Mysteries

You can research these mysteries on the Web. Most of these mysteries aren't even electronic, they are just electrical. Transistors have plenty of mysteries but the following, non-electronic mysteries are plenty.

Four-armed Bridge

With diodes, it turns a transformer secondary into full-wave-rectified DC. Center tap not needed. Search term: bridge rectifier.

With resistors, allows very sensitive comparison of resistors such as for temperature measurement (with thermistor) or for strain gauge.

With AC stimulus, compares capacitors or inductors and lets you find the eddy loss of a core. Search term: impedance bridge.

Beware a simple, solitary rectifier on a transformer secondary, it can cause core saturation and can burn up the transformer and your house. It is better to use a bridge rectifier (four rectifiers). By the way, if you short the secondary of a transformer, excess current flows in both the secondary and the primary, and it will burn up. So, be careful with transformers. It is good to put a fuse in series with the primary.

Fuses are more complicated than you might expect. (This is true also for household circuit breakers.) A fuse is really just a wire that is thin enough to melt when the current gets too high. There are time-delay fuses that have some thermal mass so they can pass a transient, like the inrush current into power-supply capacitors. The important thing is that the fuse will blow before wiring gets hot enough to start a fire. The “wire” includes the wire in a transformer, in other words a transformer may be overloaded or it might have too much AC voltage on the primary, and after ten or twenty minutes it is going to start burning. The fuse has to be chosen so that doesn't happen. You can make your own fuse with very thin wire, but sometimes it has to be thinner than we can get our hands on. Even 30 gauge wire can take several amps.

One of the mysterious things about fuses is that they have to be built specially to handle the higher voltages. A fuse that can interrupt a 600V circuit is bigger than the common, 250V fuse because of arcing. Whenever a fuse blows, there is an arc. The arc can be much longer than the “dielectric strength of air,” which is about 83V per thousandth of an inch. Even a 120VAC arc can be .1” long. As a consequence, if a volt-ohmmeter has a fuse to protect the current ranges, that fuse may be a $10 giant fuse. Since volt-ohmmeter manufacturers lose market share if their instruments are big and expensive, it is common for VOMs to skip the fuse, and instead just put a label near the current jack, “unfused.” Some manufacturers, like the John Fluke company, make professional-grade VOMs that are safer, and they put the giant fuses in there.

Electrical safety is not the emphasis of this section about mysteries, but since you can die from 120VAC, let it be said that any voltage over 24V, DC or AC, has a risk. The higher the voltage, the more the risk of shock. And shock isn't the only risk. Any voltage source that can deliver over two watts can get combustibles hot enough to burn. So there are lots of things we do to keep ourselves safe, and when we build a project we may need to think of all the ways it may be hazardous. We often put warning labels on our projects, and we think about how other people might touch or use our project.

In the field of home and business construction, “building codes” are a big consideration, and many of them are to keep the electricity in structures from burning them up. Public safety authorities learn quickly when there is a disaster, and they are pretty quick to update building codes and zoning codes to protect people. Search terms: Shirtwaist fire, building code, NEMA electrical code. The Johnstown Flood is an example of a disaster that resulted in better dam inspections (though after over 1000 people died, the dam was rebuilt twice and collapsed twice). Search: Johnstown flood dam inspection

Mr. Engelbrecht once detected a threat of electrocution from prototype printers when he worked for IBM. The designers had rearranged the wires in the connector for 120V power, and it was going to put 120V on the metal case. Mr. E warned a coordinator, and they took care of the problem.

Even though 120VAC is hazardous, we often use it in projects to power up a stepdown transformer or to wire up a convenience outlet. Part of your learning about electronics is how to do these things safely, and we always have to keep from burning down our homes, being indicted for negligence, and going to prison. Even your average person in the home needs to select extension cords so they have thick enough wire to not melt and short out.

An electrical riddle: why do you need no more than .25W power rating in resistors above 150,000 ohms? Or even above 3600 ohms?

Duty cycle is a term you hear about. It is about pulses, such as in logic. Duty cycle is from 0% to 100%. Your duty cycle of sleeping is about 30%. The duty cycle of TV commercials is about 18%. During summer, the duty cycle of the air conditioner is about 15% at night and 70% during the afternoon. The duty cycle of the clock in logic is often 50%.

Sine waves, like the 120VAC at any power outlet, peak at 1.41*RMS, so a power outlet peaks at 170V. You can get DC voltage from an AC voltage, higher than the peak AC voltage, with just diodes and capacitors. How much higher? Like two or four or ten times as high, like 600VDC! It doesn't give much current, but it gives some. Search terms: multiplier doubler tripler diode rectifier ladder

With an inductor-capacitor “resonant” circuit, you can tune a radio station. The series version of the resonant circuit can produce a voltage that is 30 to 150 times as much as the input, though you can't pull any power out of it. (Pulling out power reduces the Q.) Search term: tank resonance tuning

Radio tuning is worth entire careers, it is a big field. The hard thing about tuning is that the station or channel you want, like 99.1MHz, is surrounded pretty closely by other, stronger stations, and you want to isolate just the one, weak station. The Q you need is about 99.1MHz/150kHz=661, and you can't do that with a copper-wound coil, not even with stagger tuning. But we know the radio or TV or WiFi is working, so how do they do it? The most sophisticated tuning I know of is quartz-crystal filters and crystal-lattice filters. There are also microwave-cavity tuners, SAW filters, and phase-locked loops. A $.60 tuning device is a ceramic resonator, and I have some from Murata. 455KHz SFULA455KU2B-B0 with Q of 46 (acutally, the passband is 10kHz and the skirts are quite steep), and 10.7MHz SFELF10M7GA00-B0. The 455kHz is for 3000 ohms input and output impedance and the 10.7MHz is for 330 ohm.

All commercial tuning is with superheterodyning, which is a sophisticated technique dreamed up by geniuses and barely understandable by regular people, but only if they are really interested.

