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| Introduction to DC Circu |
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 electronic 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.
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.
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Looking at Electrons Consider the drawing to the right. Here we see a representation of a single atom — the smallest possible unit of any given element. At the center of the atom is the nucleus, which consists of two kinds of particles: protons and neutrons. The number of protons determines exactly what element we're dealing with, and the number of neutrons is similar to (but not necessarily the same as) the number of protons. We will not be concerned with the nucleus in these discussions, except to note that each proton has a unit positive charge, while each neutron has zero charge.
In constant motion around the nucleus we find a number of electrons, shown here as black dots. Each electron has a unit negative charge, precisely balancing the positive charge on one proton in the nucleus. As you might expect, in a normal atom with no external forces applied, the number of electrons orbiting the nucleus is equal to the number of protons in the nucleus. As a result, the entire atom is electrically neutral, or uncharged.
The electrons in any given atom do not just orbit the nucleus haphazardly; rather, they occupy specific energy levels, or "shells," around the nucleus. Electrons will always try to occupy the lowest energy level available to them, dropping into a closer orbit if they can. However, scientists have found that there is a limit to the number of electrons that can occupy any shell; electrons beyond that limit must take a higher-energy position in a wider orbit around the nucleus.
The first, or innermost shell is limited to two electrons. Once those two electrons are in place, this shell is filled and will force all other electrons to occupy positions further away from the nucleus. The second shell can hold a maximum of eight electrons, while the third can hold eighteen. Mathematically,
where N is the shell number, counting out from the nucleus. This gives us maximum shell capacities of 2, 8, 18, 32, 50 and 72 electrons in the first six shells. That is enough to account for all natural elements, with room left over for those heavy elements that have so far been created in the laboratory and then some.
Of course, this discussion is very basic, and doesn't cover any of the fine details that have been gradually learned. Deeper discussions belong in the realms of chemistry and nuclear physics, and as such are beyond the scope of these pages.
The factor that becomes useful in dealing with electricity and electrical phenomena is that those elements that have only one or two electrons in their outermost shells don't hold on to these outermost electrons very strongly. Therefore it requires little external energy to pull these electrons away from their parent atoms and move them someplace else. These electrons make all electrical activities possible.
In metals such as copper and silver, these outer electrons are essentially free to move around anywhere within the body of the metal. In these metals, the outer electrons are so loosely held that the thermal energy inherent at room temperature is sufficient to free them from their parent atoms. We can cause these electrons to move from one place to another in a variety of different ways. We'll be exploring such methods and the uses to which such moving electrons can be put in these pages.
Have you ever walked across a carpet and then gotten a slight shock as you reached out to turn on a light switch? Or heard and felt all the "crackles" as you removed a load of clothes from the dryer? Or gotten a similar effect when stroking a cat?
You probably already know that these phenomena are generally known as "static," but do you know how and why they happen?
What has happend in each case is that the friction of the physical action — walking over the carpet, stroking the cat, etc. — has caused loosely-held electrons to be transferred from one surface to the other. This results in a net negative charge on the surface that has gained electrons, and a net positive charge on the surface that has lost electrons. If there is no path for the electrons to take to restore the balance of electrical charges, these charges will remain where they are (although they will gradually leak off, as they cannot easily be held forever).
If the electrical charge continues building through ongoing friction or similar action, it will eventually reach the point where it cannot be contained, and will discharge itself over any available path. Lightning is a spectacular display of electrical energies discharging themselves after being built to high values by clouds rubbing and bumping against each other. It makes no difference that clouds consist of many tiny droplets of water floating in the air; each such droplet contributes a small amount to the total charge, which can reach enormous totals.
The point about static electricity is that it is indeed static, which means that it doesn't move from one place to another. Therefore, while some interesting experiments can be performed with it, it does not serve the purpose of providing energy to do sustained work in any practical capacity. Static electricity certainly exists, and under certain circumstances we must allow for it and account for its possible presence, but it will not be the main theme of these pages.
Basic Circuit Concepts
The figure to the right shows the basic type of electrical circuit, in the form of a block diagram. It consists of a source of electrical energy, some sort of load to make use of that energy, and electrical conductors connecting the source and the load.
The electrical source has two terminals, designated positive (+) and negative (-). As long as there is an unbroken connection from source to load and back again as shown here, electrons will be pushed from the negative terminal of the source, through the load, and then back to the positive terminal of the source. The arrows show the direction of electron current flow through this circuit. Because the electrons are always moving in the same direction through the circuit, their motion is known as a direct current (DC).
The source can be any source of electrical energy. In practice, there are three general possibilities: it can be a battery, an electrical generator, or some sort of electronic power supply.
The load is any device or circuit powered by electricity. It can be as simple as a light bulb or as complex as a modern high-speed computer.
The electricity provided by the source has two basic characteristics, called voltage and current. These are defined as follows:
- Voltage
The electrical "pressure" that causes free electrons to travel through an electrical circuit. Also known as electromotive force (emf). It is measured in volts.
- Current
The amount of electrical charge (the number of free electrons) moving past a given point in an electrical circuit per unit of time. Current is measured in amperes.
The load, in turn, has a characteristic called resistance. By definition:
- Resistance
That characteristic of a medium which opposes the flow of electrical current through itself. Resistance is measured in ohms.
