Introduction to DC Circuits
The field of electronics is very broad, and applies to many aspects of our everyday life. Every radio, television receiver, VCR, and DVD player is electronic in design and operation. So are modern microwave ovens and toaster ovens. Even conventional ovens now include elewctronic sensors and controls.
Beyond that, however, are even simpler devices that are still electronic in nature. For example, a recent development is the laser pointer, which is essentially a specialized flashlight — and both of these are rather basic electronic devices.
Nor are electronic devices all that new in most households. The telephone system, including standard telephones, is a widespread electronic network designed to be rugged and reliable, with only very simple electronic components. This has changed in more recent years, as more sophisticated electronic devices and methods have enabled improved performance, but the fundamental nature of the telephone system is still pretty much the same.
Basic Electronic Components
All components used in electronic circuits have three basic properties, known as resistance, capacitance, and inductance. In most cases, however, one of these properties will be far more prevalent than the other two. Therefore we can treat components as having only one of these three properties and exhibiting the appropriate behavior according to the following definitions:
Resistance
The property of a component to oppose the flow of electrical current through itself.
Capacitance
The property of a component to oppose any change in voltage across its terminals, by storing and releasing energy in an internal electric field.
Inductance
The property of a component to oppose any change in current through itself, by storing and releasing energy in a magnetic field surrounding itself.
As you might expect, components whose main property is resistance are called resistors; those that exhibit capacitance are called capacitors, and the ones that primarily have inductance are called inductors.
In this set of pages, we will examine each type of component. We will see how they are made and what basic properties they have. Then we will see how they behave when a fixed, dc voltage is applied to them, both by themselves and in combination with other types of components.
Once we see how they behave in response to dc voltages, another set of pages will explore how these components respond to the application of ac voltages.
What is Electricity?
The modern science of electricity originated with Benjamin Franklin, who began studying and experimenting with it in 1747. In the course of his experiments Franklin determined that electricity was a single force, with positive and negative aspects. Up to that point, the prevailing theory was that there were two kinds of electricity: one positive, the other negative.
To describe his experiments and results, Franklin also coined some twenty five new terms, including armature, battery, and conductor. His famous kite-flying experiment in a thunderstorm was performed in 1752, near the end of his work in this field.
Since then, many scientists, inventors, and entrepreneurs around the world have performed their own experiments, verifying and building on Franklin's beginnings in the field. Now, some 250 years later, we use electricity in almost every aspect of our daily lives. In some cases, we may not even realize that electricity is involved as an integral part of our activities.
So just what is electricity? Let's start with the dictionary definition, to give all of us some common ground. The American Heritage Dictionary actually gives four specific definitions:
Electricity
The class of physical phenomena arising from the existence and interactions of electric charge.
The physical science of such phenomena.
Electric current used or regarded as a source of power.
Intense emotional excitement.
We will skip the fourth definition as having no useful connection to the other three, and deal with electricity as a physical phenomenon which may be studied and manipulated using the tools of science.
When Ben Franklin developed his hypotheses about electricity, he arbitrarily assumed that the actual carriers of electrical current had a positive electrical charge. All of his theories, calculations, and descriptions were based on this assumption. Fortunately, his experiments still worked even with this incorrect assumption built into them. This "conventional" assumption was used for 200 years or more, and is still built into many of the common rules and procedures used to design and analyze electrical devices and behaviors.
We now know that the actual carriers of electricity are electrons, which have a negative electrical charge as defined in our system of science. Because of this, "electron theory" has been replacing "conventional theory" in schools and in regular usage.
The obvious next questions are:
What is an electron?
Where do electrons come from?
How do they carry an electrical charge from place to place?
We'll begin answering those questions when we take a closer look at electrons
Using Schematic Diagrams
Because different electronic components have different characteristics, it is necessary to distinguish between them in any circuit diagram. Of course, we could use the block diagram approach, and just identify each component with words. Unfortunately, this takes up a lot of space and makes the overall diagram harder to recognize or understand quickly. We need a way to understand electrical diagrams far more quickly and easily.
The answer is to use schematic symbols to represent electronic components, as shown in the diagram to the right. In this diagram, we show the schematic symbol of a battery as the electrical source, and the symbol of a resistor as the load. Even without the words and arrows, the symbols define exactly what this circuit is and how it behaves.
The symbol for each electronic component is suggestive of the behavior of that component. Thus, the battery symbol above consists of multiple individual cells connected in series. By convention, the longer line represents the positive terminal of each cell. The battery voltage would normally be specified next to the symbol.
The zig-zag line represents any resistor. In most cases, its resistance is specified next to the symbol just as the battery voltage would be given. It is easier and faster to read the symbol and the legend "4.7k" next to it, than to see a box and have to read "4700-ohm resistor" inside it.
As we introduce various electronic components in these pages, we will provide their schematic symbols as well.
One of the problems that can occur with schematic diagrams is too many lines all over the page. It's not a big deal when there are only two components in the circuit, but think of what the complete diagram for a modern television receiver or even a radio receiver would look like. We need a way to reduce the number of lines showing electrical connections.
We can help reduce the problem by noting that one connection is common to all circuitry, and serves as the reference point from which all electrical measurements are made. This electrical connection is designated the "ground reference," or simply "ground," in the circuit. The modified schematic diagram is shown to the right.
This circuit is actually the same as the one above, with the voltage source designated "E" (for EMF or ElectroMotive Force) and the load designated "R" (for Resistance). The ground symbols ( ) are assumed to be electrically connected to each other without any explicit connection shown. Often a circuit will be constructed on a steel or aluminum chassis, in which case the chassis itself is commonly used as the electrical ground as well as the mechanical support for the circuitry.
Ohm's Law
One thing we need to be able to do when we see a schematic circuit diagram is to perform mathematical calculations to define the precise behavior of the circuit. All information required to perform such calculations should be included on the schematic diagram itself. That way the information is all in one place, and any required detail can be determined readily.
Consider the basic circuit shown to the right. We know immediately that the battery voltage is 6 volts and that the resistor is rated at 1000 . Now, how can we determine how much current is flowing through this circuit?
If you go back to The Basic Circuit
• R = E ÷ I
• I = E ÷ R
These three equations describe Ohm's Law, which defines this relationship.
In the circuit shown above, we see that E = 6 volts and R = 1000 . To find the current flowing in this circuit, we must select the equation that solves for I. Using that equation, we note that:
I = E ÷ R
I = 6v ÷ 1000
I = 0.006 ampere (A) = 6 milliamperes (mA)
All calculations involving Ohm's Law are handled in exactly the same way. If the circuit gets complex, the calculations must be tailored to match. However, each calculation is still just this simple.
Circuit Components: the Resistor
The resistor is the simplest, most basic electronic component. In an electronic circuit, the resistor opposes the flow of electrical current through itself. It accomplishes this by absorbing some of the electrical energy applied to it, and then dissipating that energy as heat. By doing this, the resistor provides a means of limiting or controlling the amount of electrical current that can pass through a given circuit.
