ELECTRICAL ENGINEERING

Wednesday, 1 April 2015

Capacitors

Capacitors
Why Are Capacitors Important?
The capacitor is a widely used electrical component. It has several features that make it useful and important:

A capacitor can store energy, so capacitors are often found in power supplies.
A capacitor has a voltage that is proportional to the charge (the integral of the current) that is stored in the capacitor, so a capacitor can be used to perform interesting computations in op-amp circuits, for example.
Circuits with capacitors exhibit frequency-dependent behavior so that circuits that amplify certain frequencies selectively can be built.

What Is A Capacitor?

        Capacitors are two-terminal electrical elements. Capacitors are essentially two conductors, usually conduction plates - but any two conductors - separated by an insulator - a dielectric - with conection wires connected to the two conducting plates.
        Capacitors occur naturally. On printed circuit boards two wires running parallel to each other on opposite sides of the board form a capacitor. That's a capacitor that comes about inadvertently, and we would normally prefer that it not be there. But, it's there. It has electrical effects, and it will affect your circuit. You need to understand what it does.
        At other times, you specifically want to use capacitors because of their frequency dependent behavior. There are lots of situations where we want to design for some specific frequency dependent behavior. Maybe you want to filter out some high frequency noise from a lower frequency signal. Maybe you want to filter out power supply frequencies in a signal running near a 60 Hz line. You're almost certainly going to use a circuit with a capacitor.
        Sometimes you can use a capacitor to store energy. In a subway car, an insulator at a track switch may cut off power from the car for a few feet along the line. You might use a large capacitor to store energy to drive the subway car through the insulator in the power feed.
        Capacitors are used for all these purposes, and more. In this chapter you're going to start learning about this important electrical component. Remember capacitors do the following and more.

Store energy
Change their behavior with frequency
Come about naturally in circuits and can change a circuit's behavior

Goals
You need to know what you should get from this lesson on capacitors. Here's the story.

Given a capacitor,

Be able to write and use the voltage-current relationship for the capacitor,
Be able to compute the current through a capacitor when you know the voltage across a capacitor.

Given a capacitor that is charged,

Be able to compute the amount of energy that is stored in the capacitor.

Capacitors and inductors are both elements that can store energy in purely electrical forms. These two elements were both invented early in electrical history. The capacitor appeared first as the legendary Leyden jar, a device that consisted of a glass jar with metal foil on the inside and outside of the jar, kind of like the picture below. This schematic/picture shows a battery attached to leads on the Leyden jar capacitor.

Although this device first appeared in Leyden, a city in the Netherlands sometime before 1750. It was discovered by E. G. von Kleist and Pieter van Musschenbroek. Although it has been around for about 250 years, it has all of the elements of a modern capacitor, including:

Two conducting plates. That's the metallic foil in the Leyden jar.
An insulator that separates the plates so that they make no electrical contact. That's the glass jar - the Leyden jar.

The way the Leyden jar operated was that charge could be put onto both foil elements. If positive charge was put onto the inside foil, and negative charge on the outside foil, then the two charges would tend to hold each other in place. Modern capacitors are no different and usually consist of two metallic or conducting plates that are arranged in a way that permits charge to be bound to the two plates of the capacitor. A simple physical situation is the one shown at the right.
If the top plate contains positive charge, and the bottom plate contains negative charge, then there is a tendency for the charge to be bound on the capacitor plates since the positive charge attracts the negative charge (and thereby keeps the negative charge from flowing out of the capacitor) and in turn, the negative charge tends to hold the positive charge in place. Once charge gets on the plates of a capacitor, it will tend to stay there, never moving unless there is a conductive path that it can take to flow from one plate to the other.
There is also a standard circuit symbol for a capacitor. The figure below shows a sketch of a physical capacitor, the corresponding circuit symbol, and the relationship between Q and V. Notice how the symbol for a capacitor captures the essence of the two plates and the insulating dielectric between the plates.

