Capacitores

6.1 INTRODUCTION

So far we have limited our study to resistive circuits. In this chapter, we

shall introduce two new and important passive linear circuit elements:

the capacitor and the inductor. Unlike resistors, which dissipate energy,

capacitors and inductors do not dissipate but store energy, which can be

retrieved at a later time. For this reason, capacitors and inductors are

In contrast to a resistor, which spends or dis- called storage elements.

sipates energy irreversibly, an inductor or capacitor

stores or releases energy (i.e., has a

memory).

The application of resistive circuits is quite limited. With the introduction

of capacitors and inductors in this chapter, we will be able to

analyze more important and practical circuits. Be assured that the circuit

analysis techniques covered in Chapters 3 and 4 are equally applicable to

circuits with capacitors and inductors.

We begin by introducing capacitors and describing how to combine

them in series or in parallel. Later, we do the same for inductors. As

typical applications, we explore how capacitors are combined with op

amps to form integrators, differentiators, and analog computers.

6.2 CAPACITORS

A capacitor is a passive element designed to store energy in its electric

field. Besides resistors, capacitors are the most common electrical components.

Capacitors are used extensively in electronics, communications,

computers, and power systems. For example, they are used in the tuning

circuits of radio receivers and as dynamic memory elements in computer

systems.

A capacitor is typically constructed as depicted in Fig. 6.1.

Metal plates,

each with area A

d

Dielectric with permittivity e

Figure 6.1 A typical capacitor.

A capacitor consists of two conducting plates separated

by an insulator (or dielectric).

In many practical applications, the plates may be aluminum foil while the

dielectric may be air, ceramic, paper, or mica.

When a voltage source v is connected to the capacitor, as in Fig.

6.2, the source deposits a positive charge q on one plate and a negative

charge −q on the other. The capacitor is said to store the electric charge.

The amount of charge stored, represented by q, is directly proportional

to the applied voltage v so that

q = Cv (6.1)

where C, the constant of proportionality, is known as the capacitance

of the capacitor. The unit of capacitance is the farad (F), in honor of

the English physicist Michael Faraday (1791–1867). From Eq. (6.1), we

Alternatively, capacitance is the amount of charge may derive the following definition.

storedper plate for a unit voltage difference in a

capacitor.

−

−

−

+q −q

+

+

+

+

+

+

+ −

v

Figure 6.2 A capacitor

with applied voltage v.

Capacitance is the ratio of the charge on one plate of a capacitor to the voltage

difference between the two plates, measured in farads (F).

Note from Eq. (6.1) that 1 farad = 1 coulomb/volt.

CHAPTER 6 Capacitors andInd uctors 203

Although the capacitance C of a capacitor is the ratio of the charge

q per plate to the applied voltage v, it does not depend on q or v. It

depends on the physical dimensions of the capacitor. For example, for

the parallel-plate capacitor shown in Fig. 6.1, the capacitance is given by

C = A

d

(6.2)

where A is the surface area of each plate, d is the distance between the

plates, and  is the permittivity of the dielectric material between the

plates. Although Eq. (6.2) applies to only parallel-plate capacitors, we

may infer from it that, in general, three factors determine the value of the

capacitance:

Capacitor voltage rating andca pacitance are typically

inversely ratedd ue to the relationships in

Eqs. (6.1) and(6.2). Arcing occurs if d is small

and V is high.

1. The surface area of the plates—the larger the area, the greater

the capacitance.

2. The spacing between the plates—the smaller the spacing, the

greater the capacitance.

3. The permittivity of the material—the higher the permittivity,

the greater the capacitance.

Capacitors are commercially available in different values and types.

Typically, capacitors have values in the picofarad (pF) to microfarad (μF)

range. They are described by the dielectric material they are made of and

by whether they are of fixed or variable type. Figure 6.3 shows the circuit

symbols for fixed and variable capacitors. Note that according to the

passive sign convention, current is considered to flow into the positive

terminal of the capacitor when the capacitor is being charged, and out of

the positive terminal when the capacitor is discharging.

i C i

+ v −

C

+ v −

Figure 6.3 Circuit symbols for capacitors:

(a) fixed capacitor, (b) variable capacitor.

