Electronics

Basics of Digital Recording

2008-03-28T02:14:00.000-07:00

Basics of Digital Recording

CONVERTING SOUND INTO NUMBERS

In a digital recording system, sound is stored and manipulated as a stream of discrete numbers, each number representing the air pressure at a particular time. The numbers are generated by a microphone connected to a circuit called an ANALOG TO DIGITAL CONVERTER, or ADC. Each number is called a SAMPLE, and the number of samples taken per second is the SAMPLE RATE. Ultimately, the numbers will be converted back into sound by a DIGITAL TO ANALOG CONVERTER or DAC, connected to a loudspeaker.

Fig. 1 The digital signal chain

Figure 1 shows the components of a digital system. Notice that the output of the ADC and the input of the DAC consists of a bundle of wires. These wires carry the numbers that are the result of the analog to digital conversion. The numbers are in the binary number system in which only two characters are used, 1 and 0. (The circuitry is actually built around switches which are either on or off.) The value of a character depends on its place in the number, just as in the familiar decimal system. Here are a few equivalents:

          BINARY DECIMAL
              0=0
              1=1
             10=2
             11=3
            100=4
           1111=15
1111111111111111=65535

Each digit in a number is called a BIT, so that last number is sixteen bits long in its binary form. If we wrote the second number as 0000000000000001, it would be sixteen bits long and have a value of 1.

Word Size

The number of bits in the number has a direct bearing on the fidelity of the signal. Figure 2 illustrates how this works. The number of possible voltage levels at the output is simply the number of values that may be represented by the largest possible number (no "in between" values are allowed). If there were only one bit in the number, the ultimate output would be a pulse wave with a fixed amplitude and more or less the frequency of the input signal. If there are more bits in the number the waveform is more accurately traced, because each added bit doubles the number of possible values. The distortion is roughly the percentage that the least significant bit represents out of the average value. Distortion in digital systems increases as signal levels decrease, which is the opposite of the behavior of analog systems.

Fig. 2 Effect of word size

The number of bits in the number also determines the dynamic range. Moving a binary number one space to the left multiplies the value by two (just as moving a decimal number one space to the left multiplies the value by ten), so each bit doubles the voltage that may be represented. Doubling the voltage increases the power available by 6 dB, so we can see the dynamic range available is about the number of bits times 6 dB.

Sample Rate

The rate at which the numbers are generated is even more important than the number of bits used. Figure 3. illustrates this. If the sampling rate is lower than the frequency we are trying to capture, entire cycles will be missed, and the decoded result would be too low in frequency and might not resemble the proper waveform at all. This kind of mistake is called aliasing. If the sampling rate were exactly the frequency of the input, the result would be a straight line, because the same spot on the waveform would be measured each time. This can happen even if the sampling rate is twice the frequency of the input if the input is a sine or similar waveform. The sampling rate must be greater than twice the frequency measured for accurate results. (The mathematical statement of this is the Nyquist Theorem.) This implies that if we are dealing with sound, we should sample at least 40,000 times per second.

Fig. 3 Effects of low sample rates

The Nyquist rate (twice the frequency of interest) is the lowest allowable sampling rate. For best results, sampling rates twice or four times this should be used. Figure 4 shows how the waveform improves as the sampling rate is increased.

Fig. 4 Effect of increasing sample rate

Even at high sample rates, the output of the system is a series of steps. A Fourier analysis of this would show that everything belonging in the signal would be there along with a healthy dose of the sampling rate and its harmonics. The extra junk must be removed with a low pass filter that cuts off a little higher than the highest desired frequency. (An identical filter should be placed before the ADC to prevent aliasing of any unsuspected ultrasonic content, such as radio frequency interference.)

If the sampling rate is only twice the frequency of interest, the filters must have a very steep characteristic to allow proper frequency response and satisfactorily reject the sampling clock. Such filters are difficult and expensive to build. Many systems now use a very high sample rate at the output in order to simplify the filters. The extra samples needed to produce a super high rate are interpolated from the recorded samples.

By the way, the circuits that generate the sample rate must be exceedingly accurate. Any difference between the sample rate used for recording and the rate used at playback will change the pitch of the music, just like an off speed analog tape. Also, any unsteadiness or jitter in the sample clock will distort the signal as it is being converted from or to analog form.

Recording Digital Data

Once the waveform is faithfully transformed into bits, it is not easy to record. The major problem is finding a scheme that will record the bits fast enough. If we sample at 44,100 hz, with a sixteen bit word size, in stereo, we have to accommodate 1,411,200 bits per second. This seems like a lot, but it is within the capabilities of techniques developed for video recording. (In fact, the first digital audio systems were built around VCRs. 44.1 khz was chosen as a sample rate because it worked well with them.)

To record on tape, a very high speed is required to keep the wavelength of a bit at manageable dimensions. This is accomplished by moving the head as well as the tape, resulting in a series of short tracks across the tape at a diagonal.

On a Compact Disc, the bits are microscopic pits burned into the plastic by a laser.The stream of pits spirals just like the groove on a record, but is played from the inside out.To read the data, light from a gentler laser is reflected off the surface of the plastic (from the back: the plastic is clear.) into a light detector. The pits disrupt this reflection and yield up the data.

In either case, the process is helped by avoiding numbers that are hard to detect, like 00001000. That example is difficult because it will give just a single very short electrical spike. If some numbers are unusable, a larger maximum (more bits) must be available to allow recording the entire set. On tape, twenty bits are used to record each sixteen bit sample, on CDs, twenty-eight bits are used.

Error Correction

Even with these techniques, the bits are going to be physically very small, and it must be assumed that some will be lost in the process. A single bit can be very important (suppose it represents the sign of a large number!), so there has to be a way of recovering lost data. Error correction is really two problems; how to detect an error, and what to do about it.

Fig. 5 Effects of data errors

The most common error detection method is parity computation. An extra bit is added to each number which indicates whether the number is even or odd. When the data is read off the tape, if the parity bit is inappropriate, something has gone wrong. This works well enough for telephone conversations and the like, but does not detect serious errors very well.

In digital recording, large chunks of data are often wiped out by a tape dropout or a scratch on the disk. Catching these problems with parity would be a matter of luck. To help deal with large scale data loss, some mathematical computation is run on the numbers, and the result is merged with the data from time to time. This is known as a Cyclical Redundancy Check Code or CRCC. If a mistake turns up in this number, an error has occurred since the last correct CRCC was received.

