## Capacitors

### Introduction

The purpose of this article is to collect in one place information, both practical and theoretical, on capacitance and capacitors. Capacitors are second only to resistors as a circuit element in electronics, and a large assortment is usually found in every electronics workshop. Capacitors are used as examples and illustrations in explaining electrostatics.

### Capacitance

The net charge on a capacitor is zero, but equal and opposite charges ±Q are found on the two plates, and the "charge on the capacitor" is usually the absolute value of the charges, Q. The electrodes of a capacitor are called "plates" even though they are usually not plates at all, but surfaces of various forms. If we mark one terminal of a capacitor with a polarity marking, say a +, then Q > 0 means that a positive charge Q is on the corresponding plate, and if Q < 0, then a negative charge is on that plate. If a current i = dQ/dt, then a positive current flows into the marked terminal and causes the voltage V across the capacitor to increase. The voltage V is the potential difference between the two plates, positive if the potential of the marked plate is higher than the potential of the other. These sign conventions are easy to understand, and are essential if confusion is to be avoided.

In an ideal capacitor, the voltage is proportional to the charge, and the constant of proportionality is the capacitance C. That is, Q = CV, or C = Q/V. Most capacitors are close to ideal if the voltage does not vary too rapidly, and is not excessive. Unit capacitance results when unit charge means unit voltage. In the practical system, unit charge is 1 coulomb, and unit voltage is 1 volt, or 1 joule/coulomb. The corresponding unit of capacitance is the farad, which is one coulomb per volt. This happens to be an absurdly large unit, so the microfarad, μF, is commonly used. In Gaussian or electrostatic units (esu), the unit of charge is the esu (or statcoulomb; c/10 = 3 x 109 esu = 1 C) and the unit of potential is the statvolt, which is about 300 volt. The unit of capacitance is then the centimetre, or one esu per statvolt. From these figures, it follows that 1 cm = 1.11 x 10-12 F = 1.11 pF.

A capacitor can be made from two conducting sheets, each of area A, separated by a distance d. Normally, the separation d is maintained by a sheet of dielectric of this thickness, which has a dielectric constant κ. The capacitance of this parallel-plate capacitor is given by C = εoκA/d F, where A is in m2 and d is in metres. The constant εo = 4π/c2 = 8.854 x 10-12 F/m, an embarrassment of the SI system of units. In centimetres, C = κA/4πd, where A is in cm2 and d is in cm. It is very easy to make some capacitors like this, perhaps with paper, κ = 3 and d = 0.015 cm, as dielectric, and to measure them with a capacitance meter. Most high-end DMM's now have a capacitance scale. Don't forget to subtract the reading before the capacitor is connected.

### Electrostatics of Capacitors

The principles of electrostatics are (1) no energy can be transferred in any process in which the final state of the charges are the same as the intiial state--that is, the electrostatic field is conservative, and (2) the electrostatic force is inversely proportional to the square of the distance. In terms of the vector electric field E, the first principle is stated mathematically by the vanishing of the line integral around any closed curve of the component of the electric field tangent to the curve, and the second by the equality of the integral of the normal component of the electric field over any closed surface and 4π times the net charge within the surface, which is called Gauss's Law. We also see that the field due to two charges is the sum of the fields due to each charge; that is, the principle of superposition holds for the electrostatic field.

These principles have been shown to be true to the ultimate precision of experimental investigation over many years. The restriction to equilibrium states of no motion results in great simplification. Relativity brings in many interesting aspects and shows that electrostatics cannot be considered in isolation from magnetic phenomena, but it is not necessary to appeal to relativity in any electrostatic problem in a single frame of reference. All the phenomena of electrostatics can be ascribed to tiny, mobile particles interacting by an inverse square force and having a charge with algebraic quantity. We do not know why charge is, nor what it is, in terms of simpler concepts, but there is no mystery in its workings.

The first principle permits the definition of the electrostatic potential φ such that the electric field is a vector in the direction of greatest rate of decrease of φ and of a magnitude equal to its rate of decrease in that direction. In vector calculus, E = - grad φ. With this definition, and the single-valuedness of φ, the statement in terms of the line integral is easily derived. Using the divergence theorem, div E = 4πρ, where ρ is the volume density of charge in esu/cm3. Combining these two results, we have div grad φ = 4πρ, which is Poisson's equation connecting the charge density and the potential. It is not necessary to know vector calculus for this article, but it is by far the clearest way to make general mathematical deductions about the electrostatic field. The electrostatic potential is valuable because it replaces the three components of a vector field by one scalar function, and because the potential of an assembly of charges is the sum of the potentials due to each charge separately.

Now we define a conductor as a medium in which an electric field cannot exist, because it would cause the motion of charges to positions where the field is neutralized. A metal is a good example. In its normal state, it consists of positive and negative charges in equal numbers and equal densities, so it appears macroscopically neutral. However, at least one sign of the charge is mobile, and moves freely under an electric field. Charge motion continues as long as the field is different from zero, and ceases when the field is reduced to zero. This takes place in a very short time when equilibrium is disturbed, and we consider only the final, steady state. In electrostatics, any material with an nonzero electrical conductivity will behave as a metal.

