Capacitors, Inductors, and Filters Primer

As the heading indicates, this is a primer on how capacitors and coils work in an electrical circuit. I've tried to strike a balance between over-simple and over-technical as neither benefits the seeker of basic electronics theory - those for whom this was written. The task has been akin to explaining the basics of algebra to someone that has never studied arithmetic. I am assuming here that you already understand the ideas of voltage, resistance, current flow, and the difference between a DC and an AC voltage. For further study, there is no better source of information, or a better place to start, than a first-year electronics textbook, Chapter 1, page 1. Don't worry if it is an old textbook.the laws of physics aren't likely to change anytime soon!

A capacitor is an electrical device that consists of two conductive "plates" separated by an insulator called the dielectric . The plates are usually a metal foil; aluminum is common but there are many other conductive materials commonly used. Capacitance ( C ) is the measure of a capacitor's ability (capacity) to store an electric charge on its plates. The universally accepted unit of measure for capacitance is the Farad . This is an unusually large unit of measure. A 1 Farad, 250 volt electrolytic capacitor would probably require the use of a pickup truck to haul it! The microfarad (millionths of a Farad) is the most common unit of measure for a capacitor in a practical audio-frequency circuit. The smaller nano- and pico-farad values are usually found in radio equipment that operate at much higher frequencies. The symbols " µ F" ( µ is the Greek letter "mu"), or mF are the most common abbreviations for microfarad. There are many different types of capacitors. We'll cover the basic theory of an electrolytic, non-polarized capacitor, or one that is intended for use with voltage of either polarity.

 

The surface area of the capacitor's plates is directly proportional to the capacitance, or the quantity of charge it can store. The thickness and makeup of the dielectric material is also very important. The voltage rating and the capacitance are the main specifications of a capacitor. The voltage rating of a capacitor, referred to as WVDC for W orking V oltage D irect C urrent, is the maximum DC voltage that can safely be applied to the capacitor. In an AC circuit, we would use the peak voltage for comparison.

 

The first thing you must understand is that during operation, there is no continuous current flow through a capacitor . In other words, the electrons that enter a capacitor do not and must not exit the capacitor by the other lead. This is prevented by the dielectric material between the plates, and is its reason for being there. An over-voltage condition can cause a "puncture" in the dielectric caused by arcing through the material. A punctured capacitor "leaks" electrons which reduces its effectiveness as a capacitor. So the WVDC is a very important specification when choosing a capacitor for a specific application. Leaking capacitors can upset the operation of the circuit they're in, and their condition will usually worsen with use.

 

Capacitors and DC When a DC voltage source is applied to a capacitor, two things occur. The applied voltage causes free electrons to accumulate on one plate, making it negative, which in turn repels free electrons off of the other plate, thereby inducing a positive charge there (a deficit of electrons). So the other plate becomes positively charged by electrostatic induction . For every electron forced onto one plate by the voltage, one electron is repelled off of the other plate. Therefore, the now "positive" plate receives an equal charge, but of the opposite polarity . In other words, it is lacks the same number of electrons the negative plate has gained. This action initially gives the appearance of current flow through the capacitor.

 

Max Current, Minimum Voltage At the initial application of DC, the only thing impeding the flow of current is the very small resistance in the wire leads and plate material. This is why we immediately get a large amount of current flow. Only when one plate takes on electrons and the other loses electrons, is there a "potential" difference between them. A difference in potential is, by definition , voltage! So our two things immediately happening upon the application of voltage is a high, though decreasing current flow in and out of cap, and a very low, but increasing voltage between the plates. As the capacitor voltage rises from zero, the rate of charge steadily decreases until the capacitor voltage matches the source voltage. As the capacitor's voltage (CV) rises toward the source voltage, there is less of a difference of potential between the voltage source and the plates to cause current flow on and off of them. When the source voltage and the capacitor voltage are finally equal, the capacitor is fully charged and current flow stops.

