8.1 Introduction

In this chapter, we will examine capacitors, basic electronic components very popular in all kinds of circuits.

8.2 History

The capacitor or originally known as condenser¹ is a two‐terminal electric component that can store potential energy in the form of electric field.

Technically, a capacitor is composed of two metal plates, separated by a medium. This medium, called dielectric, can be any nonconductive element like air, oil, plastic, etc.

8.3 How It Works

Capacitors are basically two metal plates in parallel. The plates are put very close to each other, without touching. These plates are at rest and have roughly the same number of electrons (see Figure 8.1).

Something magical happens when a battery is connected to the plates. Suddenly, an electric field (F) is formed between the plates, as shown in Figure 8.2.

The electric field makes the plate connected to the battery’s negative pole (A) to accumulate an excess of negative charges (electrons) and the other one to accumulate positive charges, or lack of electrons (B).

Because both plates are very close and charges with the same polarity repel each other, this excess of electrons in plate A will create a force that will repel electrons on plate B, expelling them from the plate and forcing them to migrate to the battery’s positive pole.

At this time, one may have the illusion of current flow between the plates. The truth is that the electric field generated by plate A forced electrons from plate B to move off the plate to the battery’s positive pole. No electron crossed through the dielectric from plate A to B.

As plate A receives more electrons, it saturates and an increasing repulsion force develops to prevent more electrons from arriving at plate A. The force between plates stops increasing and cannot remove electrons from the other plate anymore. At this point, current stops “flowing.”

Capacitors are normally represented on schematics by the uppercase letter C.

Figure 8.3 shows the symbols used to represent a capacitor in the American and in the international standards.

8.3.2 Construction Methods

Capacitors with simply round or rectangle plates parallel to each other are not practical. They are generally huge in size and have low capacitances.

One method for building better capacitors is using long two thin strips of metal foil (A and C) with dielectric material sandwiched between them (B) – everything wounded into a tight roll and sealed in paper or metal tubes – as shown in Figure 8.4.

This method, in combination with diverse dielectrics, allows building high capacitance capacitors in a small package.

Different dielectrics also make possible to create capacitors with large capacitances, large breakdown voltages, and better performances under certain circumstances, as high frequencies, for example.

Table 8.1 shows several materials and their permittivities.

Material	Permittivity (F/m)
Vacuum	1
Glass	3.7–10
Teflon	2.1
Polyurethane	2.25
Polyamide	3.4
Polypropylene	2.2–2.36
Polystyrene	2.4–2.7
Titanium dioxide	86–173
Strontium titanate	310
Strontium barium titanate	500
Barium titanate	1250–10,000
Diverse polymers	1.8–10,000
Calcium copper titanate	> 250,000

8.4 Electric Characteristics

These are the main characteristics of capacitors:

• They store electric charges in the form of an electric field.
• An ideal capacitor can store an electric charge indefinitely. In real life, they discharge slowly through the dielectric (leakage), because dielectrics are not perfect insulators and let a little bit of current flow.
• Capacitors block direct current.
• Capacitors let alternating current (AC) pass.

8.5 Electric Field

Considering that the plates of a capacitor have an area equal to A and are separated by a distance d, the electric field between the plates can be found by using the following equation.

8.6 Capacitance

Capacitance is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential.

Capacitance is measured in Farads³ and represented by the uppercase letter F in the SI.

There are two kinds of capacitance: one produced by the object itself and one produced by proximity to other objects.

Any object that can be electrically charged will have its own value of capacitance.

Also, any object in the proximity of others will generate mutual capacitance.

Capacitance can also be expressed by the following formula.

8.7 Stored Energy

Capacitors can store a large amount of energy in the form of electric field in a relatively short amount of time, differently from batteries that take a long time to charge.

However, capacitors cannot supply energy or hold their electric charge for a long time because they slowly discharge through their own dielectrics or through the circuit they are connected.