The figure shows the cross-section of the surface of a conductor, with a field E to the left of the surface, and a zero field to the right. The rectangle is a path of integration, with long sides 1 cm long and short sides of an indefinite short length in comparison. This presumes that the surface is plane and that E does not vary with position, but the general case is just as easily considered with an infinitesimal path of integration. The electric field can be resolved into a normal component En and a tangential component Et. The integral over the long side outside the conductor gives a contribution Et x 1 cm, while the short sides give equal and opposite contributions from En, and the contribution from the long side inside the conductor is zero. Therefore, Et = 0, which is an important result. The electric field at the surface of a conductor is normal to the conductor. Since the electric field is the rate of change of the potential, there must be no change in potential as one moves along the surface of the conductor. The surface of a conductor is an equipotential surface.

Now consider the surface of a conductor again, and apply Gauss's Law to a surface consisting of parallel areas of 1 cm2 just outside and just inside of the surface, joined by a lateral surface that is just a thin strip. This surface resembles what was called a "pillbox" that had circular halves that fit into one another, like the shell of a diatom. Again, if the surface is not plane and E is not constant, an infinitesimal pillbox will give the required result. The flux of E over the closed surface is just E x 1 cm2, since E is normal. There is no component through the later surface (which would vanish in the limit anyway), nor through the surface within the conductor. Therefore, E = 4πσ, where E is the magnitude of the electric field. Note that σ is positive if E is outwards, negative if E is inwards. The magnitude of the outward normal field is equal to 4π times the surface charge density.

Finally, let's make a parallel-plate capacitor, of which a part is shown at the right. Two flat conducting plates are a distance d apart, and connected to a battery that maintains them at a potential difference of V volt. All quantities will be independent of y and z, depending only on x. Let the voltage of the left-hand plate be taken as 0, so the voltage of the right-hand plate is +V. The electrostatic potential between the plates must satisfy div grad φ = 0, or, since we assume that the only variation is in x, d2φ/dx2 = 0. This is easily integrated twice to give φ = Ax + B, where A and B are constants. At x = 0, φ = 0, so B = 0. At x = d, φ = V, so V = Ad. Then φ = (V/d)x. The only nonzero component of the electric field is Ex = -dφ/dx = -V/d. We have just solved a simple boundary value problem, and found that the electric field in a plane-parallel capacitor is uniform, of magnitude V/d, and directed from the (+) plate to the (-) plate.

The charge density on the positive plate is σ = E/4π = V/4πd. Since E is uniform, so is σ. The charge density on the negative plate is the same in magnitude, but opposite in sign. If the total area of one plate is A, then the charge on the capacitor is Q = Aσ = VA/&4πd = CV, where C = A/4πd is the capacitance. This demonstrates that the capacitance is a constant independent of voltage, depending only on the geometry of the capacitor.

Since the charges on the two plates are opposite, the plates attract one another. There are several ways to find this force, but it is easy to make an erroneous argument. Since the field is E, and the charge density is σ, one might be tempted to argue from the definition of electric field that the force per unit area is σE, or E2/4π, but this is WRONG and twice the actual value. For security, then, we calculate the force from Coulomb's Law between charge elements. Let's find the force on an element σdA of the left-hand plate. The charge on a ring of radius r and width dr about the normal from dA is σ(2πrdr), and it is all the same distance s = √(r2 + d2) from dA. Each element of it is attracted to dA with a force df whose x-component is (d/s)df, so the x-component of the force for the whole ring is (d/s)(2πrdr)dAσ2/s2, the y and z components cancelling. The total force on dA is then (2πdσ2dA)∫(0,∞)rdr/s3. Make the substitution u = r2 + d2, after which the integration is easy. The force per unit area dA is then πσ2d∫(0,∞)u-3/2du = 2πσ2 = 2πσ2 = E2/8π. This is the force of attraction per unit area between the plates of a parallel-plate capacitor.

To get a feeling for the size of this force, consider a capacitor with plates 10 cm x 10 cm and spacing 1 mm. The capacitance is 100/4π(0.1) = 79.6 cm or 88 pF. If charged to 300 V, 1 statvolt, the force will be 100 x (100/8π) = 398 dynes (0.4 g weight). At 0.1 mm spacing, the force will be 100 times greater, or 39,800 dynes, 40 g weight. Two plates with a sheet of paper between them should noticeably stick together under these conditions, but the experiment is a difficult one to realize in practice.

Now let's find the energy stored in a capacitor. Since the force between plates of area A will be 2πAσ2, let us disconnect the battery so that σ, and so the force, will remain constant, and see how much work we get out by letting the plates come together. This will be U = 2πAdσ2. This is equal to (E2/8π)Ad, or an energy E2/8π per cubic centimetre. One can consider this as the energy density of the electrostatic field. Since E = V/d when the plates are at a distance d apart and at a potential difference of V, the total energy of the capacitor is also U = (Ad)(V2/8πd2) = (1/2)(A/4πd)V2 = CV2/2.

Since Q = CV, we can express the energy in terms of Q by U = Q2/2C = (Q2/2)(4πd/A). Then, taking the negative derivative of U with respect to d, holding Q constant, we find -dU/dd = -2πAσ2, the correct result, where the (-) sign means that the force is opposite to the direction of increasing d. On the other hand, if we took the negative derivative of U = CV2/2 with respect to d, we would get +2πAσ2, which is the same magnitude, but in the opposite direction. I have known physics instructors who concluded that the force between the plates depends on whether the battery is connected or not, which is absurd. If we hold V constant, then the battery must do some work supplying additional charge dQ = V dC. The work is V dQ = V2 dC = -(AV2/4πd2)dd = -4πAσ2 dd, so the total energy change in a displacement dd is exactly as it was before, and we find the same force. This example is often used to show the importance of considering what is held constant when taking a derivative, and of taking care in the interpretation of energy.