 

Capacitive Reactance When the DC voltage was initially applied, the capacitor acted like a virtual "short circuit" and permitted a great amount of current flow. But as the capacitor charged and its voltage (Vc) increased, the rate of charge slowed and finally stopped as the capacitor voltage reached the source voltage. This describes the capacitor's reaction to the application of a voltage between its plates: first it acted like a short circuit, but quickly changed into an "open" circuit. Understand also that the rate of change in the capacitor's going from a discharged state to a charged state is not linear! Plotted on a graph, it would represent a curved line, not a straight line. A capacitor discharges in exactly the same fashion.

 

This very specific behavior is extremely useful in electronic circuits, and becomes much more apparent in AC circuits. The non-linear reduction in current flow as the capacitor takes charge is called "reactance", the symbol is X . Capacitive reactance is Xc, and it rises (and falls) in step with Vc, the capacitor voltage. Since its affect is to impede current flow, the unit of measure is in Ohms, just like resistance.

 

All AC circuits display "nominal" resistance and reactance in varying amounts, whether they actually contain reactive components or not. When we're considering both R and X in an AC circuit, we call it " impedance ". The symbol for impedance is Z , and the unit of measure is also Ohms.

 

As a general statement, we often say that a capacitor is able to pass alternating current (AC) and block direct current. From our example above, we see that this is a bit of an over-simplification. With the initial application of DC voltage, the cap acted like a virtual short circuit allowing a lot of direct current to flow. The actual length of time it takes for a capacitor to charge to the source voltage is quite short, and is often measured in micro-seconds or milli-seconds. Mathematically, it is 5 times the product of the circuit resistance and capacitance. For example, it would take 500 uS (micro-seconds) for a 10-uF cap to charge (or discharge) through 10 W of resistance. So a capacitor can appear to flow DC, but generally for a very short period of time. Now we'll begin to look at how a capacitor can pass and block AC.

 

Capacitors and AC: Filter Action So how does a capacitor pass or block AC? If the positive and negative alternations of an applied AC voltage reach peak before a significant Xc and Vc develop, then the capacitor will have little blocking or filtering effect on the AC circuit. Electrons are flowing nearly unimpeded as the AC voltage source forces the cap through repeated cycles of charge and discharge. Therefore, the capacitor can "pass" such an AC signal. But we're over-simplifying again if we say capacitors can pass AC. Actually, a capacitor's inability to pass some AC currents is where most of their usefulness comes from!

Passband and Stopband The effect of a capacitor on an AC signal first occurs when a significant Xc and Vc develop before the applied AC voltage reaches its peak. If we reduce the frequency of the AC voltage, it will give the capacitor more time to take on charge and develop Xc. Frequencies high enough not to create significant Xc and Vc in the capacitor are said to be in the passband of the capacitor. The voltage and current (power) are able to go past it to the circuit load.

 

So it should now stand to reason that if a capacitor is developing a voltage between its leads (Vc), and reducing the amount of current flow in the circuit (Xc), it is blocking power (current and voltage) from reaching the circuit load. For example, if 10VAC is being delivered to a capacitive circuit and 3 volts can be measured (is being "dropped") across the capacitor, only 7 volts are left for the circuit load. If we reduced the frequency of this AC signal, a greater capacitor voltage will have the opportunity to develop more Xc, which will further reduce current flow in the circuit.

 

The frequency where ½ of the power delivered to the circuit is blocked from the circuit load (-3dB) is said to be the "cut off frequency", or HPP (Half Power Point). This is considered to be the beginning of the stop band of the "filter". Understand that a capacitor does not actually dissipate the power of the AC voltages in the stopband (do work or generate heat with it). It merely blocks it by reducing current and "dropping" the voltage of those frequencies that create sufficient reactance.

 

Capacitor as a Crossover In our AC circuit example, if our circuit load had been a speaker, and our capacitor were connected in series, we would have a "highpass crossover". It passes all frequencies higher than the crossover frequency or HPP to the speaker. All frequencies for which the speaker is operating at or below ½-power are said to be "rolled off" and in the stopband. The lower the frequency, the less power that reaches the load. So the lower the frequency, the less of it you will hear in the speaker. As we reduce frequency, 6 dB of power is lost at every octave below the HPP. This is called the "slope" of the filter, and is characteristic of a "First Order" filter, or one that uses just one reactive device.