Resistors, such as the two pictured to the right, have two ratings, or values, associated with them. First, of course is the resistance value itself. This is measured in units called ohms and symbolized by the Greek letter Omega ( ). The second rating is the amount of power the resistor can dissipate as heat without itself overheating and burning up. Typical power ratings for modern resistors in most applications are ½ watt and ¼ watt, which are the two sizes shown in the figure. High-power applications can require high-power resistors of 1, 2, 5, or 10 watts, or even higher.
A general rule of thumb is to always select a resistor whose power rating is at least double the amount of power it will be expected to handle. That way, it will be able to dissipate any heat it generates very quickly, and will operate at normal temperatures.
For purposes of physical comparison, the larger resistor to the right is rated at ½ watt; its body is a cylinder 3/8" long and 1/8" in diameter. The smaller resistor, rated at ¼ watt, is of the same shape but is only 1/4" long and 1/16" in diameter.
The traditional construction of ordinary, low-power resistors is as a solid cylinder of a carbon composition material. This material is of an easily-controlled content, and has a well-known resistance to the flow of electrical current. The carbon cylinder is molded around a pair of wire leads at either end to provide electrical connections. The length and diameter of the cylinder are controlled in order to define the resistance value of the resistor — the longer the cylinder, the greater the resistance; the greater the diameter, the less the resistance. At the same time, the larger the cylinder, the more power it can dissipate as heat. Thus, the combination of the two determines both the final resistance and the power rating.
A newer, more precise method is shown to the left. The manufacturer coats a cylindrical ceramic core with a uniform layer of resistance material, with a ring or cap of conducting material over each end. Instead of varying the thickness or length of the resistance material along the middle of the ceramic core, the manufacturer cuts a spiral groove around the resistor body. By changing the angle of the spiral cut, the manufacturer can very accurately adjust the length and width of the spiral stripe, and therefore the resistance of the unit. The wire leads are formed with small end cups that just fit over the end caps of the resistor, and can be bonded to the end caps.
With either construction method, the new resistor is coated with an insulating material such as phenolic or ceramic, and is marked to indicate the value of the newly finished resistor.
High-power resistors are typically constructed of a resistance wire (made of nichrome or some similar material) that offers resistance to the flow of electricity, but can still handle large currents and can withstand high temperatures. The resistance wire is wrapped around a ceramic core and is simply bonded to the external connection points. These resistors are physically large so they can dissipate significant amounts of heat, and they are designed to be able to continue operating at high temperatures.
These resistors do not fall under the rule of selecting a power rating of double the expected power dissipation. That isn't practical with power dissipations of 20 or 50 watts or more. So these resistors are built to withstand the high temperatures that they will produce in normal operation, and are always given plenty of physical distance from other components so they can still dissipate all that heat harmlessly.
Regardless of power rating, all resistors are represented by the schematic symbol shown to the right. It can be drawn either horizontally or vertically, according to how it best fits in the overall diagram.
The Color Code
To the right is an image of a ½-watt resistor. Due to variations in monitor resolution, it may not be precisely to scale, but it is close enough to make the point. You can see that there are four colored stripes painted around the body of this resistor, and that they are grouped closer to one end (the top) than to the other. To someone who knows the color code, these stripes are enough to identify this as a 470 , 5% resistor. Imagine putting all of that in numbers on something that small! Or worse, on a ¼-watt resistor, which is even smaller.
The use of colored stripes, or bands, allows small components to be accurately marked in a way that can be read at a glance, without difficulty or any great possibility of error. In addition, the stripes are easy to paint onto the body of the resistor, and so do not add unreasonably to the cost of manufacturing the resistors.
Starting with the color band or stripe closest to one end of the resistor, the bands have the following significance: The first two bands give the two significant digits of the resistance value. The third gives a decimal multiplier which is some power of 10, and generally simply defines how many zeroes to add after the significant digits. The fourth band identifies the tolerance rating of the resistor. If the fourth band is missing, it indicates the original default tolerance of 20%. The bands may take on colors according to the following figure and table:
Color Significant Digits (1 and 2) Multiplier (3) Tolerance (4)
Black 0 1
Brown 1 10
Red 2 100
Orange 3 1000
Yellow 4 10,000
Green 5 100,000
Blue 6 1,000,000
Violet 7
Grey 8
White 9
Gold 0.1 5%
Silver 0.01 10%
(None) 20%
Standard resistors may be obtained in values ranging from 0.24 to 22 Megohms (22,000,000 ). However, they are not available in just any value; only the following combinations of first and second significant digits are used:
10 * 11 12 13 15 * 16 18 20 22 * 24 27 30 33 * 36 39 43 47 * 51 56 62 68 * 75 82 91
All values above may be obtained in 5% tolerance, while the boldface entries are available in 10% tolerance. Only the ones marked with an asterisk (*) are available in 20% tolerance, and you probably won't be able to find even them on today's market.
If you'd like to practice reading the color code, follow this link
Circuit Components: the Capacitor
We have said that an electrical current can only flow through a closed circuit. Thus, if we break or cut a wire in a circuit, that circuit is opened up, and can no longer carry a current. But we know that there will be a small electrical field between the broken ends. What if we modify the point of the break so that the area is expanded, thus providing a wide area of "not quite" contact?
The figure to the right shows two metal plates, placed close to each other but not touching. A wire is connected to each plate as shown, so that this construction may be made part of an electrical circuit. As shown here, these plates still represent nothing more than an open circuit. A wide one to be sure, but an open circuit nevertheless.
Now suppose we apply a fixed voltage across the plates of our construction, as shown to the left. The battery attempts to push electrons onto the negative plate (blue in the figure), and pull electrons from the positive plate (the red one). Because of the large surface area between the two plates, the battery is actually able to do this. This action in turn produces an electric field between the two plates, and actually distorts the motions of the electrons in the molecules of air in between the two plates. Our construction has been given an electric charge, such that it now holds a voltage equal to the battery voltage. If we were to disconnect the battery, we would find that this structure continues to hold its charge — until something comes along to connect the two plates directly together and allow the structure to discharge itself.
Because this structure has the capacity to hold an electrical charge, it is known as a capacitor. How much of a charge it can hold is determined by the area of the two plates and the distance between them. Large plates close together show a high capacity; smaller plates kept further apart show a lower capacity. Even the cut ends of the wire we described at the top of this page show some capacity to hold a charge, although that capacity is so small as to be negligible for practical purposes.
The electric field between capacitor plates gives this component an interesting and useful property: it resists any change in voltage applied across its terminals. It will draw or release energy in the form of an electric current, thus storing energy in its electric field, in its effort to oppose any change. As a result, the voltage across a capacitor cannot change instantaneously; it must change gradually as it overcomes this property of the capacitor.