The relationship between voltage, current, and resistance in an electrical circuit is fundamental to the operation of any circuit or device. Verbally, the amount of current flowing through a circuit is directly proportional to the applied voltage and inversely proportional to the circuit resistance. By explicit definition, one volt of electrical pressure can push one ampere of current through one ohm of resistance. Two volts can either push one ampere through a resistance of two ohms, or can push two amperes through one ohm. Mathematically,
E = I × R,
where
- E = The applied voltage, or EMF
I = The circuit current
R = The resistance in the circuit
Using SchemUsing Schematic 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 ( 
Ohm'Laws
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
If you go back to The Basic Circuit, you'll note that the relationship between voltage, current, and resistance is given as E = I × R. Using basic algebra we can also rewrite this as:
- 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
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 ResistorThe 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 (
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.
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| 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
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:
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| 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
| 10 * | 11 | 12 | 13 | 15 * | 16 | 18 | 20 | 22 * | 24 | 27 | 30 | 33 * | 36 | 39 | 43 | 47 * | 51 | 56 | 62 | 68 * | 75 | 82 | 91 |
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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.
We've looked at Ohm's Law for a circuit containing a single resistor, and found it to be quite simple. But what happens if we have multiple resistors in the circuit? How can we deal with that, and still perform accurate calculations on the overall circuits?
On this page, we'll consider a circuit using two resistors, connected so that the same current flows through both, and the two resistors must share the applied voltage. This is known as a series connection.
In the circuit shown to the right, we see two resistors instead of one. To distinguish them without assigning fixed values to them, we designate them "R1" and "R2."
In this circuit, we don't know the specific voltage across each resistor, but we do know that the same current must be flowing through both of them. We will use that fact to help us with our calculations.
We have already said that the current must be the same everywhere in a series circuit. If it were not so, we would have to allow current to be created or destroyed at random, which is not possible. Therefore, the same current, I, flows through both resistors.
We can also apply Ohm's Law separately to each resistor, by noting that the voltage E1 across R1 must be:
E1 = I × R1
Similarly,E2 = I × R2
But we also know that the total battery voltage E = E1 + E2. Therefore we can say that:
E = E1 + E2
E = (I × R1) + (I × R2)
E = I × (R1 + R2)
That last equation has a serious consequence throughout all of electronics: resistors in series add their values together to get the total resistance.
In addition, we can now solve for the total circuit current:
I = E ÷ (R1 + R2)
Additional resistors in series simply continue to add to the total circuit resistance.
In the circuit to the right, the total resistance R1 + R2 = 1.8k + 5.6k = 7.4k. Incidentally, the designation "k" specifies units of kilohms, or thousands of ohms. The "k" stands for kilo, meaning 1000. Most practical circuits involve resistances of this magnitude, so they are often specified this way.
In any case, we can now calculate the circuit current:
I = E ÷ (R1 + R2)
I = 9 ÷ (1.8k + 5.6k)
I = 9 ÷ 7.4k
I = 1.216 mA
We can now also calculate the voltage across each resistor:
E1 = I × R1
E1 = 1.216 mA × 1.8k
E1 = 2.19 v.
and,E2 = I × R2
E2 = 1.216 mA × 5.6k
E2 = 6.81 v.
Finally,E = E1 + E2
E = 2.19 v. + 6.81 v.
E = 9 v.
Note that we can make our calculations directly and without converting units if we match milliAmperes (mA) with volts (v) and kilohms (k). This works because the changes in orders of magnitude cancel each other out. This is not true of all kinds of calculations, however, so you have to be careful of how you handle your units.
In addition, we did round off our results above. However, if you make the calculations with greater precision, you will find that the results still add up correctly.
Resistors in Parallel
When two resistors are connected in parallel, as shown to the right, the same voltage appears across each resistor. However, each resistor provides its own path for the flow of current. If the resistors have different resistance values, they will carry different amounts of current, each in accordance with Ohm's Law.
As a result, we can calculate the currents through each resistor, and the total current I, as:
I1 = E ÷ R1
I2 = E ÷ R2
I = I1 + I2
Now let's apply Ohm's Law again, and solve the above equation for total resistance:
| E | = | E | + | E |
| RT | R1 | R2 |
Since E is the same everywhere in the circuit, we can multiply both sides of the equation by 1/E and thus remove it. Then we solve for RT, the total circuit resistance:
| 1 | = | 1 | + | 1 |
| RT | R1 | R2 | ||
| RT = | 1 | |||
| 1 | + | 1 | ||
| R1 | R2 | |||
| RT = | 1 | |||
| R2 | + | R1 | ||
| R1R2 | R2R1 | |||
| RT = | 1 | |||
| R1 + R2 | ||||
| R1 × R2 | ||||
| RT = | R1 × R2 | |||
| R1 + R2 | ||||
Voltage Dividers
In many circuits, it is necessary to obtain a voltage not available from the main power source. Rather than have multiple power sources for all needed voltages, we can derive other voltages from the main power source. In most cases, the needed voltage is less than the voltage from the main source, so we can use resistors in an appropriate configuration to reduce the voltage from the power source, for use in a small circuit.
If we know precisely both the voltage and current required, we can simply connect a resistor in series with the power source, with a value calculated in accordance with Ohm's Law. This resistor will drop some of the source voltage, leaving the right amount for the actual load, as shown to the right.
Usually, however, this doesn't work too well. The required value of the series dropping resistor will almost never be a standard value, and the cost of having special values manufactured for specific circuits is prohibitive. For example, suppose we have a 9 volt battery as your main power source, and want to operate a load that requires 5 volts at 3.5 milliamperes. Our series resistor, R, must drop 4 volts at 3.5 mA. Using Ohm's Law to calculate the required resistance value, we find that we need a resistance of 4/0.0035 = 1142.8571
A more practical solution to the problem is to use two resistors in series, and use the voltage appearing across one of them. This configuration is known as a voltage divider because it divides the source voltage into two parts. The basic circuit is shown to the right.
