FM Equation: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

FM Equation

Our plan is to come up with a mathematical representation of a frequency modulated oscillations of carrier waves that incorporate given sound waves.

Sound waves are changes in the air pressure and can be represented by some function of time m(t) (m stands for modulation, as we want this signal to modulate the frequency of sinusoidal carrier signal).

Practically, m(t) is the changes in the current of some circuit that contains a microphone, that converts the air pressure changes into synchronous changes in the electric current.

Let's discuss the concept of angular frequency of the carrier signal in more mathematical terms.
Frequency f, if constant, is the number of periods of oscillations per unit of time (second).
It's often expressed in radians per second ω=2πf and unmodulated oscillations of an electric current in the carrier LC circuit that produces base oscillations would be
I(t) = A·cos(ω·t)

This sinusoidal function of time I(t) that represents oscillations of electric current can be viewed as an X-coordinate of a point on a circle of radius A (amplitude of oscillations) with a center at the origin of coordinates, that rotates along a circle counterclockwise with constant angular speed ω radians per second, assuming that at t=0 it is located on the X-axis at point (A,0).

So, angular frequency in more mechanical terms is angular speed.

From now on we will consider the above presented rotation of a point as mathematical representation of sinusoidal oscillations.

Another important characteristic of an oscillation is its phase φ(t).
The definition of a phase is an angle a rotating point has rotated to during the time of rotation from its start up to a current moment t, which can be expressed as a time-dependent function φ(t).

From this definition immediately follows the analogy between kinematics terms distance, speed and concepts phase, angular frequency used in radio electronics.

This analogy is complete in a sense that the relationship between a phase φ(t) and angular frequency ω(t) is similar to that between a distance S(t) covered by a moving object and its instantaneous speed V(t).

The instantaneous speed at moment in time t in, generally, a non-uniform movement, as a function of time, is a derivative of a distance covered by a moving object from the start of movement up to a position at time t, as a function of time:
V(t) = dS/dt = S'(t)

Similarly, angular frequency (sometimes called instantaneous angular frequency or simply instantaneous frequency) is the first derivative of a phase (angle of rotation) as a function of time:
ω(t) = dφ/dt = φ'(t)

Knowing speed V(t) of a moving object at every moment of time from start to t, we can restore the distance S(t) covered by this object as a function of time
S(t) = ∫_[0,t]V(τ)·dτ

Similarly, we can restore the phase φ(t) (that is, an angle a point has rotated from the start of its rotation to a moment in time t), knowing the instantaneous angular frequency ω(t) at each moment of time.
φ(t) = ∫_[0,t]ω(τ)·dτ

Consider the main equation of oscillations of an electric current in the LC circuit of a carrier without any sound modulation
I(t) = A·cos(ω₀·t)

The argument to a function cos() is a product of a constant angular frequency (speed) ω₀ by time, which is an angular distance of rotation or, using terminology introduced above, a phase of the rotation at time t
φ(t) = ω₀·t

Therefore, our representation of carrier signal can be expressed in a more general form, suitable even for non-uniform rotation:
I(t) = A·cos(φ(t))

In a non-uniform rotation with variable instantaneous angular frequency ω(t) we can always derive this frequency from the phase:
ω(t) = φ'(t)

If we want to combine the carrier signal I(t) with some frequency modulating signal m(t) in such a way that the resulting variable instantaneous frequency ω_mod(t) of a modulated signal reflected the modulation, we need to satisfy the following equation:
ω_mod(t) = ω₀ + m(t)
where
ω₀ is the carrier own unmodulated constant frequency determined by it main LC circuit,
m(t) is a modulating add-on to reflect the sound waves to be transmitted.

