# CAPACITANCE

In mechanical systems, if an elastic body such as a rubber band is stretched, there is little or no opposition during the first moment of movement. The opposition increases as the rubber band is extended. If the rubber band is stretched and then fixed in position, the work done in stretching it is stored in the band and is returned when the rubber band is released. In the realm of electricity, a capacitor behaves like the rubber band. Applying electrical energy to the capacitor is like stretching the rubber band--electrical energy is stored on the two conductive plates. When the source of electrical energy is then removed from the capacitor, the energy remains stored on the conductive plates--like the energy stored in our rubber band. Finally, when a path for conduction is provided between the charged plates of the capacitor, the stored energy is released--just like releasing the stretched-out rubber band .

This mechanical property of elastic bodies (rubber band, in our example) illustrates the electrical property called capacitance.

A capacitor is any two conductors separated by an insulating material. The conductors mentioned above are called the plates of the capacitor, and the insulating material is called the dielectric. More precisely, a dielectric is a material that is a good insulator (incapable of passing electrical current), but is capable of passing electrical fields of force.

 Some Examples Dielectric Materials air aluminum oxide various ceramics Barium titanate glass  water mica oil paper polyetheylene tantalum oxide calcium titanate waxed paper

When a DC voltage is first applied across a capacitor, considering no losses, the current flow is maximum the first instant and then rapidly decreases to zero when the voltage developed across the capacitor equals the applied voltage. Energy was consumed in building up the voltage. Removal of the applied voltage leaves the capacitor in charged condition, and energy is stored in the capacitor.

The plate with the larger number of electrons has the negative polarity. The opposite plate then has the positive polarity.

When a capacitor is charged, energy is stored in the dielectric material in the form of an electrostatic field.

 Charge Effect The source of current in this circuit is the DC voltage supply, Vs. Current flows through this circuit in a counter-clockwise direction. The current charges the plates of the capacitor, but does not flow through the capacitor, itself. Current flows in this circuit until the capacitor is completely charged--until the voltage across the plates of the capacitor is equal to the voltage source, Vs. Discharge Effect The source of current in this circuit is the energy that has been stored in the capacitor. Current flows through this circuit in a clockwise direction. The current discharges the plates of the capacitor, but does not flow through the capacitor, itself. Current flows in this circuit until the capacitor is completely discharged--until the voltage across the plates of the capacitor is zero

 ImportantCurrent never flows through the dielectric material. Current flows only through the external circuit while the capacitor is charging or discharging. Electrons pile up on the negative plate during a charging operation, and they remain stored on the negative plate until a discharge occurs. During discharge, electrons are removed from the negatively charged plate.

### Units of Capacitance

Electrical charge, which is symbolized by the letter Q, is measured in units of coulombs. The coulomb is given by the letter C, as with capacitance. Unfortunately this can be confusing. One coulomb of charge is defined as a charge having 6.28 × 1018 electrons. The basic unit of capacitance is the farad and is given by the letter f. By definition, one farad is one coulomb of charge stored with one volt across the plates of the capacitor. The general formula for capacitance in terms of charge and voltage is:

 C = Q V

where:

C = capacitance in farads (F)

V = voltage between the plates of the capacitor in volts (V)

Q = charge on the capacitor in coulombs (C)

This is one of those equations that is rarely used in everyday work. Rather, it serves the purpose of defining and demonstrating how capacitance, voltage, and the electrical charge are related. So we are not providing any exercises for the equation; but of course you are encouraged to play around with it  in wqays that might be of interest to you.

In practical terms, one farad is a large amount of capacitance. Typically, in electronics, much smaller units are used:

microfarad (μF), which is 10-6
nanofarad (nF) which is `0-9 F
picofarad (pF), which is 10-12 F

### Factors Affecting Capacitance

The actual value of a capacitor is determined by its geometry and the dielectric constant of the material that separaes the plates. Except for variable capacitors, the geometry and dielectric constant are fixed at the time of manufacture and do not change as long as the capacitor is operated with its rated voltage and ambient temperature.

 C = 8.85 x 10-12 kA s

where

C = value of capacitance in farads, F

k = dielectric constant of the material separating the plates

A = common area between the plates in square meters, m2

s = spacing between the plates in meters, m

The dielectric constant, k, is an expression of how much effect dielectric materials between the plates affect the value of a capacitor. The term has no units of measure. Here are a few examples:

 Some Examples  Dielectric Constants (k) vacuum air glass polyethylene film tantalum pentoxide ceramic 1 8 8 3 27 12 - 400

Again, this is one of those equations that technicians rarely use for crunching numbers. The equation, however, does demonstrate some important concepts:

 The value of a capacitor is proportional to the area shared by the plates. Increasing the area of the plates increases the value of capacitance. Decreasing the area of the plates decreases the value of capacitance. The value of a capacitor is inversely related to the spacing between the plates. Increasing the spacing between the plates reduces the value of capacitance. Decreasing the spacing between the plates increases the value of capacitance.