Once an error is detected, the system must deal gracefully with the problem. To make this possible, the data is recorded in a complex order. Instead of word two following word one, as you might expect, the data is interleaved, following a pattern like:

words 1,5,9,13,17,21,25,29,2,6,10,14,18,22,26,30,3,7,15,19,27 etc.

With this scheme, you could lose eight words, but they would represent several isolated parts of the data stream, rather than a large continuous chunk of waveform. When a CRC indicates a problem, the signal can be fixed. For minor errors, the CRCC can be used to replace the missing numbers exactly. If the problem is more extensive, the system can use the previous and following words to reconstruct a passable imitation of the missing one. One of the factors that makes up the price difference in various digital systems is the sophistication available to reconstruct missing data.

The Benefits of Being Digital

You may be wondering about the point of all of this, if it turns out that a digital system is more complex than the equivalent analog circuit. Digital circuits are complex, but very few of the components must be precise; most of the circuitry merely responds to the presence or absence of current. Improving performance is usually only a matter of increasing the word size or the sample rate, which is achieved by duplicating elements of the circuit. It is possible to build analog circuits that match digital performance levels, but they are very expensive and require constant maintenance. The bottom line is that good digital systems are cheaper than good analog systems.

Digital devices usually require less maintenance than analog equipment. The electrical characteristics of most circuit elements change with time and temperature, and minor changes slowly degrade the performance of analog circuits. Digital components either work or don't, and it is much easier to find a chip that has failed entirely than one that is merely 10% off spec. Many analog systems are mechanical in nature, and simple wear can soon cause problems. Digital systems have few moving parts, and such parts are usually designed so that a little vibration or speed variation is not important.

In addition, digitally encoded information is more durable than analog information, again because circuits are responding only to the presence or absence of something rather than to the precise characteristics of anything. As you have seen, it is possible to design digital systems so that they can actually reconstruct missing or incorrect data. You can hear every little imperfection on an LP, but minor damage is not audible with a CD.

The aspect of digital sound that is most exciting to the electronic musician is that any numbers can be converted into sound, whether they originated at a microphone or not. This opens up the possibility of creating sounds that have never existed before, and of controlling those sounds with a precision that is simply not possible with any other technique.

For further study, I recommend Principles of Digital Audio by Ken C Pohlmann, published by McGraw-Hill inc ISBN number0-07-050469-5.

Peter Elsea 1996

Basic Electronics

2008-03-28T02:07:00.000-07:00

Basic Electronics

The goal of this chapter is to provide some basic information about electronic circuits. We make the assumption that you have no prior knowledge of electronics, electricity, or circuits, and start from the basics. This is an unconventional approach, so it may be interesting, or at least amusing, even if you do have some experience. So, the first question is ``What is an electronic circuit?'' A circuit is a structure that directs and controls electric currents, presumably to perform some useful function. The very name "circuit" implies that the structure is closed, something like a loop. That is all very well, but this answer immediately raises a new question: "What is an electric current?" Again, the name "current" indicates that it refers to some type of flow, and in this case we mean a flow of electric charge, which is usually just called charge because electric charge is really the only kind there is. Finally we come to the basic question:

What is Charge?

No one knows what charge really is anymore than anyone knows what gravity is. Both are models, constructions, fabrications if you like, to describe and represent something that can be measured in the real world, specifically a force. Gravity is the name for a force between masses that we can feel and measure. Early workers observed that bodies in "certain electrical condition" also exerted forces on one another that they could measure, and they invented charge to explain their observations. Amazingly, only three simple postulates or assumptions, plus some experimental observations, are necessary to explain all electrical phenomena. Everything: currents, electronics, radio waves, and light. Not many things are so simple, so it is worth stating the three postulates clearly.

Charge exists.

We just invent the name to represent the source of the physical force that can be observed. The assumption is that the more charge something has, the more force will be exerted. Charge is measured in units of Coulombs, abbreviated C. The unit was named to honor Charles Augustin Coulomb (1736-1806) the French aristocrat and engineer who first measured the force between charged objects using a sensitive torsion balance he invented. Coulomb lived in a time of political unrest and new ideas, the age of Voltaire and Rousseau. Fortunately, Coulomb completed most of his work before the revolution and prudently left Paris with the storming of the Bastille.

Charge comes in two styles.

We call the two styles positive charge, $+$ , and (you guessed it) negative charge, $-$ . Charge also comes in lumps of $1.6 ×10$ ^-19C , which is about two ten-million-trillionths of a Coulomb. The discrete nature of charge is not important for this discussion, but it does serve to indicate that a Coulomb is a LOT of charge.

Charge is conserved.

You cannot create it and you cannot annihilate it. You can, however, neutralize it. Early workers observed experimentally that if they took equal amounts of positive and negative charge and combined them on some object, then that object neither exerted nor responded to electrical forces; effectively it had zero net charge. This experiment suggests that it might be possible to take uncharged, or neutral, material and to separate somehow the latent positive and negative charges. If you have ever rubbed a balloon on wool to make it stick to the wall, you have separated charges using mechanical action.

Those are the three postulates. Now we will present some of the experimental findings that both led to them and amplify their significance.

Voltage

First we return to the basic assumption that forces are the result of charges. Specifically, bodies with opposite charges attract, they exert a force on each other pulling them together. The magnitude of the force is proportional to the product of the charge on each mass. This is just like gravity, where we use the term "mass" to represent the quality of bodies that results in the attractive force that pulls them together (see Fig. 4.1).

**Figure 4.1:** Opposite charges exert an attractive force on each other, just like two masses attract. External force is required to hold them apart, and work is required to move them farther apart.

Electrical force, like gravity, also depends inversely on the distance squared between the two bodies; short separation means big forces. Thus it takes an opposing force to keep two charges of opposite sign apart, just like it takes force to keep an apple from falling to earth. It also takes work and the expenditure of energy to pull positive and negative charges apart, just like it takes work to raise a big mass against gravity, or to stretch a spring. This stored or potential energy can be recovered and put to work to do some useful task. A falling mass can raise a bucket of water; a retracting spring can pull a door shut or run a clock. It requires some imagination to devise ways one might hook on to charges of opposite sign to get some useful work done, but it should be possible.

The potential that separated opposite charges have for doing work if they are released to fly together is called voltage, measured in units of volts (V). (Sadly, the unit volt is not named for Voltaire, but rather for Volta, an Italian scientist.) The greater the amount of charge and the greater the physical separation, the greater the voltage or stored energy. The greater the voltage, the greater the force that is driving the charges together. Voltage is always measured between two points, in this case, the positive and negative charges. If you want to compare the voltage of several charged bodies, the relative force driving the various charges, it makes sense to keep one point constant for the measurements. Traditionally, that common point is called "ground."