        Now, consider a capacitor that starts out with no charge on either plate. If the capacitor is connected to a circuit, then the same charge will flow into one plate as flows out from the other. The net result will be that the same amount of charge, but of opposite sign, will be on each plate of the capacitor. That is the usual situation, and we usually assume that if an amount of charge, Q, is on the positive plate then -Q is the amount of charge on the negative plate.
        The essence of a capacitor is that it stores charge. Because they store charge they have the properties mentioned earlier - they store energy and they have frequency dependent behavior. When we examine charge storage in a capacitor we can understand other aspects of the behavior of capacitors.
        In a capacitor charge can accumulate on the two plates. Normally charge of opposite polarity accumulates on the two plates, positive on one plate and negative on the other. It is possible for that charge to stay there. The positive charge on one plate attracts and holds the negative charge on the other plate. In that situation the charge can stay there for a long time.
        That's it for this section. You now know pretty much what a capacitor is. What you need to learn yet is how the capacitor is used in a circuit - what it does when you use it. That's the topic of the next section. If you can learn that then you can begin to learn some of the things that you can do with a capacitor. Capacitors are a very interesting kind of component. Capacitors are one large reason why electrical engineers have to learn calculus, especially about derivatives. In the next section you'll learn how capacitors influence voltage and current.

Voltage-Current Relationships In Capacitors
There is a relationship between the charge on a capacitor and the voltage across the capacitor. The relationship is simple. For most dielectric/insulating materials, charge and voltage are linearly related.

Q = C V

where:

V is the voltage across the plates.

You will need to define a polarity for that voltage. We've defined the voltage above. You could reverse the "+" and "-".

Q is the charge on the plate with the "+" on the voltage polarity definition.
C is a constant - the capacitance of the capacitor.

        The relationship between the charge on a capacitor and the voltage across the capacitor is linear with a constant, C, called thecapacitance.
Q = C V         When V is measured in volts, and Q is measured in couloumbs, then C has the units of farads. Farads are really coulombs/volt.
        The relationship, Q = C V, is the most important thing you can know about capacitance. There are other details you may need to know at times, like how the capacitance is constructed, but the way a capacitor behaves electrically is determined from this one basic relationship.

       Shown to the right is a circuit that has a voltage source, V_s, a resistor, R, and a capacitor, C. If you want to know how this circuit works, you'll need to apply KCL and KVL to the circuit, and you'll need to know how voltage and current are related in the capacitor. We have a relationship between voltage and charge, and we need to work with it to get a voltage current relationship. We'll look at that in some detail in the next section.
        The basic relationship in a capacitor is that the voltage is proportional to the charge on the "+" plate. However, we need to know how current and voltage are related. To derive that relationship you need to realize that the current flowing into the capacitor is the rate of charge flow into the capacitor. Here's the situation. We'll start with a capacitor with a time-varying voltage, v(t), defined across the capacitor, and a time-varying current, i(t), flowing into the capacitor. The current, i(t), flows into the "+" terminal taking the "+" terminal using the voltage polarity definition. Using this definition we have:
i_c(t) = C dv_c(t)/dt         This relationship is the fundamental relationship between current and voltage in a capacitor. It is not a simple proportional relationship like we found for a resistor. The derivative of voltage that appears in the expression for current means that we have to deal with calculus and differential equations here - whether we want to or not.

Energy In Capacitors
        Storing energy is very important. You count on the energy stored in your gas tank if you drove a car to school or work today. That's an obvious case of energy storage. There are lots of other places where energy is stored. Many of them are not as obvious as the gas tank in a car. Here are a few.

You're reading this on a computer, and the computer keeps track of the date and time. It does that by keeping a small part of the computer running when you think that the computer is turned off. There's a small battery that stores the energy to keep the clock running when everything else is turned off.
If you own a stereo or television that you have to plug into the wall plug, then you should realize that the wall plug voltage becomes zero 120 times a second. When that happens, the system keeps running because there are capacitors inside the system that store energy to carry you through those periods when the line voltage isn't large enough to keep things going!

Capacitors can't really be used to store a lot of energy, but there are many situations in which a capacitor's ability to store energy becomes important. In this lesson we will discuss how much energy a capacitor can store.
Capacitors are often used to store energy.

When relatively small amounts of energy are needed.
Where batteries are not desired because they might deteriorate.
For larger power/short duration applications - as in power supply filters, or to keep power up long enough for a computer to shut down gracefully when the line power fails.

To calculate how much energy is stored in a capacitor, we start by looking at the basic relationship between voltage and current in a capacitor.
i(t) = C dv(t)/dt

Once we have this relationship, we can calculate the power - the rate of flow of energy into the capacitor - by multiplying the current flowing through the capacitor by the voltage across the capacitor.
P(t) = i(t)v(t)

Given the expression for the power:

P(t) = i(t)v(t)

And given the expression for the current:

i(t) = C dv(t)/dt

We can use the expression for current in the power expression:

P(t) = (C dv(t)/dt) v(t)

We can recognize that power is simply rate of energy input.