### Capacitor Voltage Rating

The choice and thickness of the dielectric material determines not only the capacitance but also the voltage rating of the capacitor. Since it is usually desirable to have relatively large capacitance in a small space, the dielectric is quite thin. It must have high dielectric strength to avoid puncture by the applied voltage.

Capacitors are commonly rated as to capacitance, working voltage, and peak voltage. DC working voltage is the maximum DC voltage the capacitor will stand satisfactorily under certain applied conditions (in its normal temperature range). AC working voltage rating is lower than DC working voltage rating for a given capacitor because the voltage used is the effective line voltage. For example, a capacitor with an AC working voltage rating of 400 V must be able to withstand a peak voltage of 566 V because that is the peak voltage of the 400 VAC. Peak voltage ratings are generally higher than working voltage since most dielectrics can withstand higher voltages for a short time than they can stand continuously.

### Some Examples of Capacitors

 Here is an assortment of ceramic capacitors with values from 1pF to 100nF and voltage rating of 50 wvdc. (They are called "ceramic" capacitors because they use a ceramic dielectric).

 These are 4.7mF tantalum surface-mount capacitors rated at 25VDC. Their dimensions are approximately 0.1in x 0.05in x 0.005in.

 2300mF electrolytic capacitor. 1000mF, 25V electrolytic capacitors

Electrolytic capacitors provide large capacitance in a small space. It is two metal plates separated by an electrolyte. Most electrolytic capacitors are polarized and thus suitable only for DC circuits. The dielectric is a very thin oxide film formed on the surface of the positive capacitor plate. This allows a high capacitance in a small space because of the thin dielectric. Electrolytics are marked to indicate polarity, which you must carefully observe--a reversal of applied voltage would destroy the dielectric film.

### Variable Capacitors

Schematic symbol
for a variable capacitor

Trimmers. Variable capacitors are mostly as "trimmers" -- in circuits where it is necessary to manually trim a capacitance to a precise value. Trimmers use ceramic or plastic as a dielectric, and the capacitance is varied by using a screw mechanism to vary the effective thickness of the dielectric material.,

 Note Recall that the value of capacitance is inversely related to the spacing between the plates.

Varactors. A voltage-variable capacitor or varactor is also known as a variable capacitance diode or a varicap. This device utilizes the variation of the barrier width in a reversed-biased diode. Because the barrier width of a diode acts as a non-conductor, a diode forms a capacitor when reversed biased. Essentially the N-type material becomes one plate and the junctions are the dielectric. If the reversed-bias voltage is increased, then the barrier width widens, effectively separating the two capacitor plates and reducing the capacitance.

Tuning Capacitor. The tuning capacitor (shown below) was the king of variable capacitors during most of the 20th century. You can actually see groups of capacitor plates (the rotor plates and the stator plates) that would mesh with a second set of plates. The dielectric is the surrounding air.

Turning the knob caused the area common to the rotor and stator plates to chage, thus changing the capacitance.

The great popularity of tuning capacitors was due to the fact they were the analog tuning mechanism for radio transmitters and receivers through most of the early history of radio.

Figure 4-4. Capacitors in Series.

When the upper plate of C1 is made negative, the lower plate of C1 must become equally positive. The same holds true for C2. Since the upper plate of C2 is connected to the lower plate of C1, these two points are electrically the same and effectively bring the two dielectrics together forming a single capacitor with a doubled thickness of dielectric. In this series circuit, as in all series circuits, the current flow is the same in all parts of the circuit. As a result, the same amount of charge will accumulate on each capacitor, thus the voltage is equally divided between the two capacitors.

Calculation of total series capacitance becomes more difficult when capacitors of unequal value are connected in series. In figure 4-4, two capacitors of equal value are connected in series. By Kirchhoff's law, the sum of the individual voltages must equal the applied voltage. The current is the same in all parts of the circuit. Since the current is the same, the charge deposited on each capacitor will be the same (Q=It). Charge, voltage, and capacitance are related by

This shows that the voltage on a capacitor is directly proportional to the charge and inversely proportional to the capacitance. The smallest capacitor will have the greatest voltage across it.

Substituting the value Q/C in place of E, in Kirchhoff's law for series circuits (Et = E1 + E2) you get

Dividing both sides of this equation by Q, you get

You can use this equation for determining the total series capacitance of any number of capacitors in series.

You can rearrange equation as follows:

This equation is a useful and timesaving device, but may be used only for two capacitors in series.

