A WIZARD'S ELECTRONICS COMPANION
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A WIZARD'S ELECTRONICS COMPANION |
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Electricity is the existence of charges. Charge is the quality of electrons (minus charge) and protons (plus charge) to repel like charges and attract unlike charges. Electrons are usually the most mobile charges in our electronics circuits. Electrons can be thought of as identical microscopic blobs of energy clinging to the nuclei of atoms. When an electron is displaced from an atom, that atom has a charge imbalance, becoming positively-charged with a voltage which tries to pull a negative charge toward it and push positive charges away. In a metal wire, where electrons are easily displaced, an electrical current of charges occurs in the form of electrons hopping from one metal atom to the next, displacing electrons already there like dominoes. The actual electron flow is slow, less than a millimeter per second, but the "domino effect", the charge imbalance, is felt at near the speed of light. In a battery or in your body, electricity is in the form of charged ions of atoms and molecules.
For a more detailed, but still simple, exposition on this issue, check out Bill Beaty's Which Way Does Electricity Really Flow?
The other properties of the electron can be understood better from three perspectives (see diagram): It manifests a static electrical field in all directions while at rest, a magnetic field is magically unveiled when moving uniformly, and a ripple in the static electrical and moving magnetic fields around it is created when accelerating called an electromagnetic wave (radio waves, infrared, light waves, ultraviolet, x-rays, etc. They differ only in their frequency of vibration).
Electrical charge must be unbalanced and harnessed in order for it to do work for us. Also, one electron isn't much electricity; we need to move huge numbers of them for household appliances. Some materials are good conductors for electricity (metals, salt-water, moist skin, that have relatively moveable charges), and some are insulators (air, plastic, glass, wood, pure water, dry skin, which do not have particularly moveable charges); thus we can create a channel to contain the electricity whether it sits or flows. The water pipe (figure 2) is a useful analogy. The water (charge-balanced free electrons) in the circuitous path is constant and at rest. It is free to move withing the space inside the pipe (conductor). The water path is kept from leakage by the pipe itself (insulator). If we want it to do work for us we must pump it, creating a flow of charges (electrical current). High water pressure (electrical voltage) at the pump output is pushing against our mill-wheel load (electrical load), and low pressure is created on the other side as the pump pulls it in (electrical ground or circuit return). The difference between the high and low pressure multiplied by how much pressure (quantity of electrical charge) determines how much work can be done. Work (electrical energy) is expended on the load. The electrical circuit is thus created. You may also discuss subcircuits, which are incomplete circuits or functional blocks of a larger circuit.
To make an electrical pump or generator, a magnet moving past a metal wire (or any conductor) was discovered to cause electrons to move in the wire; we build electrical generators based on this principle. For everyday use, we let the power company do it for us. If we can't use line power, the battery is our choice. We can think of it as a pump, but it's really a chemical storage device for electricity and what little energy has been pumped into it will soon be drained back out. Solar batteries are genuine electricity pumps which convert sunlight directly into electricity.
If our electrical pump moves the electrons steadily in one direction only, we say that the current is a direct current (DC). If the electrons are being pumped first one direction, then the other (typical of an AC generator in a hydroelectric dam), we call it an alternating current (AC). If the pressure (or potential) is there, whether or not a current flows, we call it a voltage (V), which can be direct (DCV) or alternating (ACV).
For the electrical channel, copper metal wire is used almost exclusively and is often plated to reduce the non-conductive tarnish; gold plating is sometimes seen on electrical connectors, and is always used for microscopic electrical channels because it will not tarnish. As we enter the age of nanotechnology (the construction of atomically-perfect machines), we take advantage of subatomic phenomena that is only useful over distances of a few atomic diameters. Superconductivity is perfect conductance without lost energy in the form of heat, and can take place in tubular carbon structures, among other things. The curious phenomena of electrons tunnelling across an insulating barrier a few atoms thick can be understood if one imagines the electron as a somewhat diffuse energy field. Part of the field is on the other side of the insulator. If it "bumps" too hard against it, it might momentarily be mostly on the other side where it will get stuck (unless it tunnels back.) The probability of tunnelling is often drawn graphically with little balls rolling on a surface of smooth little hills and valleys of various heights.