Early workers, like Coulomb, also observed that two bodies with charges of the same type, either both positive or both negative, repelled each other (Fig. 4.2). They experience a force pushing

**Figure 4.2:** Like charges exert a repulsive force on each other. External force is required to hold them together, and work is required to push them closer.

them apart, and an opposing force is necessary to hold them together, like holding a compressed spring. Work can potentially be done by letting the charges fly apart, just like releasing the spring. Our analogy with gravity must end here: no one has observed negative mass, negative gravity, or uncharged bodies flying apart unaided. Too bad, it would be a great way to launch a space probe. The voltage between two separated like charges is negative; they have already done their work by running apart, and it will take external energy and work to force them back together.

So how do you tell if a particular bunch of charge is positive or negative? You can't in isolation. Even with two charges, you can only tell if they are the same (they repel) or opposite (they attract). The names are relative; someone has to define which one is "positive." Similarly, the voltage between two points $A$ and $B$ , $V$ _AB , is relative. If $V$ _AB is positive you know the two points are oppositely charged, but you cannot tell if point $A$ has positive charge and point $B$ negative, or visa versa. However, if you make a second measurement between $A$ and another point $C$ , you can at least tell if $B$ and $C$ have the same charge by the relative sign of the two voltages, $V$ _AB and $V$ _AC to your common point $A$ . You can even determine the voltage between $B$ and $C$ without measuring it: $V$ _BC = V_AC - V_AB . This is the advantage of defining a common point, like $A$ , as ground and making all voltage measurements with respect to it. If one further defines the charge at point $A$ to be negative charge, then a positive $V$ _AB means point $B$ is positively charged, by definition. The names and the signs are all relative, and sometimes confusing if one forgets what the reference or ground point is.

Current

Charge is mobile and can flow freely in certain materials, called conductors. Metals and a few other elements and compounds are conductors. Materials that charge cannot flow through are called insulators. Air, glass, most plastics, and rubber are insulators, for example. And then there are some materials called semiconductors, that, historically, seemed to be good conductors sometimes but much less so other times. Silicon and germanium are two such materials. Today, we know that the difference in electrical behavior of different samples of these materials is due to extremely small amounts of impurities of different kinds, which could not be measured earlier. This recognition, and the ability to precisely control the "impurities" has led to the massive semiconductor electronics industry and the near-magical devices it produces, including those on your RoboBoard. We will discuss semiconductor devices later; now let us return to conductors and charges.

Imagine two oppositely charged bodies, say metal spheres, that are being held apart, as in Fig. 4.3.

**Figure 4.3:** Two spheres with opposite charges are connected by a conductor, allowing charge to flow.

There is a force between them, the potential for work, and thus a voltage. Now we connect a conductor between them, a metal wire. On the positively charged sphere, positive charges rush along the wire to the other sphere, repelled by the nearby similar charges and attracted to the distant opposite charges. The same thing occurs on the other sphere and negative charge flows out on the wire. Positive and negative charges combine to neutralize each other, and the flow continues until there are no charge differences between any points of the entire connected system. There may be a net residual charge if the amounts of original positive and negative charge were not equal, but that charge will be distributed evenly so all the forces are balanced. If they were not, more charge would flow. The charge flow is driven by voltage or potential differences. After things have quieted down, there is no voltage difference between any two points of the system and no potential for work. All the work has been done by the moving charges heating up the wire.

The flow of charge is called electrical current. Current is measured in amperes (a), amps for short (named after another French scientist who worked mostly with magnetic effects). An ampere is defined as a flow of one Coulomb of charge in one second past some point. While a Coulomb is a lot of charge to have in one place, an ampere is a common amount of current; about one ampere flows through a 100 watt incandescent light bulb, and a stove burner or a large motor would require ten or more amperes. On the other hand low power digital circuits use only a fraction of an ampere, and so we often use units of $1/1000$ of an ampere, a milliamp, abbreviated as ma, and even $1/1000$ of a milliamp, or a microamp, $µa$ . The currents on the RoboBoard are generally in the milliamp range, except for the motors, which can require a full ampere under heavy load. Current has a direction, and we define a positive current from point $A$ to $B$ as the flow of positive charges in the same direction. Negative charges can flow as well, in fact, most current is actually the result of negative charges moving. Negative charges flowing from $A$ to $B$ would be a negative current, but, and here is the tricky part, negative charges flowing from $B$ to $A$ would represent a positive current from $A$ to $B$ . The net effect is the same: positive charges flowing to neutralize negative charge or negative charges flowing to neutralize positive charge; in both cases the voltage is reduced and by the same amount.

Batteries

Charges can be separated by several means to produce a voltage. A battery uses a chemical reaction to produce energy and separate opposite sign charges onto its two terminals. As the charge is drawn off by an external circuit, doing work and finally returning to the opposite terminal, more chemicals in the battery react to restore the charge difference and the voltage. The particular type of chemical reaction used determines the voltage of the battery, but for most commercial batteries the voltage is about 1.5 V per chemical section or cell. Batteries with higher voltages really contain multiple cells inside connected together in series. Now you know why there are 3 V, 6 V, 9 V, and 12 V batteries, but no 4 or 7 V batteries. The current a battery can supply depends on the speed of the chemical reaction supplying charge, which in turn often depends on the physical size of the cell and the surface area of the electrodes. The size of a battery also limits the amount of chemical reactants stored. During use, the chemical reactants are depleted and eventually the voltage drops and the current stops. Even with no current flow, the chemical reaction proceeds at a very slow rate (and there is some internal current flow), so a battery has a finite storage or shelf life, about a year or two in most cases. In some types of batteries, like the ones we use for the robot, the chemical reaction is reversible: applying an external voltage and forcing a current through the battery, which requires work, reverses the chemical reaction and restores most, but not all, the chemical reactants. This cycle can be repeated many times. Batteries are specified in terms of their terminal voltage, the maximum current they can deliver, and the total current capacity in ampere-hours.