Capacitors are often connected in series to allow their being used in circuits having a higher voltage than that for which they are rated. For instance, four 500-V, 4-mfd units in series are the equivalent of a single 2,000-V, 1-mfd unit. Series operation of capacitors of different values should be avoided. Since all units receive the same amount of charge, the voltage is divided in inverse proportion to the capacitance of each unit. One thousand V applied to 1 mfd and 4 mfd in series will produce 200 V across the larger capacitor and 800 V across the smaller. If both were rated at 500 V, the 1-mfd capacitor would promptly break down putting the entire 1,000 V on the 4-mfd capacitor, which would also be punctured.

Parallel Connection

Two equal capacitors in parallel are the equivalent of a single capacitor having doubled plate area and the same dielectric thickness (figure 4-5). The total capacitance, therefore, is the sum of the parallel capacitance, while the voltage applied must be no higher than the rating of the individual units. For any number of parallel capacitors, total capacitance is

CT = C1 + C2 + C3 + C4 + ........ + Cn.

Figure 4-5. Capacitors in Parallel.

Capacitive Reactance

Since capacitors are used in AC circuits, it is necessary to define the farad in terms of a changing voltage. When a steady voltage is applied to a capacitor, a momentary charging current flows. This current ceases almost immediately.

If the voltage were increased or decreased, another momentary flow of current would take place. If the voltage could be constantly changed, a continuous current would flow. A more rapidly changing voltage would cause a larger average current. For a given rate of voltage change, a larger capacitance causes a larger current to flow. The unit of capacitance (fd) may then be defined in terms of a changing voltage.

A capacitor is said to have a capacitance of 1 fd when a change of 1 V per second produces a current of 1 amp. This can be expressed as the equation,

where Iav is the average current that is forced to flow through a capacitance of C fd by a change of E V in t sec.

In figure 4-6, a sine wave has been divided into 4 equal parts along the time base. The time required for one cycle is 1/f. The time required for a change of voltage, from zero to Emax, is a quarter cycle, or 1/4f. If a sine wave of voltage with a peak voltage of Emax and a frequency f is applied across a capacitor of C fd, the average current during a quarter cycle is

Figure 4-6. Relationship Between Time and Frequency.

In fact, it is true for the entire cycle because each quarter cycle has the same amount of voltage change per second. This equation should be simplified to eliminate having to work with both average and maximum terms in one operation. The relationship between average current and maximum current of a sine wave is then

You should recognize 2/p as the familiar 0.637. Substituting this value for Iav in its equation,

Imax = 2pfCEmax

Both sides of the above equation are in terms of maximum values and may be changed to effective values, giving

I = 2p fCE,

which indicates that the current flowing through a capacitor is proportional to the frequency of the applied sine wave and the capacitance. As in the case of resistive circuits, you find the opposition to the flow of current by dividing the voltage by the current. Then by equation I = 2pfCE.

where f (frequency) is the Hz and C is capacitance in farads. The angular velocity 2pf is often written w (omega), making reactance equal to 1/wC. This equation shows that capacitive reactance is inversely proportional to both capacitance and frequency. The manner in which the opposition to current by a capacitor (Xc) changes with frequency is shown in the graph in figure 4-7. As f increases, Xc decreases along with the vertical axis. Ohm's Law is applied to capacitive reactance by substituting Xc for R,

Figure 4-7. Relationship Between Capacitive Reactance (Xc) and Frequency (f).

The capacitive reactance (Xc) is the opposition offered to the flow of the current by a capacitor. Since the formula for capacitive reactance was found by working with a sinusoidal voltage applied, it follows that the formula

is true only for sine wave voltages.

You can use the two methods discussed above for determining capacitive reactance by applying them to the circuit shown in figure 4-8. If you place AC voltmeters across R and C and insert an AC ammeter in the circuit you get the readings as indicated in the figure.

Figure 4-8. AC Voltmeters Across an RC Circuit.

These values are actual meter readings. Now since you have the circuit current measured, you can calculate the capacitive reactance of C by

Without knowing the circuit current, you can calculate capacitive reactance with

Notice in figure 4-8 that more voltage is dropped across R and C than is applied from the generator. The introduction of the capacitor in the circuit has produced a strange circuit behavior. This is because the capacitor causes a phase shift between the voltage and current. Therefore you must use vectors to calculate circuit voltage and current. Vectors for series RC circuits will be discussed later.

Phase Relationship in a Pure Capacitive Circuit

The current flowing through a capacitor at any instant depends on the rate at which the voltage across it is changing. Therefore, with a sine wave voltage applied, the current is maximum when the voltage is crossing the zero axis, because the voltage is then climbing or falling most rapidly. When the applied voltage is at its maximum, it is no longer climbing or falling. As it levels off, the current falls to zero. The relationship between current and voltage is shown in figure 4-9.

Figure 4-9. Phase Relationship Between Current and Voltage in a Pure Capacitive Circuit.