However, their charge and discharge characteristics can be utilized to create very interesting circuits, like audio filters, blinkers, and timers.

The energy stored inside a capacitor can be expressed in terms of work to move charges from a plate to another. In other words, the energy stored inside a capacitor is equivalent to the work necessary to keep positive and negative charges, +q and −q, in their respective plates.

Mathematically, this work and its equivalent charge can be related by the following equation.

To find the total energy, we can integrate the previous equation:

Hence,

We know that Q = CV, and thus we can convert the previous formula into the following equation.

8.8 Voltage and Current

Like mentioned before, when a discharged capacitor is connected to a continuous voltage source, such as a battery, one of its plates will be filled with an excess of charges and the other will have a lack of charges. This excess of charges in one of the plates forms a strong electric field that expels electrons from the other plate and makes them move to the battery. Over time, the first negative plate will become saturated with charges and the process will stop.

For all intents and purposes, we can say that a current flowed through the capacitor.

8.8.1 Current on a Charging Capacitor

During the charge process of an initially discharged capacitor, the charging current will start at the maximum value possible and will exponentially decay to 0 as the capacitor charges.

Figure 8.5 shows how the current behaves during charge.

Current across a capacitor follows the following equation.

The previous equation tells us that current in a capacitor is proportional to the ratio between voltage variations over time.

We know that charge is also equal to Q = CV.

Therefore, if we derive both sides,

Upon substituting this relation in the previous formula, we obtain the following equation.

8.8.2 Voltage on a Charging Capacitor

During the charge process of an initially discharged capacitor, voltage will start at 0 and increase exponentially until it reaches the charging level. As the voltage across the capacitor approaches the charging one, the charging speed will decrease until it reaches 0 and the capacitor is fully charged.

Figure 8.6 shows how the voltage across a capacitor behaves over time in a charging capacitor.

Rearranging the previous equation, we see that the voltage across a capacitor can be written as

We can integrate both sides and obtain the equation of voltage across a capacitor during charge.

If we integrate for a specific interval, we get the following equation.

8.9 Examples

8.9.1 Example 1

Suppose we apply a current of 5 mA for 4 ms across an ideal 10 μF capacitor. The capacitor is initially discharged. We will call this sudden current a “current pulse.”

What is the voltage across the capacitor before and after the current pulse?

8.9.1.1 Solution

We know that integrals always calculate the area below the functions they represent.

If we want to find the voltage across a capacitor at a specific instant in time, we must consider the equation for the voltage across the capacitor during charge:

This equation describes voltage in terms of current as a function of time. In other words, this equation describes the current’s area for a specific interval.

For this example, the area is the one shown on Figure 8.7.

8.9.1.2 Before the Pulse

Before the pulse, the capacitor is completely discharged. Hence, its voltage is 0.

8.9.1.3 During the Pulse

During the pulse, the capacitor charges exponentially and therefore its voltage increases.

Voltage across a capacitor is ruled by the following equation:

The current pulse we have applied is constant. Hence, we can substitute the function i(t) by a fixed value i and remove it from the integral. Also, V₀ is equal to 0, because the capacitor is initially discharged.

Upon substituting these values, we get

Solving the integral,

Upon substituting the values in the formula, we obtain the voltage across the capacitor after 4 ms:

8.9.1.4 After the Pulse

After the pulse, current drops to 0 and the capacitor stops charging.

An ideal capacitor will never discharge and hold its charge indefinitely.

Consequently, the capacitor will keep its voltage equal to 2 V forever.

8.9.2 Example 2

We repeat the same example as before but using a 1000 μF capacitor.

What is the voltage across the capacitor after the same current pulse is applied?

8.9.2.1 Solution

Upon substituting the values in the formula, we obtain the voltage across the capacitor after 4 ms:

For the same pulse, the voltage across the capacitor is very small now. This is expected because the capacitance is bigger and, for that reason, the capacitor takes more time to charge.