A radio tuning capacitor from 35 years ago. This is an "air variable" capacitor. There is a section for AM tuning and a section for FM tuning. The two leads are the steel frame and the stationary plates. (The moving plates are connected to the frame via the brass shaft.) The range of capacitance for the larger section is about 35 to 365 pF.

A TV tuner from 2002, before digital broadcasts started. This intricate module includes amplification, tuning, and demodulation of the video and sound. The tuning is done digitally by a voltage-variable diode that has variable capacitance. (The entire, modern tuner is smaller and less expensive than just the old tuning capacitor.) The front of the module has the tuning inductors. The inset at the left shows two inductors that have been adjusted with a plastic blade. Spreading the turns lowers the nanohenries slightly. The back of the module is all surface mount. There is one amplifier integrated circuit. The inset at the right has gray-bodied capacitors. The red cylinders are likely diodes. The cylinders with color may be resistors. You can see copper traces on the PC board. (Traces have a light green coating.) Radio-frequency designers worked for four months to design this module. They had to make about five revisions to get it working--you can't really breadboard an RF circuit like this with a nylon breadboard, you have to do a full PC layout for each revision and get it manufactured, then see how it is working. Robots manufacture this module. The wholesale cost is only about $4. Manufacturing is done in Mexico, probably, to lower the costs and the cost of adjusting the inductors.

Inductors, resistors, and capacitors have two terminals. Transformers are inductors with three or more terminals. Working with transformers can get really complicated, but it is easy to go to Radio Shack and buy a simple stepdown transformer for your project. Just use it with a bridge or full-wave rectification (don't use it with just one rectifier, see the warning above). When you first turn it on with secondary rectification, stay with it for ten minutes and see if there is overheating.

As evidence of how versatile and complex transformers can get, just see how big this transformer section is.

An isolation transformer “floats,” it has no ground on the secondary. You may connect the secondary to any voltage “x,” DC or AC, and the secondary will give you power riding on x.

The photo shows two transformers that mount onto printed circuit boards. The large one has a copper foil band that reduces stray magnetic flux that was interfering with other circuits on the PC board. The little one has separated primary and secondary, insulated with red and blue tape. Blue is the primary, identified by the two pins. The secondary has a center tap. The white nylon bobbin has a separator to keep the electronic product safe from lightning surge that can hit the primary. The ferrite core on the right is particularly handy for DIY. You can wind your homemade, paper bobbin and then slip the cores into it. You can even wind two paper bobbins and put one on each leg of the cores. The place you find these cores is in old, cathode-ray-tube TVs. If you wanted to buy cores brand new, you would need to make a minumum order of $200, which would buy 70 core pairs.

More about the ferrite core above--these are common in the larger consumer products like DVD player and TV, or in an old computer power supply. If you have a junk product and see that there is a ferrite-core transformer that you want to salvage, get it loose from the printed circuit board. (A little propane torch can be used but don't burn down your house.) Hacksaw off the old wire and bobbin. Be gentle with the ferrite, it is very hard but quite brittle. You will probably find the the two parts of the core are epoxied together. Put it in a pot with a lid and half an inch of water. Boil gently for fifteen minutes. See if that loosens up the epoxy. Even if you crack the ferrite, it may be perfectly usable, just apply some glue on the break when you do the final transformer assembly. As for winding your new, 10kHz to 50kHz transformer, it is complicated to choose the wire gauge and number of turns. Without test equipment, you don't know the saturation amp-turns or the inductance of 10 turns.

An autotransformer is a three-terminal transformer that can give reduced voltage or increased voltage, but no isolation. They are available for 120VAC to give variable voltage, 0 to 125VAC. (But they are expensive.) They used to be used for stage lighting.

The “impedance” of the secondary is the primary impedance times the turns ratio squared. This is especially used for radio frequency. You can “match” 50 ohms to 75 ohms or 300 ohms. For tube-type audio, there is an output transformer that matches the high tube impedance to 8 ohm speaker impedance.

Transformer secondaries often have a “center tap.” It is merely a wire connected at the middle of the secondary. It lets you get “full wave rectification” without doing a bridge. As pointed out above at “Beware a simple, solitary rectifier,” you must not use one rectifier on a transformer secondary. You either have to do a bridge or use the center tap and two rectifiers. Hint: if you buy a transformer with a center-tapped secondary and intend to do two rectifiers for full-wave rectification, you must know that the rated secondary voltage means end-to-end, not center to end. So, you probably need a secondary voltage twice what you were thinking.

Transformers need a core. The core can be air! (especially for radio frequency) Magnetic cores are laminated steel for 60Hz and ferrite for audio. Ferrite is used for 20kHz to 500kHz “switch mode power supplies” where you can use the integrated circuit TL494.

The old TVs with the picture tube have a “flyback transformer” that can generate 20,000VDC. A big transistor turns on and stores energy in the ferrite (like real quick, 60us). When the transistor turns off, a tremendous pulse of voltage appears on the secondary and can be rectified onto a capacitor.

Inductors produce “inductive kick” just like water hammer in piping. This is the same phenomenon you see when you have a coiled-up water hose outside. You get the water flowing fast, then jerk off the faucet. Do you see the hose wiggle or jump? The inductive kick of an inductor is so powerful that if you take any 120VAC transformer and put a fresh AA battery on a winding for a fraction of a second with your fingers across that same winding, when the battery comes out of contact, you will get a shock. This can be painful if you apply the battery to the low side and are feeling the high side.

Transformers often have multiple secondary windings. They are used to get an assortment of DC voltages. If your transformer has two secondaries, they can be seriesed to add voltages. You can even series the secondaries of two or more transformers. But it isn't good to parallel secondaries. If there are multiple primaries, it is fine to series or parallel them. (But you have to be careful that the turns are adding, not subtracting.)