A practical capacitor is not limited to two plates. As shown to the right, it is quite possible to place a number of plates in parallel and then connect alternate plates together. In addition, it is not necessary for the insulating material between plates to be air. Any insulating material will work, and some insulators have the effect of massively increasing the capacity of the resulting device to hold an electric charge. This ability is known generally as capacitance, and capacitors are rated according to their capacitance.
It is also unnecessary for the capacitor plates to be flat. Consider the figure below, which shows two "plates" of metal foil, interleaved with pieces of waxed paper (shown in yellow). This assembly can be rolled up to form a cylinder, with the edges of the foil extending from either end so they can be connected to the actual capacitor leads. The resulting package is small, light, rugged, and easy to use. It is also typically large enough to have its capacitance value printed on it numerically, although some small ones do still use color codes.
The schematic symbol for a capacitor, shown below and to the right of the rolled foil illustration, represents the two plates. The curved line specifically represents the outer foil when the capacitor is rolled into a cylinder as most of them are. This can become important when we start dealing with stray signals which might interfere with the desired behavior of a circuit (such as the "buzz" or "hum" you often hear in an AM radio when it is placed near flourescent lighting). In these cases, the outer foil can sometimes act as a shield against such interference.
An alternate construction for capacitors is shown to the right. We start with a disc of a ceramic material. Such discs can be manufactured to very accurate thickness and diameter, for easily-controlled results.
Both sides of the disc are coated with solder, which is compounded of tin and lead. These coatings form the plates of the capacitor. Then, wire leads are bonded to the solder plates to form the structure shown here.
The completed construction is then dipped into another ceramic bath, to coat the entire structure with an insulating cover and to provide some additional mechanical protection. The capacitor ratings are then printed on one side of the ceramic coating, as shown in the example here.
Modern construction methods allow these capacitors to be made with accurate values and well-known characteristics. Also, different types of ceramic can be used in order to control such factors as how the capacitor behaves as the temperature and applied voltage change. This can be very important in critical circuits.
Circuit Components: the Transformer
In our discussions of inductors in series and in parallel, we noted that the mutual inductance between coils could have a profound effect on the total inductance, depending on how much of the magnetic field of each coil overlaps the other coil. However, it is also possible to have two coils with interacting magnetic fields, but not connected electrically in the same circuit. The question then is, how does such a construction behave?
Before we address that question, however, we must consider that the amount of interaction between coils is not fixed. Therefore we must introduce the concept of coupling between coils. Coupling is the extent to which the magnetic field of each coil overlaps the other coil. Coupling can range from 0% (no interaction at all) to 100% (full interaction). In practice, 100% coupling is not possible, as some of the magnetic field will remain outside of the opposite coil. However, we can get close to it.
Qualitatively, coils with more than 50% coupling are said to be tightly coupled, while coils with less than 50% coupling are loosely coupled.
The schematic symbol for an iron-core transformer is shown to the right. It shows two coils sharing a common iron core. Because of the core, coupling between the two coils is as close to 100% as it can get. This is the standard arrangement for power transformers.
It is also possible to have two coils with a ferrite core, or with no core at all. These are still transformers and have the same basic properties. Only their design and construction varies, in accordance with their intended application.
Because the two coils are not electrically connected, only the magnetic field between them has any effect here. Therefore, let's take a look at what the magnetic field does.
In this circuit, the lefthand coil in the transformer is connected to the source of energy. Therefore, it is known as the primary or primary winding of the transformer. ("Winding" because the coils are wound around the core.) The righthand coil receives energy magnetically, so it is known as the secondary winding.
As long as switch S is open, the battery is not connected to the lefthand winding and no current flows. Therefore, there is no magnetic field around either coil of the transformer, and nothing happens.
When the switch closes, current begins to flow through the primary winding. This creates an expanding magnetic field around the primary winding, which also affects the secondary winding. The expanding magnetic field induces a voltage across the secondary winding, which causes current to flow through resistor R. The magnitude of the current depends on the induced voltage and the value of R, in accordance with Ohm's Law.
As switch S remains closed, the circuit current eventually reaches its maximum value and remains there, no longer changing. Therefore the magnetic field stops expanding and remains constant. Since the induced voltage in the secondary winding depends on a changing magnetic field, that has now ended and no current flows through resistor R.
Finally, when switch S is opened again, current stops flowing through the primary. The magnetic field collapses as it induces a voltage in both windings that tries to keep current flowing. Therefore current again flows through R, this time in the opposite direction from when S was first closed.
Once the magnetic field has completely collapsed, all current stops flowing, and the circuit remains in its original quiescent state as long as S remains open.
Since a transformer only works with changing currents, you may be wondering why we would even use a circuit like this one. However, there's a very practical application that people use every day. The number of turns of wire in the secondary does not have to be the same as the number of turns in the primary, and indeed generally is not the same. If the secondary has more turns of wire, it will step up the voltage generated in the secondary winding (and use up the energy in the magnetic field faster). This makes for an easy way to generate the high-voltage impulse needed to fire the spark plugs in your car's engine. It requires only a very slight adaptation of the above circuit to accomplish this.
What is Alternating Current?
Alternating Current vs. Direct Current
The figure to the right shows the schematic diagram of a very basic DC circuit. It consists of nothing more than a source (a producer of electrical energy) and a load (whatever is to be powered by that electrical energy). The source can be any electrical source: a chemical battery, an electronic power supply, a mechanical generator, or any other possible continuous source of electrical energy. For simplicity, we represent the source in this figure as a battery.
At the same time, the load can be any electrical load: a light bulb, electronic clock or watch, electronic instrument, or anything else that must be driven by a continuous source of electricity. The figure here represents the load as a simple resistor.
Regardless of the specific source and load in this circuit, electrons leave the negative terminal of the source, travel through the circuit in the direction shown by the arrows, and eventually return to the positive terminal of the source. This action continues for as long as a complete electrical circuit exists.
Now consider the same circuit with a single change, as shown in the second figure to the right. This time, the energy source is constantly changing. It begins by building up a voltage which is positive on top and negative on the bottom, and therefore pushes electrons through the circuit in the direction shown by the solid arrows. However, then the source voltage starts to fall off, and eventually reverse polarity. Now current will still flow through the circuit, but this time in the direction shown by the dotted arrows. This cycle repeats itself endlessly, and as a result the current through the circuit reverses direction repeatedly. This is known as an alternating current.
This kind of reversal makes no difference to some kinds of loads. For example, the light bulbs in your home don't care which way current flows through them. When you close the circuit by turning on the light switch, the light turns on without regard for the direction of current flow.
Of course, there are some kinds of loads that require current to flow in only one direction. In such cases, we often need to convert alternating current such as the power provided at your wall socket to direct current for use by the load. There are several ways to accomplish this, and we will explore some of them in later pages in this section.
