Why math? The main reason is if you try to read any technician-level books to repair something you can't get away from it. Math gives very concise descriptions, if you know how to read it. In most cases, you won't need to do the calculations I'm going to talk about here, but I'm using it as a way to explain things you need to know about. I know a lot of people are terrified of math, and the more you put in, the more readers you loose, but a decent scientific calculator is dirt cheap and all you need to know is how to enter the equations.

If you're going to design things, not just fix them, you will need to calculate answers. If your pump blows its capacitor or a wire fuses open, and you need to replace it, you need to figure out what value capacitor or what size wire was in there and get a replacement – after you figure out why it blew, if you can. If you need to design a solar electric system, like we're doing in a side project, you need to do a bunch of calculations.

As I say in my mast head, my purpose here is to help get people resilient to the nastiness that is coming. Growing food is essential. Filtering water is essential. Beans, bullets, band-aids and bullion are all essential. But so is keeping technology alive and keeping up what we can of what we have, even if it's just a few of us with some form of power. Make it, make it work or do without. With that out of the way, onward we go:

Everything you learned about DC applies to AC, and there's way more stuff to keep in mind.

To begin with, let me remind (or tell) you that AC stands for alternating current. The electrons vibrate back and forth at the frequency the circuit is excited by. For now, I'll talk about the power line frequency, which is 60 Hz (or 60 “cycles per second” as they used to say), and tends to be either 60 or 50 Hz in various places around the world. The line voltage coming to your house is produced by a generator, which produces a very clean sine wave. Since it goes from zero to maximum, to minimum and then back to zero 60 times per second, the period of one of those cycles is 1/60 second, or about 16 2/3 milliseconds (16.667 thousandths of a second). Many light sources and other things around you flicker at that rate, chosen to be faster than most people can see.

There is never any problem with using parts rated for 220V AC in place of those rated for 120. In the case of the plastic outlets and plugs you buy at your local store, 220V plugs and outlets are shaped differently, just to keep you from plugging 120V stuff into 220V – which would probably blow it into the next county. A lot of safety and sense is built into the National Electric Code (NEC) and the things you buy that are compliant with it.

In addition to resistance, there are two other characteristics of circuits that you need to know about. In a way they are mirror images of each other. The first of these is inductance and the second is capacitance. Components designed to exhibit these characteristics are called inductors, or coils, and capacitors.

Inductance is a property of electrical circuits that causes energy to be stored in a magnetic field – however big or small, long or briefly. Inductors are usually coils of wire, frequently wound on a core material that increases the amount of inductance. In a 60 Hz component, that core is usually laminated steel sheet (thin to prevent losses from the magnetic fields' "eddy currents"), made from alloys that tune this effect. They can also be cores made from ferrite, a ceramic composite made with iron powder. A coil of wire wound on a steel core? Doesn't that sound like part of an electric motor? Yes it is and motors are inductors, too. A coil of wire is just a longer wire wound up to save space, and they conduct DC. The unit of inductance is called a Henry (after Joe Bob Henry), and is very large for practical inductors. Instead, you'll encounter inductors in milliHenries (thousandths of a Henry) microhenries and nanohenries (millionths and billionths, respectively).

The “mirror image” of the inductor is the capacitor, and a similar definition is that capacitance is a property or electrical circuits that causes energy to be stored in an electric field. The simplest capacitors are just two metal plates alongside each other with an air gap or other insulator (also called a dielectric) between them. Because they don't present a DC connection, DC can't flow through them; they block DC but conduct AC. The unit of capacitance is called a Farad (after Joe Bob Farad), and like the Henry, is extremely large for practical capacitors. Instead, you'll encounter capacitors in most often microfarads, and picofarads (millionths and trillionths, respectively). A capacitor measuring a Farad or more was a lab curiosity until the mid 1980s, when someone introduced the “super capacitor” to take the place of a small battery for computer memory backup. Today, super capacitors are being looked at to take the place of batteries in electric vehicles.

A circuit with the symbols for an inductor (L) and capacitor (C) looks like this:

In addition to resistance to DC, these parts have new property called reactance, an “AC only” resistance. Their reactances are opposites of each other; for an inductor, the reactance is

Xl = 2*pi*freq*L

where pi is the famous irrational number which we can shorten to 3.142, freq is the frequency of the AC in Hertz (Hz), and L is the inductance in Henries. Result in ohms.