P(t) = dE/dt = (C dv(t)/dt) v(t)

Now, the derivative of energy can be integrated to find the total energy input.

P(t) = dE/dt = (C dv(t)/dt) v(t)

gives

Now, assuming that the initial voltage is zero (there is no energy stored in the capacitor initially, we find that the energy stored in a capacitor is proportional to the capacitance and to the square of the voltage across the capacitor.

E_c = (1/2)CV²

The expression for the energy stored in a capacitor resembles other energy storage formulae.

For kinetic energy, with a mass, M, and a velocity, v.

E_M = (1/2)MV²

For potential energy, with a spring constant, K, and an elongation, x.

E_Spring = (1/2)Kx²

Since the square of the voltage appears in the energy formula, the energy stored is always positive. You can't have a negative amount of energy in the capacitor. That means you can put energy into the capacitor, and you can take it out, but you can't take out more than you put in.

Power in to the capacitor can be negative. Voltage can be positive while current is negative. Imagine a capacitor that is charged. You could charge a capacitor by putting a battery across the capacitor, for example. Then, if you placed a resistor across the capacitor, charge would leave the capacitor - current would flow out of the capacitor - and the energy in the capacitor would leave the capacitor only to become heat energy in the resistor. When energy leaves the capacitor, power is negative.

When you use capacitors in a circuit and you analyze the circuit you need to be careful about sign conventions. Here are the conventions we used, and these conventions were assumed in any results we got in this lesson.

Frequency Dependent Behavior For A Capacitor
We start with a capacitor with a sinusoidal voltage across it.

where:

v_C(t) = Voltage across the capacitor
i_C(t) = Current through the capacitor
C = Capacitance (in farads)

We will assume that the voltage across the capacitor is sinusoidal:

v_C(t) = V_max sin(wt)

Knowing the voltage across the capacitor allows us to calculate the current:

i_C(t) = C dv_C(t)/dt = wC V_max cos(wt) = I_max cos(wt)

where I_max = wC V_max
Comparing the expressions for the voltage and current we note the following.

The voltage and the current are both sinusoidal signals (a sine function or a cosine function) at the same frequency.
The current leads the voltage. In other words, the peak of the current occurs earlier in time than the peak of the voltage signal.
The current leads the voltage by exactly 90^o. It will always be exactly 90^o in a capacitor.
The magnitude of the current and the magnitude of the voltage are related:

V_max/I_max = 1/ wC Now, with these observations in hand, it is possible to see that there may be an algebraic way to express all of these facts and relationships. The method reduces to the following.

If we have a circuit with sinusoidally varying voltages and currents (as we would have in a circuit with resistors, capacitors and inductors and sinusoidal voltage and current sources) we associate a complex variable with every voltage and current in the circuit.
The complex variable for a voltage or current encodes the amplitude and phase for that voltage or current.
The voltage and current variables can be used (using complex algebra) to predict circuit behavior just as though the circuit were a resistive circuit.

We need to do two things here. First, we can illustrate what we mean with an example. Secondly, we need to justify the claim above. We will look at an example first, and we will do two examples. The first example is jsut the capacitor - all by itself. The second example will be one that you have considered earlier, a simple RC low-pass filter.
Example 1 - The Capacitor
In a capacitor with sinusoidal voltage and currents, we have:

where:

v_C(t) = Voltage across the inductor

v_C(t) = V_max sin(wt)

i_C(t) = Current through the inductor

i_C(t) = wC V_max cos(wt) = I_max cos(wt)

C = Capacitance (in farads)

We represent the voltage with a complex variable, V. Considering this as a complex variable, it has a magnitude of V_maxand and angle of0^o. We would write:

V = V_max/0^o

Similarly, we can get a representation for the current. However, first note:

i_C(t) = wC V_max cos(wt) = I_max cos(wt) = I_max sin(wt + 90^o)

(Here you must excuse the mixing of radians and degrees in the argument of the sine. The only excuse is that everyone does it!) Anyhow, we have:

I = I_max/90^o = j I_max = jwC V_max

Where j is the square root of -1.
Then we would write:

V/I = V_max/jwC V_max = 1/jwC

and the quantity 1/jwC is called the impedance of the capacitor. In the next section we will apply that concept to a small circuit - one you should have seen before.
Before moving to the next section, a little reflection is in order. Here are some points to think about.