Let's say you have a stepdown transformer, 120VAC to 15VAC. Disconnect it from 120V. To the low-voltage side, apply 15VAC. Something comes out the primary! It is 120VAC. So, transformers don't have a defined input. You can input power into the high side or the low side. One way is stepup, the other way is stepdown. But if a winding is designed for some certain voltage, it can only be used for the rated voltage or lower. (If you put 120V onto a 15V winding, hoping to get a stepup on the 120V winding to give 120*120/15=960V, the 15V winding will saturate the core and it will burn up.)

Often when people hear that a transformer can be a stepup transformer, they think it is a way to generate free power, because you are getting out more than you put in! However, a stepup transformer produces less current on the secondary. Any transformer's input power = output power + winding loss due to resistance + core loss due to eddy current.

With a switch-mode power-supply (SMPS) transformer, there is a strange way to “model” it, but it works and it lets you think about how the primary current works. The model is the primary inductance in parallel with (secondary load multiplied by turns squared). Here is how this model predicts transistor current for SMPS. If the secondary isn't loaded, when the transistor turns on, the current ramps up at a rate of I/t=V/L, where L is the primary inductance. If the secondary is loaded, that means extra current in the transistor.

Inductance for ferrite cores tends to be .25uH to 2uH for small cores and maybe 2uH to 10uH for big cores, all for one turn. Inductance increases by turns squared!

The “cross-sectional area” of the copper in the primary is the same as the sum of the cross-sectional areas of the secondaries, when you design a transformer that has lowest heating of the copper due to the current. To fill out this idea more, a good transformer that is unloaded can just sit there, connected to the power source, and will heat up maybe 10 or 20 degrees F, but not more. The heating is partly from the current in the primary's wire resistance, due to the primary constituting a two-leaded inductor, where you can calculate the current from XL=2 pi f L and I=V/XL. The rest of the heating is from some current, called eddy current, in the steel laminations. Going ahead from this, ferrite is high resistance and doesn't have eddy-current loss. Search term: AWG wire table.

Buy ferrite toroids from Palomar Engineers. SMPS EE cores are hard to find, new. Try scavenging them from old electronic equipment.

For “laminated,” iron-core, 120V transformers, power capability tends to be about 30W per pound. To be sure, look up the dimensions of a transformer on a supplier's web site (like mouser.com) and get the current*voltage for the secondary. Any iron-core transformer with similar dimensions will give out the same power.

If you want to scavenge the wire from a laminated-core transformer, it is almost impossible. The E laminations are inserted in opposite directions and then varnished in place, and you can't get them out.

The troubles with making your own inductor or transformer boil down to getting the core and the wire. If you are working at radio frequency and don't need a magnetic core, that is one less problem. If you need a laminated iron core or a ferrite EE or pot core, it is hard to find a source that doesn't have a minimum order of $100. The wire, which is called magnet wire and is insulated with thin polyurethane or nylon, can be purchased but may be $140 per pound, where 30 years ago it would have been $15, without inflation. I don't know why magnet wire has gone up so much. If you need gauge 26 or thicker, you might be able to buy some wire at a motor winding shop. If you want to wind a 60Hz, iron-core transformer, you are going to need a “bobbin” to wind on. These also are available with a minimum order of maybe $150, so the net of it is that hobbyists just don't wind their own 60Hz transformers. And that is just as well, because you have to get just the right number of turns of the correct gauge of wire so it won't burn up due to saturation of the core.

If you are working with magnet wire, strip the tough insulation with a little piece of sandpaper. Don't use any solvent to dissolve off the insulation, the insulation is just too tough.

If you see a power rating of VA, volt-amps, instead of watts, it means about the same.

Inductors (just one coil with a core) can be used to store energy, E=.5L(I squared). The core usually needs to have a .001” to .007” “gap” so it won't saturate. The gap can be some tape or paper, nothing fancy.

If you want to make a Tesla coil, the first problem is getting enough of the right gauge of wire. You should know that the coil gives out only AC, not DC, and there are no practical uses for the output. On the other hand, a Tesla coil puts out a lot of radio interference, and you are likely to aggravate your neighbors.

Capacitor energy is .5CV², which is the same form as inductor energy, .5LI². The same form is mechanical kinetic energy, .5mv². Equations in physics that are of the same form are likely to have some underlying reason why they are the same form. It would be neat if E=.5mc², but that isn't the case, is it?

The form, .5xu², actually comes from calculus (from integration of a simpler equation). If you take calculus, you will see this type of equation in class and in your textbook.

The first electrical component used by experimenters in the 1700s was the capacitor, not the resistor. The experimenters really liked how they could store energy on capacitors. They would compete with each other about how severe a shock their capacitors would give. It was common for their arm to be numbed by the shock. We don't mess around like this, anymore. You can get nerve damage from shocks.

A similar story comes from the early 1900s when experimenters were making the early X-rays. They didn't know that the X-rays were killing the tissue in their hands as they looked at the moving hand bones with fluoroscopes. They had a convention in Europe, and they gathered for a chicken-dinner banquet. They were surprised and saddened that so many experimenters in the dining room couldn't cut their chicken because they had had to have their atrophied, gangrenous hands amputated.

Plastic-dielectric capacitors are rolled, literally, with two strips of plastic (or wax paper) and two strips of aluminum foil. (Commercially, it is often done with the aluminum evaporated onto the surface of the plastic.) You can roll your own, homemade capacitors but there are problems. 1) The little bit of air that gets rolled in there turns out to dominate the microfarads, and air reduces the microfarads a lot and makes it variable depending on whether the cap is being squeezed. 2) If you use your cap with much voltage, it just might short out, so you dare not use it where this could be a hazard. (Commercially, they might use two plastic strips everywhere they need one, and where the one strip has a weakness, the other one will back it up.) 3) Anywhere there is air or even a tiny air bubble, where AC voltage is applied, the air bubble gets corona in it, which is a hot plasma, and that will melt the plastic. (Commercially, they make AC capacitors in a vacuum so that there are zero air bubbles.)