You should handle batteries carefully, especially the ones we use in this course. Chemicals are a very efficient and compact way of storing energy. Just consider the power of gasoline or explosives, or the fact that you can play soccer for several hours powered only by a slice of cold pizza for breakfast. Never connect the terminals of a battery together with a wire or other good conductor. The battery we use for the RoboBoard is similar to the battery in cars, which uses lead and sulphuric acid as reactants. Such batteries can deliver very large currents through a short circuit, hundreds of amperes. The large current will heat the wire and possibly burn you; the resulting rapid internal chemical reactions also produce heat and the battery can explode, spreading nasty, reactive chemicals about. Charging these batteries with too large a current can have the same effect. Double check the circuit and instructions before connecting a battery to any circuit. More information on batteries can be found in Chapter 7.

Circuit Elements

Resistors

We need some way to control the flow of current from a voltage source, like a battery, so we do not melt wires and blow up batteries. If you think of current, charge flow, in terms of water flow, a good electrical conductor is like big water pipe. Water mains and fire hoses have their uses, but you do not want to take a drink from one. Rather, we use small pipes, valves, and other devices to limit water flow to practical levels. Resistors do the same for current; they resist the flow of charge; they are poor conductors. The value of a resistor is measured in ohms and represented by the Greek letter capital omega. There are many different ways to make a resistor. Some are just a coil of wire made of a material that is a poor conductor. The most common and inexpensive type is made from powdered carbon and a glue-like binder. Such carbon composition resistors usually have a brown cylindrical body with a wire lead on each end, and colored bands that indicate the value of the resistor. The key to reading these values is given in Chapter 2.

There are other types of resistors in your robot kit. The potentiometer is a variable resistor. When the knob of a potentiometer is turned, a slider moves along the resistance element. Potentiometers generally have three terminals, a common slider terminal, and one that exhibits increasing resistance and one that has decreasing resistance relative to the slider as the shaft is turned in one direction. The resistance between the two stationary contacts is, of course, fixed, and is the value specified for the potentiometer. The photoresistor or photocell is composed of a light sensitive material. When the photocell is exposed to more light, the resistance decreases. This type of resistor makes an excellent light sensor.

Ohm's Law

Ohm's law describes the relationship between voltage, $V$ , which is trying to force charge to flow, resistance, $R$ , which is resisting that flow, and the actual resulting current $I$ . The relationship is simple and very basic: . Thus large voltages and/or low resistances produce large currents. Large resistors limit current to low values. Almost every circuit is more complicated than just a battery and a resistor, so which voltage does the formula refer to? It refers to the voltage across the resistor, the voltage between the two terminal wires. Looked at another way, that voltage is actually produced by the resistor. The resistor is restricting the flow of charge, slowing it down, and this creates a traffic jam on one side, forming an excess of charge with respect to the other side. Any such charge difference or separation results in a voltage between the two points, as explained above. Ohm's law tells us how to calculate that voltage if we know the resistor value and the current flow. This voltage drop is analogous to the drop in water pressure through a small pipe or small nozzle.

Power

Current flowing through a poor conductor produces heat by an effect similar to mechanical friction. That heat represents energy that comes from the charge traveling across the voltage difference. Remember that separated charges have the potential to do work and provide energy. The work involved in heating a resistor is not very useful, unless we are making a hotplate; rather it is a byproduct of restricting the current flow. Power is measured in units of watts (W), named after James Watt, the Englishman who invented the steam engine, a device for producing lots of useful power. The power that is released into the resistor as heat can be calculated as $P=VI$ , where $I$ is the current flowing through the resistor and $V$ is the voltage across it. Ohm's law relates these two quantities, so we can also calculate the power as The power produced in a resistor raises its temperature and can change its value or destroy it. Most resistors are air-cooled and they are made with different power handling capacity. The most common values are 1/8, 1/4, 1, and 2 watt resistors, and the bigger the wattage rating, the bigger the resistor physically. Some high power applications use special water cooled resistors. Most of the resistors on the RoboBoard are 1/8 watt.

Combinations of Resistors

Resistors are often connected together in a circuit, so it is necessary to know how to determine the resistance of a combination of two or more resistors. There are two basic ways in which resistors can be connected: in series and in parallel. A simple series resistance circuit is shown in Figure 4.4.

**Figure 4.4:** Two Resistors in Series

Determining the total resistance for two or more resistors in series is very simple. Total resistance equals the sum of the individual resistances. In this case, $R$ _T=R₁+R2 . This makes common sense; if you think again in terms of water flow, a series of obstructions in a pipe add up to slow the flow more than any one. The resistance of a series combination is always greater than any of the individual resistors.

The other method of connecting resistors is shown in Figure 4.5, which shows a simple parallel resistance circuit.

**Figure 4.5:** Two Resistors in Parallel

Our water pipe analogy indicates that it should be easier for current to flow through this multiplicity of paths, even easier than it would be to flow through any single path. Thus, we expect a parallel combination of resistors to have less resistance than any one of the resistors. Some of the total current will flow through R1 and some will flow through R2, causing an equal voltage drop across each resistor. More current, however, will flow through the path of least resistance. The formula for total resistance in a parallel circuit is more complex than for a series circuit:

R

_T={1{1R₁}+{1R₂}...+{1R_n}}

(1)

Parallel and series circuits can be combined to make more complex structures, but the resulting complex resistor circuits can be broken down and analyzed in terms of simple series or parallel circuits. Why would you want to use such combinations? There are several reasons; you might use a combination to get a value of resistance that you needed but did not have in a single resistor. Resistors have a maximum voltage rating, so a series of resistors might be used across a high voltage. Also, several low power resistors can be combined to handle higher power. What type of connection would you use?

Capacitors

Capacitors are another element used to control the flow of charge in a circuit. The name derives from their capacity to store charge, rather like a small battery. Capacitors consist of two conducting surfaces separated by an insulator; a wire lead is connected to each surface. You can imagine a capacitor as two large metal plates separated by air, although in reality they usually consist of thin metal foils or films separated by plastic film or another solid insulator, and rolled up in a compact package. Consider connecting a capacitor across a battery, as in Fig. 4.6.

**Figure 4.6:** A simple capacitor connected to a battery through a resistor.

As soon as the connection is made charge flows from the battery terminals, along the wire and onto the plates, positive charge on one plate, negative charge on the other. Why? The like-sign charges on each terminal want to get away from each other. In addition to that repulsion, there is an attraction to the opposite-sign charge on the other nearby plate. Initially the current is large, because in a sense the charges can not tell immediately that the wire does not really go anywhere, that there is no complete circuit of wire. The initial current is limited by the resistance of the wires, or perhaps by a real resistor, as we have shown in Fig. 4.6. But as charge builds up on the plates, charge repulsion resists the flow of more charge and the current is reduced. Eventually, the repulsive force from charge on the plate is strong enough to balance the force from charge on the battery terminal, and all current stops. Figure 4.7 shows how the current might vary with

**Figure 4.7:** The time dependence of the current in the circuit of Fig. 4.6 for two values of resistance.

time for two different values of resistors. For a large resistor, the whole process is slowed because the current is less, but in the end, the same amount of charge must exist on the capacitor plates in both cases. The magnitude of the charge on each plate is equal.