A practical parallel-plate capacitor cannot be infinite in area. At the edges of the plates, there is fringing of the field as it bows out into space. The charge density is somewhat higher at the edges, and charge can even appear on the outer surface. If the separation d of the plates is considerably smaller than the smallest dimension of the area of the plates, then it is a pretty good approximation to assume that the field is uniform and confined to the space between the plates. For a more accurate estimate of the capacitance, add the gap d to each of the plate dimensions to get a larger effective area A. A more exact result can be obtained by numerical analysis, but this is usually not worth the trouble. If the separation is anywhere close to a transverse dimension of a plate, a numerical analysis is necessary.

Some fields due to infinite uniform plane charge distributions are shown in the figure at the right. These fields can be found by using Gauss's Law and superposition. Note the difference between a charge density on the surface of a conductor and a sheet of charge in space. These are also the fields close enough to surfaces that they can be considered approximately plane. Charge will naturally distribute itself unifomly over a plane surface. There is no field outside a double sheet of equal and opposite charge densities, but a strong field inside. If the separation of the sheets d is reduced, but σ is increased, so that σd = p = constant, we have a uniform double sheet of dipoles, which produces no external field. The potential, however, jumps by 4πσd = 4πp from the negative side to the positive side. The surface of a metal is such a double sheet, with the potential jump creating a potential well that traps the electrons inside.

The reader can consider other geometries as exercises. Simple results can be obtained for cylindrical and spherical geometries, and these sometimes can be useful approximations to practical cases. The potential of a point charge Q is Q/r, and from this the capacitance of concentric spheres of radii a and b can be found as 1/C = κ(1/a - 1/b), or C = κab/(b - a), where b > a. Note that C → a as b → ∞. For an isolated conductor, the other plate is assumed to be at infinity. The field of a line charge of λ esu/cm can be found from Gauss's Law, and from it the potential φ = 2λ ln(b/r) for r < b. The capacitance per unit length of a coaxial cylindrical capacitor is C = λ/V = 1/2 ln(b/a). This result can be applied to a coaxial cable. There are no fringing fields for a spherical capacitor (except for the supports of the inner sphere), but there are fringing fields at the ends of a cylindrical capacitor.

### Dielectrics

Most practical capacitors have a material between the plates, called the dielectric, from dia-electric, "that which lets electricity through." The electric field appears to penetrate dielectrics, while it is completely stopped by a conductor, because charges are induced that terminate the lines of force. Since matter is electrical in nature, a dielectric must have an effect on the electrostatic field. Instead of charges coming loose and roaming around, the charges seem to be bound elastically to their equilibrium positions, and only move slightly in response to the electrical forces, the positive charges in the direction of the field, and the negative charges oppositely. This is actually an effect on the crystal lattice or the molecules or the atoms of the material, so the results are by no means simple. An approximation of the general behavior is, however, reasonably simple and serves well in most ordinary situations. An account of the connection between molecular properties and macroscopic properties of dielectrics can be found in Polarization.

Consider a 1 cm cube of uniform charge density ρ superimposed on a 1 cm cube of uniform charge density -ρ. There is no unbalanced charge, and no electrostatic field. Physically, think of small "molecules" in random directions consisting of a postive q and negative charge -q very close together. If the distance from the centers of gravity of the negative to the positive charge is r, then the molecule will have a dipole moment p = qr, which has a field falling off as r-3, plus small higher moments whose fields decrease even more rapidly with distance. Although some molecules have a dipole moment even in the absence of an external field, they are usually randomly oriented so that the macroscopic material is electrically neutral and is not accompanied by any electrical fields, so long as we do not add or take away any charge, which would upset the balance and give a field falling off as r-2, the usual long-distance Coulomb field. Let the dielectrics we discuss here be electrically neutral.

Now suppose the dielectric is placed in an electric field E in the direction of a cube edge. The cube of positive charge will be strained in the direction of E, while the cube of negative charge will be strained in the opposite direction. Suppose the cubes are displaced relatively by an amount s = kE, where k is a constant. This is just the usual Hooke's Law assumption of strain proportional to stress, which is an approximation. A thin sheet of positive charge ρs will become uncompensated at x = 1, while a thin sheet of negative charge -ρs will become uncompensated at x = 0. The dipole moment of the whole system will be P = ρkE, which, since we considered a unit cube, is the dipole moment per unit volume, the polarization P. It is also the amount of the charge uncovered. The product ρk is denoted by χ, and called the electric susceptibility. We then have P = χE. In an isotropic material, χ is independent of direction. Generally χ depends on direction. This can be handled without much trouble, but we shall assume the dielectrics considered here to be isotropic.

Now we put the cube in the parallel-plate capacitor we have already analyzed. The electric field E = V/d causes the dielectric to polarize, as we have just described. For simplicity, assume that the dielectric fills the capacitor exactly. To maintain the field at V/d, the extra charge uncovered at the surfaces of the dielectric must be compensated by equal and opposite charges on the plates. At the positive plate, which had a charge density E/4π before the dielectric was emplaced, we must add a charge density P, to get a total charge density of E/4π + P = (E + 4πP)/4π. We see that the combination E + 4πP gives us the total charge per unit area on the plate in the same way that E gave the charge per unit area in the absence of the dielectric. The combination D = E + 4πP is called the electric displacement. Its flux through a closed surface is 4π times the amount of "free" charge enclosed. The free charge is the total charge less the charge contributed by polarization, and amounts to that part of the charge that can move, while polarization charge cannot. The electric displacement is composed of two different things, and is best considered a mathematical artifice that makes it easier to express certain problems.