Since this is a linear relationship, for a particular coil, as the frequency goes up, the reactance goes up, or for a fixed frequency, as the inductance goes up, the reactance goes up.

Capacitors behave the opposite way.

Capacitive reactance is expressed by

Xc = - 1/(2*pi*freq*C)

where pi and freq are as before, and C is the capacitance in Farads. Result in ohms, and notice that this is a negative number.

In this inverse relationship, for a particular capacitor, as the frequency goes up, the reactance goes down, or for a fixed frequency, as the capacitance goes up, the reactance goes down.

They're inverse in yet another way. The voltage and current shift out of phase to each other, or the current in a resistor, and inductors behave differently than capacitors. In an inductor, the voltage zips on through, but as the magnetic field starts to grow, it opposes current flow. In an RL circuit, the current peaks ¼ cycle, 90 degrees, after the voltage. In a capacitor, by contrast, the voltage zips in, hits that open plate and stops. At this point the charge on that plate causes an opposite electric field to grow on the other plate of the capacitor, which then effects the circuit on that side of the gap. Electrons moving on one side of the insulator gap are causing them to move on the other side. It sounds a lot like – and is – the same as current flowing, so in the capacitor the current leads the voltage by 90 degrees.

They combine oppositely, too. Inductors combine like resistors: in series they add (think of sticking more wire on the end of wire in the coil - it makes the coil bigger); in parallel, they get smaller, with the reciprocal of the sum of reciprocals like resistors. Capacitors in parallel add (think of making the plates a larger area). In series, capacitors get smaller by the reciprocal of the sum of reciprocals.

While reactance is stated in ohms, it doesn't cause real power to be lost in a circuit. Power put into the reactance can come back out, but there's still a product of volts times amps in the reactance. Since it isn't really lost (like power in a resistor turns into heat and goes away) this is called apparent power. To distinguish apparent power from real power, it isn't referred to in Watts, but VAR – Volts*Amps Reactive.

This probably sounds pretty awkward, but I'm trying to dance around invoking trigonometry. I'd really rather not get into a ton of vector diagrams and triangles, but the fact is reactance is at right angles to resistance, so that you don't just add ohms of reactance and ohms of resistance. If you have 10 ohms of resistance and 10 ohms of reactance in series, the total isn't 20 ohms. Instead, we define a new “super resistance” that includes both reactance and resistance called Impedance, denoted by Z, (pronounced with a long “E” - it impedes the flow of current). In a simple series RL or RC circuit, Z is the square root of the sum of the squares.

Z= Sqrt(R^2+X^2)

If you are familiar with "8 ohm speakers" or "600 ohm headset", that's impedance they're referring to.

What if the circuit has both kinds of reactance, an RLC circuit? Series or parallel L and C? That's for later, when we get into “real” electronics.

Have you ever seen a motor with a starting capacitor on it? Maybe you can see why it's there now: the inductance of the motor windings causes large positive reactance. The capacitor introduces an offsetting negative reactance to cancel out some or all of that positive reactance. If you connect a large inductor (motor) to the AC line without that capacitor, due to that right angle relationship of impedance, the inductance of the motor can actually dump power back into the AC generator. Most generators don't take this joke very well. The capacitor cancels out the inductance and prevents the motor from dumping power back into the generator.

The concept I'm describing here is called the power factor, after all the inductors and capacitors in a circuit have done their stuff, the circuit has a net reactance; either inductive or capacitive. This gives a phase angle between the voltage and current. Power factor is numerically the cosine of the angle between voltage and current, always between -1 and +1. Power factor is important in the design of power systems. Even in electronic boxes, where inductors and capacitors are used all over, the power supply designer might impose limits on the amount of excess capacitance or inductance allowed. (source)

One of the advantages of AC over DC, and why our power grid is almost exclusively AC, is that the voltage can be changed at will, almost completely without loss, by components called transformers. A typical transformer has two windings, primary and secondary, wrapped on some sort of bobbin (often paper) and wrapped on a metallic core to increase the inductance. The important characteristic is the turns ratio, N, the ratio of the number of turns of wire on the secondary divided by the number on the primary sets the output voltage. This is usually just called N.