A phasor summarizes information about a sinusoidal signal. Magnitude and phase information are encoded into the phasor. Frequency information is not encoded, and there is a tacit assumption that all signals are of the same frequency, which would be the case in a linear circuit with sinusoidal voltage and current sources.
We looked at a case where we encoded a signal V_max sin(wt) into a phasor of V_max/0^o. That was completely arbitrary, and many others would have encoded V_max cos(wt) into a phasor of V_max/0^o.
Phasors are intended only to show relative phase information, and it doesn't matter which way you go.

Using Impedance
        In the last section we began to talk about the concept of impedance. Let us do that a little more formally. We begin by defining terms.
        A sinusoidally varying signal (v_C(t) = V_max sin(wt) for example) will be represented by a phasor, V, that incorporates the magnitude and phase angle of the signal as a magnitude and angle in a complex number. Examples include these taken from the last section. (Note that these phasors have nothing to do with any TV program about outer space.)
v_C(t) = V_max sin(wt) is represented by a phasor V = V_max/0^o
i_C(t) = I_max sin(wt + 90^o) is represented by a phasor I = I_max/90^o
v_a(t) = V_A sin(wt + f) is represented by a phasor V_a = V_A/f
        Next, we can use the relationships for voltage and current phasors to analyze a circuit. Here is the circuit.

        Now, this circuit is really a frequency dependent voltage divider, and it is analyzed differently in another lesson. However, here we will use phasors. At the end of this analysis, you should compare how difficult it is using phasors to the method in the other lesson.
        We start by noting that the current in the circuit - and there is only one current - has a phasor representation:
I = I_max/0^o We will use the current phase as a reference, and measure all other phases from the current's phase. That's an arbitrary decision, but that's the way we will start.
        Next we note that we can compute the voltage across the capacitor.
V_C = I/jwC This expression relates the current phasor to the phasor that represents the voltage across the capacitor. The quantity 1/jwC is the impedance of the capacitor. In the last section we justified this relationship.
        We can also compute the phasor for the voltage across the resistor.

V_R = IR

This looks amazingly like Ohm's law, and it is, in fact, Ohm's law, but it is in phasor form. For that matter, the relationship between voltage and current phasors in a capacitor - just above - may be considered a generalized form of Ohm's law!
Now, we can also apply Kirchhoff's Voltage Law (KVL) to compute the phasor for the input voltage.
V_IN = V_R + V_C = IR + I/jwC = I(R + 1/jwC) You should note the similarities in what happens here and what happens when you have two resistors in series.

If you have a resistor, R, and a capacitor, C, in series, the current phasor can be computed by dividing the input voltage phasor by the sum of R and 1/jwC.
If you have two resistors in series (call them R₁and R₂), the current can be computed by dividing the input voltage by the sum of R₁and R₂.

Example
Consider a series circuit of a resistor and capacitor. The series impedance is:
Z = R + 1/jwC That's the same as we showed just above. The impedance can be used to predict relationships between voltage and current. Assume that the voltage across the series connection is given by:
v_Series(t) = V_max cos(wt) That corresponds to having a voltage phasor of:
V = V_max/0^o We also know that the impedance establishes a relationship between the voltage and current phasors in the series circuit. In particular, the voltage phasor is the product of the current phasor and the impedance.
V = I Z For our particular impedance, we have:
V = I*(R + 1/jwC) So, we can solve for the current phasor:
I = V / (R + 1/jwC) Now, we know the voltage phasor and we know the impedance so we can compute the current phasor. Let us look at some particular values.
Assume:

R = 1.0 kW
C = .1mf = 10^-7 f
f = 1 kHz, so w = 2p 10³
V_max = 20 v

Then:

Z_R = 1.0 kW
Z_C = 1/(jwC) = 1/(j2p 10³10^-7 ) = j 1.59 kW

And, the total impedance is:

Z = Z_R + Z_C = (1.0 + j 1.59) kW

This impedance value can also be expressed in polar notation:

Z = 1.878 /62^o

Now, compute the current phasor:

I = V / (R + 1/jwC)

Substituting values, we find:

I = V / Z = V_max/0^o / 1.878 /62^o=20/0^o / 1.878 /62^o
I = V / Z = (20 / 1.878) /-62^o = 10.65 /-62^oamps

And, we need to examine exactly what this means for the current as a function of time. But that isn't very difficult. We can write out the expression for the current from what we have above.

i_C(t) = 10.65 cos(wt - 62^o) amps

Combinations of Resistors

Combinations of Resistors
Resistors do not occur in isolation. They are almost always part of a larger circuit, and frequently that larger circuit contains many resistors. It is often the case that resistors occur in combinations that repeat.