The generator and electrical loads are called transducers because they change electrical energy into other forms of energy and vice-versa to get our jobs done. Light bulbs, hydroelectric dams, electric motors, speakers, TV screens, electric kitchen appliences, and computer touch-screens are all examples of transducers.
We were introduced to electrical generators, conductors, and transducers. Defining these tools gives us the medium in which we can express our electronic art. Transducers perform many useful and startling energy conversions, but what if we need to alter the electricity itself first? Since electrons interact mainly via their electric and magnetic fields (figure 1), we can only build devices to take advantage of these two properties, and a third one of the constriction of current flow. Table 3 compares the three core devices based on these three effects. No device is pure; every device made has some combination of these three effects. Electronic parts are intended to create lumped effects, that is, to try to concentrate in a single device primarily the desired properties. (For completeness, I should mention here that scientists like symmetry and having four core effects is much cooler than three. A fourth could be thought of as a negative-resistance device (i.e., one which passes less current as you increase voltage) like the tunnel diode. However, another candidate for this mystery-symmetry position is the memristor which links charge and flux, just as a resistor links voltage and current, covering all four fundamental circuit variables. The capacitor, then, covers only charge, and the inductor covers only flux.)
The resistor is the component which restricts current flow like a tight spot in a water pipe. It converts electrical energy into heat in the process. The capacitor stores static charges (energy stored in an electrical field) much like filling a barrel full of water. The inductor stores moving charges (energy stored in a magnetic field) much like a long pipe full of rushing water; if you try to shut it off suddenly, the momentum of all that water will burst the pipes. The final type of component I would like to introduce is called the semiconductor which acts like an electrically-controllable resistor. The right half of table 3 is mostly semiconductors, and even a few of the transducers. The integrated circuits use mostly semiconductors and some resistors and capacitors.
Resistance is an energy dissipative effect that converts electrical energy to heat. A broader category for this might be transductance, which is an energy conversion effect which changes energy to or from electrical energy. If the device which uses resistance converts electrical energy as dissipated heat, then we call it a resistor. Any device which converts to or from electrical energy in a more useful manner we call a transducer (like a motor). The property of materials to conduct less than perfectly means they have resistance. More exotic materials like the metalloids silicon and germanium have been used to create non-linear resistances (like a diode) and even matrices in which there is more than one current path (like a transistor).
Capacitance is an energy storage effect which stores electrical energy as static charges. Any two conductors separated by a non-conducting space will exhibit capacitance as unbalanced charges push or pull at other charges in the space around them. Our bodies have an intrinsic capacitance of roughly 0.1µF with respect to the air and materials around us. A capacitor is made by rolling or stacking many conducting plates interspersed with many insulating plates, in order to concentrate the effect (a so-called "lumped-capacitance" effect--this term would be used more in circuit designs where significant stray capacitance exists as well.)
Inductance is an energy storage effect which stores electricity energy as a magnetic field around the moving electrons.When a switch is opened to provide current to a load, there are only two things which limit that current, resistance and inductance. Resistance is pure loss as heat, and that is what I was using. The inductor provides electrical momentum. When you apply a voltage to an inductor, in the first instant no current at all is flowing. Then, current gradually rises, storing energy in a magnetic field around the inductor. It seems to act like a resistor for a short time, and indeed we will use this simplification many times, but this is like modeling pushing a big boulder floating in space like a very small boulder being shoved along the ground. Perhaps they both required the same effort to get from place to place, but the energy of the one on the ground was lost as heat and stopped moving when the pushing ceased, whereas the energy of the one floating in space was efficiently stored as kinetic energy so it took just as much effort to stop it moving when it got to its destination. That movement of the stone is very much like the movement of current, which is why I use the potential versus kinetic energy model. The stored energy in the coil's magnetic field is directly dependent on the level of current flow. When the current tries to be reduced, some of the magnetic field energy will be converted into a current which will directly oppose the external change in current flow. If you wound trillions of turns of superconducting nanowire on a bobbin and tried to use it in a circuit, it would take years to get any appreciable current to flow in it, the inductance would be so high. There would be no resistance so the current would continue to climb as long as you continued to apply a voltage. The magnetic field would be so strong it would be frightfully dangerous to be near it with anything ferrous. Then, when you tried to completely shut off the current, the collapsing magnetic field would keep piling charges up along the windings causing the voltage to climb as high as necessary to dissipate that energy, creating arcs between the windings (internal short ciruits) and resulting in a cascade failure which would manifest as a rather violent explosion as all the stored energy was dissipated at once. Internal shorts often do occur in real coils because of that flyback voltage.
