June 2013

Capacitors And Inductors
Article By Grey Rollins

You know that thing with metal balls hanging from strings? The one where you pull back a ball on one end, let go, and it hits the others, knocking the one on the opposite end loose? (It is called a Newton's Cradle if you want to Google up a picture.) Okay, now imagine slipping a piece of paper between the middle balls. That's actually a pretty good visualization of a capacitor at work. The paper represents the insulator between the plates and the balls are electrons. When an electron enters one side of the capacitor, one leaves the other side. If two enter on one side, two will leave on the other. After a period of time, the balls that left return, and an equal number of balls from first side leave.

In the real world, the balls in a Newton's Cradle are obeying Newton's Laws of Motion. An object at rest will remain at rest unless acted upon by an outside force and all that. In a real world capacitor the motivating force is electrostatic repulsion. Opposite charges attract, and like charges repel. When an electron enters one side of the capacitor, it carries with it a negative charge. That charge drives a like negative charge (in other words, an electron) off the other side and balance is restored. A signal only moves through a capacitor as long as electrons are in motion, otherwise the charges reach equilibrium and electrons cease moving out of the other end of the capacitor. If you apply DC to a capacitor, you'll get a surge out of the other end, representing the change from 0V to whatever your DC voltage might be — technically a short-term AC signal — then things stop. But if you apply an AC signal, you'll see motion for as long as you care to keep the AC going.

Capacitors are easy to make. In fact, you can easily make one at home. Take two sheets of aluminum foil, place a piece of paper between them, and attach one lead to each sheet of foil and...Voila! Your first capacitor!

Practical capacitors are rolled up with an additional sheet of insulation — technically called dielectric — to keep the two sheets of foil from shorting together. This also makes them less fragile and more compact. That's really all there is to the physical design of a basic capacitor. You can change the dielectric from paper to plastic film (think Saran Wrap for your kitchen capacitor), or you can use vacuum, or air, or most anything else that won't conduct electricity.

The parameters that matter most in a capacitor are the surface area of the two sheets of metal (called plates — bigger gives you more capacitance) and the thickness of the dielectric (thinner gives you more capacitance). The composition of the plates and dielectric matters, too, and this is where we begin to run into problems. For instance, copper conducts electricity better than aluminum, so if you make the plates out of copper, electrons will be able to get in and out of the capacitor more easily. This is a good thing…but copper is more expensive than aluminum and heavier too, so companies tend to use aluminum foil when making capacitors. The increase in resistance means that two otherwise identical caps — one made with copper, the other with aluminum — will not perform the same. The resistance is, electrically speaking, in series with the capacitance, and this means that the capacitor no longer functions as a pure capacitance. It has now become a filter. If you look on the spec sheets for capacitors, you'll see an entry for Equivalent Series Resistance (ESR), which takes into account the resistance of the plates and leads.

Something spooky happens when you charge a cap with a DC voltage, then discharge it…if you've got a meter hooked to it, after its discharged you'll see voltage come out of nowhere, slowly recharging the cap, even if it's not hooked to a circuit. (The effect is particularly pronounced with electrolytic caps.) Those electrons were hiding in the dielectric. When they think nobody's looking, they sneak back out into the plates. Some dielectrics are better than others; plastics and mica tend to be very good, so when possible use plastic or mica dielectric caps.

Capacitors are prone to another ill, which is that the rolled-up plates are a pretty good imitation of an inductor, which, like resistance, introduces another imperfection into what would otherwise be a theoretically perfect capacitor.

Capacitors, unlike resistors, change behavior with frequency. High frequencies pass with ease, whereas low frequencies face increasing difficulty. Direct current cannot pass at all, which leads to one of the most important uses for a capacitor — a gatekeeper that is able to discriminate between DC and AC. Circuits, particularly tubed circuits, often make use of this function. In addition, the size of the capacitor determines how close to 0 Hz (i.e. DC) the frequency response will go. This property is useful in filters.

Schematic diagram of a parallel plate capacitor. Two plates with area $A$ are separated by a distance $d$. When a charge $\pm{}Q$ is moved between the plates, an electric field $E$ exists in the region between the plates.

Capacitors do not perform their magic in isolation; they need a dance partner, that being the impedance of the circuit they find themselves in. Impedance is similar to resistance, but takes into account the AC-related "resistance" of the other parts nearby. Technically this is called reactance because it varies with frequency. Impedance is the sum of the resistance and reactance.

The formula for the reactance of a capacitor is:

X = 1/(2πfC)

Where:
X = the reactance of the capacitor
f = the frequency of the signal

In practical use, a Farad is an awful lot of capacitance. Most capacitors used in audio circuits are in the picofarad (10-12) to microfarad (10-6) range. That's quite sufficient for day to day use.

Inductors are, functionally, the mirror image of capacitors. Envision an inductor as an electromagnet. If you ever wound a wire around a nail and hooked it to a battery, you found that it picked up other nails. Having a magnet that you can turn on and off is very useful, but there are other things going on that are of more interest to us at the moment.