The existence of the separated charges on the plates means there must be a voltage between the plates, and this voltage be equal to the battery voltage when all current stops. After all, since the points are connected by conductors, they should have the same voltage; even if there is a resistor in the circuit, there is no voltage across the resistor if the current is zero, according to Ohm's law. The amount of charge that collects on the plates to produce the voltage is a measure of the value of the capacitor, its capacitance, measured in farads (f). The relationship is $C = Q/V$ , where Q is the charge in Coulombs. Large capacitors have plates with a large area to hold lots of charge, separated by a small distance, which implies a small voltage. A one farad capacitor is extremely large, and generally we deal with microfarads ( $µf$ ), one millionth of a farad, or picofarads (pf), one trillionth $(10$ ^-12) of a farad.

Consider the circuit of Fig. 4.6 again. Suppose we cut the wires after all current has stopped flowing. The charge on the plates is now trapped, so there is still a voltage between the terminal wires. The charged capacitor looks somewhat like a battery now. If we connected a resistor across it, current would flow as the positive and negative charges raced to neutralize each other. Unlike a battery, there is no mechanism to replace the charge on the plates removed by the current, so the voltage drops, the current drops, and finally there is no net charge left and no voltage differences anywhere in the circuit. The behavior in time of the current, the charge on the plates, and the voltage looks just like the graph in Fig. 4.7. This curve is an exponential function: $exp (-t/RC)$ . The voltage, current, and charge fall to about 37% of their starting values in a time of $R ×C$ seconds, which is called the characteristic time or the time constant of the circuit. The $RC$ time constant is a measure of how fast the circuit can respond to changes in conditions, such as attaching the battery across the uncharged capacitor or attaching a resistor across the charged capacitor. The voltage across a capacitor cannot change immediately; it takes time for the charge to flow, especially if a large resistor is opposing that flow. Thus, capacitors are used in a circuit to damp out rapid changes of voltage.

Combinations of Capacitors

Like resistors, capacitors can be joined together in two basic ways: parallel and series. It should be obvious from the physical construction of capacitors that connecting two together in parallel results in a bigger capacitance value. A parallel connection results in bigger capacitor plate area, which means they can hold more charge for the same voltage. Thus, the formula for total capacitance in a parallel circuit is:

C

_T=C₁+C₂...+C_n ,

(2)

the same form of equation for resistors in series, which can be confusing unless you think about the physics of what is happening.

The capacitance of a series connection is lower than any capacitor because for a given voltage across the entire group, there will be less charge on each plate. The total capacitance in a series circuit is

C

_T={1{1C₁}+{1C₂}...+{1C_n}}.

(3)

Again, this is easy to confuse with the formula for parallel resistors, but there is a nice symmetry here.

Inductors

Inductors are the third and final type of basic circuit component. An inductor is a coil of wire with many windings, often wound around a core made of a magnetic material, like iron. The properties of inductors derive from a different type of force than the one we invented charge to explain: magnetic force rather than electric force. When current flows through a coil (or any wire) it produces a magnetic field in the space outside the wire, and the coil acts just like any natural, permanent magnet, attracting iron and other magnets. If you move a wire through a magnetic field, a current will be generated in the wire and will flow through the associated circuit. It takes energy to move the wire through the field, and that mechanical energy is transformed to electrical energy. This is how an electrical generator works. If the current through a coil is stopped, the magnetic field must also disappear, but it cannot do so immediately. The field represents stored energy and that energy must go somewhere. The field contracts toward the coil, and the effect of the field moving through the wire of the coil is the same as moving a wire through a stationary field: a current is generated in the coil. This induced current acts to keep the current flowing in the coil; the induced current opposes any change, an increase or a decrease, in the current through the inductor. Inductors are used in circuits to smooth the flow of current and prevent any rapid changes.

The current in an inductor is analogous to the voltage across a capacitor. It takes time to change the voltage across a capacitor, and if you try, a large current flows initially. Similarly, it takes time to change the current through an inductor, and if you insist, say by opening a switch, a large voltage will be produced across the inductor as it tries to force current to flow. Such induced voltages can be very large and can damage other circuit components, so it is common to connect some element, like a resistor or even a capacitor across the inductor to provide a current path and absorb the induced voltage. (Often, a diode, which we will discuss later, is used.)

Inductors are measured in henrys (h), another very big unit, so you are more likely to see millihenries, and microhenries. There are almost no inductors on the RoboBoard, but you will be using some indirectly: the motors act like inductors in many ways. In a sense an electric motor is the opposite of an electrical generator. If current flows through a wire that is in a magnetic field (produced either by a permanent magnet or current flowing through a coil), a mechanical force will be generated on the wire. That force can do work. In a motor, the wire that moves through the field and experiences the force is also in the form of a coil of wire, connected mechanically to the shaft of the motor. This coil looks like and acts like an inductor; if you turn off the current (to stop the motor), the coil will still be moving through the magnetic field, and the motor now looks like a generator and can produce a large voltage. The resulting inductive voltage spike can damage components, such as the circuit that controls the motor current. In the past this effect destroyed a lot of motor controller chips and other RoboBoard components. The present board design contains special diodes that will withstand and safely dissipate the induced voltages -- we hope.

Combinations of Inductors

You already know how inductors act in combination because they act just like resistors. Inductance adds in series. This makes physical sense because two coils of wire connected in series just looks like a longer coil. Parallel connection reduces inductance because the current is split between the several coils and the fields in each are thus weaker.

Semiconductor Devices

The Truth About Charge

Our statements above about charge are not wrong, but they are simple and incomplete. In order to understand how semiconductor devices work one needs a more complete description of the nature of charge in the real world. Charge does not exist independently; it is carried by subatomic particles. For this discussion we will be concerned primarily with electrons, which carry a negative charge of $1.6 × 10$ ^-19 C , the minimum amount of charge that can exist in isolation. At least, no one has found any smaller amount than this fundamental quantum of charge.