Whereas resistors or inductors in series add, this is not true with capacitors. However, the voltage rating of seriesed, equal-uF caps adds. Another fact: if you roll your own cap, and you use twice the thickness of plastic, the uF is half.

If you build a cap with glass plates and foil squares, and just keep stacking foil-glass-foil-glass for ten layers, it will have a value of, maybe, 200pF from top to bottom. If you leave all the glass in place and slip out the intermediate foil, leaving the top and bottom foil, the pF will stay the same!

For tungsten-filament lamps, if you measure the filament resistance with an ohmmeter, you get ohms that is about 10% to 20% of the “hot” resistance. This is because tungsten, and almost all metals, are higher resistance at high temperature. The hot resistance isn't necessarily measured, it is calculated from P=V*V/R, like for a 60W lamp used at 120V. You do the math.

Going ahead with the idea of hot wire having higher resistance, a transformer that is hot has more resistance and gets even hotter.

The flux in magnets was visualized hundreds of years ago using iron filings. Around 1850, the electrical experimenters did iron filings with electromagnets. It is fun to do this today, and all you need is a coil of wire, not necessarily with a core, and a DC supply of several amps (or a car battery). Instead of iron filings, just get coarse steel wool and make particles with tin snips or old scissors. The experimenters found that the highest field came when there were many amps, many turns, magnetic material like non-hardened steel, and a “short flux path.” A short flux path is a closed loop through the coil and around the outside. This means, for a helix coil, a small diameter and a short helix, but that makes the coil run hot. (If you do this on your own and try a car battery, be careful that you don't start a fire or burn yourself.) You can try a toroid instead of a helix. Steel “yokes” help increase flux. Search term: research magnet. An electrical way of measuring flux is to use a linear Hall effect sensor, like from Jameco Electronics, but Hall sensors are not very sensitive. (A magnetometer, used to measure the earth's magnetic field, is sensitive.)

Magnetic field strength is measured in amp-turns, whether you have a permanent magnet or an electromagnet. For an electromagnet of 100 turns and .5A, the field strength is 50 amp-turns. (There are other units like gilberts and webers, and there are conversion factors between them.) A different magnetic measure is flux density, which is what we think of when we think about "powerful magnets." It is more complex than amp-turns.

The funny thing about magnetic flux is that the flux lines behave as if they shy away from each other. For an electromagnet with a core, you can put little nails on the "pole pieces" and visualize the flux. You will see that as the flux comes out the iron, the lines slant away from each other. The lines spread out in the air to lessen their flux density. If you put your nails all on one pole and push nail A toward nail B, nail B will shy away from nail A. This is a curious thing to see. There is a practical consequence of this in powerful electromagnets, like $1000 magnets that use 300 Amps. The flux in the middle of the turns is so powerful that it actually pushes out on the copper turns. The biggest electromagnets have to be built with metal bands around the outside to keep the coils from exploding outward. In superconducting electromagnets for particle accelerators, the magnetism is so intense and the outward forces on each turn are so great that the coils will push out into any tiny space they can find. This changes the magnetic field by a tiny amount and the researcher has to adjust the current to get the field back to what he needs, which amounts to the particle beam wandering a little from the center of the beam tube, which is high vacuum. What they do is vary the current a little for a couple of hours and let the coils settle down into stable positions. This is a very mysterious thing.

Another mystery thing about flux is about lightning rods, which is what Benjamin Franklin worked on. (He didn't like the way lightning was setting houses on fire.) You have to "ground" lightning rods to a big stake driven into dirt. The wires that connect the rods to the ground are sometimes found to be ruptured by a lightning strike. Here is what happens. Lightning current is like 20,000 amps, and it lasts maybe 10 microseconds. The current is so high that the magnetic flux is terrific. Any right-angle turn of the "down conductor" amounts to concentrating the flux in the 90-degree turn, and flux doesn't like to be concentrated. The flux tries to straighten out the turn to be straight! It can put hundreds of pounds of force on the poor wire, and sometimes it breaks the wire.

If you have an electromagnet with AC current, it is an inductor and it has AC flux. A simple sensor for AC flux is merely a single loop of wire, like 4" diameter, going to an earphone, or through an amplifier to an earphone. You can use ten turns or 100 turns if you wish, and it will be that much more sensitive. The loop doesn't even have to be round, it can be any shape. The wire can be tiny, no need for 18 gauge. If you have a loop going to an earphone, you can put your loop near a power supply (such as the supply in any electronic device that is powered by 120VAC) and you are likely to hear the AC flux from the transformer and the rectifiers. It will be directional--you will be able to "null out" the sound by orienting your loop so the flux doesn't "cut through" the coil. By the way, your loop must be one wire, not two or three like in a consumer earphone set.

Insulation is a fundamental concept for currents. You need insulation so voltages don't short out. Extension cords have insulation so you don't get shocked. But magnetism is different. There is no magnetic insulation. Nothing can stop a flux line, flux lines always form closed loops. If there were magnetic insulation, it would be useful in the record heads of hard disk drives and tape drives, to push the flux out into the tape or the platter.

What is available for flux is shielding. Non-hardened steel, ferrite, and other magnetic materials can be arranged to take in flux and shunt it away from where you don't want the flux, but this is very different from having a magnetic insulator.

Spectrum analysis, harmonics, and Fourier

The simplest sound is a sine wave. (See Wikipedia, sines and cosines are very special because their maximum slope is 1 or -1 and the differentiation or integration of a sine is a cosine, and vice versa.) A sine wave is a pure tone with no "harmonics." A sine is merely how much above or below the origin a dot is, where it moves counterclockwise in a circle of radius 1, starting at x=1, y=0. A cosine is how far right or left the dot is. A sine and a cosine are shifted by 90 degrees.

If you have a "resonant circuit" with an inductor and capacitor (or a mechanical resonance of a mass on something that is springy), the voltage and current are sine and cosine, exactly. Most rattly things in cars are resonance.