The voltage and current are seen to be one-quarter cycle, or 90° apart. Since the positive alternation of current reaches maximum before that of voltage, current leads the voltage. The phase or time relationship between these two sine waves is also shown by vectors in figure 4-9. The general statement can be made that when a sine wave of voltage is applied to a pure capacitance circuit, the current leads the voltage by 90°.

Types of Capacitors

The major types of capacitors used in electronics and other electrical devices are, mica, ceramic, impregnated paper, air, and electrolytic.

Mica. The mica capacitor is thin metal plates separated by sheets of mica. The smaller sizes may be only two plates; larger sizes are numbers of plates and mica sheets. Alternate plates are connected together to provide a large plate area. The general arrangement is shown in figure 4-10. For electrodes, some mica capacitors use thin layers of silver deposited directly on the surface of the mica. A mica capacitor with molded bakelite case and wire terminals is shown in figure 4-11. Mica capacitors are made in small capacitances up to about 0.5 mfd. They can be made for very high voltages, to a high degree of accuracy, and very stable under temperature changes. They are highly suitable for use in high frequency circuits.

Figure 4-10. Mica Capacitor Showing Mica and Metal Sheets.

Figure 4-11. Molded Mica Capacitor (Enlarged).

Ceramic. Ceramic capacitors have most of the desirable qualities of mica capacitors and, in addition, may be made with a positive or negative temperature coefficient of capacitance. That is, the capacitance may be made to increase or decrease with changes in temperature. Such capacitors are very useful in compensating for temperature changes in nearby circuit components. The ceramic capacitor is made by depositing silver on the surfaces of a ceramic tube (figure 4-12).

Figure 4-12. Ceramic Capacitor.

Ceramics are used in transmitters because they meet the exacting requirements of rigid frequency control. They are the only kind of capacitor used for holding oscillator frequencies to the close limits obtained by crystal control. (Such capacitors are usually negative temperature coefficients.) Hermetically sealed ceramic capacitors have been developed for the precision circuitry of electronic instruments. These are hermetically sealed to withstand changes in atmospheric moisture. Also, they will withstand the vibration and shock normally encountered in military electronic equipment. These capacitors permit capacitance tolerances within plus or minus 1 percent and temperature coefficient tolerances within plus or minus 10 parts per million, per degree Celsius.

Impregnated Paper. Probably the most used capacitor has impregnated paper as the dielectric. The paper is saturated with any number of resins, waxes, oils, or synthetic compounds. For conservation of space and for ease of manufacture, these capacitors are made by winding up two strips of metal foil separated by sheets of paper. To reduce the danger of breakdown because of flaws in the paper, several laminated sheets of paper are used, instead of a single, thicker sheet. The units must be carefully sealed to prevent the paper from absorbing moisture. When made for high voltages, the case of the capacitor is often filled with a high-grade mineral oil. Typical paper capacitors are shown in figure 4-13. The paper capacitor has a greater ratio of capacitance to weight than the mica or ceramic type. It is suitable for power and audio frequencies but has excessive losses at higher radio frequencies.

Figure 4-13. Impregnated Paper Capacitors.

Air. Air is used as the dielectric to make the capacitor variable. Figure 4-14 shows the construction of two kinds. The rotor plates are all fixed to a common movable shaft. The stationary (stator) plates are fastened to fixed terminals. There may be one or two sets of stator plates. Because of the large spacing between plates and the low dielectric constant, air capacitors are rather bulky and are rarely made larger than 500 pfd. They have low losses and are highly suitable for radio frequencies. An advantage is that the dielectric is self-healing. That means, that after an excessive voltage has been applied, the capacitor can be used again.

Figure 4-14. Variable Air Capacitor.

Electrolytic. The electrolytic capacitor is important because it provides a very large capacitance in a small space. It is two metal plates separated by an electrolyte. The electrolyte is the negative electrode. The dielectric in an electrolytic capacitor is a very thin oxide film formed on the surface of the positive capacitor plate. This allows a high capacitance in a small space because of the thin dielectric. Electrolytic capacitors can be used in AC or DC circuits, but the construction for each differs. DC capacitors are commonly marked to indicate polarity, which you must carefully observe. A reversal of applied voltage would destroy the dielectric film. AC electrolytic capacitors are usually used only for intermittent operation. A good example is the electric refrigerator's starting motor. Capacitance ratings of electrolytic capacitors are quite high; the smallest readily available size being 8 mfd, while larger values run into thousands of microfarads. Voltage ratings are limited to about 450 working V. They are not suitable for use in critical circuits, since the capacitance varies greatly with temperature and age. In addition, they have considerable current leakage. Electrolytic capacitors are self-healing if the breakdown current is not too large. A dry electrolytic capacitor is shown in figure 4-15. Cheaper units are also supplied in cardboard containers with wire leads.

Figure 4-15. Electrolytic Capacitor.

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