In this circuit, the output voltage, VOUT, can be set accurately as a fraction of the source voltage, E. Using our example above, we want to select R1 and R2 such that we will drop 4 volts across R1, leaving 5 volts across R2. Since VOUT is the voltage across R2, this will give us the voltage we want. But how do we find the correct values of resistance to do this?
The first step is to note that, with no external connection to VOUT, this is simply a series connection and the same current must flow through both resistors. (We'll deal with the load current shortly.) Therefore, in accordance with Ohm's Law, the ratio of voltage across these resistors will be equal to the ratio of the resistance values themselves. In this case, the voltage ratio we want is 4:5, or 0.8:1. Therefore, we want this resistor ratio as well.
But there are 20 different standard 5% resistance values in each decade range, so there are lots of possible resistance ratios. Most will be wrong for this purpose, of course. So how do we find two standard resistance values that will give the ratio we want? We could do it manually, testing each possible combination. But a better way is to let the computer do the tedious work and present all options to us. Then we can select the values we want from the list of valid possibilities.
To this end, remember that ratio of 0.8, and insert it into the table in the Resistance Ratio Calculator on these pages. Press
The table shows the significant digits of standard 5% resistor values. When you type in a ratio, it calculates the corresponding significant digits that would be required to complete that ratio. All you need to do is pick out ratios of valid significant digits. In this case, the table shows that you can use resistance values of 15:12, 20:16, or 30:24 to obtain the ratio of 1:0.8. If you had specified 1:1.25 (the inverse of 0.8:1), you would have gotten the ratios of 12:15, 16:20, and 24:30. Either way, these are workable choices, while all other choices fail to match standard values.
We can get a VOUT of 5 volts, then, if we set R1 = 1.2k and R2 = 1.5k. We can also get the same VOUT if we make R1 = 12k and R2 = 15k. The exact resistor values don't matter, so long as their ratio is correct.
The one thing we haven't accounted for as yet is the current drawn by the load. This will necessarily upset the resistance balance, since any load current will flow through R1, but not through R2. As a result, the load will reduce output voltage of the voltage divider by some amount. Appropriately, this effect is called loading.
To calculate the effect of loading and its extent in any given instance, we must realize that the voltage divider circuit behaves in exactly the same way as a battery of voltage VOUT with a series resistor whose value is equal to the parallel combination of R1 and R2. The figure to the right shows the equivalent circuit for our example voltage divider.
Now, we noted earlier that our example load draws 3.5mA at 5 volts. In accordance with Ohm's Law, this current will drop a voltage of 2.33333 volts across that 667
If we reduce the resistors in the voltage divider to 120
The drawback of this is that such low resistance values will draw a significant amount of current from the original source. This is probably acceptable if the original source is an electronic power supply, but not if it's an actual battery. Thus, this use of a voltage divider is reasonable and appropriate in some circumstances, but not in all cases.
The voltage divider is a very simple circuit that can be highly accurate if not loaded down. In many cases it cannot be used directly, as we have seen. However, in such cases it can either be adapted, or augmented with other components to preserve its operation while avoiding the problems that can occur. Thus, even in those cases where a voltage divider by itself is not sufficient to meet the need, it can serve as the basis of a circuit that will perform as required.
Three-Terminal Resistor Configuration
The Wye ("Y") Configuration
Consider the schematic diagram to the right. Because the resistors are shown schematically in a way that resembles the letter "Y," this arrangement is known as a "Y" (or "Wye") configuration. At first glance, it may seem a bit strange, and perhaps not very useful. Is it a voltage divider with an extra resistor in series with the output? A kind of mixer with two inputs and one output? Or maybe a splitter with one input and two outputs? Or something else entirely?
In fact, this circuit is not generally used in any of the suggested ways. Nevertheless, it is found in a wide variety of devices and larger circuits, often with capacitors or inductors substituted for one or more of the resistors shown here.
Even with just resistors, this configuration is useful in a number of different ways. It is often used to provide a specific amount of signal attenuation in a transmission line, while still matching the characteristic impedance of the transmission line in both directions. It can also be used to match two transmission lines of different characteristic impedances, so that both transmission lines operate at maximum efficiency.
A key point to note about this configuration is that the resistance between any two of the three external connections will be the series combination of two of the three resistors. Thus, the resistance between points X and Y, which we can call RXY will be R1 + R2. As a consequence of this, the effective resistance of this circuit will always be substantially greater than the values of the individual resistors. This means relatively small resistor values can be used to obtain greater resistance effects.
It is often easier to understand this circuit if it is rearranged as shown in the second figure to the right. Now we see a clear input and output, although the actual electrical connections among the resistors are unchanged. This schematic arrangement also takes up much less room on paper or a Web page, so it is often preferred for display purposes. When it is displayed like this, it is generally identified as a "T" configuration rather than "Y." However, this does not change its behavior in any way.
The Delta Configuration
A second configuration involving three resistors is shown in the schematic diagram to the right. Since the modern English/Latin alphabet has no symbol that resembles this triangle, we use the Greek letter "Delta" (
This configuration is just as important as the "Y" configuration above. An essential difference, however, is that in the Delta configuration, the resistance between any two points is a series-parallel combination of all three resistors. Therefore, the effective resistance of the circuit will be less than the values of the individual resistors involved. This can be very useful in situations where we want to be able to use larger resistance values than the circuit would normally require. Mathematically,
As you might expect, it is usually convenient to redraw this circuit as shown in the second figure to the right. The circuit is electronically the same, but now has clear input and output connections. Because this layout resembles the Greek letter "pi" (
Converting Between Wye and Delta Configurations
Although these two configurations look very different, it is quite possible for them to exhibit exactly the same resistance between each pair of external connections. Thus, RXY can be the same for both, as can RXZ and RYZ. In such a case, if you don't know the configuration, you have no way to tell which configuration is being used.