Figure 6.4 showscommontypes of fixed-value capacitors. Polyester

capacitors are light in weight, stable, and their change with temperature is

predictable. Instead of polyester, other dielectric materials such as mica

and polystyrene may be used. Film capacitors are rolled and housed in

metal or plastic films. Electrolytic capacitors produce very high capacitance.

Figure 6.5 shows the most common types of variable capacitors.

The capacitance of a trimmer (or padder) capacitor or a glass piston capacitor

is varied by turning the screw. The trimmer capacitor is often placed

in parallel with another capacitor so that the equivalent capacitance can

be varied slightly. The capacitance of the variable air capacitor (meshed

plates) is varied by turning the shaft. Variable capacitors are used in radio

(a) (b) (c)

Figure 6.4 Fixed capacitors: (a) polyester capacitor, (b) ceramic capacitor, (c) electrolytic capacitor.

(Courtesy of Tech America.)

204 PART 1 DC Circuits

receivers allowing one to tune to various stations. In addition, capacitors

are used to block dc, pass ac, shift phase, store energy, start motors, and

suppress noise.

(a)

(b)

Figure 6.5 Variable capacitors:

(a) trimmer capacitor, (b) filmtrim

capacitor.

(Courtesy of Johanson.)

To obtain the current-voltage relationship of the capacitor, we take

the derivative of both sides of Eq. (6.1). Since

i = dq

dt

(6.3)

differentiating both sides of Eq. (6.1) gives

i = C

dv

dt

(6.4)

According to Eq. (6.4), for a capacitor to carry

current, its voltage must vary with time. Hence,

for constant voltage, i = 0 .

This is the current-voltage relationship for a capacitor, assuming the positive

sign convention. The relationship is illustrated in Fig. 6.6 for a

capacitor whose capacitance is independent of voltage. Capacitors that

satisfy Eq. (6.4) are said to be linear. For a nonlinear capacitor, the

plot of the current-voltage relationship is not a straight line. Although

some capacitors are nonlinear, most are linear. We will assume linear

capacitors in this book.

Slope = C

0 dv ⁄dt

i

Figure 6.6 Current-voltage

relationship of a capacitor.

The voltage-current relation of the capacitor can be obtained by

integrating both sides of Eq. (6.4). We get

v = 1

C

t

−∞

i dt (6.5)

or

v = 1

C

t

t0

i dt + v(t0) (6.6)

where v(t0) = q(t0)/C is the voltage across the capacitor at time t0.

Equation (6.6) shows that capacitor voltage depends on the past history

of the capacitor current. Hence, the capacitor has memory—a property

that is often exploited.

The instantaneous power delivered to the capacitor is

p = vi = Cv

dv

dt

(6.7)

The energy stored in the capacitor is therefore

w =

t

−∞

p dt = C

t

−∞

v

dv

dt

dt = C

t

−∞

v dv = 1

2

Cv2



t

t=−∞

(6.8)

We note that v(−∞) = 0, because the capacitor was uncharged at t =

−∞. Thus,

w = 1

2

Cv2 (6.9)

Using Eq. (6.1), we may rewrite Eq. (6.9) as

w = q2

2C

(6.10)

CHAPTER 6 Capacitors andInd uctors 205

Equation (6.9) or (6.10) represents the energy stored in the electric field

that exists between the plates of the capacitor. This energy can be retrieved,

since an ideal capacitor cannot dissipate energy. In fact, the word

capacitor is derived from this element’s capacity to store energy in an

electric field.

We should note the following important properties of a capacitor:

1. Note from Eq. (6.4) that when the voltage across a capacitor is

not changing with time (i.e., dc voltage), the

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