If you examine the current flow as the magnet is first switched on, you will find that it doesn't simply leap from zero current to full current in an instant. There's a time delay. That delay results from the magnetic field. As the magnetic field builds, it pushes back — an electronic application of the law that every action has an equal and opposite reaction. If you leave the current going, the growing magnetic field will prevail over the pushback and the current will reach its full value. When the electromagnet is turned off, the collapsing magnetic field creates an electronic current even though there is no longer current being supplied by the external circuit.

The important thing to note is that inductors resist change. Low frequencies (DC being the ultimate low frequency, since its frequency is zero) pass through the inductor easily. The faster the change (i.e. higher frequency AC), the more the inductor fights back. This is opposite from the behavior of a capacitor, which is quite content to allow high frequencies to pass, yet blocks DC.

Like simple capacitors, inductors are easily built. The nail in the electromagnet above is optional, but increases the efficiency. The essential element is the coil of wire. The more turns of wire, the more inductance.

You won't be surprised to learn that inductors are imperfect. The DC resistance of the wire used to build the coil is a problem, although it can be offset to some degree by using larger gauge wire. There is also capacitance between the adjacent turns in the coil. Not much, but it adds up as the number of turns increases. Assuming that you use an empty coil — this is called an air-core inductor — you'll have no problems with the magnetic properties of air, but if you use some sort of magnetic core in an effort to increase the efficiency of the inductor, you'll run into all sorts of odd things that you'll need to account for. And finally, inductors are heavy and expensive when compared to resistors and capacitors. More than anything else, weight and expense have contributed to the comparative scarcity of inductors in today's audio designs.

The formula for an inductor looks like this:

X = 2πfL

Where:
X = the reactance of the inductor
f = the frequency of the signal
L = the inductance of the inductor in Henrys (named after Joseph Henry)

Real world resistors approach the theoretical model very closely, indeed. For our purposes, you can ignore their imperfections. Capacitors and inductors... well... let's just say that the problems are manageable, but it won't do to ignore them. Generally, the higher the frequency, the worse they will behave.

With the description of resistors, capacitors, and inductors behind us, we can add them together, the simplest way being to put like parts together in series or in parallel. Series means that one part follows another and that a single electron will travel through both parts. Parallel means that two or more parts are side-by-side and that any given electron will travel through one, and only one, of the parts. Think of the rungs on a ladder. If you place an electron on one side, it has the option of traveling through any rung to get to the other side but once there, it's time to move on.

Resistors in series add their value together, increasing the total resistance:
Rtot = R1 + R2 + R3...

Resistors in parallel decrease total resistance:
Rtot = 1/(1/R1 + 1/R2 + 1/R3...)

Like resistors, inductors increase inductance if wired in series:
Ltot = L1 + L2 + L3...

And decrease inductance if wired in parallel:
Ltot = 1/(1/L1 + 1/L2 + 1/L3...)

Capacitors, being opposite of inductors, decrease capacitance if wired in series:
Ctot = 1/(1/C1 + 1/C2 + 1/C3...)

And increase if wired in parallel:
Ctot = C1 + C2 + C3...

Note that a curious thing happens if you wire capacitors in series: The voltage across the capacitors varies with the value of the capacitor. If the capacitors are all the same voltage rating and capacitance rating, and if they are of reasonably tight tolerance, then this can be done safely, assuming that the voltage rating has been chose carefully. However, if you're using two unlike values, the voltage across the capacitors will vary, possibly exceeding the rating of one of the parts. This is easy enough to cure, though, in that a string of high value resistors in parallel with the capacitors will force the voltage to divide in a predictable manner. The downside is that there will be a small loss of current through the resistor stack. If in doubt, don't do it, as capacitor failure can be a dangerous business; they tend to explode... yes, as in BANG! Electrolytic capacitors have a vent system that — in theory — allows them to release pressure safely. Do not trust this system to function properly. No matter how small the capacitor, never exceed the rated voltage. "But it was just a teeny little capacitor!" is not going to convince the doctors in the emergency room as they try to save your eye. Although you sometimes see capacitors in circuits run at, or even above their rated voltage, don't do it. Should you run into a capacitor with too much voltage on it, either troubleshoot the circuit to determine why the voltage is high or replace the part with one rated for more voltage. Your best case scenario is that the part will simply fail at some point in the future. The worst case is that it will explode, potentially hurting you or your system.

So how much leeway should you give, voltage-wise? I like at least 5%, minimum. If you're running, say, 16V, use a 25V part, not 16V, even though 16V on a 16V part seems reasonable. Anything more than 25V is overkill. The extra cost is minimal. Go ahead and pay the pennies.

Inductors are rated for how much current they will take. Here, you have more leeway, but don't get stupid. In small signal applications the current is so minor that it won't be a problem. In larger current applications, such as a filter in a power supply, make sure the inductor is well ventilated and you can run pretty close to the rated current. That said, I still suggest some elbow room. Just because inductors can take some punishment doesn't mean that you should abuse them.

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