This is important, because sometimes it is necessary to convert from one configuration to the other. This can be because the calculated resistor values for one configuration are non-standard, or because the requirements of the circuit demand that resistance values be neither too large nor too small.
As a simple example, consider a case where such a circuit must use standard 5% resistors, and must also exhibit a resistance of 10K between any two connection points. Since this allows all three resistors to be the same, we can easily determine that the Wye confiuration would require three resistors of 5K each.
Unfortunately, this is a non-standard value. We could use 5.1K resistors instead, and hope that the result is close enough to be acceptble. But if we convert to a Delta configuration, we find that we can get the same result using three resistors of 15K each. This is a standard value, easily obtained. Therefore, we can get exactly the results we want without difficulty and without hoping we're "close enough."
Of course, the resistors in either configuration do not have to be of the same value (and usually aren't). Therefore, we need the appropriate mathematical equations to perform the conversions in either direction. For those interested in the derivation of these equations. The end results, using the resistor designations in the figures above, are:
R2 = (RA×RC)/(RA + RB + RC)
R3 = (RA×RB)/(RA + RB + RC)
RA = R2 + R3 + (R2×R3/R1)
RB = R1 + R3 + (R1×R3/R2)
RC = R1 + R2 + (R1×R2/R3)
You will encounter both configurations throughout the field of electronics, used in many different ways. You should be ready to recognize them when you see them.
Converting Between Delta and Wye Configurations
Because both the Delta and Wye configurations are used often throughout electronics, they appear in the design of new circuits as well as in older ones. Therefore, it is important to be able to convert back and forth between the two. Usually it is enough to know the formulas or equations required to perform these conversions. However, for complete understanding, it is a good idea to know and understand how these formulas are derived and why they work.
To do this, we will first determine the initial equations describing both configurations. Then we will be able to solve them simultaneously to determine the conversion formulas.
Deriving the Initial Equations
We derive the initial equations by noting that for the two configurations to appear the same externally, it is necessary for the externally measured resistances, RXY, RXZ, and RYZ, to be the same for either configuration. Therefore, we can determine these and set them equal to each other. This gives us our three initial simultaneous equations:
| <=====> | |
|---|
| (1) RXY = R1 + R2 = | (RA + RB) × RC | = | RA×RC + RB×RC |
| | | ||
| (RA + RB) + RC | RA + RB + RC | ||
| (2) RXZ = R1 + R3 = | (RA + RC) × RB | = | RA×RB + RB×RC |
| | | ||
| (RA + RC) + RB | RA + RB + RC | ||
| (3) RYZ = R2 + R3 = | (RB + RC) × RA | = | RA×RB + RA×RC |
| | | ||
| (RB + RC) + RA | RA + RB + RC | ||
Solving For R1, R2, and R3
From the basic equations above, it looks much easier to solve for the numbered resistors, so we'll do that first. That in turn may make it easier to solve these equations in the other direction. We'll isolate R1 by subtracting Equation (3) from Equation (1) to get R1 - R3, and then add Equation (2) to that result. The expressions for R2 and R3 can be derived the same way, and will be quite similar.
| (1) - (3) | = | R1 + R2 - (R2 + R3) |
| = | R1 - R3 | |
| (1) - (3) + (2) | = | R1 - R3 + R1 + R3 |
| = | 2R1 | |
Applying these to the lettered resistors, we get:
| R1 - R3 | = | RA×RC + RB×RC | - | RA×RB + RA×RC |
| | | |||
| RA + RB + RC | RA + RB + RC | |||
| = | RB×RC - RA×RB | |||
| | ||||
| RA + RB + RC | ||||
| 2R1 | = | RB×RC - RA×RB | + | RA×RB + RB×RC |
| | | |||
| RA + RB + RC | RA + RB + RC | |||
| = | 2(RB×RC) | |||
| | ||||
| RA + RB + RC | ||||
| R1 | = | RB×RC | ||
| | ||||
| RA + RB + RC | ||||
| R2 | = | RA×RC | ||
| | ||||
| RA + RB + RC | ||||
| R3 | = | RA×RB | ||
| | ||||
| RA + RB + RC | ||||
Solving for RA, RB, and RC
To solve these expressions for the lettered resistors, we first note that the equation for R1 above contains only a single instance of RA. Therefore we will rearrange that equation and solve it for RA. Then we will substitute that value for RA in the denominator of the equation for R2, and simplify the result as much as possible. This will give us simplified relationships that we can more easily apply to these expressions.
| R1 | = | RB × RC | |||
| | |||||
| RA + RB + RC | |||||
| R1(RA + RB + RC) | = | RB × RC | |||
| RA + RB + RC | = | RB × RC | |||
| | |||||
| R1 | |||||
| RA | = | RB × RC | - RB - RC | ||
| | |||||
| R1 | |||||
| R2 | = | RA × RC | |||
| | |||||
| RA + RB + RC | |||||
| R2(RA + RB + RC) | = | RA × RC | |||
| R2( | RB × RC | - RB - RC + RB + RC) | = | RA × RC | |
| | |||||
| R1 | |||||
| R2 × RB × RC | = | R1 × RA × RC | |||
| [ R3 × RC = ] R2 × RB | = | R1 × RA | |||
| RB | = | R1 × RA | |||
| | |||||
| R2 | |||||
| RC | = | R1 × RA | |||
| | |||||
| R3 | |||||
Now we can replace RB and RC with very simple expressions involving RA, so that we will be able to solve for RA in terms of only numbered resistors. RB and RC can then be found in the same way, and will have similar expressions.