Goals
What are our goals for this lesson? Here are some.

   Given a circuit with a number of resistors,
   Be able to determine resistor combinations within the circuit where two or more resistors can be combined.
   Be able to replace series and parallel resistor combinations with the correct equivalent resistors.

Combinations of Resistors
In this lesson we will look at two recurring resistor combinations, series combinations and parallel combinations. Those are common combinations, not only for resistors but other elements as well. (For example, we can speak of "a resistor in series with a capacitor".)
We'll start by examining series and parallel combinations and then move on to identifying those combinations when they are "buried" within a larger circuit. What we're doing is learning how to recognize small portions of larger circuits. Experts do that. You can click hereto see how experts are able to recognize larger combinations in many situations. It is a part of the basic "tool box" that an expert in an area acquires as s/he becomes an expert.

Series Combinations of Resistors
Two elements are said to be in series whenever the same current physically flows through both of the elements. The critical point is that the same current flows through both resistors when two are in series. The particular configuration does not matter. The only thing that matters is that exactly the same current flows through both resistors. Current flows into one element, through the element, out of the element into the other element, through the second element and out of the second element. No part of the current that flows through one resistor "escapes" and none is added. This figure shows several different ways that two resistors in series might appear as part of a larger circuit diagram.

You might wonder just how often you actually find resistors in series. The answer is that you find resistors in series all the time.

An example of series resistors is in house wiring. The leads from the service entrance enter a distribution box, and then wires are strung throughout the house. The current flows out of the distribution box, through one of the wires, then perhaps through a light bulb, back through the other wire. We might model that situation with the circuit diagram shown below.

In many electronic circuits series resistors are used to get a different voltage across one of the resistors. We'll look at those circuits, called voltage dividers, in a short while. Here's the circuit diagram for a voltage divider.

Besides resistors in series, we can also have other elements in series - capacitors, inductors, diodes. These elements can be in series with other elements. For example, the simplest form of filter, for filtering low frequency noise out of a signal, can be built just by putting a resistor in series with a capacitor, and taking the output as the capacitor voltage.
As we go along you'll have lots of opportunity to use and to expand what you learn about series combinations as you study resistors in series.

Let's look at the model again. We see that the wires are actually small resistors (small value of resistance, not necessarily physically small) in series with the light bulb, which is also a resistor. We have three resistors in series although two of the resistors are small. We know that the resistors are in series because all of the current that flows out of the distribution box through the first wire also flows through the light bulb and back through the second wire, thus meeting our condition for a series connection. Trace that out in the circuit diagram and the pictorial representation above.
Let us consider the simplest case of a series resistor connection, the case of just two resistors in series. We can perform a thought experiment on these two resistors. Here is the circuit diagram for the situation we're interested in.

Imagine that they are embedded in an opaque piece of plastic, so that we only have access to the two nodes at the ends of the series connection, and the middle node is inaccessible. If we measured the resistance of the combination, what would we find? To answer that question we need to define voltage and current variables for the resistors. If we take advantage of the fact that the current through them is the same (Apply KCL at the interior node if you are unconvinced!) then we have the situation below.

Note that we have defined a voltage across each resistor (V_a and V_b) and current that flows through both resistors (I_s) and a voltage variable, V_s, for the voltage that appears across the series combination.
Let's list what we know.

The current through the two resistors is the same.
The voltage across the series combination is given by:

V_s= V_a + V_b

The voltages across the two resistors are given by Ohm's Law:

V_a = I_s R_a
V_b = I_s R_b

We can combine all of these relations, and when we do that we find the following.

V_s= V_a + V_b
V_s= I_s R_a + I_s R_b
V_s= I_s (R_a + R_b)
V_s= I_s R_series

Here, we take R_series to be the series equivalent of the two resistors in series, and the expression for R_series is:
R_series = R_a + R_b What do we mean by series equivalent? Here are some points to observe.

If current and voltage are proportional, then the device is a resistor.
We have shown thatV_s= I_s R_series, so that voltage is proportional to current, and the constant of proportionality is a resistance.
We will call that the equivalent series resistance.

There is also a mental picture to use when considering equivalent series resistance. Imagine that you have two globs of black plastic. Each of the globs of black plasic has two wires coming out. Inside these two black plastic globs you have the following.

