The 10 Percent Rule: This rule is a standard method for selecting R 1 and R 2 that takes into account
2.23 Capacitors
If you take two oppositely charged parallel conducting plates separated a small dis- tance apart by an insulator—such as air or a dielectric such as ceramic—you have created what’s called a capacitor. Now, if you apply a voltage across the plates of the capacitor using a battery, as shown in Fig. 2.91, an interesting thing occurs. Electrons
are pumped out the negative battery terminal and collect on the lower plate, while electrons are drawn away from the upper plate into the positive battery terminal. The top plate becomes deficient in electrons, while the lower plate becomes abundant in electrons.
Very quickly, the top plate reaches a positive charge +Q and the negative plate reaches a negative charge −Q. Accompanying the charge is a resultant electric field between the plates and a voltage equal to the battery voltage.
The important thing to notice with our capacitor is that when we remove the voltage source (battery), the charge, electric field, and corresponding voltage (pres- ently equal to the battery voltage) remain. Ideally, this state of charge will be main- tained indefinitely. Even attaching an earth ground connection to one of the plates—
doesn’t matter which one—will not discharge the system. For example, attaching an earth ground to the negative terminal doesn’t cause the electrons within that plate to escape to the earth ground where neutral charge is assumed. (See Fig. 2.92.)
It might appear that the abundance of electrons would like to escape to the earth ground, since it is at a lower potential (neutral). However, the electric field that exists
within the capacitor acts like a glue; the positive charge on the upper plate “holds”
onto the abundance of electrons on the negative plate. In other words, the positive plate induces a negative charge in the grounded plate.
In reality, a real- life charged capacitor that is charged and removed from the volt- age source would eventually lose its charge. The reason for this has to do with the
FIGURE 2.91
FIGURE 2.92
imperfect insulating nature of the gas or dielectric that is placed between the plates.
This is referred to as leakage current and, depending on the construction of the capaci- tor, can discharge a capacitor within as little as a few seconds to several hours, if the source voltage is removed.
To quickly discharge a capacitor you can join the two plates together with a wire, which creates a conductive path for electrons from the negative plate to flow to the positive plate, thus neutralizing the system. This form of discharge occurs almost instantaneously.
The ratio of charge on one of the plates of a capacitor to the voltage that exists between the plates is called capacitance (symbolized C):
C Q
=V (2.32) Capacitance related to charge and voltage C is always taken to be positive, and has units of farads (abbreviated F). One farad is equal to one coulomb per volt:
1 F = 1 C/1 V
Devices that are specifically designed to hold charge (electrical energy in the form of an electric field) are called capacitors. Figure 2.93 shows various symbols used to represent capacitors, along with a real capacitor model that we’ll discuss a bit later.
The equation C = Q/V is a general one; it really doesn’t tell you why one capacitor has a larger or smaller capacitance than another. However, in practice, when you buy a capacitor all you’ll be interested in is the capacitance value labeled on the device. (A voltage rating and other parameters are important, too, but we’ll talk about them later.) Most commercial capacitors are limited to a range from 1 pF (1 × 10−12 F) to 4700 µF (1 × 10−6 F), with typical values for the first two digits of the capacitance of 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82, 100. (Examples: 27 pF, 100 pF, 0.01 µF, 4.7 µF, 680 µF).
Having a wide range of capacitances allows you to store different amounts of charge for a given potential difference, as well as maintaining different potential dif- ferences for a given charge. With the appropriate capacitor, you can therefore control the storage and delivery of charge, or control potential differences.
FIGURE 2.93
Example 1: Five volts are applied across a 1000- µF capacitor until the capacitor is fully charged. How much charge exists on the positive and negative plates?
Answer: Q = CV = (1000 × 10−6 F)(5 V) = 5 × 10−3 C. This is the charge on the positive plate. The charge on the negative plate is the same, but opposite in sign.
Example 2: A 1000- µF capacitor and a 470- µF capacitor are arranged in the circuit shown in Fig. 2.94, with a 10- V dc supply. Initially, the switch is at position B then thrown to position A, and then thrown to position B, and then to position A, and finally to position B. Assuming the capacitors have enough time to fully charge or discharge during the interval between switches, what is the final voltage across each
capacitor after the last switch takes place?