Properties of Alternating Current
A DC power source, such as a battery, outputs a constant voltage over time, as depicted in the top figure to the right. Of course, once the chemicals in the battery have completed their reaction, the battery will be exhausted and cannot develop any output voltage. But until that happens, the output voltage will remain essentially constant. The same is true for any other source of DC electricity: the output voltage remains constant over time.
By contrast, an AC source of electrical power changes constantly in amplitude and regularly changes polarity, as shown in the second figure to the right. The changes are smooth and regular, endlessly repeating in a succession of identical cycles, and form a sine wave as depicted here.
Because the changes are so regular, alternating voltage and current have a number of properties associated with any such waveform. These basic properties include the following list:
• Frequency. One of the most important properties of any regular waveform identifies the number of complete cycles it goes through in a fixed period of time. For standard measurements, the period of time is one second, so the frequency of the wave is commonly measured in cycles per second (cycles/sec) and, in normal usage, is expressed in units of Hertz (Hz). It is represented in mathematical equations by the letter 'f.' In North America (primarily the US and Canada), the AC power system operates at a frequency of 60 Hz. In Europe, including the UK, Ireland, and Scotland, the power system operates at a frequency of 50 Hz.
• Period. Sometimes we need to know the amount of time required to complete one cycle of the waveform, rather than the number of cycles per second of time. This is logically the reciprocal of frequency. Thus, period is the time duration of one cycle of the waveform, and is measured in seconds/cycle. AC power at 50 Hz will have a period of 1/50 = 0.02 seconds/cycle. A 60 Hz power system has a period of 1/60 = 0.016667 seconds/cycle. These are often expressed as 20 ms/cycle or 16.6667 ms/cycle, where 1 ms is 1 millisecond = 0.001 second (1/1000 of a second).
• Wavelength. Because an AC wave moves physically as well as changing in time, sometimes we need to know how far it moves in one cycle of the wave, rather than how long that cycle takes to complete. This of course depends on how fast the wave is moving as well. Electrical signals travel through their wires at nearly the speed of light, which is very nearly 3 × 108 meters/second, and is represented mathematically by the letter 'c.' Since we already know the frequency of the wave in Hz, or cycles/second, we can perform the division of c/f to obtain a result in units of meters/cycle, which is what we want. The Greek letter (lambda) is used to represent wavelength in mathematical expressions. Thus, = c/f. As shown in the figure to the right, wavelength can be measured from any part of one cycle to the equivalent point in the next cycle. Wavelength is very similar to period as discussed above, except that wavelength is measured in distance per cycle where period is measured in time per cycle.
• Amplitude. Another thing we have to know is just how positive or negative the voltage is, with respect to some selected neutral reference. With DC, this is easy; the voltage is constant at some measurable value. But AC is constantly changing, and yet it still powers a load. Mathematically, the amplitude of a sine wave is the value of that sine wave at its peak. This is the maximum value, positive or negative, that it can attain. However, when we speak of an AC power system, it is more useful to refer to the effective voltage or current. This is the rating that would cause the same amount of work to be done (the same effect) as the same value of DC voltage or current would cause. We won't cover the mathematical derivations here; for the present, we'll simply note that for a sine wave, the effective voltage of the AC power system is 0.707 times the peak voltage. Thus, when we say that the AC line voltage in the US is 120 volts, we are referring to the voltage amplitude, but we are describing the effective voltage, not the peak voltage of nearly 170 volts. The effective voltage is also known as the rms voltage.
When we deal with AC power, the most important of these properties are frequency and amplitude, since some types of electrically powered equipment must be designed to match the frequency and voltage of the power lines. Period is sometimes a consideration, as we'll discover when we explore electronic power supplies. Wavelength is not generally important in this context, but becomes much more important when we start dealing with signals at considerably higher frequencies.
Why Use Alternating Current?
Since some kinds of loads require DC to power them and others can easily operate on either AC or DC, the question naturally arises, "Why not dispense entirely with AC and just use DC for everything?" This question is augmented by the fact that in some ways AC is harder to handle as well as to use. Nevertheless, there is a very practical reason, which overrides all other considerations for a widely distributed power grid. It all boils down to a question of cost.
DC does get used in some local commercial applications. An excellent example of this is the electric trolley car and trolley bus system used in San Francisco, for public transportation. Trolley cars are electric train cars with power supplied by an overhead wire. Trolley busses are like any other bus, except they are electrically powered and get their power from two overhead wires. In both cases, they operate on 600 volts DC, and the overhead wires span the city.
The drawback is that most of the electrical devices on each car or bus, including all the light bulbs inside, are quite standard and require 110 to 120 volts. At the same time, however, if we were to reduce the system voltage, we would have to increase the amount of current drawn by each car or bus in order to provide the same amount of power to it. (Power is equal to the product of the applied voltage and the resulting current: P = I × E.) But those overhead wires are not perfect conductors; they exhibit some resistance. They will absorb some energy from the electrical current and dissipate it as waste heat, in accordance with Ohm's Law (E = I × R). With a small amount of algebra, we can note that the lost power can be expressed as:
Plost = I²R
Now, if we reduce the voltage by a factor of 5 (to 120 volts DC), we must increase the current by a factor of 5 to maintain the same power to the trolley car or bus. But lost power is a function of the square of the current, so we will lose not five times as much power in the resistance of the wires, but twenty-five times as much power. To offset and minimize that loss, we would have to use much larger wires, and pay a high price for all that extra copper. A cheaper solution is to mount a motor-generator set in each trolley car and bus, using a 600 volt dc motor and a lower-voltage generator to power all the equipment aboard that car.
The same reality of Ohm's Law and resistive losses holds true in the country-wide power distribution system. We need to keep the voltage used in homes to a reasonable and relatively safe value, but at the same time we need to minimize resistive losses in the transmission wires, without bankrupting ourselves buying heavy-gauge copper wire. At the same time, we can't use motor-generator pairs all across the country; they would need constant service and would break down far too often. We need a system that allows us to raise the voltage (and thus reduce the current) for long-distance transmission, and then reduce the voltage again (to a safe value) for distribution to individual homes and businesses. And we need to do this without requiring any moving parts to break down or need servicing.
The answer is to use an AC power system and transformers. (We'll learn far more about transformers in a later page; for now, a transformer is an electrical component that can convert incoming AC power at one voltage to outgoing power at a different voltage, higher or lower, with only very slight losses.) Thus, we can generate electricity at a reasonable voltage for practical AC generators (sometimes called alternators), then use transformers to step that voltage up to very high levels for long-distance transmission, and then use additional transformers to step that high voltage back down for local distribution to individual homes.
In practice, this is done in stages. The really high-voltage transmission lines hanging from long glass insulators on the arms of tall steel towers carry electricity cross-country at several hundred thousand volts. This is stepped down to about 22,000 volts for distribution to multiple neighborhoods — these are the wires you see at the top of the telephone poles in many areas. Additional transformers mounted on some of these telephone poles step this voltage down again for distribution to several homes each.