In the first glob you have two resistors in series. Only the leads of the series combination are available for measurement externally. You have no way to penetrate the box and measure things at the interior node.
In the second box you have a single resistor that is equal to the series equivalent. Only the leads of this resistor are available for measurement externally.

Then, if you measured the resistance using the two available leads in the two different cases you would not be able to tell which black plastic glob had the single resistor and which one had the series combination.
Here are two resistors. At the top are two 2000W resistors. At the bottom is single 4000W resistors. (Note, these are not exactly standard sizes so it took a lot of hunting to find a supply store that sold them!). You can click the green button to grow blobs around them.
After you have grown the blobs around the resistors there is no electrical measurement you can make that will allow you to tell which one has two resistors and which one has one resistor. They are electrically indistinguishable! (Or, in other words, they are equivalent!)

Parallel Resistors
The other common connection is two elements in parallel. Two resistors or any two devices are said to be in parallel when the same voltage physically appears across the two resistors. Schematically, the situation is as shown below.

Note that we have defined the voltage across both resistor (V_p) and the current that flows through each resistor (I_a and I_b) and a voltage variable, V_p, for the voltage that appears across the parallel combination.
Let's list what we know.

The voltage across the two resistors is the same.
The current through the parallel combination is given by:

I_p= I_a + I_b

The currents through the two resistors are given by Ohm's Law:

I_a = V_p /R_a
I_b = V_p /R_b

We can combine all of these relations, and when we do that we find the following.

I_p= I_a + I_b
I_p= V_p /R_a + V_p /R_b
I_p= V_p[ 1/R_a + 1/R_b]
I_p= V_p/R_parallel

Here, we take R_parallel to be the parallel equivalent of the two resistors in parallel, and the expression for R_parallel is:
1/R_parallel = 1/R_a + 1/R_b There may be times when it is better to rearrange the expression for R_parallel. The expression can be rearranged to get:
R_parallel = (R_a*R_b)/(R_a + R_b) Either of these expressions could be used to compute a parallel equivalent resistance. The first has a certain symmetry with the expression for a series equivalent resistance.

Parallel Resistors - A Point to Remember

It is important to note that the equivalent resistance of two resistors in parallel is always smaller than either of the two resistors.

RESISTORS

Resistors
There are many different types of electrical components. Shown below is a photograph of part of a circuit board. On this circuit board are many electrical components including some resistors.

There are resistors, capacitors, diodes and integrated circuits. Using plated copper and solder on the reverse side of the board, these components are interconnected and when a supply voltage is provided, these components can interact. Electrical engineers need to know how to design larger circuits like this and to control those interactions to achieve some purpose.
In this lesson you are going to learn about the simplest sort of electrical component, the resistor. What you learn about resistors is a starting point. That knowledge helps you as you begin to learn about all of the other kinds of electrical/electronic components. Still, although resistors are basic elements, they occur everywhere. On the board below, all of the resistors are marked with a yellow dot.

Goals

This lesson introduces you to some simple concepts about resistors and resistance. At the end of the lesson, you want to be able to do the following.

Given a common electrical device,

Know when the device is a resistor

   Given a resistor

   Be able to use Ohm's Law to compute resistance when voltage and current are given,
   Be able to use Ohm's Law to compute voltage when resistance and current are given,
   Be able to use Ohm's Law to compute current when voltage and resistance are given.

Almost every electrical product is constructed from a collection of fundamental electrical components. For example,

a radio,
a portable tape player,
a television,
a telephone,
a cellular phone,
the electronic ignition in a car,
a computer circuit board

are all built from smaller electrical components. There are many different kinds of components, including

resistors,
capacitors,
inductors,
transistors,
microwave generating tubes,
and many more.

These devices can be combined to form many different kinds of circuits and devices including all of the appliances listed earlier as well as all the various forms of computers we use daily. If you want to be able to design electrical and electronic circuits to perform a useful function you need to start by learning about components, and we will start with resistors.

The array of electrical components now available includes a vast and diverse number of components with varying shapes and numbers of leads. Here are some more electrical components you may be familiar with:

light bulbs,
computer chips,
solar cells,
inductors,
heating elements on stoves,
Light Emitting Diode (LED) displays on calculators,
operational amplifiers,
thermistors,
and many others.

Some of the electrical devices on our list have something in common. There are many ways of categorizing them. For example, you might look for elements with two leads or those with three leads. You might also look for those elements which interact with other physical variables (for example, speakers which produce sound in a stereo, or a light bulb which generates light).