Sounds are rarely sine waves, though. There are "harmonics" or overtones. For example, if your "fundamental" is 262Hz, which happens to be middle C, that could be the fundamental for a guitar string or your vocal cords, and there are going to be small amplitudes of harmonics at integer multiples, 524Hz, 786Hz, 1048Hz, etc. But there won't be anything at 300Hz or any other non-integer-multiple of the fundamental. This is a curious thing, but it was known in the 1700s! People looked at the nodes of a vibrating guitar string and could see that the sounds were integer multiples of the fundamental.

The only thing that distinguishes a guitar sound from a voice or a cornet or an organ pipe or engine noise or when you whistle or any other steady sound is the harmonics! If you can use a computer and a digital to analog converter to make a fundamental and some harmonics, you can come up with the sound of anything that makes a steady sound. You have to take care of harmonics up to about 5000Hz, anything above that doesn't matter very much. All this applies, as I mentioned, to steady sounds, which rules out explosions, white noise, varying frequencies, sounds that start and stop randomly, etc.

Dr. Fourier, in the early 1800s, studied harmonics. People were theorizing that integer multiples of a fundamental could make up sounds. Dr. Fourier made some ingenious apparatus that made low frequencies and traced the vibrations on sooty glass plates, like a vinyl record. He proved what people were suspecting, and the technique was named Fourier analysis.

A short QBasic program can be written to find the harmonics in a digitized sound. If the data file has a number of samples that is an integer power of 2 (128, 256, 2048, etc.), the Fast Fourier Transform can do a quick analysis, so quick that a computer can analyze sound in real time and do amazing things.

An instrument called a spectrum analyzer can take audio or radio frequencies and show the harmonics, where y is amplitude and x is frequency. A spectrum analyzer can show amplifier distortion and many other neat things.

When you have rectifiers and a filter capacitor on a 60Hz transformer secondary, there is no current until the AC voltage on the secondary exceeds the filter-cap voltage. Then a big pulse of current flows to charge up the cap. There are lots of harmonics in the secondary current, up to the tenth harmonic, and that causes extra eddy current in the core, and the pulse of current heats the wire a lot more than a steady current. All in all, rectification heats a transformer a lot more than a resistive load, and this may be cause to buy a transformer that can put out more current than you will need. (Transformers are rated for resistive load.)

More about harmonics

Human hearing cannot distinguish inversion; if you switch wires on the speaker and listen again for your sine wave, music, or voice, it sounds the same even though the speaker cone pulls in where it used to push out. Furthermore, for various harmonics sounding together, inverting one particular harmonic or changing a sine to a cosine does not make much difference.

On the other hand, human hearing can distinguish harmonics easily. If a sound has a fundamental of 262 Hz and harmonics of 524, 786, and 1048 Hz, the 786 Hz can be much quieter but will easily be heard. (This is not too surprising if you know that the cochlea has thousands of frequency sensors that are each sensitive to fairly narrow frequency bands.)

If you know what a speaker crossover is, it produces phase shift in the crossover region between speakers, but it does not degrade the sound.

The old type of telephones (with the wires that go to a “central office”) only send 300Hz to 3000Hz. The harmonics of your voice are mainly in that range, and you can be easily understood with that limited bandwidth. It seems strange that Middle C is 262Hz, below the telephone bandwidth. Children can hear 20Hz to 20kHz.

In multispeaker installations (crossovered system or stereo), you must get speaker phases correct (all cones move out at the same time) or there will be distortion or a sense that sounds are coming from strange locations.

Your two ears can tell where a sound comes from merely by the difference in time of the sound reaching the ears, even though it is only .8ft / 1080 ft per second = .0007 seconds, a very brief time. The design of your brain and the learning of the neurons is that amazing!

Waveforms above are Fourier analysis of a voice. Each waveform shows one cycle of the sound. Various numbers of harmonics are added together to get more and more accurate replications of the sound that was spoken into a microphone. (At 4 harmonics, ignore the kinks, these waveforms were traced and accidentally gained some corners.) At 4 harmonics, the f marks show the fourth harmonic, which is pretty strong for the "a" sound. If you could hear the "4 harmonics," it might sound like an "a," or you might not really hear it as an "a" until you add in more harmoinics, like at 14 harmonics.

The original voice spoke the "a" at about 247 Hz. That makes the fourth harmonic 988 Hz. The 28th harmonic is 6916 Hz, which is about as high a frequency as adults can hear. Adding 14 harmonics, you would clearly hear the "a" sound, but it might take 28 harmonics to tell it was said by a man, and to distinguish between different speakers.

To emphasize the point, these waveforms are composed only of sine waves that are integer multiples of 247 Hz. There are some sines and some cosines.

The DIY equipment to do this work is not trivial. You need a $3 ADC, a $1.70 static RAM memory, a DIY 10kHz oscillator, and a digital chip (such as an Atmel CPLD $6) to generate the addresses for the memory. You also need some way to feed the memory contents into a personal computer, such as into the serial port. The CPLD can do this when aided by an RS-232 transmit chip. The PC needs a program to accept the data, and this can be written in Visual Basic, which can also do the Fourier analysis. To hear the sound played back, you have to send the digital waveform back out to a memory and use a DAC to get back to analog.

A spectrum analyzer, as mentioned above, can find any harmonics in a waveform. There is a qualification to this statement: the waveform must be periodic, it can't be a "one shot" waveform. A basic spectrum analyzer is an analog instrument. It is strange that this analog instrument can be used to evaluate digital waveforms, in a particular sense, namely whether a digital clock has 50% duty cycle and how fast the "edges" of the clock are. It is true that an oscilloscope is more convenient to do these measurements, but oscilloscopes have limited "bandwidth," like 20 MHz or 100 MHz or 200 MHz. If your clock is, say, 1.8 GHz in a personal computer, or a gigabit Ethernet signal, there aren't many oscilloscopes that can evaluate the "squareness" of such fast signals. But spectrum analyzers are available that can look at frequencies to 100 GHz and more. This type of spectrum analyzer can show the harmonics of fast digital signals, and you can compare to the harmonics that are characteristic of 50%-duty-cycle "square waves." You can judge the transistion rate of the digital signal (the steepness of the edges in volts per nanosecond) based on harmonics. The only requirement is that if your digital signal is 100MHz (period 10 ns) and has edges that transition in .5 ns, there are significant harmonics at 1/(2*.5 ns) = 1 GHz, and maybe five times 1 GHz, so your spectrum analyzer ought to cover 5 GHz. Let it be said that spectrum analyzers cost upwards of $1000 and $50,000, so your average hobbyist isn't going to have a spectrum analyzer.