The design of the system minimizes the overall cost by balancing the cost of transformers against the cost of heavier-gauge copper wire, as well as the cost of maintaining the system and repairing damage. This is how the cost of electricity delivered to your home or business is kept to a minimum, while maintaining a very high level of service.
A Note on Nomenclature
When we examined DC circuit theory and discussed DC power losses above, we used capital letters to represent all quantities. This is standard nomenclature; capital letters are used to represent fixed, static values. Thus, we use a capital I to represent DC current and a capital V to represent DC voltage.
By the same token, we use the capital letters R, L, and C to represent the values of circuit components containing resistance, inductance, and capacitance, respectively. These are fixed values that are set at the time the component is manufactured.
On the other hand, AC circuits use constantly-changing voltages and currents. In addition, some circuits contain both AC and DC components, which must often be considered separately. Therefore, we typically use lower-case letters to designate instantaneous values of voltage, current and power. Thus, if you see an equation written as:
P = I × E
you know it refers to DC power, current, and voltage. On the other hand, an equation written as:
p = i × e
refers to the instantaneous power, current, and voltage of some AC signal at some specified instant in time.
There will also be cases where we must refer to an overall AC signal rather than one instantaneous value from it. In such cases, the use of upper- and lower-case letters may be adjusted so we don't have too many of one or the other in a single discussion. We will try to avoid unnecessary confusion by including subscripts in equations, or specifying in the text exactly what we are discussing.
What is Alternating Current?
Alternating Current vs. Direct Current
The figure to the right shows the schematic diagram of a very basic DC circuit. It consists of nothing more than a source (a producer of electrical energy) and a load (whatever is to be powered by that electrical energy). The source can be any electrical source: a chemical battery, an electronic power supply, a mechanical generator, or any other possible continuous source of electrical energy. For simplicity, we represent the source in this figure as a battery.
At the same time, the load can be any electrical load: a light bulb, electronic clock or watch, electronic instrument, or anything else that must be driven by a continuous source of electricity. The figure here represents the load as a simple resistor.
Regardless of the specific source and load in this circuit, electrons leave the negative terminal of the source, travel through the circuit in the direction shown by the arrows, and eventually return to the positive terminal of the source. This action continues for as long as a complete electrical circuit exists.
Now consider the same circuit with a single change, as shown in the second figure to the right. This time, the energy source is constantly changing. It begins by building up a voltage which is positive on top and negative on the bottom, and therefore pushes electrons through the circuit in the direction shown by the solid arrows. However, then the source voltage starts to fall off, and eventually reverse polarity. Now current will still flow through the circuit, but this time in the direction shown by the dotted arrows. This cycle repeats itself endlessly, and as a result the current through the circuit reverses direction repeatedly. This is known as an alternating current.
This kind of reversal makes no difference to some kinds of loads. For example, the light bulbs in your home don't care which way current flows through them. When you close the circuit by turning on the light switch, the light turns on without regard for the direction of current flow.
Of course, there are some kinds of loads that require current to flow in only one direction. In such cases, we often need to convert alternating current such as the power provided at your wall socket to direct current for use by the load. There are several ways to accomplish this, and we will explore some of them in later pages in this section.
Properties of Alternating Current
A DC power source, such as a battery, outputs a constant voltage over time, as depicted in the top figure to the right. Of course, once the chemicals in the battery have completed their reaction, the battery will be exhausted and cannot develop any output voltage. But until that happens, the output voltage will remain essentially constant. The same is true for any other source of DC electricity: the output voltage remains constant over time.
By contrast, an AC source of electrical power changes constantly in amplitude and regularly changes polarity, as shown in the second figure to the right. The changes are smooth and regular, endlessly repeating in a succession of identical cycles, and form a sine wave as depicted here.
Because the changes are so regular, alternating voltage and current have a number of properties associated with any such waveform. These basic properties include the following list:
• Frequency. One of the most important properties of any regular waveform identifies the number of complete cycles it goes through in a fixed period of time. For standard measurements, the period of time is one second, so the frequency of the wave is commonly measured in cycles per second (cycles/sec) and, in normal usage, is expressed in units of Hertz (Hz). It is represented in mathematical equations by the letter 'f.' In North America (primarily the US and Canada), the AC power system operates at a frequency of 60 Hz. In Europe, including the UK, Ireland, and Scotland, the power system operates at a frequency of 50 Hz.
• Period. Sometimes we need to know the amount of time required to complete one cycle of the waveform, rather than the number of cycles per second of time. This is logically the reciprocal of frequency. Thus, period is the time duration of one cycle of the waveform, and is measured in seconds/cycle. AC power at 50 Hz will have a period of 1/50 = 0.02 seconds/cycle. A 60 Hz power system has a period of 1/60 = 0.016667 seconds/cycle. These are often expressed as 20 ms/cycle or 16.6667 ms/cycle, where 1 ms is 1 millisecond = 0.001 second (1/1000 of a second).
• Wavelength. Because an AC wave moves physically as well as changing in time, sometimes we need to know how far it moves in one cycle of the wave, rather than how long that cycle takes to complete. This of course depends on how fast the wave is moving as well. Electrical signals travel through their wires at nearly the speed of light, which is very nearly 3 × 108 meters/second, and is represented mathematically by the letter 'c.' Since we already know the frequency of the wave in Hz, or cycles/second, we can perform the division of c/f to obtain a result in units of meters/cycle, which is what we want. The Greek letter (lambda) is used to represent wavelength in mathematical expressions. Thus, = c/f. As shown in the figure to the right, wavelength can be measured from any part of one cycle to the equivalent point in the next cycle. Wavelength is very similar to period as discussed above, except that wavelength is measured in distance per cycle where period is measured in time per cycle.
• Amplitude. Another thing we have to know is just how positive or negative the voltage is, with respect to some selected neutral reference. With DC, this is easy; the voltage is constant at some measurable value. But AC is constantly changing, and yet it still powers a load. Mathematically, the amplitude of a sine wave is the value of that sine wave at its peak. This is the maximum value, positive or negative, that it can attain. However, when we speak of an AC power system, it is more useful to refer to the effective voltage or current. This is the rating that would cause the same amount of work to be done (the same effect) as the same value of DC voltage or current would cause. We won't cover the mathematical derivations here; for the present, we'll simply note that for a sine wave, the effective voltage of the AC power system is 0.707 times the peak voltage. Thus, when we say that the AC line voltage in the US is 120 volts, we are referring to the voltage amplitude, but we are describing the effective voltage, not the peak voltage of nearly 170 volts. The effective voltage is also known as the rms voltage.
When we deal with AC power, the most important of these properties are frequency and amplitude, since some types of electrically powered equipment must be designed to match the frequency and voltage of the power lines. Period is sometimes a consideration, as we'll discover when we explore electronic power supplies. Wavelength is not generally important in this context, but becomes much more important when we start dealing with signals at considerably higher frequencies.