Electrons are one component of atoms and molecules. Atoms are the building blocks out of which all matter is constructed. Atoms bond with each other to form substances. Substances composed of just one type of atom are called elements. For example, copper, gold and silver are all elements; that is, each of them consists of only one type of atom. More complex substances are made up of more than one atom and are known as compounds. Water, which has both hydrogen and oxygen atoms, is such a compound. The smallest unit of a compound is a molecule. A water molecule, for example, contains two hydrogen atoms and one oxygen atom.

Atoms themselves are made up of even smaller components: protons, neutrons and electrons. Protons and neutrons form the nucleus of an atom, while the electrons orbit the nucleus. Protons carry positive charge and electrons carry negative charge; the magnitude of the charge for both particles is the same, one quantum charge, $1.6 ×10$ ^-19 C . Neutrons are not charged. Normally, atoms have the same number of protons and electrons and have no net electrical charge.

**Figure 4.8:** Structure of an Atom

Electrons that are far from the nucleus are relatively free to move around under the influence of external fields because the force of attraction from the positive charge in the nucleus is weak at large distances. In fact, it takes little force in many cases to completely remove an outer electron from an atom, leaving an ion with a net positive charge. Once free, electrons can move at speeds approaching the speed of light (roughly 670 million miles per hour) through metals, gases and vacuum. They can also become attached to another atom, forming an ion with net negative charge.

Electric current in metal conductors consists of a flow of free electrons. Because electrons have negative charge, the flow of electrons is in a direction opposite to the positive current. Free electrons traveling through a conductor drift until they hit other electrons attached to atoms. These electrons are then dislodged from their orbits and replaced by the formerly free electrons. The newly freed electrons then start the process anew. At the microscopic level, electron flow through a conductor is not a steady stream, like water flowing from a faucet, but rather a series of short bursts.

**Figure 4.9:** A Simple Model of Electron Flow

Silicon

Semiconductor devices are made primarily of silicon (silicon's element symbol is "Si"). Pure silicon forms rigid crystals because of its four valence (outermost) electron structure -- one Si atom bonds to four other Si atoms forming a very regularly shaped diamond pattern. Figure 4.10 shows part of a silicon crystal structure.

**Figure 4.10:** A Silicon Crystal Structure

Pure silicon is not a conductor because there are no free electrons; all the electrons are tightly bound to neighboring atoms. To make silicon conducting, producers combine or "dope" pure silicon with very small amounts of other elements like boron or phosphorus. Phosphorus has five outer valence electrons. When three silicon atoms and one phosphorus atom bind together in the basic silicon crystal cell of four atoms, there is an extra electron and a net negative charge. Figure4.11 shows the crystal structure of phosphorus doped silicon.

**Figure 4.11:** Silicon Doped with Phosphorus

This type of material is called n-type silicon. The extra electron in the crystal cell is not strongly attached and can be released by normal thermal energy to carry current; the conductivity depends on the amount of phosphorus added to the silicon.

Boron has only three valance electrons. When three silicon atoms and one boron atom bind with each other there is a "hole" where another electron would be if the boron atom were silicon; see Fig. 4.12. This gives the crystal cell a positive net charge (referred to as p-type silicon), and the ability to pick up an electron easily from a neighboring cell.

**Figure 4.12:** Silicon Doped with Boron

The resulting migration of electron vacancies or holes acts like a flow of positive charge through the crystal and can support a current. It is sometimes convenient to refer to this current as a flow of positive holes, but in fact the current is really the result of electrons moving in the opposite direction from vacancy to vacancy.

Diodes

Both p-type and n-type silicon will conduct electricity just like any conductor; however, if a piece of silicon is doped p-type in one section and n-type in an adjacent section, current will flow in only one direction across the junction between the two regions. This device is called a diode and is one of the most basic semiconductor devices.

A diode is called forward biased if it has a positive voltage across it from from the p- to n-type material. In this condition, the diode acts rather like a good conductor, and current can flow, as in Fig. 4.13.

**Figure 4.13:** A Forward Biased Diode

There will be a small voltage across the diode, about 0.6 volts for Si, and this voltage will be largely independent of the current, very different from a resistor.

If the polarity of the applied voltage is reversed, then the diode will be reverse biased and will appear nonconducting (Fig. 4.14). Almost no current will flow and there will be a large voltage across the device.

**Figure 4.14:** A Reverse Biased Diode

The non-symmetric behavior is due to the detailed properties of the pn-junction. The diode acts like a one-way valve for current and this is a very useful characteristic. One application is to convert alternating current (AC), which changes polarity periodically, into direct current (DC), which always has the same polarity. Normal household power is AC while batteries provide DC, and converting from AC to DC is called rectification. Diodes are used so commonly for this purpose that they are sometimes called rectifiers, although there are other types of rectifying devices. Figure 4.15 shows the input and output current for a simple half-wave

**Figure 4.15:** A Half-Wave Rectifier

rectifier. The circuits gets its name from the fact that the output is just the positive half of the input waveform. A full-wave rectifier circuit (shown in Figure 4.16) uses four diodes arranged so that both polarities of the input waveform can be used at the output.

**Figure 4.16:** A Full-Wave Rectifier

The full-wave circuit is more efficient than the half-wave one.

Analog Recording of Sound

2008-03-28T02:02:00.000-07:00

ANALOG RECORDING OF SOUND

Records

The basic principle of disk recording is very simple. Displacement of the microphone diaphragm is transformed into a wiggley groove on a moving piece of vinyl. A stylus tracing the wiggles exactly reproduces the motion of the diaphragm at the time the recording was made. Electricity is really incidental to the process, used as a convenient way to connect the microphone to the cutter and the pickup to the speaker.

Most of the development in record technology has been devoted to putting a lot of music on a single record. The obvious approach, slow speed and a narrow groove, reached a practical limit in the middle of the century with the 33 1/3 rpm microgroove record. At that speed, (9 inches per second in the inner part of the groove) a 20 khz signal has a wavelength of .0004 inch. It is very difficult to manufacture a stylus that would handle wavelengths smaller than that.

The major consumer of real estate on the record is low frequency content. This is because the amplitude of the electrical signal produced is proportional to the side to side velocity of the stylus. Given equal velocities, a low frequency wiggle will swing wider than one of high frequency because at low frequencies the cutter will not turn around as often as it does at high frequency. To counteract this effect, the low frequency content of the record is deliberately reduced, and this low end rolloff has to be corrected by a bass boost in the playback system.