We can even vary the degree by which the modulating affects the output signal by adding a modulating index λ as a factor to a modulator m(t):
ω_mod(t) = ω₀ + λ·m(t)

Knowing the target instantaneous frequency ω_mod(t) and the above expression of a phase in terms of this frequency
φ(t) = ∫_[0,t]ω(τ)·dτ
we can express the modulated phase as
φ_mod(t) = ∫_[0,t]ω_mod(τ)·dτ

This modulated phase will be an argument to a modulated signal of a carrier
I_mod(t) = A·cos[φ_mod(t)] =
= A·cos[∫_[0,t]ω_mod(τ)·dτ] =
= A·cos[∫_[0,t](ω₀+λ·m(τ))·dτ] =
= A·cos[ω₀·t+λ·∫_[0,t]m(τ)·dτ]

The role of modulation index λ in this formula is to define how significantly base carrier frequency should change with a change in sound waves frequency.

Below is a picture of a rather complicated air pressure oscillations (red) and the resulting modulated signal (purple) that represents the carrier frequency modulation by this sound together with equations that represent all the components of this frequency modulation

(you can click the right mouse button and open this picture in another tab for better view)

Frequency Modulation: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

Frequency Modulation

Unfortunately, amplitude modulation (AM), with its carrier frequency in the range from 540kHz to 1,600kHz, does not fit to transmit good quality (Hi-Fi) sound, especially on higher pitch notes, as was explained in the lecture "Amplitude Modulation" of this chapter.
The maximum audio frequency that is practical to transmit through AM radio is about 4.4kHz, while the sensitivity of the human ear goes up to 20kHz.
In addition, this type of modulation is too sensitive to radio noise.

The necessity to accommodate high quality transmission of sound waves caused new ideas and designs solutions.
A very useful invention was frequency modulation (FM).

First of all, frequency modulation is implemented within a carrier frequency range from 88mHz to 108mHz, which by itself helps to transmit a higher pitch audio signal of up to 15kHz.

Secondly, the basic principle of FM transmission is to represent sound waves, which are just changes in the air pressure, by a carrier signal's deviations from the base frequency assigned to it. Higher air pressure is represented by higher frequency of the carrier signal, lower air pressure is represented by a lower carrier frequency.

Amplitude of the carrier signal frequency remains always the same, only its frequency is fluctuating in synch with changes of the air pressure that carries a sound, as converted by a microphone into electric current.

All the deviations from the base frequency of the carrier signal must be within certain range assigned for each base frequency, so different radio stations, having different base frequencies of signal transmission, do not step over each other.

In the US the FM transmission uses the frequencies from 88mHz to 108mHz divided into 100 channels of 0.2mHz wide. That means, the frequency deviation from the base frequency of each FM transmitter should not exceed 0.1mHz up or down.

A picture below shows how the frequency of the carrier's signal (blue line) is changing in response to the most simple sound waves that represent a single note of a constant intensity (red line). Mathematically, the red line is a graph of an air pressure at some point in space as a function of time - a sinusoid for a single note of constant intensity.

(you can click the right mouse button and open this picture in another tab for better view)

What remains to discuss is a physical implementation of the frequency modulation.
This implementation involves changing the frequency of the transmitting oscillations of an electric current in the LC circuit connected to an antenna in synch with sound waves input from a microphone.

This can be accomplished by adding a variable capacitor into a circuit that is controlled by a current that represents the sound waves on the output of a microphone.
Changing the capacitance C in the LC circuit will effectively change the angular frequency ω=1/√L·C.

There are many different ways to put together such a device, we will leave this to special courses on radio electronics.

Finally, as an example of a more practical case of complicated sound waves that result in superposition of many different sources of sound, here is a picture of the sound waves (red) and modulated carrier signal (purple) with its frequency in synch with the sound waves.

(you can click the right mouse button and open this picture in another tab for better view)

As you see, higher intensity of sound waves (higher pressure of air) is represented by more frequent oscillations of the carrier signal and lower intensity of sound waves is represented by lower frequency of the carrier signal.