Answer: When the switch is thrown from B to A the first time, C1 charges to:
Q1 = C1V = (1000 × 10−6 F)(10 V) = 0.01 C When the switch is then thrown to B, the circuit becomes essentially one big capacitor equal to C1
+ C2 or 1470 µF. Charge will flow from C1 to C2, since the system wants to go to the lowest energy configuration. The charge on each capacitor is the percentage of capacitance to the total capacitance for each capacitor multiplied by the initial charge on C1 before the switch was thrown to position B:
Q C
Q C
1000 F
1470 F(0.01 ) 0.0068 C 470 F
1470 F(0.01 ) 0.0032 C
1
2
= µ
µ =
= µ
µ =
The voltage at the new equilibrium is:
V1 = Q1/C1 = 0.0068/1000 µF = 6.8 V V2 = Q2/C2 = 0.0032/470 µF = 6.8 V
The rest of the results are obtained in using similar calculations—the final result yields 9.0 V, as shown in the graph to the right.
We could be content with this limited knowledge. However, if you want to build your own capacitors, as well as understand time- dependent behavior, such as displace- ment current and capacitive reactance, a deeper understanding of capacitance is needed.
2.23.1 Determining Capacitance
The capacitance of a capacitor is determined by plate area A, plate separation d, and insulating material or dielectric. If a voltage V is applied between two parallel plates,
FIGURE 2.94
an electric field equal to E = V/d will be produced. From Gauss’s law, each plate must contain an equal and opposite charge given by:
Q AE AV
d
= ε =
ε (2.33)
where ε is the permittivity of the dielectric. Free space (vacuum) has a permittivity given by:
ε0 = 8.85 × 10−12 C2/N ⋅ m2 (2.34) The constant εA/d term in the equation is the capacitance,
C A
d
=
ε (2.35)
The relative permittivity of a material referenced to the permittivity in vacuum is referred to as the dielectric constant, which is given by:
0
k= ε ε
Plugging this into the previous expression, we get the capacitance in terms of dielec- tric constant:
C k A d
k A d
(8.85 10 C /N m)
0 12 2
= ε
=
× − ⋅ × ×
(2.36) where C is in farads, A is in meters squared, and d is in meters.
The dielectric constant varies from 1.00059 for air (1 atm) to over 105 for some types of ceramic. Table 2.6 shows relative dielectric constants for various materials often used in constructing capacitors.
Capacitors often have more than two plates, the alternate plates being connected to form two sets, as shown in the lower drawing in Fig. 2.95. This makes it possible to obtain a fairly large capacitance in a small space. For a multiple- plate capacitor, we use the following expression to find the capacitance:
C k A
d n k A
d n
( 1) (8.85 10 C /N m)
( 1)
0
12 2
= ε
− = × ⋅ × ×
−
−
(2.37) where the area A is in meters squared, the separation d is in meters, and the number of plates n is an integer.
Example: What is the capacitance of a multiple- plate capacitor containing two plates, each with an area of 4 cm2, a separation of 0.15 mm, and a paper dielectric?
Answer:
C k A
d (n 1) (8.85 10 C /N m) 3.0 (4.0 10 m )
(1.5 10 m) (2 1) 7.08 10 F
70.8 pF
0
12 2 4 2
4
= 11
ε
− = × ⋅ × × ×
× − = ×
=
− −
−
−
2.23.2 Commercial Capacitors
Commercial capacitors, like those shown in Fig. 2.95, are constructed from plates of foil with a thin solid or liquid dielectric sandwiched between, so relatively large capacitance can be obtained in a small unit. The solid dielectrics commonly used are mica, paper, polypropylene, and special ceramics.
Electrolytic capacitors use aluminum- foil plates with a semiliquid conducting chemical compound between them. The actual dielectric is a very thin film of insulat- ing material that forms on one set of the plates through electrochemical action when a dc voltage is applied to the capacitor. The capacitance obtained with a given area in an electrolytic capacitor is very large compared to capacitors having other dielec- trics, because the film is so thin—much less than any thickness practical with a solid dielectric. Electrolytic capacitors, due to the electrochemical action, require that one lead be placed at a lower potential than the other. The negative lead (−) is indicated on the package, and some surface mount electrolytics mark the positive end. This polarity adherence means that, with the exception of special nonpolarized electrolyt- ics, electrolytic capacitors shouldn’t be used in ac applications. It is okay to apply a superimposed ac signal riding upon a dc voltage, provided the peak voltage doesn’t exceed the maximum dc voltage rating of the electrolytic capacitor.