Why Use Alternating Current?
Since some kinds of loads require DC to power them and others can easily operate on either AC or DC, the question naturally arises, "Why not dispense entirely with AC and just use DC for everything?" This question is augmented by the fact that in some ways AC is harder to handle as well as to use. Nevertheless, there is a very practical reason, which overrides all other considerations for a widely distributed power grid. It all boils down to a question of cost.
DC does get used in some local commercial applications. An excellent example of this is the electric trolley car and trolley bus system used in San Francisco, for public transportation. Trolley cars are electric train cars with power supplied by an overhead wire. Trolley busses are like any other bus, except they are electrically powered and get their power from two overhead wires. In both cases, they operate on 600 volts DC, and the overhead wires span the city.
The drawback is that most of the electrical devices on each car or bus, including all the light bulbs inside, are quite standard and require 110 to 120 volts. At the same time, however, if we were to reduce the system voltage, we would have to increase the amount of current drawn by each car or bus in order to provide the same amount of power to it. (Power is equal to the product of the applied voltage and the resulting current: P = I × E.) But those overhead wires are not perfect conductors; they exhibit some resistance. They will absorb some energy from the electrical current and dissipate it as waste heat, in accordance with Ohm's Law (E = I × R). With a small amount of algebra, we can note that the lost power can be expressed as:
Plost = I²R
Now, if we reduce the voltage by a factor of 5 (to 120 volts DC), we must increase the current by a factor of 5 to maintain the same power to the trolley car or bus. But lost power is a function of the square of the current, so we will lose not five times as much power in the resistance of the wires, but twenty-five times as much power. To offset and minimize that loss, we would have to use much larger wires, and pay a high price for all that extra copper. A cheaper solution is to mount a motor-generator set in each trolley car and bus, using a 600 volt dc motor and a lower-voltage generator to power all the equipment aboard that car.
The same reality of Ohm's Law and resistive losses holds true in the country-wide power distribution system. We need to keep the voltage used in homes to a reasonable and relatively safe value, but at the same time we need to minimize resistive losses in the transmission wires, without bankrupting ourselves buying heavy-gauge copper wire. At the same time, we can't use motor-generator pairs all across the country; they would need constant service and would break down far too often. We need a system that allows us to raise the voltage (and thus reduce the current) for long-distance transmission, and then reduce the voltage again (to a safe value) for distribution to individual homes and businesses. And we need to do this without requiring any moving parts to break down or need servicing.
The answer is to use an AC power system and transformers. (We'll learn far more about transformers in a later page; for now, a transformer is an electrical component that can convert incoming AC power at one voltage to outgoing power at a different voltage, higher or lower, with only very slight losses.) Thus, we can generate electricity at a reasonable voltage for practical AC generators (sometimes called alternators), then use transformers to step that voltage up to very high levels for long-distance transmission, and then use additional transformers to step that high voltage back down for local distribution to individual homes.
In practice, this is done in stages. The really high-voltage transmission lines hanging from long glass insulators on the arms of tall steel towers carry electricity cross-country at several hundred thousand volts. This is stepped down to about 22,000 volts for distribution to multiple neighborhoods — these are the wires you see at the top of the telephone poles in many areas. Additional transformers mounted on some of these telephone poles step this voltage down again for distribution to several homes each.
The design of the system minimizes the overall cost by balancing the cost of transformers against the cost of heavier-gauge copper wire, as well as the cost of maintaining the system and repairing damage. This is how the cost of electricity delivered to your home or business is kept to a minimum, while maintaining a very high level of service.
A Note on Nomenclature
When we examined DC circuit theory and discussed DC power losses above, we used capital letters to represent all quantities. This is standard nomenclature; capital letters are used to represent fixed, static values. Thus, we use a capital I to represent DC current and a capital V to represent DC voltage.
By the same token, we use the capital letters R, L, and C to represent the values of circuit components containing resistance, inductance, and capacitance, respectively. These are fixed values that are set at the time the component is manufactured.
On the other hand, AC circuits use constantly-changing voltages and currents. In addition, some circuits contain both AC and DC components, which must often be considered separately. Therefore, we typically use lower-case letters to designate instantaneous values of voltage, current and power. Thus, if you see an equation written as:
P = I × E
you know it refers to DC power, current, and voltage. On the other hand, an equation written as:
p = i × e
refers to the instantaneous power, current, and voltage of some AC signal at some specified instant in time.
There will also be cases where we must refer to an overall AC signal rather than one instantaneous value from it. In such cases, the use of upper- and lower-case letters may be adjusted so we don't have too many of one or the other in a single discussion. We will try to avoid unnecessary confusion by including subscripts in equations, or specifying in the text exactly what we are discussing.
Electrical Circuits
An electrical circuit is any combination of wires and electrical devices (also called circuit elements) through which current can flow. We will begin with a typical flashlight powered by two 1.5 V AA batteries.
Below is diagram of the actual flashlight next to a schematic diagram. Note the symbols used to denote the batteries and the switch. The general symbol for any resistor is just , the oval shown below indicates that it is also a light bulb. (Check out the standard electrical symbols listing.) The switch is shown in the "open" position. That is, the flashlight is off and the circuit is broken or not complete. No current can flow until there is a path. Charge will not flow from one side of the battery unless and equal amount can flow into the other side! When the switch is in the "closed" position, the flashlight is on and there is a path for the current to flow. (Although there is no set rule, switches are usually shown in the open position in schematics.)
The batteries are called DC sources. DC or "Direct Current" means that current flows just one way. Which way? Positive current, also called conventional current, flows out of the positive side of the battery (indicated by the longer line) and into the negative (indicated by the shorter line). And what do we mean by positive current? It is positive charge flowing as the arrows show.
Foul you cry! Yes, we now know that what actually happens is that electrons (negative charge) flow out of the negative side and into the positive. But in the late 1700's Ben Franklin, a scientist as well as a statesman, had no way of knowing that. He made a guess at what was happening and we're stuck with it. But it doesn't change any of the physics. So, from hence forth, we pretend positive charge flows out of the positive side of the battery.
Series Circuit
The two batteries are connected in series, that is, one after another. The current that flows through one must also flow through the other. What does that mean for the voltage? Think of climbing a 1.5 ft step and then following it with another 1.5 ft step. Your are now 1.5 + 1.5 = 3.0 ft higher. Similarly, the potential from the negative side of the first battery to the positive side of the second is 1.5 V + 1.5 V = 3.0 V. The potential across any two or more elements connected in series is always the sum of the individual voltages.
What potential is across the lightbulb of the flashlight shown above? The resistance in the "wires" that provide the connections between the batteries and bulb should be much lower than that of the bulb itself. Therefore, there is negligible "potential drop" across the wires. Essentially all 3.0 V provided by the batteries will be across the bulb. Later, we will address the role that resistance in the wires play. But for now, let's assume all potential drops are across the circuit elements. Note that the bulb is in series with the batteries. The current that flows throught the batteries is the same that flows through the bulb. The current has no other path.
