although one that can be useful in understanding it, is that voltage is what pushes the charges around to create current. In a sense, it quantifies the amount of force that can be exerted on an electric charge by other accumulated charge. In fact, some decades ago, voltage was commonly called electromotive force (EM), but this has fallen out of favor in most contexts for a variety of reasons, not least of which is that voltage is not a force, being dimensionally quite different. Voltage (V) is quantified using units of volts [V], and is a measure of how much work is required to move an electric charge in the vicinity of other electric charges. The unit of volt is named for Italian physicist Alessandro Volta, ( ), who possibly invented the first chemical battery, called a voltaic pile. One volt is defined as one joule per coulomb. In other words, if one joule of energy is required to move one coulomb of charge from one place to another, the voltage between those two points is one volt. Since work is required to move an electric charge in the vicinity of other charges, we might recall the definition of work in another context. Work equals force times the distance through which that force moves an object: W = d. Similarly, work equals charge times the difference in voltage through which that charge moves: W = Q V. One specific case of work is potential energy. If a force is used to raise a mass above the surface of the planet, the work done is stored as energy in the mass of the object being raised. When the object is dropped, the energy will convert form from potential energy to kinetic energy. Similarly, if a bunch of charge is moved closer to an accumulation of like charges, the work done to move the packet of charge is stored as energy in the packet of charge. If a path is provided for the packet of charge to move, the stored energy will be converted into another form, often heat, but it might also include light, sound, chemical energy, etc. Similar to the need to know whether an object is being lifted (storing potential energy) or is falling (converting the stored energy into kinetic energy), we need to know whether charges are moving toward like charges or away from them. Just like we use an arrow to denote the assumed direction of a current, we need some sort of notation to indicate the assumed polarity of a voltage which end is assumed to be more positive than the other. This allows us to keep track of whether energy is being stored or released. We do this by placing a plus sign on one side of the device through which the current is flowing and a negative sign on the other end.

6 8.11 ElECTriCal CONCEpTS ma Device 25 V We can talk about a voltage at a point, such as, the voltage at point A is 15 volts, but such statements are really based on some reference point, often the planet itself, so the statement is really equivalent to something like the voltage across (or between) point A and ground is 15 volts. I R V electrical resistance Resistance is a measure of how difficult it is to move charges through a material. In some substances, such as many metals, electrons can move quite easily. In other materials such as glass or air, considerable force, thus considerable voltage, is required to make electrons move therein. Resistance (R) is quantified using units of ohms [Ω], where one ohm is defined as one volt per ampere. or example, if a 1 volt battery were connected to a device having a resistance of one ohm, one ampere of current would flow through it, assuming the chemical reaction could replenish the charge rapidly enough to maintain such a current. The ohm is named for Georg Simon Ohm, the German physicist who first described the relationship linking voltage, current, and resistance. Resistance relates the voltage across a device to the current through the device. Take particular note of the choice of prepositions across and through. Understanding this choice will help you understand voltage and current. Electric current is the movement of charge, typically electrons moving THROUGH a substance. Voltage is to some extent a measure of the force being exerted on the moving charges by forces at either end of the device. This is where it can be a little confusing, particularly without using electric fields in the discussion. However, imagine that on each side of a device is an accumulation of charge, each exerting a force on the electrons inside the device. Each of those forces might be a push or a pull, and the total force on the electrons in the device is the difference in these forces. The difference in the forces from one side of the device to the other is referred to as the voltage ACROSS the device. As an analogy, if you are trying to push a sofa across the room, but someone else is trying to push the sofa in the opposite direction with the same force, the net force is zero and the sofa does not move. If one person pulls and the other pushes, however, the sofa will move quicker than with either person alone. Resistance is related to current and voltage by Ohm s Law: V = IR. Note the following implications of Ohm s Law: To maintain a specific current through a resistance requires a voltage proportional to the resistance. A larger resistance makes it harder to push the electrons through the device, thus a larger voltage is required. Similarly, current is inversely proportional to resistance. or a given voltage, if the resistance increases, the voltage cannot push as many electrons through the device per second, so the current must decrease. In some contexts, it is computationally simpler to use conductance instead of resistance. Conductance (G) is measured in siemens [S] and is simply the reciprocal of resistance: G = 1>R. An older unit for conductance that you might find, particularly in older references, is the mho (ohm spelled backwards) and is represented by an upsidedown omega []. The unit siemens is named for the German inventor Ernst Werner von Siemens who, among other things, built the first electric elevator and founded the company known today as Siemens AG.

7 246 Chapter 8 UNiVErSal UNiTS example 8-27 The voltage across a resistor is 15 volts, and the current through it is 6 milliamps [ma]. What is the value of the resistance? V = I R so R = V I = 15 V 1000mA 6 ma` A = 2500 V A` A V = 2500 = 2.5 k comprehension check 8-19 The current through a 12 kilo-ohms [kω] resistor is 25 microamps [µa]. What is the voltage across the resistor? A volt times an ampere is a watt. V = J>C A = C>s V A = J>s = W electric Power Conceptually, electric power is perhaps easiest to understand by examining the formula for gravitational potential energy. A mass has its potential energy increased by expending energy to lift it higher above the surface of the planet, since the mass of the object and the mass of the planet are mutually attracting each other. Similarly, forcing electrons closer to other electrons stores potential energy since they are mutually trying to repel each other. Recall our discussion of batteries. or each electron that is transferred to the negative terminal of the battery by the chemical reaction, a little bit of energy is stored in the battery. This is effectively electrical potential energy. The more electrons per second that are jammed together, the more energy per second is stored. Current is measured in charge (electrons) per second, power is proportional to current: P I or P = XI, where X is the proportionality constant. Now think back to voltage. Voltage is a measure of how much energy is used to move a given amount of charge: one volt is one joule per coulomb. Therefore, voltage is the proportionality constant and P = VI. example 8-28 A semiconductor diode has 500 millivolts [mv] across it and 700 microamps [µa] of current through it. How much power is the diode absorbing? 500 m P = VI = a V` 1V 700 b a ma` 1000 mv = 3.5 * 10-4 W = 350 mw 1A 1*10 6 ma b or resistors, the electrical power absorbed is usually converted to heat, and we can use Ohm s Law to replace either the voltage or the current in this power relationship: P = VI = (IR) I = I 2 R or P = VI = V (V>R) = V 2 >R Note that these two relationships expressed in terms of resistance are only valid for resistors, not for other electrical components. However, it gives us a means to quickly calculate the power absorbed by a resistor when we know only the voltage or current, but not both.