For the sake of completeness and as a reminder to myself I will briefly mention it here, but not attempt to expand on at this time as it won't be available to the hobbyist for some time as people develop the technology and rewrite the textbooks. The memristor was postulated by Chua in a 1971 paper (due to a gap in the theory) as a device which changed resistance by the amount of charge which flowed through it. When the current is reversed, the effect is gradually reversed with resistance decreasing with quantity of charge flowed. In 2008, HP was able to build a real device in the lab which exhibited this effect, though it may stir up a firestorm of patent litigation (see some history.) The effect in this particular process is quite small and appears to only be significant on the nanometer scale, which is perhaps why it wasn't noticed before. It allows for much smaller, non-volatile computer memory. Hopefully, we will eventually see discrete components built to play with (and additions to this tutorial.) The variables for the fundamental mathematics apparently should not have been voltage and charge, but flux and charge:
| Current | Charge | |
|---|---|---|
| Voltage | R = dV*dt / dI*dt voltage and current |
C = dv/dt voltage and charge |
| Flux | L = dI*dt flux and current |
MEM = dt/dI = dt*dv flux and charge |
An abstract shorthand has been developed to describe electrical components called schematic symbols. Table 1 contains these symbols, which one arranges together to form a schematic diagram. There are many others, such as vacuum tubes, but these are the most common at our current level of technology that we will use here. Some of these are my variant on the standard, and, just like variations in hand-drawn letters, anyone familiar with schematic symbols will recognize them. Don't feel you have to memorize all these symbols at once. You will learn them as you use them, and I haven't yet used all of them. It is more important for creativity that you become familiar with them as tools of the trade that shape what you can do in electronics.
Putting our alphabet soup of parts symbols together yields a schematic diagram of how the circuit is wired to shuffle electrons around. Figure 3a is an example showing the overall look of a schematic diagram. Because circuits can get astronomically complicated, even greater simplifications are often necessary to understanding what is going on, leading us to the concept of the signal, which is the effect we wish to achieve in that part of the circuit. If the electricity and parts are like the water in a pond, then the signal is like the ripples on the pond that we are trying to make. Sometimes, we draw a diagram showing just the ripples called a flow chart for the electrical signal, or a block diagram (figure 3b) showing how interconnected functional blocks of common circuits will create the desired signal. Just as single, lumped components help simplify a complex reality, functional circuit blocks provide a higher level of abstraction to further simplify visualizing circuit operation. This abstraction technique is used in all engineering fields.
As you can see in figure 3a, the signal we desire to create (shown dashed) flows left-to-right -- our reading direction in the West, to show forward progress. Components are drawn showing circuit wiring to build the signal as it progresses along through the manipulation of electron flows between the power supply connections. The power supply is shown abstracted as "battery" because we're not concerned with its details, and is shown at the top and bottom of the schematic to constantly support the construction of the signal without impeding its progress. One of the two power supply rails (another common name, shown fairly literally here, like parallel railroad rails) is considered the low-energy power supply return part of the circuit, or ground (so-called because many early circuits used the Earth as the low-energy return path), and is typically placed at the bottom of your schematic diagram. The high-energy side of the power supply is usually placed at the top. The parts are almost always placed strictly horizontally or vertically with wiring drawn in a similar rectangular manner, minimizing wire cross-overs. The parts list (figure 3c) reduces schematic clutter and the identifying numbers (R1, etc.) are supposed to uniquely identify every part in the circuit.
Don't feel overwhelmed by the complexity of figure 3a. The drawing patterns are repeated, and certain conventions used, in building even complex diagrams making them much easier to read. Nobody wants to read a "rat's nest" which introduces unnecessary new, one-of-kind patterns to be deciphered.