V out = N* Vin

For example, if the primary has 250 turns of wire and the secondary has 1000 turns, the ratio is 4, so if you put in 120 V, you'll get 480 V out. Now a transformer is just a hunk of iron and wire; it can't increase the amount of power; that stays almost constant (there's a little bit of loss). When the voltage goes up, the current goes down by the same ratio, so that the amount of power is conserved. The advantage to the power grid is that loss in wire is current squared times resistance, and smaller wires have higher resistance; going to higher voltages allows them to reduce the losses and keep the same small wire. Bigger wire is more expensive than smaller and harder to work with in every way, so stepping the voltage up allows “normal” sized wire to be used to link power all around the country. Similarly, if the ratio were opposite, 250 turns on the secondary and 1000 on the primary, the ratio is ¼: if you put 120 on the primary the output is 30 V at 4 times the current.

Let's take a breather from a bunch of theory and talk about more practical stuff for a while.

This is as good a place as any to say this: the common 120 house current kills more people than all the high voltage systems combined. That's mostly because people who work on the higher voltage systems get more training but partly because a 120 shock causes your muscles to paralyze and you can't let go of it. Think of your house 120 as an infinite power source. No, nothing is infinite, but it will deliver enough power to kill you and burn your house down around you. How much more do you need?

What is ground? As I said before, all voltages are measured across something. What that means is that potential (voltage) can only exist between different points and a voltage only exists with respect to another potential. The reference that has the most historical use is ground, and the British specifically call it "Earth". What we call grounding something - sinking an 8' long rod into wet soil - they call "earthing". In most systems that use a single polarity supply, like your car, a solar panel, or millions of others, the chassis is connected to the negative terminal of the battery and the whole large mass of metal is called ground. It could be connected to the earth, but usually isn't (how do you tie a moving airplane to a ground rod?). I must note the exception: there were positive ground cars. I think that stopped in the 1950s, but car collectors can correct me.

The AC and DC worlds have decided to use very different color codes. DC systems that just have a single voltage use black for ground and red for the voltage. AC systems use a Neutral (white) and Hot (black) - ground is green (the voltage is from white to black). This is a good way to get killed. If you think it's safe to touch the black wire in an AC system, you die. In DC systems with lots of voltages (a lot of it uses positive and negative voltages) color codes aren't standardized. If you wire stuff that only you will ever work on, feel free to use any connector and any color code you want. I have actually seen a home made 12 V supply that used plain 120V outlets for the connector. I guess if you plugged a 120V device into it, it just wouldn't run, but still... If you care about the folks who will work on your project after it kills you, use the standards.

**Edit 1708 3/27**to correct typo pointed out in comments.

## 10 comments:

"It sounds a lot like – and is – the same as current flowing, so in the capacitor the current leads the voltage by 90 degrees."

Is that why a condensor (capacitor) in a points type ignition system works?

"One of the advantages of AC over DC, and why our power grid is almost exclusively AC, is that the voltage can be changed at will, almost completely without loss, by components called transformers. "

Isn't this advantage purely a consideration due to looses incerred with central genaration and distribution? As in, with point generation and use step up/down is typically avoided?

If I remember correctly, it's because the capacitor worked with the ignition coil to build up a large peak voltage. I haven't had a system with points and cap since about 1990.

That's a good point, but whenever you have to distribute voltage around, even over smaller distances, you might save money on wiring going to a higher voltage. I noticed some solar power systems use 24V batteries instead of 12V to reduce the current. When you double voltage, current halves, so power lost in the wire goes down to 1/4 of the original.

"I'm trying to dance around invoking trigonometry."

IIRC there are imaginary numbers involved too.

Good stuff SiG. We live in "The Electric Age" and anyone interested in getting off the grid so to speak should study this stuff if they want to install solar, wind, hydro or some other such to keep their toys working: refrigerators, computers, lights, radios ...

DC systems are often at 12 volts simply because most of the development was done for motor vehicle application using the battery that was already there. When people started adding a deep cell just to run electrics it was easy to then take advantage of this to move to 24 volts. This happened first in sailboats, which were early adopters of solar, and then the RVers caught on, so there is now a good deal of 24 volt solar.

But the generally preferred in a "serious" home installation is 48 volts. It's as close to optimum in cost/efficiency/safety as makes no never mind, at least for a conventional central power source.

I prefer distributed redundancy with the shortest possible paths, but that's just me.

"Inductors are usually cores of wire, frequently wound on a core material that increases the amount of inductance."

Should read...are usually coils of wire,...

Thank you! Obviously thinking too many words ahead.

Hi,

It was very informative and I also digg the way you write! Keep it up and I’ll be back to read more soon mate.

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