While changing the amplitude of a radio signal is the most obvious method to modulate it, it is by no means the only way. It is also possible to change the frequency of a signal to give frequency modulation or FM. Frequency modulation is widely used on frequencies above 30 MHz, and it is particularly well known for its use for VHF FM broadcasting.

Although it may not be quite as straightforward as amplitude modulation, nevertheless frequency modulation, FM, offers some distinct advantages. It is able to provide near interference free reception, and it was for this reason that it was adopted for the VHF sound broadcasts. These transmissions could offer high fidelity audio, and for this reason, frequency modulation is far more popular than the older transmissions on the long, medium and short wave bands.

In addition to its widespread use for high quality audio broadcasts, FM is also sued for a variety of two way radio communication systems. Whether for fixed or mobile radio communication systems, or for use in portable applications, FM is widely used at VHF and above.

What is frequency modulation, FM?

To generate a frequency modulated signal, the frequency of the radio carrier is changed in line with the amplitude of the incoming audio signal.

Frequency Modulation, FM

When the audio signal is modulated onto the radio frequency carrier, the new radio frequency signal moves up and down in frequency. The amount by which the signal moves up and down is important. It is known as the deviation and is normally quoted as the number of kilohertz deviation. As an example the signal may have a deviation of ±3 kHz. In this case the carrier is made to move up and down by 3 kHz.

Broadcast stations in the VHF portion of the frequency spectrum between 88.5 and 108 MHz use large values of deviation, typically ±75 kHz. This is known as wide-band FM (WBFM). These signals are capable of supporting high quality transmissions, but occupy a large amount of bandwidth. Usually 200 kHz is allowed for each wide-band FM transmission. For communications purposes less bandwidth is used. Narrow band FM (NBFM) often uses deviation figures of around ±3 kHz. It is narrow band FM that is typically used for two-way radio communication applications. Having a narrower band it is not able to provide the high quality of the wideband transmissions, but this is not needed for applications such as mobile radio communication.

Advantages of frequency modulation, FM

FM is used for a number of reasons and there are several advantages of frequency modulation. In view of this it is widely used in a number of areas to which it is ideally suited. Some of the advantages of frequency modulation are noted below:

• Resilience to noise: One particular advantage of frequency modulation is its resilience to signal level variations. The modulation is carried only as variations in frequency. This means that any signal level variations will not affect the audio output, provided that the signal does not fall to a level where the receiver cannot cope. As a result this makes FM ideal for mobile radio communication applications including more general two-way radio communication or portable applications where signal levels are likely to vary considerably. The other advantage of FM is its resilience to noise and interference. It is for this reason that FM is used for high quality broadcast transmissions.



• Easy to apply modulation at a low power stage of the transmitter: Another advantage of frequency modulation is associated with the transmitters. It is possible to apply the modulation to a low power stage of the transmitter, and it is not necessary to use a linear form of amplification to increase the power level of the signal to its final value.



• It is possible to use efficient RF amplifiers with frequency modulated signals: It is possible to use non-linear RF amplifiers to amplify FM signals in a transmitter and these are more efficient than the linear ones required for signals with any amplitude variations (e.g. AM and SSB). This means that for a given power output, less battery power is required and this makes the use of FM more viable for portable two-way radio applications.

Summary

Frequency modulation is widely used in many areas of radio technology including broadcasting and areas of two way radio communication. In these applications its particular advantages can be used to good effect. For the future, other forms of digital modulation are becoming more widely used - DAB for radio broadcasting and a number of other formats such as TETRA for two-way radio communication systems. Despite these changes, FM will remain in use for many years to come as there are many advantages of frequency modulation for the areas in which it has gained a significant foothold in recent years.

(Original article on http://www.electronics-radio.com/articles/radio/modulation/frequency_modulation/fm.php)

Amplitude modulation or AM as it is often called, is a form of modulation used for radio transmissions for broadcasting and two way radio communication applications. Although one of the earliest used forms of modulation it is still in widespread use today.

The first amplitude modulated signal was transmitted in 1901 by a Canadian engineer named Reginald Fessenden. He took a continuous spark transmission and placed a carbon microphone in the antenna lead. The sound waves impacting on the microphone varied its resistance and in turn this varied the intensity of the transmission. Although very crude, signals were audible over a distance of a few hundred metres, although there was a rasping sound caused by the spark.

With the introduction of continuous sine wave signals, transmissions improved significantly, and AM soon became the standard for voice transmissions. Nowadays, amplitude modulation, AM is used for audio broadcasting on the long medium and short wave bands, and for two way radio communication at VHF for aircraft. However as there now are more efficient and convenient methods of modulating a signal, its use is declining, although it will still be very many years before it is no longer used.