2.23.3 Voltage Rating and Dielectric Breakdown
The dielectric within a capacitor acts as an insulator—its electrons do not become detached from atoms the way they do in conductors. However, if a high enough voltage
FIGURE 2.95
is applied across the plates of the capacitor, the electric field can supply enough force on electrons and nuclei within the dielectric to detach them, resulting in a breakdown in the dielectric. Failed dielectrics often puncture and offer a low- resistance current path between the two plates.
The breakdown voltage of a dielectric depends on the dielectric’s chemical com- position and thickness. A gas dielectric capacitor breakdown is displayed as a spark or arc between the plates. Spark voltages are typically given in units of kilovolts per centimeter. For air, the spark voltage may range from 100 kV/cm for gaps as narrow as 0.005 cm to 30 kV/mm for gaps as wide as 10 cm. Other things that contribute to the exact breakdown voltage level are electrode shape, gap distance, air pressure or density, the voltage, impurities within the dielectric, and the nature of the external circuit (air humidity, temperature, etc.).
Dielectric breakdown can occur at a lower voltage between points or sharp- edged surfaces than between rounded and polished surfaces, since electric fields are more concentrated at sharp projections. This means that the breakdown voltage between metal plates can be increased by buffing the edges to remove any sharp irregularities. If a capacitor with a gas dielectric, such as air, experiences break- down, once the arc is extinguished, the capacitor can be used again. However, if the plates become damaged due to the spark, they may require polishing, or the capacitor may need to be replaced. Capacitors with solid dielectrics are usually permanently damaged if there is dielectric breakdown, often resulting in a short or even an explosion.
Manufacturers provide what’s called a dielectric withstanding voltage (dwv), expressed in voltage per mil (0.001 in) at a specified temperature. They also provide a dc working voltage (dcwv) that takes into account other factors such as tempera- ture and safety margin, which gives you a guideline to the maximum safe limits of dc voltage that can be applied before dielectric breakdown. The dcwv rating is most useful in practice.
As a rule of thumb, it is not safe to connect a capacitor across an ac power line unless it is designed for it. Most capacitors with dc ratings may short the line. Special ac- rated capacitors are available for such tasks. When used with other ac signals, the peak value of ac voltage should not exceed the dc working voltage.
2.23.4 Maxwell’s Displacement Current
An interesting thing to notice with our parallel- plate capacitor is that current appears to flow through the capacitor as it is charging and discharging, but doesn’t flow under steady dc conditions. You may ask: How is it possible for current to flow through a capacitor, ever, if there is a gap between the plates of the capacitor? Do electrons jump the gap? As it turns out, no actual current (or electron flow) makes it across the gap, at least in an ideal capacitor.
As we calculated a moment ago using Gauss’s law, the charge on an air- filled capacitor plate can be expressed in terms of the electric field, area, and permittivity of free space:
0 0
Q AE AV
d
= ε =
ε (2.38)
Some time ago, Scottish physicist James Clerk Maxwell (1831–1879) noted that even if no real current passed from one capacitor plate to the other, there was never- theless a changing electric flux through the gap of the capacitor that increased and decreased with the magnitude and direction of the electric flux. (Electric flux for a parallel- plate capacitor is approximated by ΦE = EA, while a changing electric flux is expressed as dΦE/dt). Maxwell believed the electric flux permeated the empty space between the capacitor plates and induced a current in the other plate. Given the state of knowledge of electrodynamics at the time, he envisioned a displacement current (which he coined) crossing the empty gap, which he associated with a kind of stress within the ether (accepted at the time)—the “stress” being electric and magnetic fields.
(The displacement current helped supply Maxwell with the missing component to complete a set of electromagnetic formulas known as Maxwell’s equations.) He asso- ciated the displacement current with displacements of the ether. With a bit of theoret- ical reckoning, as well as some help from some experimental data, he came up with the following equation, known as the displacement current, to explain how current could appear to enter one end of a capacitor and come out the other end.
( 0 ) 0
I dQ dt
d
dt AE d
d dtE
= = ε = ε Φ (2.39)
Maxwell’s displacement expression appears to provide the correct answer, even though his notion of the ether has lost favor in the realm of physics. Modern physics provides a different model for displacement current than that envisioned by Maxwell and his ether. Nevertheless, the results obtained using Maxwell’s equation closely correlate with experiment.
As a side note, there also exists a magnetic field due to the displacement current, as shown in the drawing on the right in Fig. 2.96. You can calculate the magnetic field using what’s called Maxwell’s generalized form of Ampere’s law; however, the size of the magnetic field turns out to be so small that it essentially has no practical influ- ence when compared to the electric field.