Table 2 lists the part number prefixes I like to use. These are seen as prefixes to the numbered parts in a parts list (the "R1", "L1", and the like in fig.3c), and often, on the schematic diagram (as in fig.3a). Do not feel that this list is either exhaustive or cast in concrete, just try to be consistent. I have seen Q, T, XSTR, and TR all represent "transistor". I've probably taken a few more liberties here than I have with the schematic symbols. If you have electronics modeling software, note the differences in symbols and prefixes between theirs and mine.
Electronics modeling with math, like modeling any part of the real world, can grow quickly to impossible complexity. In practice, few circuits warrant precise calculations and many components simply aren't manufactured to precise tolerances. Some circuit values may be off by a factor of two or even ten times and still allow it to operate, so you will need few, if any, calculations. I've included the most important simplified formulae to handle the majority of your calculations in table 3. A four-function calculator is helpful, preferably with square-roots (Hint: If a result is 32.4978012, just 32 is likely meaningful). We'll summarize table 3's formulae and all the most used formulae in the next several sections, but defer the application until Circuits section.
This is a fairly complete list of all the units you'll likely encounter in electronics.
Electronics uses a wide range of prefixes for various units. One should become familiar with seeing them appended to units. This list probably contains more than you'll ever use.
| Power-of-ten | prefix | abbrev. |
|---|---|---|
| -24 | yocto- | y |
| -21 | zepto- | z |
| -18 | atto- | a |
| -15 | femto- | f |
| -12 | pico- | p |
| -9 | nano- | n |
| -6 | micro- | u * |
| -3 | milli- | m |
| 0 | none | none |
| +3 | kilo- | k |
| +6 | mega- | M |
| +9 | giga- | G |
| +12 | terra- | T |
| +15 | peta- | P |
| +18 | exa- | E |
| +21 | zetta- | Z |
| +24 | yotta- | Y |
| * "u" will stand for Greek mu. | ||
Here we summarize most of the formulas you'll likely ever use.
There's a confusing pile of related terms that should be clarified, though you likely won't use them all. There are six terms, three are resistance terms (in Ohms) and three are conductance terms (in Mhos -- the inverse of the three resistance terms.) Mhos is ohms spelled backwards, symbolized by an upside-down Greek omega. Cute, huh?
DC resistance is RESISTANCE (R) in Ohms = 1 / CONDUCTANCE (G) in Mhos (DC conductance)
AC resistance is REACTANCE (X) in Ohms = 1 / SUSCEPTANCE (B) in Mhos (AC conductance)
DC resistance mixed with AC resistance is IMPEDENCE (Z) in Ohms = 1 / ADMITTANCE (Y) in Mhos (DC conductance mixed with AC conductance)
Alternating current can take any number of shapes of voltage and current over time, but the most common is the sinusoidal waveform, or sinewave (fig.4). This is the shape of the voltage waveform from a common AC power outlet. Anyone who has had trigonometry has the math already down, the rest will just have to bear with me. Geometrically, you can create a sinewave by starting with a circle and imagine a point travelling around counterclockwise at constant speed. Create another long, horizontal drawing to the right of it with a center line going through the center of the circle. On this long drawing you will have a second point travelling to the right at the same speed as the first point is travelling around the circle. Now, at every instant in time, draw a vertical line up from the point on the long drawing, and draw a horizontal line right from the point on the circle. All the places where each pair of lines intersect in the long drawing fall on a sinusoidal wave-shape.
The "unrolled" portion of the graph in figure 4 containing the sinewave horizontal position is equivalent to a measure of the time domain. The vertical offsets from the centerline are equivalent to a measure of amplitude, or how tall the wave is. The instrument known as an OSCILLOSCOPE graphs this sort of time domain information. The unique feature of a sinewave is that it is a single frequency -- a pure tone unmixed with any other pitches. The instrument known as a SPECTRUM ANALYZER would display this same signal in the frequency domain in the horizontal direction, with the amplitude once again as a vertical offset, but with only a single, sharp spike to indicate the one frequency present (fig.6).