What is amplitude modulation?

In order that a radio signal can carry audio or other information for broadcasting or for two way radio communication, it must be modulated or changed in some way. Although there are a number of ways in which a radio signal may be modulated, one of the easiest, and one of the first methods to be used was to change its amplitude in line with variations of the sound.

The basic concept surrounding what is amplitude modulation, AM, is quite straightforward. The amplitude of the signal is changed in line with the instantaneous intensity of the sound. In this way the radio frequency signal has a representation of the sound wave superimposed in it. In view of the way the basic signal "carries" the sound or modulation, the radio frequency signal is often termed the "carrier".

Amplitude Modulation, AM

When a carrier is modulated in any way, further signals are created that carry the actual modulation information. It is found that when a carrier is amplitude modulated, further signals are generated above and below the main carrier. To see how this happens, take the example of a carrier on a frequency of 1 MHz which is modulated by a steady tone of 1 kHz.

The process of modulating a carrier is exactly the same as mixing two signals together, and as a result both sum and difference frequencies are produced. Therefore when a tone of 1 kHz is mixed with a carrier of 1 MHz, a "sum" frequency is produced at 1 MHz + 1 kHz, and a difference frequency is produced at 1 MHz - 1 kHz, i.e. 1 kHz above and below the carrier.

If the steady state tones are replaced with audio like that encountered with speech of music, these comprise many different frequencies and an audio spectrum with frequencies over a band of frequencies is seen. When modulated onto the carrier, these spectra are seen above and below the carrier.

It can be seen that if the top frequency that is modulated onto the carrier is 6 kHz, then the top spectra will extend to 6 kHz above and below the signal. In other words the bandwidth occupied by the AM signal is twice the maximum frequency of the signal that is used to modulated the carrier, i.e. it is twice the bandwidth of the audio signal to be carried.

Amplitude demodulation

Amplitude modulation, AM, is one of the most straightforward ways of modulating a radio signal or carrier. The process of demodulation, where the audio signal is removed from the radio carrier in the receiver is also quite simple as well. The easiest method of achieving amplitude demodulation is to use a simple diode detector. This consists of just a handful of components:- a diode, resistor and a capacitor.

AM Diode Detector

In this circuit, the diode rectifies the signal, allowing only half of the alternating waveform through. The capacitor is used to store the charge and provide a smoothed output from the detector, and also to remove any unwanted radio frequency components. The resistor is used to enable the capacitor to discharge. If it were not there and no other load was present, then the charge on the capacitor would not leak away, and the circuit would reach a peak and remain there.

Advantages of Amplitude Modulation, AM

There are several advantages of amplitude modulation, and some of these reasons have meant that it is still in widespread use today:

• It is simple to implement



• it can be demodulated using a circuit consisting of very few components



• AM receivers are very cheap as no specialised components are needed.

Disadvantages of amplitude modulation

Amplitude modulation is a very basic form of modulation, and although its simplicity is one of its major advantages, other more sophisticated systems provide a number of advantages. Accordingly it is worth looking at some of the disadvantages of amplitude modulation.

• It is not efficient in terms of its power usage



• It is not efficient in terms of its use of bandwidth, requiring a bandwidth equal to twice that of the highest audio frequency



• It is prone to high levels of noise because most noise is amplitude based and obviously AM detectors are sensitive to it.

Summary

AM has advantages of simplicity, but it is not the most efficient mode to use, both in terms of the amount of space or spectrum it takes up, and the way in which it uses the power that is transmitted. This is the reason why it is not widely used these days both for broadcasting and for two way radio communication. Even the long, medium and short wave broadcasts will ultimately change because of the fact that amplitude modulation, AM, is subject to much higher levels of noise than are other modes. For the moment, its simplicity, and its wide usage, mean that it will be difficult to change quickly, and it will be in use for many years to come.

(Original article on http://www.electronics-radio.com/articles/radio/modulation/amplitude_modulation/am.php)

What is a quadratic equation?

The a,b, and c here represent real number coefficients. So this means we are talking about an equation that is a constant times the variable squared plus a constant times the variable plus a constant equals zero, where the coefficient a on the variable squared can't be zero, because if it were then it would be a linear equation.

Examples

All these equations are equivalent to equations of the above form. The first one is already in that form. The second one can be put into it by subtracting 2x+3 from both sides. The third one can be put into it by multiplying out and then subtracting 5 from both sides.

Standard Form

The form

is the standard form for a quadratic equation, and for future reference, here the letter a will always mean the coefficient on the square of the variable, and b will be the coefficient on the variable, and c will be the constant term. To get a quadratic into standard form you must remove all parentheses and combine all like terms and add or subtract something from both sides so that the right side will be zero. Once you have your equation in standard form you can identify a,b, and c.