| RA | = | RB × RC | - RB - RC | |||||||
| | ||||||||||
| R1 | ||||||||||
| R1 × RA | R1 × RA | |||||||||
| | × | | ||||||||
| RA | = | R2 | R3 | - | R1 × RA | - | R1 × RA | |||
| | | | ||||||||
| R1 | R2 | R3 | ||||||||
| RA | = | R1 × RA × R1 × RA | - | R1 × RA | - | R1 × RA | ||||
| | | | ||||||||
| R1 × R2 × R3 | R2 | R3 | ||||||||
| 1 | = | R1 × RA | - | R1 | - | R1 | ||||
| | | | ||||||||
| R2 × R3 | R2 | R3 | ||||||||
| R1 × RA | = | 1 | + | R1 | + | R1 | ||||
| | | | ||||||||
| R2 × R3 | R2 | R3 | ||||||||
| R1 × RA | = | R2 × R3 | + | R1 × R3 | + | R1 × R2 | ||||
| RA | = | R2 × R3 | + | R3 | + | R2 | ||||
| | ||||||||||
| R1 | ||||||||||
| RB | = | R1 × R3 | + | R1 | + | R3 | ||||
| | ||||||||||
| R2 | ||||||||||
| RC | = | R1 × R2 | + | R1 | + | R2 | ||||
| | ||||||||||
| R3 | ||||||||||
You don't really need to know these derivations, although understanding them will help you with your understanding of electronics in general. However, you should know the conversion formulas themselves, and be able to apply them when designing or analyzing electronic circuits.
| |
History
The circuit we now know as the Wheatstone Bridge was actually first described by Samuel Hunter Christie (1784-1865) in 1833. However, Sir Charles Wheatstone invented many uses for this circuit once he found the description in 1843. As a result, this circuit is known generally as the Wheatstone Bridge.
To this day, the Wheatstone bridge remains the most sensitive and accurate method for precisely measuring resistance values.
The Basic Bridge Circuit
The fundamental concept of the Wheatstone Bridge is two voltage dividers, both fed by the same input, as shown to the right. The circuit output is taken from both voltage divider outputs, as shown here.
In its classic form, a galvanometer (a very sensitive dc current meter) is connected between the output terminals, and is used to monitor the current flowing from one voltage divider to the other. If the two voltage dividers have exactly the same ratio (R1/R2 = R3/R4), then the bridge is said to be balanced and no current flows in either direction through the galvanometer. If one of the resistors changes even a little bit in value, the bridge will become unbalanced and current will flow through the galvanometer. Thus, the galvanometer becomes a very sensitive indicator of the balance condition.
Using the Wheatstone Bridge
In its basic application, a dc voltage (E) is applied to the Wheatstone Bridge, and a galvanometer (G) is used to monitor the balance condition. The values of R1 and R3 are precisely known, but do not have to be identical. R2 is a calibrated variable resistance, whose current value may be read from a dial or scale.
An unknown resistor, RX, is connected as the fourth side of the circuit, and power is applied. R2 is adjusted until the galvanometer, G, reads zero current. At this point, RX = R2×R3/R1.
This circuit is most sensitive when all four resistors have similar resistance values. However, the circuit works quite well in any event. If R2 can be varied over a 10:1 resistance range and R1 is of a similar value, we can switch decade values of R3 into and out of the circuit according to the range of value we expect from RX. Using this method, we can accurately measure any value of RX by moving one multiple-position switch and adjusting one precision potentiometer.
Applications of the Wheatstone Bridge
It is not possible to cover all of the practical variations and applications of the Wheatstone Bridge, let alone all types of bridges, in a single Web page. Sir Charles Wheatstone invented many uses himself, and others have been developed, along with many variations, since that time. One very common application in industry today is to monitor sensor devices such as strain gauges. Such devices change their internal resistance according to the specific level of strain (or pressure, temperature, etc.), and serve as the unknown resistor RX. However, instead of trying to constantly adjust R2 to balance the circuit, the galvanometer is replaced by a circuit that can be calibrated to record the degree of imbalance in the bridge as the value of strain or other condition being applied to the sensor.
A second application is used by electrical power distributors to accurately locate breaks in a power line. The method is fast and accurate, and does not require a large number of field technicians.
Other applications abound in electronic circuits. We'll see a number of them in action as these pages continue to expand.
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 fluorescent 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.
The basic unit of capacitance is the farad, named after British physicist and chemist Michael Faraday (1791 - 1867). For you physics types, the basic equation for capacitance is:
- where:
- C is the capacitance in farads.
- q is the accumulated charge in coulombs.
- V is the voltage difference between the capacitor plates.
Verbally, a capacitance of one farad will exhibit a voltage difference of one volt when an electrical charge of one coulomb is moved from one plate to the other through the capacitance.
To help put this in perspective, one ampere of current represents one coulomb of charge passing a given point in an electrical circuit in one second.
In practical terms, the farad (f) represents a extremely large amount of capacitance. Real-world circuits require capacitance values very much smaller. Therefore, we use microfarads (µf) and picofarads (pf) to represent practical capacitance values. The use of the micro- and pico- prefixes is standard. 1 µf = 1 × 10-6 f and 1 pf = 1 × 10-6 µf. Sometimes you will see the designation µµf in place of pf; they have the same meaning.