The high frequency content is given a treatment opposite to that of the lows. High frequency information is emphasized during recording, and reduced during playback. This is an attempt to reduce the noise generated by the roughness of the vinyl. That noise is white noise, and as such sounds like a high frequency phenomenon. When the playback system reduces the high frequency content to its proper level, the noise in that range is reduced by the same amount.

The combination of bass roll-off and treble boost is called the recording characteristic, and the complementary response of the playback system is called RIAA equalization after the manufacturer's association which standardized this feature in 1956.

The LP is an endangered species with the advent of Compact Disk technology, but it will not disappear overnight. Even if no new records are produced, there are hundreds of millions in existence, including many unique performances and compositions.

Analog tape

Tape deck mechanism

The principle of tape recording is just as simple as that of disk recording. The tape is a strip of plastic which has been coated with a material that is easily magnetized. (The most commonly used material is highly refined rust, or iron oxide.) The capstan is a spinning post. The tape is held tightly against the capstan by the pinch roller and dragged across the three heads at a steady rate.

All three heads are essentially the same in construction: a C-shaped piece of metal with the very narrow gap of the "C" near the tape. A coil of wire around the metal can serve to either detect or produce magnetic fields at the gap. If a strong current is passed through the coil, a field is produced which creates a magnetic spot on the nearby tape. The amount of magnetism will be proportional to the amount of current. If the tape is moved and the current varied in a periodic way, a "track" of magnetic areas will be imprinted on the tape. All of this happens at the record head. When the tape subsequently passes the play head, the varying magnetic field on the tape produces a varying current in the play head coil, which can be detected by some sensitive electronic circuitry. (The erase head works just like the record head, but at a super high frequency which will not be recorded but which will obliterate any existing information.)

The signal applied to the record heads is equalized in a manner similar to LPs, for essentially the same reasons. The equalization is varied according to speed and type of tape. That is automatic on most decks, but must be set manually on some cassette decks.

This drawing shows how the magnetic fields are oriented on a stereo tape. You can see four tracks, two of which are played when the tape is going one direction, the other two when the tape is reversed. (Multi-track tape decks use all four tracks or more at one pass. There are also formats which use only one or two tracks, recorded on the entire width of the tape.) The width of a single track is an important factor in the strength of signal that can be recorded, which ultimately limits the noise of the system. The width of a track depends on the width of the tape as well as the number of tracks; various sizes ranging from 1/8 inch to 2 inches are used, recording up to 24 tracks.

The distance between spots in a single track is the wavelength of the signal, which depends on the frequency of the signal and the speed of the tape. The higher the frequency of the signal, the shorter the wavelength (a familiar formula). There is a limit to how small the shortest spot of magnetism can be; namely the width of the gap in the recording head. The practical result of this limitation is that the highest frequency that can be recorded is limited by the speed of the tape. Again, various standards are in use, from 1⁷/8 to 30 inches per second.

This drawing shows another feature of the magnetic spots. At high frequencies, the magnetic domain is a rather slender, tall shape, almost a line. If that line is not exactly the same angle as the gap in the play head (which is supposed to be perpendicular to the tape), the energy represented by the magnetism will not be accurately detected, resulting in a reduced output signal. Low frequency signals are not affected by this factor (known as azimuth adjustment), so the net result is a loss of highs. In a cassette system, this alignment is a function of the plastic shell, and is generally rather sloppy in all but the most expensive tapes.

AC Bias

Tape recording would be a very low fidelity business without AC bias. The process of magnetization is linear only when applied to fields of medium strength. There is a limit to the strength of field that the tape can accept. Once the tape is completely magnetized, no amount of extra current in the head will increase the resultant field. That condition is called saturation. As the strength approaches saturation, there is a gradual falloff in the effectiveness of the magnetization process. This results in a phenomenon called "soft clipping", which is definitely distortion, but as distortions go is reasonably unobjectionable. Incidentally, since the record equalization increases the high frequency content of the signal, this clipping will happen to the highs first: that is why a cassette seems to lose its top end response when it is recorded "hot".

Clipping is something we live with in all electronic systems, and is easy to avoid; simply keep the gain down. Another region of nonlinearity is more difficult to deal with. The magnetization process produces field regions that alternate in polarity: one north, one south, north again, and so forth. In between there are regions where the field strength is zero. When the oxide is not magnetized, a fair amount of current is required to produce any magnetization at all; this leaves a flat spot in the middle of the waveform, as illustrated in the second waveform in the diagram below.

We avoid the effects of this nonlinear region by adding a very high frequency (over 100 khz) bias signal to the signal we are trying to record. The result is the third waveform. The center of the bias frequency is distorted, but the original signal, which is the shape of the overall waveform, is clean. The playback head cannot respond to the bias signal, and simply returns the original. The amplitude of the bias signal has to be carefully adjusted to provide a distortion free recording. Many tape decks (especially cassettes) offer a switch to make a coarse change in bias for different tape types, but a finer calibration is really required for optimum results.

Peter Elsea 1996

Basic Electronics

2008-03-28T01:57:00.001-07:00

SOME BASIC ELECTRONICS

It is adequate for a music class to take a rather simplistic view of the precise nature of electricity, so the following discussion is not complete or rigorous.

The phenomenon known as electricity involves the exchange of very small particles (electrons) between atoms. Now a particular atom generally possesses a constant number of electrons, and there are some rather potent forces within the atom working to keep the correct number of electrons, so if an atom looses an electron because of some mechanical or chemical process, it will very quickly pull in a replacement. There are seldom spare electrons around, so the replacement likely comes from an adjacent atom, which, shy one electron, steals from its neighbor, and so forth. The exchange of electrons propagates in much the same way as the pressure disturbance I discussed in the section on sound, and exhibits all of the associated wave effects. The speed of propagation is so high however, (almost the speed of light) that we may ignore the wave characteristics of electricity for all but the briefest events.

The electron exchange has two important side effects: it produces a tiny amount of heat or light, and supports a weak magnetic field. These two effects represent an energy loss, and must be compensated for by the original electron moving mechanism. They can also be thought of as energy transference from the point of generation to any place where these effects are apparent. Most of the science of electrical engineering deals with getting these effects to happen at the right place.

Since the heat and magnetism effects are small, in order to produce any useful work we must move a lot of electrons and ensure that it is a continuous process. To get a steady flow of electricity, we must have something that absorbs electrons (called a sink) something that supplies electrons (a source), and some path in between for the electrons to follow. If we want the process to be continuous forever, we must realize the sink will eventually fill and the source will be depleted, so the two should be combined and the path from one to another be made circular. The source/sink combination can be realized chemically with a battery or mechanically with a generator, or in a variety of ways I'll lump together and call the power supply.