In the mechanical theory of the electromagnetic field, E is looked upon as similar in nature to P, the polarization of the ether instead of the polarization of matter. It was then convenient to think of E as a cause, and D as an effect, and even in a vacuum to set D = εoE, where the permittivity εo was like an elastic constant. This view was totally wrecked by relativity, but it survives in the system of electromagnetic units based on the practical units, where it expresses nothing of physical meaning, and confuses many calculations, especially in magnetism, and makes relativistic electrodynamics of materials very confusing indeed. It is much easier to work in Gaussian units, as I have done here, and convert to practical units when required.

Now, D = E + 4πχE = (1 + 4πχ)E = κE, which defines the dielectric constant κ. The charge on the capacitor plate in the presence of the dielectric is then σ = D/4π = κE/4π. The total charge Q is increased by the factor κ, so the capacitance is now C = κA/4πd. If we have a system of conductors charged to various potentials V, then if it is all immersed in a dielectric of dielectric constant κ, the charges on each conductor are increased by the factor κ. If the conductors hold various charges Q, then on introducing a uniform dielectric the potentials are reduced by a factor 1/κ, since the electric fields must be reduced by this factor to keep the same charges Q. This is all very simple, but in fact dielectrics are seldom uniform, and we have charges appearing on the surfaces of the dielectrics that affect the fields.

Dielectric constants vary widely, and are a function of the frequency of the applied electric field. We are mainly concerned here with the low-frequency or static dielectric constant. Air has a dielectric constant 1.0006, and it usually may be assumed to be 1 with little error. Polyethylene is 2.26, polystyrene 2.54-2.56, teflon 2.1, lucite 2.8, and plexiglas 3.12. Polyvinylchloride has 4.55, pyrex glass 5.00, window glass 7.6-8, and steatite 5.5-7.5. Mica can have from 2.5-8, but good mica is 6.4-7.5. The titanate ceramics give 15 - 12,000, while paraffin wax gives only 2.0 to 2.5. Paper (cellulose) has a dielectric constant of 2-3. Quartz has a dielectric constant 4.34 perpendicular to the optic axis, and 4.27 parallel. The amorphous silica in integrated circuits has dielectric constant 3.85, crystalline silicon 11.8. The corresponding figures for calcite are 8.5 and 8.0. Water has the very high dielectric constant of 80, ice even higher, 105 parallel and 92 perpendicular, for an average of about 96, for polycrystalline ice.

Suppose, for example, we put our dielectric cube in an external field E. Charges of +P and -P appear at the end faces of the cube, and these charges make a field opposite to the applied field E. If the dielectric is in the form of a thin plate normal to E, then the field created by the polarization charges is E' = -4&piP;, and the actual field in the dielectric is E" = E - 4πP = E - 4πχE". Solving for E", E" = E/(1 + 4πχ). The field due to the polarization charge is called the depolarizing field, so the actual field in the dielectric is less than the external field E. In the dielectric, D = E" + 4πχE" = (1 + 4πχ)E/(1 + 4πχ) = E. Since D = E outside the dielectric, we see that the normal component of D is continuous at the boundary. This is generally true, not just for this special case, as can be proved by considering a Gaussian surface at the boundary. Taking a path of integration like the one we used to prove that the tangential component of E is zero at the surface of a conductor, we now find that the tangential component of E is continuous at the boundary between dielectrics.

Considering a dielectric body in the form of a slender cylinder parallel to the electric field, the continuity of the tangential component of E means that E is the same inside and outside the body. Polarization charge can appear only at the ends of the cylinder, and so only in small amounts, so the depolarizing field is small and confined to the ends of the body. This is familiar from permanent magnets, but is less familiar in electrostatics. It happens that if the dielectric body is in the shape of an ellipsoid with an axis in the direction of the applied field, then the field inside the body is uniform. This is found by solving the boundary value problem, a mathematical exercise that there is no space for here.

The polarization of a dielectric is due to the polarization of its crystal lattice or molecules. Although higher moments than the dipole moment are created, their effect is very small in comparison with the dipole effects, and so is usually neglected. The actual intermolecular fields are so great that they swamp the usual applied fields, so the susceptibility is very accurately a constant. In fact, dielectric breakdown occurs before any nonlinearity is perceived.

If the external field is applied suddenly at t = 0, the polarization may behave according to P(t) = Po(1 - et/τ), where τ is a relaxation time. The dielectric relaxation time in ice is about 20 μs. The total polarization may be the sum of components of different relaxation times. If a frequency f in the neighborhood of 1/τ is applied, there may be significant energy loss because of this. In fact, microwave ovens generate most of their heat from the relaxation of the dipole moments in the water that foods contain. If you charge a capacitor, and then discharge it rapidly to 0 V, the terminal voltage will be observed to rise with time. These are the phenomena of dielectric absorption, which are very deleterious in capacitors that should hold accurate voltages.