Example

Problem: Write the equation in standard form and identify a,b, and c.

Solving

Quadratic equations are harder to solve than linear equations, because once you have them in standard form it is hard to simplify them any further, and in this form there are still two occurrences of the variable, so it's hard to see what we can do to get the variable alone.

So we have to find some clever tricks to get around this problem.

Solving by Factoring

If A and B are real numbers and AB=0, then either A=0 or B=0.

This is a very special property that only zero has. For other numbers there are lots of ways to multiply and get them, but not for zero. For zero, the only way to multiply numbers and get it, is if one of the numbers is zero.

The principle of zero products allows us to reduce a complicated equation to simpler equations provided the right side of the equation is zero, and the equation is factored, because we can set each of the factors equal to zero.

• To solve a quadratic by factoring, first you must make sure it is in standard form. It is especially important that it is set equal to zero, because remember, the principle of zero products only works for zero.

• Then you must factor the left side.

• Then you set each of your factors equal to zero and solve the equations you get to find the solutions to your equation.

Example

What if you can't factor?

Consider the following equation.

This looks like a nice simple friendly equation, but we can't solve it by factoring, because we can't find two numbers to multiply and get -1 and add and get -2, so we are going to have to find another method.

But if only that minus sign on the 1 weren't there, then we could factor it really easily, in fact it would be a perfect square. How can we make that minus sign go away?

Well, one thing we could do is add 2 to both sides of the equation, and then the equation would become

Now, I know what you're saying. You are saying, "But you said that you have to get it set equal to 0 to solve it by factoring, because the principle of zero products only works for 0. What good does it do to have something factored and set equal to 2?"

And if you are saying this to yourself, you are absolutely right. But this isn't just any old factorization. It is aperfect square, and maybe you can do something with a perfect square set equal to 2.

If we could figure out how to solve equations like

that is, perfect squares set equal to numbers, then we could solve an equation like

And if we could find a way to add a number to both sides of other quadratics so that we can put them into the form perfect square equals constant, then maybe we could be able to solve them too.

This means that to help us solve quadratic equations, we need to learn two skills.

To work our way up to the task of solving equations of the form

let's first start with the slightly easier task of solving equation of the form

How do we solve an equation of the form

If x is greater than 0 then the obvious answer is

but this is not quite right because it only gives you the positive square root of d, and all positive numbers have two square roots, a positive one and a negative one. So to be sure that you are getting all solutions to an equation of this form, your answer must be

Now let's look at the more general equation of the form

This is really not much harder since anything you can do with x you should be able to do with x+k. x+k represents a number too. So solve for x+k and then add something to both sides of the equation to get x alone.

Example 1

Example 2

Completing the Square

This is what is meant by completing the square, and the secret to it is to expand out the expression

and see what makes perfect squares tick. Applying our formula for squaring a binomial, we get

The key here is to look at the relationship between the coefficient on x and the constant coefficient. The coefficient on x is 2k and the constant term is k². This means that if we know the coefficient on x, and we want to know what the constant term has to be for the expression to be a perfect square, then we need to divide the coefficient on x by 2 to get k, and then square to get k².

So if you have an expression of the form

and you want to find something to add to it to make it a perfect square, then you need to

Example

I hope the above has helped you understand the process of completing the square. If not, there is another approach to it that I have written an article about that you might find interesting for further understanding. It is a geometrical approach based on the method that many earlier mathematician used.

Solving by Completing the Square

One thing we left out. So far all of the equations we have solved have had a coefficient of 1 on x². What do we do if we have a coefficient other than 1 on there?

Well, we don't really have any method of completing the squares to deal with that situation, so the easiest thing to do is just divide both sides by it and put up with the fractions. With completing the squares, fractions are not so bad to deal with because there is no guess work.

Example

The Quadratic Formula

You see, with such a mechanical method like the method of completing the squares, why not just apply it to the general quadratic equation and solve all quadratics in the world at once, and be done with it, and never have to use algebra to solve a quadratic again.

Problem: Solve the equation:

Just do it the same way with the letters as you did with the numbers. First divide both sides by a. Then subtract c/a from both sides to be able to see more easily what to add to the left side to make it a perfect square. Then complete the square on the scratch paper and add what you get to both sides. Write the left side as the perfect square that it is. Instead of doing arithmetic on the right side you have to do a little bit of algebra, using 4a² as a common denominator. Then take square roots and subtract b/2a from both sides and use 2a a common denominator to get the final answer.