The key ingredient of a spectrum analyzer is a low-distortion sine wave that is variable in frequency and of known frequency, up to the "coverage frequency" of the instrument. Generating such frequencies is a fundamental job of electrical engineering and physics, and it isn't easy. There are whole books written by professors about this.

A related field is phase-locked loops. A PLL can generate any integer division (sub-multiple) of a frequency. PLLs are common in computers and are a key part of the tuning of cell phones. PLLs use fast digital logic, so whereas a spectrum analyzer can be a purely analog instrument, a PLL is going to be partly digital. A PLL in a cell phone is a tiny part of the chip and might have a dollar value of ten cents. So we see that there is a spectrum of cost in insturments: PLLs can be ten cents (though as lab instruments will be thousands of dollars), oscilloscopes are $50 to $2000, and spectrum analyzers are $100 to $50,000.

Any person who is wanting to design instruments such as these must have a lot of math training. Also, if you are going to just understand or appreciate these devices and things like cell-phone tuning, you will be limited in your understanding if you have limited math training. Countries that have excellent universities have a foundation for designing cell-phone infrastructure, computer networking, defense industry, nuclear power, and high-tech manufacturing. One way that China has gained militarily and in its ability to manufacture (such as computer motherboards) is by sending young people to study at leading U.S. and European universities. I saw this from 1985 to 2003 in Austin at the University of Texas. As China's universities advance, they will be more and more able to make their own radar, nuclear, and space industries, and this poses a challenge to the U.S. On the other hand, a Communist country that sends its young people to free cultures will have those people pushing for freedom when they return home. Iran wants to dominate its area of the world, but Iran is held back by cultural effects and limited technology in its universities, but Iran gets technology from China and Russia. China crippled its technology push during Mao's political Great Leap Forward when the Red Guard, a purification movement of young people that is related to the 2011 Occupy Movement in the West, sent university professors to plant rice and work in sewage treatment plants.

Oscillation in amplifiers is a bad thing. An amplifier is supposed to amplify, not oscillate. (If you are intending to make an oscillator, that is another thing.) But any amplifier that amplifies more than 100 is prone to oscillate.

There are some reasons for amplifier oscillation.

The common emitter transistor stage is quite sensitive to supply noise. On the other hand, a differential pair can have good “supply noise rejection” if you know how to design for that.

An amplifier that has feedback is very prone to oscillate. The feedback may be in a Class B power amplifier, where the feedback is supposed to reduce the crossover dead spot. Or the feedback may be in an op-amp circuit.

Oscillation in microphone systems (public address systems) is common. The speakers feed back into the microphone and you get the irritating squeal. If you move the mic toward or away from the speaker, the squeal often changes pitch; this shows the delay of the speed of sound working with phase shift in the amplifier to produce 360 degrees of phase shift.

A bullhorn is made to reduce sound-wave feedback from speaker to mic: the mic is in the back where the sound is less likely to go.

Churches and auditoriums often have an audio computer that looks for any oscillation and instantly adjusts a filter bank to reduce gain at the frequency that is building up.

If you are making a high-gain RF amplifier, like to amplify a radio station or data link (gain is often 100,000), it is easy for the high frequency to “couple” from the output to the input. If the loop gain is more than one, it will oscillate. That is why RF amplifiers in TVs have a lot of metal shielding, and why coaxial cable is used on the inputs and outputs of RF amplifiers. Radio-frequency grounding is especially strange because a metal part you intend to be ground might be a quarter wave (or integer multiple) long, and that transforms it into an inductance so that it might not be ground after all.

If you have an oscillating amplifier, you always think about adding capacitors from supplies to ground. This is called a decoupling cap or bypass cap. In fact, any circuit that has a transistor needs at least one capacitor from each supply to ground. The cap is often 100uF or other electrolytic and/or a ceramic of .01uF or .1uF. If you put an electrolytic on a negative supply, be sure that the minus end of the cap is on the “more negative” point.

An oscillating amplifier may stop oscillating if it is redone with a “ground plane,” a layer on the printed circuit board that is solid, grounded copper except for cutouts for through-board wires (vias or leaded components). Professionally made PCBs usually have two hidden “innerplanes,” one for ground and one for 5V or 3.3V. Computer motherboards always have a ground plane. The idea of a ground plane is that a wide trace has less inductance than a narrow trace, and the ultimate, wide, grounded trace is a rectangular ground plane the size of the board. DIY PCBs can have a ground plane by using a double-sided board and making the back the ground.

“Transducers” usually have low voltages on their outputs. A microphone may have 50uV or 2mV. A thermocouple is at the 100uV level. Strain gauges have tiny outputs. Solar cells are usually below 1V. The photodiodes that receive 30km fiber-optic transmissions have current outputs in the microamp region. All these transducers normally must be amplified.

An amplifier has one input and one output. A stereo amp is just two amplifers. An oscillator has no inputs. A differential amplifier has two inputs, one plus and one minus. An XLR audio amp has two inputs (differential) and will have either one output (single ended) or two outputs (differential).