If one were to begin the sinewave by placing the first point at a different starting location on the circle (fig.5), the shape of the resulting waveform would be the same, except it would appear shifted to the left or right by the angle you started from, a phase shift away from the original waveform. One can see that once you phase shift all the way around the circle (360°), the waveform will once again line up with the original.
As you might expect if you give a little thought to the storage and momentum concepts I described in the section The Tools of Electronics , A direct current applied to a circuit with capacitors and inductors will only have dynamically changing voltages and currents until the circuit reaches some steady state, at which point the capacitors are essentially an open circuit and the inductors are essentially a short circuit (the resistors stay the same.) With AC applied to such a circuit, however, those changing values will continue to change.
Capacitors and inductors, as a group, are known as reactors because they store and return energy. In a resistor, there are no energy storage effects, and the current waveforms are always exactly following the voltage waveforms.
The capacitor, being like a fat water pipe of fixed diameter that you can fill up from the bottom, presents little resistance to a sudden change in voltage (pressure) or flow (current). There is only the static force of charge voltage pushing back on more charge being pumped into it. When discharged, the capacitor can dump its charge very rapidly if desired because of very low momentum in static charge storage. After establishing an initial steady voltage level (at which point the current through the capacitor is zero), the first change of voltage applied to a capacitor sees no resistance to a change in current flow (for an instant, a short circuit.)
Applying the capacitor concepts to that of phase, it is discovered that the current waveform leads the voltage waveform by 90° in a capacitor.
The inductor, being like a very long horizontal pipe full of water, doesn't have that static push, but it does have a great deal of momentum. This is manifested in the storage of energy in a magnetic field which appears around any moving charges. It's hard to get them to start flowing, but it is by the same token, hard to get them to stop. The momentum which can burst your water pipes in a "water hammer" is analogous to the rise in voltage in an inductor in which you are attempting to suddenly stop current flow. The magnetic field will start collapsing as energy is drawn from it, which is turned into an actual voltage throughout the inductor which is causing current to want to continue to flow. Inductors do not so much care what voltage is across them as the voltage will rise as high as necessary in an attempt to cause some current to flow in order to dissipate the energy of charge motion stored in the magnetic field. Not unusually, an arc may occur in a coil between windings, damaging the insulation and dissipating the energy in the the arc. After establishing an initial steady current level (at which point the voltage across the inductor is nearly zero), the first change of voltage applied to the inductor causes no change in current flow (for an instant, an open circuit.)
Applying the inductor concepts to that of phase, it is discovered that the current waveform lags the voltage waveform by 90° in an inductor.
Capacitors store potiential energy as voltage, and coils store kinetic energy as current. In their pure form, they have no losses, no resistance. L and C are complementary, yet polar opposite effects in a circuit. Resistance cannot store energy, and is neither L or C. When there is a single-frequency sinusoidal waveform, creating an alternating current in your circuit, a useful construct to help model these relationships is a phasor diagram (fig.7). The units can all be currents, voltages, or reactances. It is merely a convenient geometric construct to help you visualize two or more vectors from your circuit being added together in an algebraic sum. The phasor (V1 in this case) is the resultant vector you get. All the factors in your circuit are sorted into three categories: L, R, and C. Each one can be described as a vector (a magnitude with a direction) pointing in the appropriate direction. The frequency is generally known to start with, so what V1 tells you in this figure is that there is some resistance in the circuit, and that, at the given frequency, the capacitive reactance dominates over the inductive reactance (if we're talking about reactances.) Also, the resulting reactance (summing L and C) is about the same as the resistance. (Of course, we may not care about the phase shifts at all in rough-and-tumble circuit design.)
One of the most unique features of these effects is that of RESONANCE, the natural vibration frequency. If the frequency causes the reactances of both L and C to be equal, they will cancel each other out, leaving only R. This is the principle behind the Tesla coil and some filter designs. Here's the brain bender once again: In reactive series circuits with different types of components, you cannot simply add the voltages across L, R, and C to equal the total, because the voltage waveforms under AC conditions are out-of-phase with one another. In fact, they aren't even in the same mathematical dimension! They are drawn at right angles to one another. This is graphically shown with phasor diagrams. Note the lengths of V(R) and V[X(C)] cannot simply be added together, but must be added algebraically. The total equals the length of the hypotenuse of the right triangle.