8.10 AC Analysis

We will examine, now, what happens when a capacitor is connected to an AC source.

Capacitors have the property to block direct current and let pass AC under certain conditions.

Capacitors react when subjected to AC by offering a resistance to the current flow. This resistance, known as reactance, varies with the wave’s frequency. This property can be used to create filters and manipulate, for example, audio⁴ signals.

8.11 Capacitive Reactance

Capacitors resist to any sudden changes in voltage. This resistance is called capacitive reactance, represented, normally, by the symbol X_c and measured in Ohms (Ω).

Capacitive reactance is represented by the following formula.

or by

considering that ω = 2πf.

Capacitive reactance and frequency are inversely proportional; if frequency increases, the reactance decreases.

If frequency reduces gradually toward 0,⁵ capacitive reactance will skyrocket to infinity. This explains why capacitors block direct current.⁶

8.12 Phase

Like explained before, capacitors resist sudden changes in voltage. This resistance represents a kind of inertia that prevents capacitors from reacting instantly to voltage changes.

When a capacitor starts to charge, current will be maximum and voltage minimum. When the capacitor is charged, voltage will be maximum and current will be 0.

It is clear the existence of a delay between voltage and current and that voltage is 90° behind current.

Technically speaking, voltage and current are 90° out of phase.

8.12.1 Mathematical Proof

Like examined before, the voltage across a capacitor can be described in terms of current by this formula:

If we apply an alternated current across a capacitor, this current follows a sinusoidal form:

where I_P is the peak current and I is the instantaneous current.

Substituting (8.3) in (8.2), we get

To solve this integral we must use the substitution rule.

Hence,

Substituting this on the integral,

By applying the previous concept, we get

By applying this other concept, we obtain the final equation for the voltage across a capacitor.

What do we see when we compare equations (8.3) and (8.5)?

We see that the first equation describes current in terms of sin(ωt) and the second describes the voltage in terms of sin(ωt − π/2).

The voltage equation is telling us to subtract π/2, equivalent to 90°, to the angle, meaning that the voltage is delayed 90° in relation to current.

Figure 8.8 shows a diagram of voltage curve (second curve) that is 90° behind the current curve (first curve).

8.13 Electrolytic Capacitor

To increase the capacitance of a capacitor, we can do a few things:

• Increase the plate areas, but this is not practical and will make capacitors huge.
• Create new dielectrics that can increase the capacitance by increasing the insulation between the plates.
• Reduce the distance between the plates, but this is a problem, because the distance between the plates depends on how thin we can make the dielectric.

To overcome these problems, a new kind of capacitor was invented, called electrolyte capacitor, where the plates are formed by an aluminum foil with an oxide layer, a tissue soaked in electrolyte liquid, and a regular aluminum foil.

The oxide layer on the first aluminum foil insulates it from the electrolyte liquid, becoming the dielectric and making the electrolyte liquid itself the other “plate,” a kind of virtual plate.

For that reason, the electrolyte capacitor is not formed by two metal plates, but rather by an oxidized metal plate and an electrolyte element. The oxidized layer serves as a dielectric and the second regular aluminum foil serves just to supply voltage to the electrolyte.

Therefore, an electrolyte capacitor is composed of a metal foil (A) with an aluminum oxide layer (B), a tissue soaked in electrolyte liquid (C), and a regular metal foil (D), as shown in Figure 8.9.

The electrolyte capacitor is far from being a traditional capacitor. In fact, the process that creates the oxide layer on one of the plates makes the electrolyte capacitor only able to conduct current in one direction. If an electrolyte capacitor is subjected to a reverse current flow, it will be destroyed and probably explode.

Therefore, these electrolyte capacitors, also known as “polarized capacitors,” have a minus sign on their bodies to identify their polarity.