Like resistors, capacitors are generally manufactured with values to two significant digits. Also, small capacitors for general purposes have practical values greater than 1 pf and less than 1 µf. As a result, a useful convention has developed in reading capacitance values. If a capacitor is marked "47," its value is 47 pf. If it is marked .047, its value is .047 µf. Thus, whole numbers express capacitance values in picofarads while decimal fractions express values in microfarads. Any capacitor manufactured with a value of 1 µf or greater is physically large enough to be clearly marked with its actual value.
A newer nomenclature has developed, where three numbers are printed on the body of the capacitor. The third digit in this case works like the multiplier band on a resistor; it tells the number of zeros to tack onto the end of the two significant digits. Thus, if you see a capacitor marked "151," it is not a precision component. Rather, it is an ordinary capacitor with a capacitance of 150 pf. In this nomenclature, all values are given in picofarads. Therefore you might well see a capacitor marked 684, which would mean 680000 pf, or 0.68 µf.
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| | |
| | |
| | Capacitors in Series |
| | |
Connecting capacitors in series is no more difficult than connecting resistors in series. After all, as an electronic component a capacitor has two leads, so capacitors can be connected to each other or to other types of components very easily. But what is the effect of such a connection?
We have already seen resistors connected in series and in parallel. Clearly, capacitors can be connected in series and in parallel as well, or they can be interconnected with resistors. We'll look at the latter possibilities elsewhere in this series of pages. For the moment, we'll concentrate on capacitors in series.
The figure to the right shows two capacitor symbols connected in series. As a starting point, let's assume that these are two identical capacitors. The connection between them is assumed to have no resistance, and therefore no effect on the behavior of these to capacitors or any circuit in which they may be connected. Therefore, this connection may be any length, covering any distance, without having any noticeable effect.
This being the case, let's shorten the distance between capacitors to zero. This means that the connected plates of the two capacitors will actually touch, as shown in the second image to the right.
Next, we recognize that the thickness of that center plate is unimportant; it's simply a broad conductor between the two capacitors. Therefore we can make this center plate as thin as we want. Therefore, at least in theory, we can reduce it to atomic thickness without any effect on the capacitance of the series combination.
But that center plate is nothing more than an equipotential plane in the middle of an electric field. Since the outer plates are still parallel to each other, removing the center plate won't change the total electric field. This leaves us with a single capacitor, but with the plates spaced twice as far apart as for either of the original capacitors. As a result of this, the combined capacitance of the two identical capacitors in series is just half the capacitance of either one.
Capacitors of differing values may also be connected in series. In such cases, we can note that such capacitors may all be constructed with the same plate size and shape, with only the plate spacing determining the capacitance value. Connecting them in series, we note that the effective spacing between the end plates is equal to the sum of the spacings of the individual capacitors. This applies to any number of such capacitors in series.
Since the capacitance of any capacitor is inversely proportional to the distance between the plates, we can express the total capacitance, CT, of any number of capacitors connected in series as:
| 1 | = | 1 | + | 1 | + | 1 | + ··· |
| CT | C1 | C2 | C3 |
For two capacitors in series, this can be written as:
| CT = | C1 × C2 |
| C1 + C2 |
You probably noticed that this formula looks just like the formula for resistors in parallel. This is correct, and it is no accident. Remember that resistance is a property that reduces current flow, while capacitance is a property that enables current flow, at least briefly. Since these are inverse properties, it is only reasonable that they would behave in opposite ways when connected in series or parallel. This is in fact the case.
Capacitors in Parallel
When we connect two capacitors in parallel, as with resistors in parallel, the same source voltage is applied to each capacitor. The figure to the right shows such a parallel connection.
When this is done with capacitors, each capacitor charges to the same voltage, without regard to the behavior of the other capacitor. Logically, then, it would seem that the total capacitance would simply be the sum of the capacitance values of the individual capacitors connected in parallel. This is in fact the case, and the equation to determine the total capacitance, CT, of any number of capacitors in parallel is:
In essence, by connecting capacitors in parallel we have increased the surface area of the plates without changing the distance between them. If we do this with two identical capacitors, we effectively double the surface area of the plates while leaving the spacing between them unchanged. This doubles the capacitance of the combination.
Circuit Components: the Inductor
One characteristic of electricity is that as current flows it generates a magnetic field. The greater the current, the stronger the magnetic field it generates. However, this magnetic field is generally small and weak, and can't be used for very much. Indeed, most of the time it doesn't have a noticeable effect on anything less sensitive than a small compass needle. Is there a way we can intensify this field so we can experiment with it and study its properties?
In the figure to the right, electrons are moving through a wire from left to right, as shown by the blue arrows. This motion of electrically charged electrons generates a circular magnetic field around the wire, and extending along the entire length of the wire, as indicated by the green lines. The direction of the magnetic lines of force shown here is upwards on the "front" side of the wire, and downwards behind it.
You can always determine the direction of the magnetic field by applying the Left Hand Rule: Grasp the wire in your left hand, with your thumb pointing along the wire in the direction of electron flow. Your fingers will curl around the wire, pointing in the direction of the magnetic field.
Note: Under the original assumptions of conventional current, this was stated as the Right Hand Rule, because current carriers were assumed to be positive. Since we are using the more modern electron current specifications, we must switch to a Left Hand Rule to correctly describe the direction of the magnetic field.
If we have two wires close together, with the same current flowing through them but in opposite directions as shown to the left, the magnetic field between the two wires will be the sum of the two separate fields, and therefore will be stronger than the field around a single wire. However, this doesn't help much — adding a third wire must reinforce one of these two, but oppose the other. Hmmmm. Maybe we can make use of this phenomenon, but clearly it won't work by itself.