### Dielectric Breakdown

The maximum voltage at which a capacitor can be used is determined by the formation of an electrical discharge in the dielectric. The heat produced by the discharge usually damages the capacitor, except in the case of an air or liquid dielectric, when little permanent damage may be done. Only a capacitor with a dielectric of a vacuum is not subject to this limitation, but the vacuum must be nearly absolute, since low pressures will encourage a discharge. Breakdown in gases has been studied extensively, and a great deal is known about this complex phenomenon. Less is known theoretically about breakdown in liquids and solids, but there is a also a great deal of empirical knowledge in this case, since it is technologically important. The dielectric strength is the maximum voltage difference a certain thickness of a dielctric can sustain without electrical breakdown.

Spark breakdown in air occurs when the field strength becomes sufficient to accelerate electrons to a speed which makes them capable of ionizing gas molecules. The ions, accelerated in the field, release electrons from the electrodes by collision, and breakdown occurs when this process becomes self-sustaining. If the current is limited, the discharge becomes a glow discharge, and if not, an arc is formed. The voltage at which breakdown occurs depends on the shape and material of the electrodes, the gas pressure, the distance between the electrodes, and other incidental conditions such as ultraviolet illumination of the electrodes. The most important dependence is on gas pressure p and electrode spacing d, and the breakdown voltage Vs is a function of the product pd (Paschen's Law). At lower pressures, breakdown occurs at greater distances. The curve of Vs vs. pd has a minimum, so there is a minimum breakdown voltage. For air, and usual electrode materials, Vs(min) = 327 V, and occurs for pd = 0.567 mmHg-cm. This means that under any conditions there is no danger of breakdown at household voltages, whether 120 in the US or 240 in the UK.

The breakdown field strength for air is often stated to be 30 kV/cm. This is true only for an electrode spacing around 2 cm. The breakdown field strength is greater for smaller spacings, and smaller for greater spacings. One common formula is Vs = 30d + 1.35 kV, good for d in the region of 1 mm. This formula may be used to estimate the voltage rating of air-dielectric capacitors. The electrode shape has no great influence, so long as it is a reasonably rounded or flat body. Sharp points are a different matter, and may show corona discharge at lower voltages. The breakdown voltage is not a linear function of the electrode separation, but may be approximately so over modest ranges.

Mica has excellent characteristics, with a dielectric strength of 3800-5600 V/mil (1 mil = 0.001" = 0.0254 mm). Teflon is also very good, with 1000-2000 V/mil. Polystyrene has a strength of 500-700 V/mil. Pyrex and soft glass can resist 335 V/mil and 200-250 V/mil, respectively. Use these figures with caution; they can, however, serve for rough estimates. Remember that breakdown can occur through other routes than across the dielectric. The edges of plates, or the leads, can flash over, which is equally undesirable. A high-voltage capacitor must have a suitable geometry and packaging. Apparatus that may not flash over at atmospheric pressure may do so at reduced pressures, as in aircraft or space vehicles.

Temperature has no effect on gas dielectrics, but a strong effect on ceramic dielectrics (titanates), and the temperature dependence is a function of the composition. Thermal expansion of a dielectric changes the capacitance, and may have an effect on the breakdown voltage. Present-day capacitors can resist elevated temperatures much better than the wax-impregnated paper capacitors of the past. Electrolytic capacitors are probably the most sensitive to temperature extremes. They are generally limited to temperatures less than about 105°C.

Electrolytic capacitors are available with voltage ratings up to about 450 V. For higher voltages, capacitors can be connected in series (with, of course the resultant decrease in capacitance). The leakage current may be sufficient to equalize the voltage on the capacitors, but parallel high-value resistors may be added to make sure. For other types of capacitors, voltage-equalizing resistors are required.

### Capacitors as Circuit Elements

Capacitors are used in electronic circuits to block a DC voltage, but to allow varying signals to pass ("coupling" capacitors). They are also used to prevent the voltage at a circuit node from changing ("bypass" capacitors). These two applications do not depend on the exact value of the capacitance. Capacitors are also used in LC resonant circuits, in RC bridges, and as timing elements, and here the exact value is significant. They may be used to store charge accurately, as in sample-and-hold circuits or frequency-to-voltage converters, and here the quality of the dielectric is more important than the value, so long as it is stable. Capacitors store charge in a gross way for discharge through lamps, as in photographic flash lamps, or stroboscopes. Capacitors are used in filters, both in power-supply filters and in signal filters, often in connection with operational amplifiers. In power circuits, capacitors may be used to shift phase, as in capacitor-start motors, or for power factor compensation in the case of inductive loads. Capacitors are important in integrated circuits, where they may be metal or polysilicon films with silicon dioxide dielectrics on silicon. In all their applications, they store energy in the form of an electrostatic field.

The terminal characteristic of an ideal capacitor is Q = CV. The time derivative gives dQ/dt = i = C dV/dt. In a DC circuit, a capacitor charges to the voltage established between its terminals by the rest of the circuit, according to this equation. By Thévenin's theorem, the rest of the circuit can be replaced so far as the capacitor is concerned by a voltage source V in series with a resistance R. The resulting circuit is the familiar series RC circuit. Kirchhoff's Law gives V = VC + iR, or dV/dt = 0 = i/C + Rdi/dt. This can be written di/dt + i/RC = 0, with i(0) = V/R. The solution of this differential equation is i(t) = (V/R)e-t/RC. The capacitor voltage is then given by the solution of dV/dt = i(t)/C with V(0) = 0 (if the capacitor was initially uncharged; the modifications in any other case are obvious). The result is V(t) = V[1 - e-t/RC]. All this is probably very familiar to the reader, but it does have a large number of interesting applications, mostly to timing circuits. The constant RC = τ, the time constant or relaxation time. The important thing to note is that the R is the Thévenin R.