VOLTAGE, CURRENT, RESISTANCE

If we set up a power supply and a wire connecting its source and sink terminals, we can make some observations about the flow of electricity in this simple circle, or circuit.

Fig. 1 A simple circuit.

The amount of heat or light produced is determined by the number of electrons moving around. The concept of "number of electrons" is called CURRENT and is measured in units called AMPERES. The symbol used to stand in for current in mathematical formulas is I.

The amount of heat produced for a given current depends on the material used for the wire. The heat producing mechanism opposes the flow of electrons, so it is termed the RESISTANCE. It is measured in units called OHMS, and the symbol is R. Many materials have such a high resistance that they will not carry measurable current. Such materials are termed insulators, whereas materials that will carry current are conductors.

If you want to move a current against a resistance, you must supply some ELECTROMOTIVE FORCE. This force is measured in VOLTS, is symbolized by V or E, and is a property of the power supply or battery. Electromotive force is a bit of a mouthfull, so it is also called VOLTAGE. Since this force acts in moving current from one spot to another (from source to sink) a voltage measurement must be made between two places. If the two places are on the same piece of wire, the voltage will be unmeasurable. Usually, we designate a convenient spot as being "0 volts" and use that as a common point in all measurements.

These three properties are everything we need to know about a constant current flow. They are related in the following way:

If the voltage is held constant, and the resistance reduced, the current will increase.

If the resistance is not changed, but the voltage is reduced the current will decrease.

If we know two of the quantities, we can calculate the third, because the units are defined that way.

The mathematical statement of these relationships is:

I=E/R

Where (to review),

I is current in Amperes,

Eis electromotive force in Volts, and

R is resistance in Ohms.

This relationship is the cornerstone of electronics, and is known as OHM'S LAW.

There are many uses for current flow, of course. In fact, our modern civilization is pretty dependent on it. The two most important are the transmission of power and the transmission of information. In electronic music, we use electricity mostly to transmit and modify information; the flow of current represents the pressure of air as it changes to produce sound. To do this, we use devices that generate current from the energy of the sound waves, and ultimately, devices that generate sound from changes in current.

IMPEDANCE

Ohm's law is true for any steady state condition but if the current is changed, Ohm's law is not true while the change is taking place. This is because there are effects that oppose any change in the amount of current flow. Opposition to current change is called REACTANCE, and is measured in ohms because it must be combined with resistance in order to describe the current to voltage relationship. (For the time being, think of reactance as the electrical equivalent of momentum.)

This combination is not a simple addition because it must take into account the rate of change. The rate correction is usually computed on the basis of frequency of a sine wave. Some structures have a reactance which increases with frequency, and some devices have a reactance that decreases as frequency rises. There are complex devices that favor certain frequency regions but react against frequency content above or below the magic value. The combination of resistance and reactance is called IMPEDANCE (symbolized by Z).

Ohm's law for impedance is stated:

I=E/Z

Where I= the current amplitude of the signal[*]

E= the voltage amplitude of the signal*

Z= the impedance of the circuit at the frequency of the signal.

The moral of all this is that for a given voltage, low impedance circuits require large currents, and high impedance circuits require small currents.

POWER

There is little point in moving these electrons around unless the current can make something happen. How much can happen is expressed by the POWER of the circuit. In a resistive circuit (i.e. no reactance in any of the components) the power is the product of the voltage and the current, and is expressed in watts.

P=EI

For instance, a light bulb that passes a half ampere of current if one hundred volts is applied to it will produce 50 watts worth of light and heat. If the bulb were redesigned to pass a whole ampere, it would provide 100 watts. If 200 volts were applied to the original bulb, it would also provide 100 watts, but probably not for long.

Power calculations in reactive or partially reactive circuits are complex. The ratings on light bulbs and appliances assume you are going to use the resistive power formula to determine whether a fuse will blow, even if the device is reactive. (In the latter case the label should read VA for volt-amperes instead of W for watts.)

INFORMATION

When electricity is used to transmit information, we lose interest in voltage and current values (as long as there is enough of both for the circuit to work) and pay attention to how much information can be represented in a second and how accurate that representation is. We use two basic systems to represent information. In analog systems, the amount of current varies directly with the represented quantity, such as air pressure. In digital systems, the information is coded into binary numbers, and the value of each bit of the number is represented by the presence or absence of current.

For analog systems, the accuracy of representation is measured as percent of distortion; which is the difference between whatever went into the system and the actual output compared to the "perfect" output. The amount of information that may be represented is termed the bandwidth. For instance, a high fidelity sound system that can reproduce sound from 20hz to 20,000hz has a bandwidth of 19980hz.

In digital systems, the amount of information is determined by the number of bits that may be transmitted per second, also known as the baud rate. The accuracy is the percentage of bits that are transmitted wrong or the error rate. Translation of baud rate to bandwidth and error rate to distortion is a function of the encoding scheme used; some are fast and sloppy, others are accurate and slow. Low distortion, high bandwidth data transmission is expensive no matter what system is used.

In both analog and digital audio systems, the representation of sound is usually called the signal. We will often trace the signal path through various devices, which, because they modify the information, are called signal processors. Any output the system produces which is not the signal is considered noise.

Peter Elsea 1996

Basic Electronics - Part 2

2008-03-28T01:50:00.001-07:00

Kirchhoff's Laws

Kirchhoff's Current Law

The algebraic sum of currents entering and leaving any point in a circuit must equal zero.

Stated another way

No matter how many paths into and out of a single point all the current leaving that point must equal the current arriving at that point.

Kirchhoff's Voltage Law

The algebraic sum of the voltages around any closed path is zero.

Stated another way

The voltage drops around any closed loop must equal the applied voltages.

When voltages are opposing as seen at the right, the difference is the voltage applied to the circuit. In this case 4 volts must be dropped by the resistors to equal the applied voltage

Basic Electronics

2008-03-28T01:06:00.000-07:00

Basic Electronics

Here are some basic formulas for wiring capacitors, inductors, and resistors in series or parallel. These are useful when you cannot find a component with the exact value that you are looking for. You can also wire resistors in parallel so that the wattage rating of each resistor only needs to be half (or less) of the total necessary value.

The formulas shown are for inductors (L) and capacitors (C). Resistors (R) act in the same way as inductors, so all inductor formulas can be applied to resistors.