One way of categorizing the items on the previous page is that all of the items above produce heat as a by-product of their operation. Here are some more devices that produce heat when a voltage is applied - a stove heating element, a toaster and two light bulbs.

In fact, all of the items on the pictures above are resistors.

There are numerous places where resistors are used. Here are some places where resistors are used.

Heating Applications:

Home baseboard heaters
Toasters
Clothes Dryers

Measurement Applications:

Photoresistors to measure light intensity (Photgraphic light meters in a camera)
Thermistors to measure temperature
Strain gages to measure strain (Used in bathroom scales, measurements on bridges)

Resistors are ubiquitous today. They are literally everywhere, not only in electronic equipment but are they are in many of the ordinary devices we rely on like toasters, irons, stoves and light bulbs.

To really understand resistors you need to understand the law they obey - Ohm's Law - which gives a relationship between the current through a resistor and the voltage across the resistor. That's something you may not have considered yet. A resistor is a device that establishes a unique relationship between a current and a voltage.

Ohm's Law - What You Need To Know About Resistors.

The Genesis of Electrical Devices and Theories

Work on electrical devices all started with resistors and resistance. The concept of resistance was first enunciated by Georg Simon Ohm who was the first to point out that voltage and current in a wire were related mathematically. That's an important idea. Since Ohm's time many different devices have been discovered and generalizations of Ohms' Law are used to describe those devices.

Ohm found that mathematics could be applied to what was going on in resistors, and the mathematical relation he found was just the start of the application of mathematics to electrical phenomena.

The Application of Mathematics to Electrical Science

You are probably accustomed to the idea that physical phenomena can be described using the language of mathematics. That's an idea that wasn't always accepted, and when Ohm first proposed to describe electrical phenomena using mathematics he was taken to task vehemently. (That happened even though Newton had described mechanical phenomena mathematically two hundred years earlier, and had, in fact, invented calculus in order to do that!)

Ohm didn't even have the concepts of current and voltage to work with. He had to invent the concepts and then show that there was an experimental relationship between voltage and current, and that that relationship could be described mathematically.

It all sounds simple today, especially since we have nice clean concepts of voltage and current as we discussed in the first chapter. In Ohm's time, things were not so clear, and his discoveries really include clarification of the concepts of voltage and current as well as resistance.

Working with very primitive instruments that he designed and constructed himself, Ohm discovered that voltage and current were linearly related in wires. That means that if you measure voltage across a wire and plot that against the current through the wire you get a straight line in the plot.

Working with very imprecise measurements, Ohm was able to determine that voltage and current for any fixed geometrical structure built from conducting material satisfied a relationship:

V = I R

where

V is the voltage across the device,
I is the current flowing through the device,
R is a constant.
R depends upon the material from which the device is constructed and the geometry of the material.

There are several different ways that Ohm's Law can be represented.

Graphically:

We can draw/sketch/diagram/plot the relationship.

Mathematically:

We can represent the relationship mathematically with anequation:

V_r = I_r R

Symbolically:

We can devise a symbol for the resistor and use it in circuit diagrams to stand for any element that satisfies Ohm'sLaw. The standard symbol is shown below along with polarity definitions for the voltage and current.

We will use the symbol R for any resistor, although there will occasionaly be devices (like light bulbs) that are really resistors but which have a special symbol that can be used for them. Still, you can use this symbol for an resistive device.

Ohm's Law - Continued

Electrical engineers communicate with symbols. Circuit diagrams are abstract representations of real circuits and are composed of symbols for the various elements in the circuit. Here is the circuit symbol for a resistor.

The symbol for a resistor should include definitions for the voltage across the resistor and the current through the resistor. The definitions include:

A symbolic name for the variable ( V_r for the voltage and I_r for the current for the symbol shown above).
A definition of the polarity (Shown by "+" and "-" for the voltage and an arrow for the current).

Ohm's Law gives us a relationship between the voltage across a piece of conducting material (a resistor) and the current flowing through that resistor. Those two variables are represented algebraically with symbols, V_r for the voltage across the resistor, and I_r for the current flowing through the resistor.

It's important to remember that the voltage across the resistor, and the current through the resistor are related by Ohm's Law: V_r = RI_r when the polarities are as shown below.

We need to be very precise when we consider Ohm's law because the polarities are very important.

Ohm's law holds - that is V_r = RI_r - only when the current is defined positive flowing into the terminal that is labelled positive for the voltage.
We paraphrase that by saying that Ohm's law holds in the usual form whenever the arrow defining positive current flows into the "+" terminal (referring to voltage polarity definition).