 

Again, this is very much simplified, but should give you the basic idea about how capacitors work.

 

 

 

 

Inductors

 

The terms inductor , coil , and choke are synonymous. A coil is a winding of insulated wire wrapped around a core . The unit of measure for Inductance is the Henry ; the symbol is L . Inductors used in the audio range of frequencies are in the lower millihenry (thousandths of a Henry, mH) value.

 

The most common core materials are air and laminated iron. The core material, even if it is air, has a definite affect on the performance of a coil. In fact, all core materials are compared to air to calculate their effect upon inductance. Other specifications affecting inductance are the thickness of the wire used, the diameter and length of the coil, the diameter of the core, and the number of turns and layers of wire. The wire used in a coil, often called "magnet wire", is always insulated, usually with a thin but very robust coating. This prevents current from flowing from turn to turn, which would be little more than a short circuit.

 

Coils and capacitors are very similar in their behavior in an AC circuit, though a coil's response to an applied voltage is the opposite ! The basic ideas of charge and discharge rates, reactance, stopband and passband, etc., apply to coils much as they do to capacitors. If you removed a capacitor from a circuit that was acting has a highpass crossover and replaced it with a coil, it would then be a lowpass crossover. As we have already discussed, capacitors pass AC and block DC. A coil blocks AC and passes DC . There is also another axiom: coils resist a change in voltage, capacitors resist a change in current.

 

Inductors and DC When we applied DC voltage to the capacitor, it initially acted like a virtual short circuit (zero ohms) as a large amount of current flowed. A coil responds in exactly the opposite fashion: it will initially act like an open circuit (infinite ohms), flowing virtually no current. At the first instant of the application of voltage, current flow is near zero, inductive reactance (XL) and inductor voltage (VL) are at maximum values. This situation changes very quickly, just as the capacitor did not flow current for very long. Let's first look at why this initial no-current, full voltage situation occurs. Very important!

 

Induction When current passes through any conductor (though specifically wire), lines of force called magnetic flux lines form and "grow" outward and rotate around the conductor. The number and density of these lines of force around a current-carrying wire are well defined mathematically, and are directly proportional to the amount of current flowing in the wire. Also, when these flux lines cut through a conductor, they will "induce" a voltage in the conductor which can cause current to flow. This is precisely how electricity is generated! More on this in a moment.

 

When the amount of current flow in a wire is constant, a specific number and density of flux lines are present and stationary around the wire. If the current in the wire is increased, more flux lines form and move outward to create a "thicker" layer of flux lines. If the current is reduced, some of the lines collapse back into the wire. This phenomenon of moving flux lines as the result of a change of current serves a number of extremely useful purposes in electronics. It is one of the many reasons we go to the trouble of using AC!

 

With the initial application of a DC voltage to a coil, two things also happen that are analogous to how the capacitor reacted. There is the immediate appearance of magnetic flux lines growing outward and rotating along the length of the coiled wire as current begins to flow. As flux lines move outward and around each turn of wire, they cut across the adjacent winding(s) . By growing outward and cutting through the adjacent windings of wire, they induce a voltage there . Each turn of wire in the coil induces voltage in the adjacent turns. It is the moving flux lines cutting through a wire that causes this voltage to be induced.

 

It is extremely important to understand that this "self-induced" voltage created between the turns of a coil by the moving lines of flux is always of the opposite polarity of the voltage that produced it! This opposite-polarity, induced voltage that the coil creates is often referred to as back-EMF or Counter Electro-Motive Force, CEMF. Initially almost equal to our applied DC voltage, the CEMF allows very little current flow through the coil. It virtually cancels out any current that the applied voltage would otherwise create were the wire not coiled.