When capacitors are connected in parallel, with the same voltage across them, each takes its appropriate charge Q = CV, so the total charge ΣQ = (ΣC)V. This means that a single capacitor of capacitance ΣC would hold the same total charge. When capacitors are connected in series across a voltage V, the same Q is stored in each, so each contributes its appropriate voltage Q/C. Therefore, V = Q Σ(1/C) this means a single capacitor of capacitance C* such that 1/C* = Σ(1/C) is equivalent to the set. These rules are like those for resistors in series and parallel, respectively. For capacitors in parallel, note that the equivalent capacitance is always greater than the largest single capacitance, since adding a capacitor in parallel, however small, will always add some charge to Q and increase the equivalent capacitance with the same voltage. For capacitors in series, the equivalent capacitance is always less than the smallest of the individual capacitors, since adding any capacitor in series, however large, will always add some voltage with the same Q.

In a sinusoidal AC circuit of angular frequency ω = 2πf, the terminal characteristic becomes i = jωC V, where we assume time dependence ejωt and i, V are the phasor magnitudes. A phasor rotates anticlockwise, and a factor j corresponds to a phase advance of 90°. We may use either peak or rms values, so which we use should be specified. The circuit equations can now be solved algebraically. The current leads the voltage across a capacitor (remember ICE). The real quantity 1/ωC is called the capacitive reactance X, measured in ohm, while the imaginary quantity -jX is the capacitive impedance, also measured in ohm. An impedance is a ratio of voltage drop to current, so V/i = -jX, which is the same as i = (j/X)V = jωC V. The reader should investigate how the usual rules for series and parallel circuits give the same results as in the preceding paragraph for capacitors in series and parallel.

Real capacitors have losses, which depend on the frequency. Let's suppose that we can define a complex capacitance C^ = C(1 - j tan δ). Here, δ is a new parameter, and tan δ is called the loss tangent or dissipation factor of the capacitor. Now, i = jωC^V = (jωC + ωC tan δ)V. The first term is the current through an ideal capacitor C, leading the voltage by 90°. The second term is in phase with the voltage, and is the current through a resistance X/tan δ. This suggests an equivalent circuit of an ideal capacitor in parallel with a resistance. The current through the resistance gives an energy loss equal to that of the capacitor. There is no actual resistor; this is just a way to express the losses in the capacitor, from whatever source they arise. If V is an rms (effective) value, then the loss in the resistance is (V2/X) tan δ. It happens that tan δ is nearly independent of frequency, so this expression allows us to find the loss in the capacitor in a wide variety of circuits. Values of tan δ are normally supplied by the capacitor manufacturer where losses need to be evaluated.

An ordinary polyester capacitor has a specified loss tangent of 0.01. Let's suppose an 0.001 μF capacitor is used at 50V rms and 1.0 MHz. Its capacitive reactance is X = 159Ω. The parallel loss resistance is then 15.9k, and the actual loss is 157 mW. This loss will appear as heat in the capacitor. At 1 kHz, X = 15.9M, and the energy loss will be only 157 μW. We can see why we are not frying capacitors very often in our circuits. It can happen, however, and with large currents or high frequencies it is best to check the power dissipation. Polypropylene capacitors have loss tangents around 0.001, and micas about 0.0005. The parallel loss resistance may have other circuit effects, such as lowering the Q of tuned circuits.

A bypass capacitor is intended to hold a circuit node to signal ground. To find what value to use, find the equivalent Thévenin impedance Z looking into the network from the capacitor terminals, shorting all voltage sources and opening all current sources. The reactance of the bypass capacitor at the lowest frequency bypassed should be much smaller than Z, perhaps 0.1Z. Be sure to consider all paths in the network, especially looking into cathodes of amplifying elements, such as transistors, and not just the path through a cathode resistor. Another application is power supply bypassing, especially for integrated circuits. Here, Z is practically zero for DC, but what we want to bypass is the inductance in the power lead. By placing a capacitor near the load, current spikes (very common in integrated circuits) will not cause a disturbance in the power voltage that may trigger logic level changes. CMOS circuits are specially liable to this problem, because of their high impedance. A poorly chosen despiking capacitor may have more impedance than the power lead, and so be ineffective. An excellent choice is a monolithic ceramic with very short leads and about 0.01 μF in value. Such a capacitor will supply 10A for 5 ns, 10 mA for a microsecond.

Coupling capacitors generally act as a single-stage RC filter, with a time constant τ = RC. The capacitor should be chosen to make this time constant smaller than the time over which the signal varies significantly (that is, less than 1/2πf for a sinusoidal signal of frequency f). As in the case of bypass capacitors, all contributions to R should be recognized. Sometimes the filtering action is desired, and RC filters can be high-pass or low-pass. At a frequency f = 1/2πRC, the capacitive reactance equals the resistance, and this is the -3dB point of the filter. Note that there is also a phase shift of the output relative to the input, which is made use of in many applications. A thorough knowledge of the RC branch is invaluable in circuit design.