Skin Current

A very mysterious effect is the skin effect. (Search term skin effect or skin depth.) This is where high-frequency currents do not penetrate very far into the inside of a wire. This comes from physics, where it is known that electric field lines terminate onto conductors at right angles, not at shallow angles. A conductor sitting in an electric field, either DC or AC, is all at one potential. There is no internal field inside the conductor. The exception is that if the conductor is carrying a lot of DC current, there is a small field along the wire that makes the current flow. But AC electric field at a high enough frequency doesn't penetrate to the middle of the conductor, and the AC current on a wire is higher on the skin of the wire than in the middle. As frequency rises, this is more and more true. 60Hz wires more than about 2” diameter don't carry much current at the middle. At 1 MHz, the conducting skin of the wire is only xxx thick.

Wire resistance at RF frequencies is a lot higher than at DC because of skin effect. RF wires for radio transmitters are usually hollow tubing. Resonant circuits for radio tuning are lower Q because the coil's resistance is higher than the DC resistance. Food cooking in a microwave oven gets hot merely from high current on the surface of the food. The current only penetrates about a quarter of an inch. (Higher-resistance conductors like food let the RF penetrate deeper than metal.) If you have a frozen block of food in a microwave, the inside isn't getting heated at all by the RF current, it will get hot only by thermal conduction. So, from knowledge of the skin effect, we know that uniform heating in a microwave comes from thin strips or sheets of food. Chopped food in a bowl will heat better if you mash the food into a ring, up on the side of the bowl.

There is a correspondence between skin effect and magnetism. Due to eddy current, changing magnetic field takes a while to penetrate to the inside of a thick chunk of steel. The writer once made a 50-pound electromagnet using a core of nested steel pipe parts. They were not tightly packed. When the current in the coil was turned on, there was a delay of half a second, then you heard a small pipe on the inside click up onto some other pipe as the magnetism worked into the inside. This is why 60Hz-transformer cores are laminated. The laminations are insulated from each other and practically eliminate the eddy current.

RF (Radio Frequency) currents do strange things in long-wire antennas. If you were to take an RF ammeter and put it in series with the antenna, just at the feed point from the transmitter, you would measure some current. If you move the ammeter out to the midpoint of the antenna (and especially if the antenna is several wavelengths long), you would measure less current. This violates Kirchoff's Current Law, which says that DC current in a wire is the same all along the wire, until you get to a branch point with another wire (or if electrons are jumping out of the wire, like due to the photoelectric effect). There is actually no violation of Kirchoff's Current Law, because it is true for DC, not AC. With the antenna, electromagnetic energy is leaving the wire and taking away energy, and this leaves less energy, and less current, flowing down the wire. There is a common-sense feature about this mystery: if you move the ammeter out to the very end of the antenna, there is no current whatever, because at the end of the wire there is no place for the current to go anymore.

Talking about antennas, there are other mysteries. If you talk to the typical electrical engineer about radio frequencies, like 100 MHz, such as a clock in logic circuitry, engineers have a common conception that energy can just leap off a logic wire and "couple" to other wires, causing logic glitches in the logic. This is true to an extent, and it is because of the magnetic field around the current-carrying wire. There really is a transformer effect, and there is a primary and a secondary. The interference is less if there is a ground plane (see above) or if you do balanced currents with twisted pair or coax. I am not giving reasons, this is just what has been observed for decades. If you look at a photo of an old Cray supercomputer, one built in a ring, in the middle you see the twisted pairs that carried the balanced, "ECL," logic signals.

A related mystery is where you have RF currents, like in any personal computer, and the FCC requires that not too much radio interference escape to interfere with radio, cell phone, and TV communication. (Look for the label on any digital device, about Class B or Class A.) The simplest (but not cheapest) way to keep interference from escaping is to put a metal shield around your product. RF can't go through metal shielding. But if there is any wire leaving the shield, like the DC power wires, they will act like antennas. This is why so many products have the "ferrite bead" molded onto the cable, that reduces RF radiation about 70%. Another problem with metal shielding is that any seam that isn't soldered, riveted, or otherwise joined every 1/20 of a wavelength constitutes a "slot antenna." This is very strange. The reason is that the inside of the shielding has RF currents induced by the "accidental antennas" inside the shield, and the currents on the inside surface of the shield exactly balance out the RF energy that is striking the shield. Any seam interrupts where the shield currents should be flowing, and RF currents appear on the outside of the shield (they leak through the seam), right at the seam, and radiate away from the product. If you doubt this, look at the case of a tower computer. The better cases have unpainted "fingers" that establish good conduction between the various pieces of the case. This is also true of the connector mounting plate at the back of the tower, there are little dimples that connect the plate to the case to reduce the slot antenna effect.

All this RF business is described in some manuals of the ARRL, American Radio Relay League. http://www.arrl.org/shop/ARRL-Antenna-Book-22nd-Hardcover-Edition

The professional electiical engineering field for radio interference is called electromagnetic compatibility. This is an extreme specialty and many technical people are in awe of what EMC engineers do. Related EE fields are generally called RF engineering, and they get into cell phone antennas and radar. If you see a flat-screen TV with advertising about QAM, that is quadrature amplitude modulation, and it is RF engineering.

Radio hams and RF engineers speak of impedance in ways other than resistors. They talk about free space being 377 ohms. They say that coax cables are 90 ohms (for TV) or 50 ohms (for instruments). Common twisted pair, like in Ethernet cables for networking, are 100 ohms. There is no measurable resistance, not that you measure with a DMM. The RF ohms comes from a tricky concept involving inductance and capacitance per inch of cable.

Power and Energy

Power and energy are different. Energy is Joules. Power is energy per time, such as Joules per second=Watts.

A household pays for energy, not power. In other words, you pay for Joules (kilowatt hours), not Watts. You pay the same for 100 kilowatthours no matter if you use it all on three days or use a little every day for the whole month.

All appliances are rated in Watts except vacuums and extension cords, which are amps. Household breakers are sometimes 15 amp, so if you plug too many high-power appliances into one breaker, it can trip. What is a hair dryer rated in, amps or Watts? The maximum power for an appliance is usually 120V * 15A = 1800W.

Batteries are rated in amp hours. A car battery is about 80 amp hours. A digital-camera AA rechargeable cell is about 1500 mA hours.