There is a method for understanding potential drops. The total potential drop around a complete circuit must be zero. (This similar to saying that the sum of all height changes on a round-trip hike must sum to zero.) As we move across the batteries, we gain potential of 1.5 V each (for a total of 3.0 V), followed by the potential across the bulb ... then we are back to where we started. (Don't forget that potential change along the wires is essentially zero.) Therefore, the potential across the bulb must be a 3.0 V drop.
Let's apply the formulas from the previous section.
Problem: If the bulb in the flashlight above has resistance 2 , then what current flows through the bulb and what power does it consume?
Solution: The current flowing through the bulb is I = 3.0 V / 2 = 1.5 A. The power being consumed by the bulb is P = I2 R = (1.5 A)2 x 2 = 4.5 W. The power being supplied by the two batteries is P = I V = (1.5 A) x (3.0 V) = 4.5 W. It should be no surprise that the power consumed by the bulb is equal to that supplied by the batteries.
Parallel Circuit
Elements in a circuit are in parallel when they are connected in such a way that the potential across them is always the same. Essentially, the ends of each element are connected together. Let's consider two important cases.
The Car - Consider the simplified diagram for the electrical system of a car. Notice that all the elements receive a full 12 V from the car battery.
The current that flows out of the battery can split up and flow to each element individually. Each element will receive a different portion of the total current, according to its resistance, but each element receives the same 12 V. The elements are in parallel with one another when the switches are closed. Note that each switch is in series with the element it controls. Do you understand why a switch must be in series with the device it controls? (Actually, this is not absolutely true. In lab, you will have the chance to observe such an exception.)
Look at the diagram and see if you can understand the effect of each switch. Can you identify which one represents the key ignition switch? Is this consistent with your experience of what can be turned on with your car key? (A more proper diagram would show the ignition and starter switches as part of one compound switch.)
Summarizing Series and parallel circuits ...
Elements in series have the same current, but the voltage drops may be different.
Elements in parallel have the same voltage, but the currents may be different.
Here's an important difference to remember.
Elements in parallel can be operated independently, while elements in series cannot.
The Home - Are the electrical elements in your house (the fridge, the toaster, the lights, etc) in series or parallel? They better be in parallel, right? The electrical diagram below shows the wiring for a typical room in a home. In this example, a single power line, consisting of a red wire and a green wire, services the six outlets and two overhead light fixtures. You may think of the red wire as if it were connected to the positive terminal of 120 V battery while the green wire is connected to the negative side . This is not correct, but we will deal with the details of the AC power in your home in the next section.
All the outlets and the light fixtures (with their switches) are connected in parallel. However, the switches are in series with the lights they control. If the switch is closed, the red wire coming into the switch from the power line is connected to the red wire going out to the light, and the light will be on. If the switch is opened, there is no connection and the light is off. The green wire from the other side of the light fixture is connected directly to the green wire of the power line.
In the simplified diagram above, it may not appear that the outlets are in parallel, but they are. Most outlets provide two screwposts on each side of the casing, like that shown in the picture to the right. Both screwposts on a side are connect to the "slot" on that side. That makes it convenient to run a line into the outlet, say at the bottom posts, and then run a line out to the next outlet from the top posts. When an appliance is plugged into the outlet, it will be in parallel with all the other outlets, lights, etc. (Of course, the appliance may contain its own switch.)
The colors for the wires in the diagram above were chosen for clarity. The colors you are most likely to find in household wiring is discussed in AWG and household wiring.
A few notes:
• The longer slot is connected to the O V wire. This is the convention for the somewhat new "polarized" plug. The extra copper ground wire shown the the right is discussed in the "AWG and household wiring" link above.
• The outlet in the upper left corner of the room electrical diagram is at the "end of the line". One of the screwposts on either side will not be used.
• Two or more lines may service a room if the electrical needs are particularly high. The kitchen usually has separate lines to the refrigerator, the stove, and the outlets.
Problem: Let's put a few numbers to the car circuit example. Assume we have a 10 W dome light, a 50 W radio, and two headlights at 300 W each. What current do they draw?
Solution: For the dome light, P = I V, so I = P/V = 10 W / 12V = .83 A. The radio consumes 50 W, so I = 50 W / 12 V = 4.2 A. The headlights are rated at 300 W each. So they draw 300 W / 12V = 25 A each! If all these devices are on at the same time, the total current will be about 55 A. And what power is the battery supplying? P = IV = (55A) x (12V) = 660 W, exactly equal to the total (10 + 50 + 300 + 300 = 660 W) of what is being consumed by the devices.
Problem: Let's put a few numbers to the household circuit example. Assume we have a 100 W light fixture, and a 300 W television plugged into one of the outlets. What current do they draw?
Solution: For the light fixture, I = P/V = 100 W / 120 V = .83 A. Note that the 100 W light has the same current as the 10 W dome light in the previous example. Although the current is the same, the light fixture consumes 10 times the power of the domelight since it operates at 10 times the voltage.
The television consumes 300 W, so I = 300 W / 120 V = 2.5 A. Again, notice how the 300 W car headlight draws 10 times the current of the 300 W television. It must, since it operates at 1/10 the voltage.
Why do the Lights Dim?
If the dome light is on when you start the car, you'll probably notice it dims just a bit. The main reason is that the battery has internal resistance. The voltage may be 12 V when no current is being drawn. But when current flows through the battery, there is a potential drop across the internal resistance and the voltage at the battery terminals will be less than 12V. But you must be draw quite a bit of extra current to notice the change. A typical internal resistance might be only .01. Let's say you turn on the dome light, drawing .83 A. The internal potential drop in the battery would be V = I R = .83A x .01 = .0083V. You turn on the radio, creating an additional drop of 4.2A x .01 = .042 V. You won't notice it. But when you start your car, the starter motor pulls 50 A. Now the potential drop is 50 A x .01 = .5 V. So the voltage across the battery and all the devices has dropped to about 11.5 V and you will probably notice the change.
Here's another example with a slightly different cause. The fridge or the air conditioner kicks on and you notice the house lights dim. It is not the internal resistance of WAPA! You may have inadequate wiring. Both these devices draw quite a bit of current in the first second or so when they first start up. If the wiring in your house is not "large" enough (i.e. the cross sectional area is too small), it will have too much resistance. When a large current is present in the wire, there will be a significant voltage drop, leaving less voltage available to your lights. You will notice that they dim.
• Before moving to the next section, try a few electricity PROBLEMS to reinforce these new concepts.
• If you are comfortable with what you have learned so far, then you are ready to move on to AC Power Systems.
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RETURN
Electricity Problems
Let's state the two working equations we have so far.