DC circuits need only single values to describe their voltages, currents, and resistances. AC values, on the other hand, may also contain a phase angle which relates the phase difference of its waveforms to some reference phase angle. This phase angle is only useful for single-frequency AC values such as you would have with 120V 60 Hz line power or some special-purpose circuit. If the circuit is fed a signal with many frequencies, the phanse angle term is ignored and other techniques are used.
Typically, one applies a voltage to an AC circuit, then measures or calculates the amount the current waveform leads or lags the input voltage waveform in degrees of phase shift. In a circuit in which resistance dominates, the phase will be near zero degrees, where capacitance dominates the current rushes right in and leads the voltage, and in an inductive circuit momentum must be built-up gradually and the current lags the voltage.
Note that reactors are frequency dependent. If a single AC frequency signal is applied to any circuit composed of fixed inductances, capacitances, and resistances, the AC waveforms of all voltages and currents in every component will be sinewaves of the same frequency. They will differ only in their amplitudes and phase-shifts. Remember also that these phasor relationships only exist under steady-state conditions with a single AC frequency applied to the circuit. Transient conditions are much trickier to analyze.
An example of using the properties of the phase shift in an RC circuit directly is the phase-shift oscillator.
These are two ways of handling our information signal. Analog allows a continuous range of values for some signals, like a voltage changing over time, whose resolution is limited only by circuit noise. Its disadvantage is that extra speed and accuracy is costly and the circuits tend to be more sensitive to effects like temperature variations. Many useful analog integrated circuits are used daily, one of the most common is the operational amplifier. Digital, on the other hand, represents a signal as a series of on / off signals interpreted as binary numbers (such as the information in a typical computer) which may have always been digital or may represent an analog signal which has been digitized. While the circuit has to be dozens of time faster than its analog counterpart, it has the advantages of stability and accuracy.
Once in the digital realm, all signals are typically manipulated in bistable circuits biased so that only two stable states can exist: +5V or 0V, TRUE or FALSE, 1 or 0, HIGH or LOW, ON or OFF. We then use simple binary logic to make logical decisions based on the limited input states. The basic logic unit is the digital switch with which we design logic gates which react in limited and very deterministic ways to their finite nature. To make something incredibly complex like a computer's central processing unit (CPU), you only need the ability to combine two logic states together (AND, OR) and the ability to invert a logic state (NOT). You can also combine these together (NAND--NOT AND, NOR--NOT OR) in a single device.
The basic TRUTH TABLES for the common one- and two-input (A and B), one-output (C) logic gates are shown in the table below: AND, NAND (NOT AND), XOR (EXCLUSIVE OR), etc. Very complex truth tables can be constructed to accommodate a design.
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Although it is not as critical for professional work as it once was, boolean algebra can provide a personal circuit design with an elegant solution. The "+" stands for a logical OR operator between two terms. The "*" stands for a logical AND operator. The logical NOT is usually indicated by a line over the Boolean expression, which I will show as a minus sign when in plain text: -(-A*-B). A, B, C, etc. are typical variables to be used in the expression. The following table contains the more trivial rules, DeMorgan's Theorems, and those which follow the commutative, associative, and distributive rules of algebra.
| Boolean Identities | ||
| A * 0 | = | 0 |
| A + 0 | = | A |
| A * 1 | = | A |
| A + 1 | = | 1 |
| A * A | = | A |
| A + A | = | A |
| A * -A | = | 0 |
| A + -A | = | 1 |
| DeMorgan's Theorems | ||
| -(A*B) | = | -A+ -B |
| -(A+B) | = | -A* -B |
| Comm., Assoc., Distrib. | ||
| -(-A) | = | A |
| A*B | = | B*A |
| A+B | = | B+A |
| A*(B+C) | = | A*B+A*C |
| A*(B*C) | = | (A*B)*C |
| A+(B+C) | = | (A+B)+C |
| A+A*B | = | A |
| A*(A+B) | = | A |
| A*(-A+B) | = | A*B |
| A+ -A*B | = | A + B |
| -A+A*B | = | -A + B |
| -A+A*-B | = | -A + -B |
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