We have just solved all quadratics in the world at once and derived the quadratic formula, which says:

For any real numbers a,b, and c, the solutions to the equation

are

Solving by the Quadratic Formula

Since this formula is somewhat long and complicated, it is best to evaluate it in two smaller pieces by first evaluating the thing inside the radical,

and then put the result into the formula

The quantity

even has a name. It is called the discriminant. And there is another advantage to computing it first. Since it is what is in the radical, it can't be negative if there are going to be solutions to the equation, because you can't take a square root of a negative number, (unless you use the imaginary numbers, and we're not yet ready for them here) so if the discriminant comes out negative, then you don't have to do any more work, and all you have to do is write "no solution" on your paper and you are done. Sometimes you can determine this quite quickly by estimating, particularly if a and c are very large and b is small.

Example 1

Example 2

Imaginary Solutions

Once you know how to find square roots of negative numbers, you find imaginary solutions to quadratics by the completing the square or the quadratic formula pretty much like you find real ones. For the following examples the instruction is to solve the equation.

Example 1:

Solution:

Completing the Square:

Explanation:

Quadratic Formula:

Explanation:

Example 2:

Completing the Square:

Explanation:

Quadratic Formula:

Explanation:

Example 3:

Completing the Square:

Explanation:

Quadratic Formula:

Explanation:

Common Factors

Function

A transistor may be used as a switch (either fully on with maximum current, or fully off with no current) and as an amplifier(always partly on).

The amount of current amplification is called the current gain, symbol h_FE.
For further information please see the Transistor Circuits page.

Types of transistor

Transistor circuit symbols

The leads are labelled base (B), collector (C) and emitter (E).
These terms refer to the internal operation of a transistor but they are not much help in understanding how a transistor is used, so just treat them as labels!

A Darlington pair is two transistors connected together to give a very high current gain.

In addition to standard (bipolar junction) transistors, there are field-effect transistors which are usually referred to as FETs. They have different circuit symbols and properties and they are not (yet) covered by this page.

Transistor leads for some common case styles.

Connecting

If you are lucky the orientation of the transistor will be clear from the PCB or stripboard layout diagram, otherwise you will need to refer to a supplier's catalogue to identify the leads.

The drawings on the right show the leads for some of the most common case styles.

Please note that transistor lead diagrams show the view from below with the leads towards you. This is the opposite of IC (chip) pin diagrams which show the view from above.

Please see below for a table showing the case styles of some common transistors.

Crocodile clip Photograph © Rapid Electronics.

Soldering

Do not confuse this temporary heat sink with the permanent heat sink (described below) which may be required for a power transistor to prevent it overheating during operation.

Heat sink Photograph © Rapid Electronics

Heat sinks

For further information please see the Heat sinks page.

Testing a transistor

Testing an NPN transistor

1. Testing with a multimeter

Test each pair of leads both ways (six tests in total):

• The base-emitter (BE) junction should behave like a diode and conduct one way only.
• The base-collector (BC) junction should behave like a diode and conduct one way only.
• The collector-emitter (CE) should not conduct either way.

A simple switching circuit to test an NPN transistor

2. Testing in a simple switching circuit

If the transistor is OK the LED should light when the switch is pressed and not light when the switch is released.

To test a PNP transistor use the same circuit but reverse the LED and the supply voltage.

Some multimeters have a 'transistor test' function which provides a known base current and measures the collector current so as to display the transistor's DC current gain h_FE.

Transistor codes

• Codes beginning with B (or A), for example BC108, BC478
  The first letter B is for silicon, A is for germanium (rarely used now). The second letter indicates the type; for example C means low power audio frequency; D means high power audio frequency; F means low power high frequency. The rest of the code identifies the particular transistor. There is no obvious logic to the numbering system. Sometimes a letter is added to the end (eg BC108C) to identify a special version of the main type, for example a higher current gain or a different case style. If a project specifies a higher gain version (BC108C) it must be used, but if the general code is given (BC108) any transistor with that code is suitable.
• Codes beginning with TIP, for example TIP31A
  TIP refers to the manufacturer: Texas Instruments Power transistor. The letter at the end identifies versions with different voltage ratings.
• Codes beginning with 2N, for example 2N3053
  The initial '2N' identifies the part as a transistor and the rest of the code identifies the particular transistor. There is no obvious logic to the numbering system.

Choosing a transistor

To make a final choice you will need to consult the tables of technical data which are normally provided in catalogues. They contain a great deal of useful information but they can be difficult to understand if you are not familiar with the abbreviations used. The table below shows the most important technical data for some popular transistors, tables in catalogues and reference books will usually show additional information but this is unlikely to be useful unless you are experienced. The quantities shown in the table are explained below.

Darlington pair

You can make up your own Darlington pair from two transistors.
For example:

• For TR1 use BC548B with h_FE1 = 220.
• For TR2 use BC639 with h_FE2 = 40.