Ohm's Law: V = I R
Power: P = I V all devices
P = I2 R resistors
The solutions to the problems below can be found at the end of this page. Try all the problems before looking at the solutions. It's much easier to understand a solution put before you than to come up with the solution yourself. To develop the skills necessary to solve the problems yourself, you must spend the time doing it.
Problems
1. A heavy duty flashlight runs on 3 D-cell batteries, each 1.5 V. You insert the batteries one after another. The bulb in the flashlight is rated to produce 5 W. What current passes through the bulb and what is it's resistance?
2. On the back of your beatbox it says, " 6 AA batteries required. Nominal current in use is 1.5 A". You open the back and note that the batteries are placed end-to-end in two rows with three batteries each. What power might be consumed by the beatbox? Be sure to consider all the ways the batteries might be connected.
3. You have a flashlight that operates on 2 D-cell batteries loaded in series, to produce 3.0 V. The bulb is rated at 5 W. The bulb burns out (the filament breaks) and you need to replace it. You cannot find the exact bulb, but you do find one with the same shaped base that is rated 5 W at 4.5 V (perhaps designed for the flashlight in problem #1). What would happen if you used this bulb? How much current would flow and what would be the power output? Do you think the bulb would survive? Repeat the problem with a bulb rated 5 W at 1.5 V.
Here's a few problems involving household devices that run on AC voltage. As you will see in the next chapter, we can use exactly the same equations for AC circuits.
4. A 60 W light bulb operates at 120 V. What current does it draw and what is the resistance of the bulb?
5. A 1200 W hair dryer operates at 120 V. What current does it draw? Calculate its "effective" resistance. Do you think the value you calculated is the resistance of the heating element in the dryer? What else consumes power in a hair dryer?
6. A 500 W microwave oven, a 350 W toaster, two 100 W lights, and 600 W mixer are all connected to one circuit. This circuit has a 20 A breaker. Can you safely run all of these appliances at once? Assume 120 VAC.
7. WAPA must deliver 60 MW (that's 60 million Watts!) over a mile-long high tension line (also called a powerline) that has resistance .05. Assume the plant delivers the power at the 600 V that comes from its generators.
o What current is required to deliver 60 MW at 600V?
o What power is dissipated as heat in the high tension line.
Don't be suprised to find out that the wire consumes more energy than the plant delivers! In order to get around this obvious problem, power companies use transformers to increase the voltage.
Repeat the problem above assuming the plant delivers 60 MW at 14,000 V and answer one more question. What voltage drop will be across the length of the wire?
8. Here's a puzzle. No need for actual numbers. The resistance of the filament in a light bulb increases with length and decreases with cross sectional area. Asssume the filament is cylindrical. (Need a review ?) Is the filament for a 40 W bulb larger or smaller than that of a 100 W bulb? Consider both length and cross section. What other factors should you consider in your answer?
Solutions
Don't look at the answers until you've tried the problems on your own!!
1. Since the batteries are inserted one after another, they are in series. So the total potential is 1.5 + 1.5 + 1.5 = 4.5 V. The current is I = 5 W / 4.5 V = 1.1 A. R = 4.5 V/ 1.1 A = 4.1 .
2. The power is just potential times voltage. But what is the voltage? You know that you have two rows of three, each producing 3 x 1.5 = 4.5 volts. But the two rows may be either in series or parallel. (You might be able to actually see the wires and figure out which.) If they are in series, the total voltage is 9.0 V and P = 1.5 A x 9.0 V = 13.5 W. But if they are connect in parallel, then the total voltage is still just 4.5 V and the power would be about 6.8 W. The purpose of a parallel combination rather just three batteries would be to provide twice the time between changes in the batteries, since each set of three would have to supply only .75 A.
3. The proper 5 W bulb designed to run on 3.0 V will operate with a current of I = 5 W / 3.0 V = 1.7 A. Its resistance is R = 3.0 V / 1.7A =1.8 . The replacement that you are considering is designed to operate with current I = 5 W / 4.5 V = 1.1 A and has a resistance of R = 4.5 V/ 1.1 A = 4.1 . When placed in th flashlight, the current will be I = 3.0 v / 4.1 = .73 A. Since it is designed to carry 1.1 A, this should not damage the filament. But the power output will be P = .73 A x 3.0 V = 2.2 W. It will not be very bright compared to a proper replacement bulb. Can you see why using a bulb designed for a lower voltage might present a problem?
4. I = 60 W / 120 V = .5 A. R = 120 V / .5 A = 240 .
5. I = 1200 W /120 V = 10A. Effective R = 120 V / 10 A = 12 . The hairdryer also has a motor which consumes energy. If the filament and motor are connected in parallel, then less than 10 A would be flowing through the filament, hence it resistance would have to be greater thatn 240. If the motor and filament are connected in series (a very unlikely design) then the resistance of the filament would actually have to be less than 240. Why?
6. The total power consumed if all are operated at the same time would be P = 500 W + 350 W + 2 x 100 W + 600 W = 1,650 W. so I = 1650 W / 120 V = 13.8 A . This is saely below the maximum for the 20 A breaker.
7. For a power of 60 MW (6.0 x 107 W), the current flowing from the power plant would have to be I = 60 MW / 600 V = .10 MA or 1.0 x 105A. The power dissipated in the powerline would be P = (1.0 x 105A)2 x .05 = 5.0 x 108 W or 500 MW. Of course, that's impossible! 600 V could not push that much current through a wire with that resistance. Conservation of energy is not violated, and this much current never flows. But the problem is clear. WAPA must increase the voltage significantly. At 14,000 V, the current needed is I = 60 MW / 14,000 V = 4.3 x 103A and the power dissipated in the powerline would be P = (4.3 x 103A)2 x .05 = 9.1 x 105 W or .91 MW. This represents about 15% of the total power. This is a more reasonable loss, although WAPA does try to keep total losses below 10%. (They use many powerlines in parallel.) The voltage drop across the mile length of line would be V = 4.3 x103A) x .05 = 200 V. The voltage at the transformers outside your house would be very close to the full 14,000 V.
8. First, the 40 W bulb has a higher resistance than a 100 W bulb. Look at the equations. Since P = IV and both bulbs will have the same voltage (120 VAC), the 100 W bulb must have more current. And that means the 100 W bulb must have a lower resistance! So the filament for the higher resistance 40 W filament must be either longer or have a smaller cross section than then filament for the 100 W bulb. Of course, it could be both. Is it just one or both?
There is another consideration. The 100 W puts out more energy and hence needs more surface area to radiate this energy. The surface area of a cylindrical filament is the circumference (2r) times the length (L). But the resistance grows with L and decreases with area (r2). Let's start with a 40 W filament and think about what we would change to get a lower resistance while increasing the surface area. If we simply reduce the length, we will reduce the resistance, but also reduce the surface area. If we increase the filament cross section, we will reduce the resistance but increase the surface area. In general, the filaments for the higher wattage bulbs are thicker. They are sometimes also longer (with the appropriate increase in thickness) to increase surface area.
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