On the other hand, if we put two wires next to each other with each one carrying the same amount of current in the same direction (see the figure to the right), an interesting phenomenon occurs. The magnetic fields between the two wires oppose each other and cancel out, but the overall field around both wires together is strengthened. Adding more wires in this manner enhances this effect, making the overall magnetic field still stronger.
Is there an easy way to accomplish this?
The figure to the left shows a wire that has been wrapped into a spiral structure, forming a coil. This structure combines both effects of adjacent, current-carrying wires discussed above. The magnetic field through the middle of the coil is directed from left to right, and is highly intensified. This magnetic field gives the coil some interesting and useful properties, which we will cover in detail when we discuss the behavior of coils in an electrical circuit.
The property conferred on this component by the concentrated magnetic field is known as inductance. The effect of inductance is to oppose any change in current through itself. It does this by generating an EMF across its terminals which opposes the applied voltage. As a result, the current through an inductance can only change gradually; it cannot change instantaneously as it could with only resistors in the circuit. The coil will store or release energy in its magnetic field as rapidly as necessary to oppose any such change.
The unit of inductance is the henry (H). By definition, one henry is that amount of inductance that will cause a counter EMF of 1 volt to be generated when the current changes at a rate of 1 ampere/second. Practical values of inductance range from a few microhenrys (µH) up to tens of henrys.
The image to the right shows a few typical, commercially-available coils. The large one to the left is mounted on an iron core to help concentrate the magnetic field and thus augment the inductance of the component. It has an inductance of 1 henry. To its right is a small coil with a movable core made partly of powdered iron. This allows the core to be adjusted to set the precise value of inductance, which is on the order of 30 microhenrys (µH). In the foreground is a 50 millihenry (mH) coil, consisting of multiple layers of wire wrapped on a non-magnetic core.
Each of these devices can be purchased directly, and each of them has practical applications in electronics.
The schematic symbols to the right represent inductors, or coils. Symbol A is used for a basic inductor with only air anywhere in the magnetic field. Symbol B shows an inductor with a core made of powdered iron (known as ferrite). Such a core helps to concentrate the magnetic field somewhat, and so increases the effective inductance of the coil. Symbol C shows a laminated iron core. This kind of core concentrates the magnetic field greatly, and therefore increases the effective inductance even more than a ferrite core.
As you can see, in each case the symbol itself suggests the multiple turns of wire that form the coil.
Inductors in Series
The coil, or inductor, has a property which forces us to treat it differently from resistors and capacitors: its magnetic field. Where a resistor generates no such field and a capacitor generates an electric field that remains internal to the capacitor, the coil's magnetic field extends beyond itself, and can easily overlap the turns of wire in an adjacent coil.
Because of this, we must deal with two separate concepts when combining inductors in a circuit. These are known as self-inductance, which is the inherent inductance of the coil under consideration; and mutual inductance, which is the inductive effect of magnetic interaction between two coils.
Mutual inductance behaves just like self-inductance in many ways, and is defined in the same way. If a change in current of 1 ampere/second in one coil causes a counter EMF of one volt to be generated in the other coil, they have a mutual inductance of 1 henry.
When we connect two inductors in series, as shown to the right, we have the question of whether or not their magnetic fields interact. If not, then their inductances simply add:
However, if they are physically placed so that they do exhibit a mutual inductance, this isn't sufficient. We must include a mutual inductance where each coil's magnetic field affects the other coil. Furthermore, we must take into account whether the magnetic fields of the two coils are aiding each other or opposing each other, since each self-inductance can be either increased or decreased by the value of the mutual inductance, designated M. Therefore, we must select one of two equations:
LT = (L1 - M) + (L2 - M)
Or,
If you try to connect three or more coils in series, you must take into account the mutual inductance between each pair of coils. That's three different mutual inductances for three coils, and six mutual inductances for four coils.
Inductors in Parallel
When we connect inductors in parallel, as with other components, we have two separate paths for current to flow. This is clear in the figure to the right. However, just as with inductors in series, we must take into account the mutual inductance between the two coils.
To do this, we first note that, as with inductors in series, the mutual inductance can either add to or subtract from the self-inductance of each coil. With this in mind, the general equation for two inductors in parallel is:
| 1 | = | 1 | + | 1 |
| LT | L1 ± M | L2 ± M |
As you would expect, the sign applied to M depends on whether the magnetic fields aid (+) or oppose (-) each other.
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.
Resistors and Capacitors Together
Now that we have looked at each of the three basic types of electronic components, we need to explore how they behave in various combinations. As we do so, remember that while each component still retains its basic properties, the combination can have its own characteristics, which may not seem intuitive at first.
On this page, we'll begin by considering a resistor and capacitor working together in a circuit, and see how the resistor affects the charging and discharging of the capacitor.
Consider the circuit shown to the right. Initially, we will have switch S in position 2. Capacitor C is fully discharged and no current flows through R and C. The circuit is quiescent at this point.
Now we move the switch to position 1. This connects the series combination of R and C to the battery. Current can flow through the circuit, and the capacitor will begin to charge. The question is, how fast?
Keep in mind that the voltage across a capacitor cannot change instantaneously. Therefore, at that first instant the entire battery voltage, E, appears across resistor R, and the charging current for C is determined by Ohm's Law: I = E/R.
However, now the capacitor voltage, VC begins to increase. This reduces the voltage, VR, that remains across the resistor. Therefore the charging current will be reduced slightly, and the capacitor will charge more slowly than before. This will continue until the capacitor has charged to the voltage E, and there is no further current flow in this circuit.