This picture intentionally presents the carrier signal of, generally, lower range of frequencies in order to visualize the individual oscillations.
In practice that range of oscillations of the carrier signal should be significantly higher to better represent each curve of sound waves.

AM Equation: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

AM Equation

Let's approach amplitude modulation (AM) from a mathematical standpoint.

High frequency oscillations of the electric current in the LC circuit with an inductor of inductance L and a capacitor of capacitance C can be described by a function
I(t) = A_c·cos(ω_c·t)
where
I(t) is the circuit's own oscillation of the current,
A_c is the amplitude of these oscillations,
ω_c = 1/√L·C is an angular frequency of these oscillations.

Simple air pressure oscillations resulting from a sound can be expressed by a similar function P(t) = A_s·cos(ω_s·t)
where
P(t) is the oscillation of the air pressure around a source of sound,
A_s is the amplitude of these air pressure oscillations,
ω_s is an angular frequency of these oscillations.

Note that our design requires ω_c to be substantially greater than ω_s.
Also note that real sound is a combination (superposition) of different overtones with different amplitudes and different angular frequencies and phases, like
P(t) = A_s1·cos(ω_s1·(t+φ_s1)) +
+ A_s2·cos(ω_s2·(t+φ_s2)) +
+ A_s3·cos(ω_s3·(t+φ_s3)) + ...

Amplitude modulation alters the amplitude of the carrier's oscillations by changing it in synch with sound oscillations.
The simplest way to achieve it mathematically is to incorporate the sound waves into an amplitude of carrier's waves:
I_m(t) = [A_c+P(t)]·cos(ω_c·t)

Here is an example of this type of modulation of a carrier's signal.

Assume, the carrier's high frequency oscillations have an amplitude A_c=4 and angular frequency ω_c=20, which can be described by an equation
I(t) = 4·cos(20t)

(you can click the right mouse button and open this picture in another tab for better view)

The sound makes air pressure oscillations that combine two different tones, one with an amplitude A_s1=2 and angular frequency ω_s1=1 and another with an amplitude A_s2=1 and angular frequency ω_s2=2, which can be described by the following equation
P(t) = 2·cos(t) + cos(2t)

Then the modulated signal can be described by an equation
I_m(t) =
= [4+2·cos(t)+cos(2t)]·cos(20t)
that graphically looks like this:


Just as a demonstration of the importance of having a high carrier frequency of a signal to properly represent a sound, here is what the transmitted signal would look like if the LC circuit of a carrier has a frequency comparable to a frequency of sound.

Assume, the carrier's oscillations have an amplitude A_c=4 and angular frequency ω_c=3, which can be described by an equation
I(t) = 4·cos(3t)
The sound waves are assumed to be as above
P(t) = 2·cos(t) + cos(2t)
Then the modulated signal can be described by an equation
I_m(t) =
= [4+2·cos(t)+cos(2t)]·cos(3t)
The following graph represents both the sound wave (purple) and modulated signal (blue):

As you see, the representation of sound waves by a modulated signal is far from exact. This lower frequency of amplitude modulation cannot be used for transmitting sound.

Amplitude Modulation: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

Amplitude Modulation

Now we have some basic understanding of how a radio signals (oscillations of electromagnetic field) are transmitted and received.
But, by themselves, these oscillations do not carry any useful information, which can be voice, image, data or any other type of information.
The oscillations of an electromagnetic field are only carriers of the information and their frequency represents the carrying frequency of the radio communication.

Transmitting and receiving information using the carrying radio waves requires certain additional design that is called modulation.
This lecture is about basic principles of amplitude modulation (AM) to transmit sound waves (oscillations of the air or other medium).

Electromagnetic oscillations are synchronous waves of electric and magnetic field forces perpendicular to each other (see "Field Waves" of this topic "Waves") that can be represented graphically as follows

(you can click the right mouse button and open this picture in another tab for better view)

Since these two kinds of forces act synchronously (in phase), we will represent the intensity of the electromagnetic field as a simple two-dimensional sinusoidal graph

At this point we'd like to state that the frequency of the carrier waves represented above should be relatively high as compared to frequency of the sound waves to accomplish successful sound transmission. The reason for this will be obvious after we explain what amplitude modulation is.