Physical Resistors

While it is true that a piece of metal, like a wire for example, is a resistor of sorts, you need to know that today resistors are made with specific values, and they often take a common form. Typically they look like the one shown at the right - only they' be pretty small, maybe a half to three quarters of an inch long for the body.

A resistor is typically formed from some sort of resistive material and put into a cylindrical form. Usually the resistor will have a brownish body with several stripes painted on the resistor body. Those stripes are in a code that will tell you the value of the resistors (in ohms). Here's what's important.

The resistor shown above is 1000 ohms = 10x10²

Notice how cleverly we put certain parts of the result in bold text and colored it. The bolded text corresponds to the stripes, and the colors are shown on the bold text. Here is the color code.

0	Black	Black
1	Brown	Brown
2	Red	Red
3	Orange	Orange
4	Yellow	Yellow
5	Green	Green
6	Blue	Blue
7	Violet	Violet
8	Gray	Gray
9	White	White

Given a resistor, to calculate the value of the resistance you use the three stripes. (If there are four stripes, just use the first three. The last stripe tells you how accurate the resistance value is.) Here is the algorithm.

The first stripe is the most significant digit, X, in XY x 10^Z.
The second stripe is the next digit, Y, in XY x 10^Z.
The third stripe is the exponent in XY x 10^Z.

Here is an example/problem.

Measuring Resistance

In this section you'll learn a little about how to measure resistances. You'll need to have an ohmmeter, a digital multimeter or a data acquisition unit. When you use any of those instruments to measure a resistance, the same thing happens. It's just that a digital multimeter can make voltage and current measurements, while a data acquisition unit can measure frequency and temperatures.

We'll also assume that you're in a lab running these lessons and that you have a lab notebook that you are using. (You should always have a lab notebook for lab work!)

An ohmmeter measures resistance, and gives you a value of the measured resistance in ohms, kilohms or megohms. Many ohmmeters look like the following diagram.

There's an internal source that provides a voltage. That source may be a battery or a small power supply. The source drives a voltage divider - two resistors in series. One of those resistors is internal to the meter, and the other resistor is the resistor being measured. An internal meter measures the voltage across the resistance being measured and converts that voltage into a resistance reading. The resistance being measured is connected to the ohmmeter terminals, and the terminals are often colored black and red.

All you have to do to measure a resistance is to connect your resistor to the ohmmeter as shown at the right, and be sure that the ohmmeter (or DVM or DAU) is set to measure resistance. Don't get uptight about which lead goes on which end of the resistor. It doesn't matter. (The resistor is a "bilateral" element and should be the same either way!)

Here's the way you connect the ohmmeter (or digital voltmeter or data acquisition unit) to the resistor. Here we're using the same resistor as was used in the questions above. The ohmmeter shown here includes all of the circuitry shown above including a power supply or battery and an internal resistance. To measure the resistance it applies a small voltage across the resistance.

At this point, you are ready to start the first laboratory exercise on resistors. Click here to go to that lab problem.

Physical Resistors - Calculating Resistance From Geometry

The resistance of a resistor is determined by several physical properties of the resistor. We're going to limit ourselves to resistors that have a constant cross section - like a wire. Here are the properties.

The geometry of the conductor, including:

The length, L
The cross-sectional area, A

A constant of the material called the resistivity, r.

If you have those quantities, then the resistance is given by:

R = rL/A

What If Questions

You may be tempted to conclude that there are no serious "What If?" questions for resistors. Actually, there are many questions about these devices. Note the following characteristics of resistors.

Resistors have voltage directly proportional to the current. That's true at every instant of time and for every frequency. Is it possible to have a situation in which voltage and current are not proportional?

In diodes - and many other devices, current and voltage are nonlinearly related. There are many devices in which the relationship is not proportional or linear.

The voltage across a resistor and the current through the resistor depend upon the values at the same time. Is it possible to have other kinds of relationships?

In a capacitor, we have i(t) = Cdv(t)/dt.
In an inductor, we have v(t) = Ldi(t)/dt.
Capacitors and inductors have voltage and current related to derivatives! That's really a different situation because it means you have to learn how to solve differential equations ot predict behavior of circuits with these components. That's a whole new kettle of fish.

A resistor is a two-terminal device. Transistors have three terminals. That means that the analysis is much more complicated.

And, if Ohm hadn't discovered his famous law - and lost his job and been blackballed for ten years - you wouldn't be reading this now.