 

This causes the second part of the coil's initial reaction to a DC voltage: the full source voltage immediately appears across the coil. T he appearance of CEMF nearly cancels any initial current flow. So much so, that the coil initially acts like a virtual "open circuit" (infinite ohms). The full source voltage shows across the coil almost as though it wasn't in the circuit at all. It is a resistance or impedance between two points in any circuit that makes a potential difference (voltage) possible. Wound into a coil and passing a changing current flow, this otherwise excellent conductor is initially creating a great impedance to current flow! The initial open circuit or infinite-ohms condition lasts for a very short time: only as long as the flux lines are moving outward and cutting adjacent turns of wire. As this movement slows, the CEMF begins to fall away, and forward current flow increases because of the applied voltage. This does cause more lines of flux to appear with resulting CEMF, but the applied voltage and current finally win out. This is the coil's reaction to a change in current flow, and is specifically "inductive" reactance, symbol XL . The current increases to the full amount that the DC voltage source and circuit load will allow. XL and the inductor voltage are initially at max, current at minimum. They fall away in step with current reaching it's maximum value.

 

Now that the current flow has stabilized at maximum, the lines of flux have stopped moving. So the coil is again just a long piece of wire offering very little resistance to current flow. The lines of flux around the coil are stationary because the amount of current is not changing. No voltage is present across the coil, other than a very small amount created by the nominal resistance of the wire itself.

 

Opposite reactions Let's stop and second and compare the capacitor's and inductor's initial reactions to the application of DC voltage: the capacitor immediately allows a maximum amount of current as it offers little internal resistance. An inductor initially stops current flow as a reaction to the change in current flow. The inductor initially showed full source voltage, the capacitor no voltage. Every step of the way, the coil's reaction to the applied DC voltage is just the opposite of how the capacitor reacts. Actually, their reactions to the application of a DC voltage occurred in the reverse order. If we were to graph the current flow in our capacitor, it would match a graph of the inductor's voltage ! The graphs would both start at maximum and have exactly the "slope" or contour down to zero!

 

Removing the DC The removal of DC voltage from a coil causes the same type reverse-reaction in the coil that applying it did. When the voltage is removed, current falls off and the lines of force collapse back into the wire, cutting through the turns just as it did when voltage was applied. But by collapsing back through the turns of wire in the opposite direction, the induced voltage is now forward, the same polarity as the original applied DC voltage. The inductor is still resisting a change in current flow, but is now trying to sustain it! If you've every unplugged a running appliance that uses a motor, you probably noticed a large spark at the plug as the connection was broken. Now you know that the forward "kick" from the collapsing fields around the motor windings is what causes that arc across the very high resistance of thin air!

 

The removal of voltage from a capacitor causes it to resist that change as well. Once fully charged to the source voltage, the capacitor is much like a battery. Removed from the circuit, the charge and voltage can remain stored in the capacitor for a long time. If the voltage begins to fall in the circuit as the DC source is removed or reduced, the capacitor then has a higher voltage. The electrons on the negative plate will move toward the less negative conductor outside the capacitor, and of course electrons will move onto the positive plate as well. The source-level voltage on the capacitor's plates tries to maintain the voltage in the conductor. But a capacitor has no means to replace its charge like a battery does, so the current and voltage will finally fall off as the charge drains away.

Frequency Response It was a change in current that caused the inductor to react "inductively" because a change, up or down, is what causes the lines of flux to move. So it should stand to reason that a coil in an AC circuit could have a great affect upon it! Connected to a speaker load in series, a coil can act as a lowpass filter. With a suitable inductance value, such a coil will pass lower frequencies just like the length of wire that it is. But at some higher frequency, the increasing rate of current change will begin to cause XL. It should make sense that the higher a frequency is, the greater the rate of flux motion, and therefore the greater the amount of XL created. So as frequency increases, inductive reactance increases. This will cause the inductor to drop voltage and block current (power) from the circuit load as frequency increases. Remember that the capacitor did the same thing, but as frequency decreased .

 

Filter Order The two filters just described used a single capacitor (highpass) or a single coil (lowpass) to block certain frequencies from an in-series circuit load such as a speaker. Such filters are referred to as "First Order" filters as they use just one reactive device. It is characteristic of First Order filters to reduce the power reaching the circuit load by -6 dB with every halving or doubling of the applied AC frequency. This is often referred to as a First Order "rolloff", First Order "slope", or 6 dB slope or rolloff. An audio signal can contain many different frequencies. Some can pass across a capacitor, while others can pass through an inductor. How very useful!

 

Comparing Capacitors and Inductors