But this isn't quite enough. We can see that the values of R and C will affect the amount of time it takes for C to become fully charged. But we'd like to be able to state the appropriate equation so we can determine not only the charging time but also the way in which VC and the circuit current will change while C is charging.
To accomplish this, we go back to some basic definitions. First, we note that by moving one coulomb of electric charge from one plate of a 1 farad capacitor to the other, we will change the voltage between plates by 1 volt. We can change the size of the capacitance, adjust the voltage, and thereby adjust the amount of charge required to make the change. However, the basic equation still holds: E = q/C, where q is the electric charge in coulombs.
The other definition is for the current flowing in the circuit: one Ampere of current consists of one coulomb of charge passing a given point in a circuit in one second.
In this circuit, however, the charging current is constantly changing as the voltage across C increases. Therefore, we must look at a steadily decreasing rate of charge. This brings us to a bit of differential calculus. If you aren't familiar with this, don't worry; you'll be able to make use of the results. But for completeness, we include the appropriate expression here:
| iC = C | dvC |
| dt |
Qualitatively, the current flowing through the capacitor is directly proportional to the value of the capacitor itself (high value capacitors charge more slowly), and is directly proportional to the change in capacitor voltage over time. The use of differential calculus allows us to track the changing current on an instant-by-instant basis.
But the current charging the capacitor is also the current flowing through R. And the voltage across R is whatever part of E that hasn't already been built up as the charge on C. Therefore, we can apply Ohm's Law here:
| iC = iR = | E - vC |
| R |
Solving differential equations is beyond the scope of this page. However, we can present the final equation that describes the capacitor voltage at any time t, for any values of R and C, and any battery voltage E:
Here, ε is the base for natural logarithms, with a value of approximately 2.7182818. Because we are using it in this fashion, the equation above is known as an exponential equation.
At the moment switch S is closed, time t = 0. Since ε0 = 1, we see that:
This is exactly what we would expect, since the capacitor is completely discharged at the start of the sequence. But what does the rest of the charging curve look like? Let's plot this expression over time, as the capacitor charges, and see how it behaves.
The figure to the right shows the capacitor voltage, vC, as a percentage of E as the capacitor charges over time. Thus, it is the plot of the equation:
| vC | = 1 - ε(-t/RC) |
| E |
Note the product RC in the exponent of ε. This is very important, because it shows that both R and C control the charging time equally. Also, because that exponent is (-t/RC), if we set RC = t, the exponent will be -1. Therefore, we show time in this graph as multiples of RC. In addition, the RC product is identified as the time constant for this circuit.
At first glance, this would seem to be very strange. How can the product of a resistance in ohms and a capacitance in farads possibly give us a time in seconds? To understand how this is possible, we go back to the basic definitions and some dimensional analysis.
- Resistance
Resistance opposes the flow of current through a circuit. By Ohm's Law, R = E/I. Thus, 1 ohm may also be expressed as 1 volt/ampere.
- Current
Current is a measure of the amount of charge flowing through a circuit in a given amount of time. By primary definition, 1 ampere is equal to 1 coulomb/second.
- Capacitance
Capacitance is the capacity to hold an electrical charge. A capacitance of 1 farad will exhibit a change of 1 volt if 1 coulomb of charge is moved from one plate to the other. Hence, 1 farad can be expressed as 1 coulomb/volt.
Putting these three basic definitions together we get the following progression:
| RC | = | ohms | × | farads |
| = | volts | × | coulombs | |
| amperes | volts | |||
| = | volts | × | coulombs | |
| coulombs/seconds | volts | |||
| = | volts × seconds | × | coulombs | |
| coulombs | volts | |||
| = | seconds | |||
Thus, we see that the RC product is indeed a measure of time, and can properly be described as the time constant of this circuit. This in turn means that this curve can be used to determine the voltage to which any capacitor will charge through any resistance, over any period of time, towards any source voltage. It is the general curve describing the voltage across a charging capacitor, over time.
Theoretically, C will never fully charge to the source voltage, E. In the first time constant, C charges to 63.2% of the source voltage. During the second time constant, C charges to 86.5% of the source voltage, which is also 63.2% of the remaining voltage difference between E and vC. This continues indefinitely, with vC continually approaching, but never quite reaching, the full value of E. However, at the end of 5 time constants (5RC), vC has reached 99.3% of E. This is considered close enough for practical purposes, and the capacitor is deemed fully charged at the end of this period of time.
Now that we have charged our capacitor, what happens if we move switch S in our original circuit to position 2 (We have repeated the circuit to the right for easier reference)? We have disconnected resistor R from the source, E, and connected it in parallel with capacitor C instead.
At this point, the capacitor has a discharge path, so it will begin discharging through R. However, as vC continues to drop, the discharge current likewise decreases, in accordance with Ohm's Law. Therefore, it is logical to assume that the capacitor discharge curve will probably follow some of the same rules as the capacitor charge curve above. However, the resistor voltage is now the same as the capacitor voltage, since R and C are now in parallel. So what is the resulting equation?
The figure to the right shows the appropriate RC discharge curve. This graph shows the function:
| vC | = ε(-t/RC) |
| E |
Here, E refers to the starting voltage on the capacitor, which need not be the same as the battery voltage E in the schematic diagram above. In fact, if you look at the capacitor voltage after it has partly discharged, the same curve applies.
As with the charge curve, the discharge curve is exponential in shape. The RC time constant still applies; the capacitor is deemed to be fully discharged at the completion of five time constants.