The salient features of the RC high-pass filter are shown at the right. The freqency is specified as the angular frequency ω = 2πf for simplicity, but the results are easily expressed in terms of the frequency in Hz. The time constant τ = RC. The output voltage v' leads the input voltage v by 90° at low frequencies, the lead decreasing to zero at high frequencies. The gain in decibels and the phase angle are easily represented by the Bode plots, which consist of straight lines that are easy to draw. The "gain," of course, is always less than 0 dB. At the break point of the gain the gain is actually down 3 dB (half-power). A slightly more accurate curve can be sketched through this point, rounding off the angle. The gain then decreases at -20 dB per decade at lower frequencies. The phase shift at the break point is 45°, and is actually 5.7° at 10ω, not zero, with a similar difference from 90° at ω/10. In some cases you will want to represent the input to the filter as a Thévenin equivalent; the output v' then includes a resistive voltage divider (if the Thévenin impedance is resistive, as it often is) reducing the gain by a factor R/(R + RT), and the time constant includes the Thévenin resistance. Of course, R includes the input resistance of the following stage.

### Commercial Capacitors

A wide variety of capacitors is available commercially, ranging in values from a few pF to F. The greatest difference between them is the dielectric, which governs the available values and voltage ratings. There are two markets, one electronics, and the other power. I will discuss capacitors for the electronics market mainly, but the same considerations hold for the power market with a different emphasis. The two most important ratings are the value in farad, and the maximum voltage in volt. The maximum voltage across the capacitor should not exceed its rating, for the danger of electrical breakdown of the dielectric. The heat generated usually destroys the capacitor, but even if this does not happen, the capacitor may become a short or open circuit. It is difficult to make a capacitor of a definite value with some dielectrics, and the nominal value is normally a minimum value, which the capacitor is guaranteed to have. For example, many ceramic capacitors are specified +80% -20% as to value. A tighter specification on the value is usually associated with a higher cost.

The principal dielectrics now used are mica, mylar, polystyrene, polypropylene, polycarbonate, polyester, ceramic, aluminium electrolytic, tantalum electrolytic, and gold double layer. Mica capacitors are an old favorite, available from 5 pF up to 10 nF at voltages from 100V to 500V (and more). They are good at radio frequencies, have small losses and are very stable. The best are made from silver-coated mica sheets and "dipped" in a protective coating. Ceramic capacitors are another old favorite, relying on the very high dielectric constants of ferroelectric ceramics. They are available over a wide range of values, up to 4.7 μF, and voltages, 25 V to 5 kV, packing a lot of capacitance in a small space. Most familiar are the "disc" ceramics of that shape, but there are also "chip" ceramics that are small and square, or come in packages that easily fit into printed circuit boards. These are usually called monolithic. Unfortunately, ceramic capacitors are not very stable and have high losses, though this is not serious in their usual applications. The plastic film dielectrics are the excellent modern replacement for the paper capacitors of yore. These were pretty good capacitors, made from a roll of foil and paper, impregnated with wax, more stable and with lower losses than ceramics, but considerably larger. The dipped polyester capacitor of today is an excellent successor, available over a wide range of values and voltages, and quite inexpensive.

For large capacitances, the thin, chemically-deposited dielectric layers of the electrolytic capacitors are the choice. When an aluminium electrode is placed in an alkaline paste electrolyte, a thin layer of insulating aluminium oxide forms, and is consolidated when a voltage is applied. These capacitors are designed to be held at a DC potential near their rating for best service. The oxide layer grows with one polarity, but is dissolved with the other polarity, so electrolytic capacitors are polarized, and must be connected the right way round in a circuit. Electrolytic capacitors are usually crimped into a metal can. The positive lead is made longer than the negative one, and the polarity is marked on a vertical stripe on the side, usually the (-) polarity, though sometimes the (+) lead is so marked. The value and the voltage rating are printed on the can. Tantalum electrolytics are smaller than aluminium electrolytics of the same rating, and are usually "dipped" instead of being packaged in a can. The value and voltage rating are printed on the body. The largest capacitances of all in one package are provided by the gold double-layer capacitors, which can be found up to 50 F (!). They are all, however, low-voltage capacitors, with ratings of 5.5, 2.5 or 2.3 V. All electrolytics have considerable leakage currents (they act like a resistor in parallel with the capacitor), but tantalums are best of all. One type of tantalum has a leakage current of I = 0.008 CV, or 0.5 μA, whichever is larger. The leakage current of a 25V, 6800 μF aluminium electrolytic is specified at 1.7 mA.

A capacitor that comes naturally in a cylindrical can may have the leads attached axially or radially. Axial leads come out of opposite ends, along the axis of the cylinder, while radial leads come out side by side at one end. Capacitors that come naturally as a flat rectangular shape (a "chiclet") usually have the leads parallel at the ends of one side. This is also called "radial."

Many capacitors are marked with the value and voltage limit in figures. Another very common scheme uses three digits. The first two digits are significant figures, and the final digit is a decimal exponent, for a value in picofarad. For example, 104 means 10 followed by 4 zeros, 100000 pf = 0.1 μF; 472 means 4700 pF or 0.0047 μF. On small ceramics, the figures give the value in pF. Other letters and numbers on disc ceramic capacitors are usually the temperature range or value tolerance. For example,.1Z Y5S means 0.1 μF, tolerance -20% +80%, useful temperature range -30°C to 85°C, ± 22% variation in value over this range. Small capacitors used to be labelled with colored stripes like the resistor code, and in the dim past mica capacitors in molded packages had six or three colored dots interpreted the same way. These AWS/JAN standard capacitors from World War II are seldom seen today. There is no uniform series of preferred values for capacitors as there is for resistors, and they occur in a wide range of shapes and sizes.