Circuit breakers are rated in amps. Household breakers will not trip until the current is 130% of the rating for a minute.

Resistors and transistors are rated in Watts, and they can tolerate spikes that are far higher than their rating.

Any spark, like static from your finger on a dry day or lightning, has an amazing power because the spark lasts only a few nanoseconds. But the static you feel has energy, not power.

A defibrillator, like one kept at the field by a football team, stores energy on a capacitor. The energy is rated in Joules and is about 70J. Another device that is rated in Joules is a battery-powered impact drill. They typically have 1.9J per impact.

If you are getting shocked in your hand because you are holding metal, there is a can't-let-go current around 0.02 amps, which is very small. If current is going through your heart with about .01A for ten seconds, that may cause cardiac arrest. Theoretically, you could electrocute 200 people simultaneously without tripping a circuit breaker.

Electrical energy in a capacitor corresponds to mechanical energy, like potential energy or kinetic energy. They all are measured in Joules.

Calories are energy. A calorie of food is actually 1000 calories, which is 4200J. That means a slice of bread is 378,000J! If you dry food thoroughly to get water out, you can burn it and turn the potential energy into kinetic energy (heat being how rapidly the atoms move). This works most convincingly on high-fat food like bacon, margarine, doughnuts, and potato chips. ( But it must be dried.)

A person can exercise or work at more than one horsepower, 746 Watts, for several seconds, then you get tired or sweaty and have to reduce your power.

Does a car's gas tank contain energy or power?

Does a battery store energy or power?

The popular rating of computer memory and processors is speed or nanoseconds, but they can also be rated as picojoules per bit or per flop (floating point math operation). Digital photographs can be rated by energy. It is typically 35J per photo (including processor and backlight but not flash) or 6uJ per pixel.

A single photon has a definite energy. It is proportional to frequency, which means it is inversely proportional to wavelength. A photon of light is much less energy than a photon of x-ray, which is much less than the gamma photon released by one plutonium atom fissioning.

Resistance is volts per amp, just like Ohm's Law says R=V/I. If you see something about thermal resistance, like for a power transistor, it is degrees per watt. It helps you decide how much “heat sinking” you need.

Thevenin Equivalent

Mr. Thevenin came up with an intriguing idea in 1883. It was about the voltage divider. He knew that when you draw current from a divider, the voltage drops off. (This is called “loading the divider.”) It causes the voltage divider to give the wrong voltage. Mr. Thevenin “modeled” the divider is a simple way, with a battery and one resistor. His model works exactly like the divider—if you put the real divider in a box and the battery-resistor model in another box, you couldn't tell which was which just by measuring with a meter. (But the real divider would heat up a little when unloaded, whereas the model wouldn't heat up at all.)

The advantage of using the “Thevenin equivalent” is that, with its one resistor, it is very simple to think about, much easier than thinking about a divider. This is such an advantage that we often use the Thevenin equivalent when we are designing circuits.

The Thevenin equivalent is easy to calculate. The battery is just the unloaded divider output. The resistor is simply the parallel of the two divider resistors.

As we are designing circuits, sometimes we design a divider by first thinking of the Thevenin equivalent, then work backwards to get the divider resistors.

When you have parallel resistors, the lowest ohms has the highest amps. A thick wire has less ohms than a thin wire. A lightweight extension cord (not the thick, orange ones) commonly has gauge 18 wire, which is only supposed to carry 3 amps. It won't get very warm at 3A, but it just isn't safe to put over 3A in 18 gauge. Bear in mind that heating is proportional to amps squared!

A wire that is 3 gauge sizes smaller in the gauge number has half the ohms. Six gauge sizes smaller is half the diameter and a fourth the ohms. All these considerations are what caused Thomas Edison's DC power distribution to lose out to the brilliant AC distribution from Nikola Tesla, in 1891.

Analog to digital and digital to analog conversion

These are abbreviated ADC and DAC. This is the basis of getting the analog world into a microprocessor. These conversions are high tech, either to get speed and accuracy or to get low cost and integration into microprocessors (which is widely done). A standalone ADC with ten bits can be had for $2. If you use audio software, you have seen selections about conversion rate and bits of resolution. Along with ADC, you usually see S/H (sample and hold), and that is often built into the ADC. You might see the Nyquist rate, which says that the sample rate needs to be at least twice the highest frequency that is in the waveform. (This is related to spectrum analysis.) It is better to sample at five times the highest frequency.

“Logic levels” used to all be 0V and 5V. The “switching threshold” for “TTL” was 1.5V. Once CMOS logic came out, it might have TTL threshold or 2.5V threshold. When microprocessors got fast, logic levels for them were 0V and 3.3V. So if you are buying some logic or memory, you have to find out what the logic level and power supply are.

Centuries ago, some smart physicist said that if you can't measure something, you don't understand it. This is very true. As measurements of various things have gotten very accurate, like to 6 decimal places or 9 places, scientists have learned a lot of things they wouldn't have learned otherwise. And they have come across amazing things like microwaves, quantum effects, nuclear magnetic resonance, tunneling, superconductivity, laser cooling of atoms, how big atoms are, and what makes up an atom. The current puzzle in astronomy, dark matter, comes from astronomers measuring the density of matter in galaxies and finding that about 80% of mass is nowhere to be found.

Lord Kelvin pointed out that “If you can’t measure it, you can’t improve it”. Albert Einstein: “If you can't explain it to a six year old, you don't understand it yourself.” H. James Harrington “Measurement is the first step that leads to control and eventually to improvement. If you can’t measure something, you can’t understand it. If you can’t understand it, you can’t control it. If you can’t control it, you can’t improve it.”

Repeating the wisdom that if you can't measure something, you don't understand it, you can understand that measuring instruments are very important in electronics, both to measure electrical things and also to measure other things, like pH, light, wavelength, magnetism, and time. Two important instruments in Solder and Circuits are the frequency counter and the oscilloscope.