The advantages of electrolytic capacitors are as follows:

• Their dielectric layer is very thin, allowing the plates to be very close, thus increasing the capacitance.
• They are extremely tough, have a high degree of insulation, and can heal themselves up to a certain degree during their lifespan.
• High capacitance electrolyte capacitors have very small sizes.
• Very cheap to produce.

Figure 8.10 shows a representation of an electrolytic capacitor where a huge minus sign can be seen on the stripe and an X is seen on the metallic end.

This X at ending is, in fact, a groove to create a weakened part of the casing, which will bulge upward and help the capacitor release its contents in case of failure. If so, they will not explode violently hurting people or causing damage to the whole equipment. These grooves are designed to prevent catastrophic failures from happening.

Figure 8.11 shows the symbols for the electrolytic capacitor in the United States (first symbol) and in the world (last two symbols).

8.14 Variable Capacitors

Some applications in the real world require capacitors that can have a variable capacitance. For that reason, a new capacitor was born, the variable capacitors or trimmers.

The first line in Figure 8.12 shows the symbols for the variable capacitors used in the United States, and the second line shows the symbols used internationally.

8.15 Capacitors in Series

If we connect two capacitors in series, like shown in Figure 8.13, the equivalent capacitance between points A and B can be found by using the following formula:

For an infinite number of capacitors, a more generic formula must be used.

If there are just two capacitors in series, a friendlier formula can be used:

8.16 Capacitors in Parallel

If we connect two capacitors in parallel, like shown in Figure 8.14, the equivalent capacitance between points A and B can be found by using the following formula:

For an infinite number of capacitors, a more generic formula must be used.

8.17 Capacitor Color Code

In the past, capacitors, as well as resistors, were identified by a color code painted on their bodies. Today this is not the case anymore.

However, it is worth to know how this code works because it is not rare to find capacitors like that on old equipment.

8.17.1 The Code

The code was composed of four or five stripes painted on their bodies.

The capacitance is represented by the four‐color code: the first two stripes represent the first digits, the third stripe is the multiplier, and the last stripe is the maximum operational voltage.

In the five‐color code, the first two stripes represent the first two digits, the third stripe is the multiplier, the fourth stripe represents the tolerance, and the last one represents maximum operational voltage (Figure 8.15).

Table 8.2 shows the values for each stripe in both codes.

A capacitor with the stripes shown next will have the following value:

Brown	1
Black	0
Orange	×1000
Brown	±1%

The first three colors mean 10 × 1000 pF or 10000 pF, which is equivalent to 10 nF.

Color	First Stripe	Second Stripe	Multiplier	Tolerance >10 pF (%)	Tolerance <10 pF
Black	0	0	×1	±20	±2 pF
Brown	1	1	×10	±1	±0.1 pF
Red	2	2	×100	±2	±0.25 pF
Orange	3	3	×1000	±3
Yellow	4	4	×10k	±4
Green	5	5	×100k	±5	±0.5 pF
Blue	6	6	×1M	+80, –20
Violet	7	7		±10	±1 pF
Gray	8	8	×0.01	±5
White	9	9	×0.1	±10
Gold			×0.1
Silver			×0.01

The last strip is the tolerance, equal to ±1%, because the capacitor is bigger than 10 pF.

Therefore, the capacitor is 10 nF ± 1% of tolerance.

8.18 Capacitor Markings

Modern capacitors use a numeric code to identify themselves like 473 J or 104 K.

The first two digits represent the capacitance, the third is the multiplier, and the letter is the tolerance.

Tables 8.3 and 8.4 show the multipliers and the tolerances.

Therefore, a capacitor marked as 473 J will be equal to 47 × 1000 pf = 47000 pf or 47 nF, and the J letter would be a tolerance of ±5%.

Capacitor will be 47 nF ± 5% tolerance.

Some capacitors may contain additional numeric codes, written apart from the main code, to represent their maximum operational voltage. Table 8.5 shows these codes.

Appendix H lists several types of capacitors produced worldwide.