Just as some people have been surprised to find inductors acting like capacitors at high frequencies, capacitors also act like inductors at high frequencies. An 0.01 μF disc ceramic capacitor is series resonant at 15 MHz if the lead length is 1/2 inch. Above this frequency it exhibits inductance, and is a series choke instead of a series bypass, as intended. One should always know the intended frequency domain in which a component is to be used, and beware when using it outside this domain.

### Variable Capacitors

There are two principal applications for capacitors whose capacitance can be varied: LC tuning circuits and small adjustments of a capacitance value. Capacitors for the second application are called trimmer capacitors and have a small range of variation. In LC circuits, the capacitance range should permit a useful tuning range, and sometimes the capacitance in related circuits should vary in step. The capacitance can be varied mechanically or electrically. Electrical variation is achieved by varying the reverse bias on a PN junction, which changes the width of the depletion layer. These varicaps will be discussed separately below. Mechanical variation is achieved by changing the plate area, spacing of the plates, or area of the dielectric.

Trimmer capacitors are of three main types. One type has two rectangular or square plates of about 1 cm2 area (typically; this, of course, may vary) separated by a mica sheet on the order of 0.1 mm thick (this, too, can vary). The capacitance with these dimensions will be about 52 pF. A typical trimmer of this type may have a capacitance range of 40-60 pF, about 40% of the mean value. The capacitance is adjusted by loosening the screw that holds the upper plate down; it springs up slightly, lowering the capacitance.

A second type has a fixed semicircular plate, a mica dielectric, and a moving semicircular plate that can be rotated with a screwdriver. These have a greater range than the first type, but generally a smaller value. A typical trimmer may vary from 6 to 40 pF.

The third type has several, usually semicircular, interleaved plates, like a miniature tuning capacitor, and an air dielectric. These are useful at higher voltages, but are more expensive than the first two types.

Tuning capacitors have a larger value than trimmers, up to several hundred pF. The traditional air-dielectric variable capacitor has a number of semicircular plates that rotate 180° on an axis to interleave with one fewer fixed plates, usually rectangular. A typical example may have 10 moving plates of 3.5 cm diameter, and 9 fixed plates, giving a capacitance range of 8-300 pF.

Ganged tuning capacitors, with the moving plates on a common axis, allow simultaneous variation of two capacitances. This is useful for the tuning of the antenna circuit and the local oscillator circuit in a superheterodyne receiver, where the capacitances must "track." The moving plates are electrically connected to the same axis, which is common with the frame of the capacitor, and usually grounded. The circuit of a two-gang capacitor is shown in the diagram at the left. The capacitor mentioned above is ganged with a capacitor with 8 smaller moving plates and 7 stationary plates, giving a capacitance of 5-130 pF. The stationary plates are insulated from the frame, and may have trimmer capacitors (of the first type). Ganged capacitors with four or more equal sections were used to tune TRF receivers, which required a series of tuned circuits to track. The elimination of this inconvenience is one of the principal benefits of superheterodyne receivers, since the IF stages are fixed-tuned.

A "butterfly" variable capacitor is a special kind of ganged capacitor with moving plates each consisting of two sectors 180° apart, interleaved with two sets of fixed plates, so the capacitance varied symmetrically from minimum to maximum. This kind of capacitor is useful in balanced circuits, for example with parallel-wire resonant lines, where neither line can be grounded.

At the present time, air-dielectric tuning capacitors are not common, having been replaced by smaller capacitors with mica or plastic dielectrics. The higher voltage rating of the air-dielectric capacitors is not required with transistors or IC's. Transmitting variable capacitors still must use an air dielectric, with a plate spacing appropriate to the voltage that is to be used.

Varactors, or tuning diodes, are PN junctions used with reverse bias. The capacitance decreases as the reverse bias increases, because of the widening of the depletion layer. The variation depends on the rate of doping variation near the junction. For a linear graded junction, C = B/V1/3, where B is a constant, and for a step junction C = B/V1/2. A hyperabrupt junction has the doping density decreasing with distance from the junction, and givea an even more rapid decrease of capacitance with voltage. Most varactors these days are hyperabrupt. A typical varactor has a reverse breakdown voltage of 25V. The capacitance ratio at 2V and 20V, C2/C20 is about 5.0 - 6.5 for a hyperabrupt varactor. For a step junction, it is about 3.2, and for a graded junction, about 2.2. Plotting log C against log V shows the type of junction from the slope of the curve. Available values of C2 range from 8.2 to 100 pF. There are hyper-hyperabrupt varactors with a reverse breakdown voltage of 12V for which 42 pF at 1 V decreases to 12 pF at 4 V. This is equivalent to a C2/C20 ratio of 10.

The use of a varactor is shown at the right. A tuning voltage of 0-20 V is applied at the left (note polarity). The symbol for a varactor is a combination of a capacitor symbol and a diode symbol. The 15 pF capacitor blocks the DC voltage from the tuned circuit, which consists of a 100 pF capacitor and an 0.25 mH inductor. The tuned circuit will resonate at around 1 MHz, and the resonant frequency will vary with the tuning capacitance. Varactors open the possibility of tuning by feedback loops. The only drawback is that a large tuning range cannot be provided.

### References

P. Horowitz and W. Hill, The Art of Electronics, 2nd ed. (Cambridge: Cambridge University Press, 1989). Many examples of the use of capacitors can be found in this classic work.