However deep you go when trying to explain the physical phenomena within a capacitor, such as using Maxwell’s equations or even modern physics, the practical equations that are useful in electronics really don’t require such detail. Instead, you can simply stick with using the following charge- based model.
FIGURE 2.96
2.23.5 Charge- Based Model of Current Through a Capacitor
Though Maxwell’s displacement current provides a model to explain the apparent current flow through a capacitor in terms of changing electric fields, it really isn’t needed to define capacitor performance. Instead, we can treat the capacitor as a black box with two leads, and define some rules relating the current entering and exiting the capacitor as the applied voltage across the capacitor changes. We don’t need to worry about the complex physical behavior within.
Now, the question remains: how do we determine the rules if we don’t understand the complex behavior within? Simple—we use the general definition of capacitance and the general definition of current, and combine the two. Though the mathematics for doing this is simple, understanding why this makes logical sense is not entirely obvi- ous. The following parallel- plate example provides an explanation. Refer to Fig. 2.97.
If we work in differentials (small changes), we can rewrite the general expression for capacitance as dQ = CdV, where C remains constant (with charge, voltage, or time).
The general expression for current is I = dQ/dt, which, when combined with the last differential expression of capacitance, provides the expression:
( )
I dQ dt
d CV
dt CdV
C C dtC
= = = (2.40) Apparent current
“through” capacitor Looking at Fig. 2.97, a “small” chunk dQ, which is equal to CdVc, enters the right plate during dt, while an equal- sized chunk exits the left plate. So a current equal to dQ/dt = CdVc/dt enters the left plate while an equal- sized current exits the right plate. (Negative electrons flow in the opposite direction.) Even though no actual cur- rent (or electrons) passes across the gap, Eq. 2.40 makes it appear that it does. After our little exercise using differentials, however, we can see that there is really no need to assume that a current must flow across the gap to get an apparent current flow
“through” the capacitor.
Moving on, we can take the capacitor current expression just derived, rearrange things, and solve for the voltage across the capacitor:
V 1
C I dt
C=
∫
C (2.41) Voltage across capacitorFIGURE 2.97
It’s important to note that these equations are representative of what’s defined as an ideal capacitor. Ideal capacitors, as the equation suggests, have several curious properties that are misleading if you are dealing with real capacitors. First, if we apply a constant voltage across an ideal capacitor, the capacitor current would be zero, since the voltage isn’t changing (dV/dt = 0). In a dc circuit, a capacitor thus acts like an open circuit. On the other hand, if we try to change the voltage abruptly, from 0 to 9 V, the quantity dV/dt = 9 V/0 V = infinity and the capacitor current would have to be infinite (see Fig. 2.98). Real circuits cannot have infinite currents, due to resistiv- ity, available free electrons, inductance, capacitance, and so on, so the voltage across the capacitor cannot change abruptly. A more accurate representation of a real capaci- tor, considering construction and materials, looks like the model shown in Fig. 2.93.
If the equations for an ideal capacitor are screwy, what do we do for real- life calcu- lations? For the most part, you don’t have to worry, because the capacitor will be sub- stituted within a circuit that has resistance, which eliminates the possibility of infinite currents. The circuit resistance is also usually much greater than the internal resistance of the capacitor, so that the internal resistance of the capacitor can typically be ignored.
In a minute, we’ll see a few resistor- capacitor circuits that will demonstrate this.
FIGURE 2.98
Under steady- state dc, a capacitor cannot pass cur- rent. It can only store or discharge charge (which it collects from current) when the voltage across it changes. Here, when the “charge switch” is closed (shorted), the 9- V battery voltage is applied instantly across the capacitor. In a real capacitor, the capacitor charges up to its maximum value practically instantly. But upon closer examination, the charge takes time to build up, due to internal resistance, meaning the displacement current cannot reach infinity. Instead, the current jumps to Vbattery/Rinternal and quickly drops exponentially as the capacitor reaches full charge, during which the voltage rises exponentially until it levels off at the applied voltage. The graph to the left shows voltage and current curves as the capacitor charges up. Note the impossible behavior that an ideal treatment of a capacitor introduces.
When the discharge switch is closed, a conduc- tive path from positive to negative plate is made, and charge electrons will flow to the plate with a deficiency of electrons. The current that results is in the opposite direction as the charging current, but resembles it in that it initially peaks to a value of Vbattery/Rinternal and decays as the charge neutralizes.
The voltage drops exponentially in the process.