Our first task is to convert sound into oscillations of electric current. This is accomplished by a microphone or any other sound capturing device.
Notice that the oscillations of the electric current must represent the sound in its frequency and amplitude. Moreover, real sound is usually a combination of different frequencies of oscillation of air or other medium with different amplitudes, and the corresponding oscillations of electric current should reflect all this multitude of different parameters.

Considering we have accomplished producing an electric current oscillations in some circuit that reflects the sound. Here is a graphical representation of these oscillations:

The next task is to transmit these oscillations.
Experiments show that these oscillations are of too low frequency and not well propagated using any kind of an antenna. Its the high frequency oscillations that needed to do it with some sort of success.

Here comes the amplitude modulation. This method is using the high frequency of the carrier waves produced by an electronic RLC oscillator (red on a picture below), superimposing on its oscillations the changes in amplitude in synch with oscillations of the sound frequency (green on a picture below).
The result is the carrier oscillations modulated by amplitude by sound waves (blue on a picture below):


It should be obvious now why the carrier frequency of radio waves oscillations must be significantly higher than the sound waves frequency. If this difference is not significant, it would be impossible to distinguish the signal of the carrier from oscillations of sound waves.

Sound waves frequencies are those in the range from 20Hz to 20,000Hz (1Hz is 1 period of oscillations per second, which is, actually, sec⁻¹), while the range of carrier frequencies for amplitude modulation (AM range) is from 540,000Hz to 1,600,000Hz (that is from 540kHz to 1,600kHz). Such a difference in ranges assures that the sound waves reflected in the amplitude of carrier waves are distinguishable from the carrier waves themselves.

For example, WCBS radio station in New York broadcasts at the carrier's frequency
f=880kHz=0.88·10⁶sec⁻¹
At the same time, the sound frequency of High A note is 440Hz. That means that for each period of this sound wave the carrier makes 2,000 oscillations.

The higher the sound's pitch (which is, higher frequency of sound waves) - the less difference there is between its frequency and the frequency of the carrier radio waves. With the highest frequency that human ear can hear, which is about 20,000Hz, the WCBS radio waves with frequency of 880kHz are only 440 times more frequent. Generally, it's still sufficient to select the sound waves from the carrier's but some small details of sound waves might be lost. That's why AM radio cannot carry really high fidelity sound, but for regular speech it works fine.

Incidentally, considering the speed of propagation of radio waves is, approximately,
c=0.3·10⁹m/sec,
the WCBS radio waves have the length of
λ = c/f ≅ 341 m

These relatively long waves allow AM radio signals to go around large objects and even be reflected from the ionosphere. As a result, AM radio can be received at a significantly more distant from a transmitter location than waves of a shorter length.

What remains to discuss is a physical implementation of the amplitude modulation.
This implementation involves reducing the amplitude of the oscillations of an electric current in the LC circuit connected to an antenna.

This can be accomplished by adding a variable resistor into a circuit that is controlled by a current that represents the sound waves on the output of a microphone.
There are many different ways to put together such a device, we will leave this to special courses on radio electronics.

Radio Transmission: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

Radio Transmission

Radio transmission is a process of generating oscillations of electromagnetic field.
In this lecture we'll attempt to put together a simple generator, kind of those created in the beginning of the radio era.

In the previous lecture we discussed a simple LC circuit and connected to it a receiving antenna as a device that can capture the oscillations of electromagnetic field and induce the oscillations of electric current of a particular frequency in the circuit.

This device accepts all electromagnetic waves of all frequencies into an antenna and, because of the resonance of a certain frequency with a circuit's own frequency, selects a particular frequency out of many received ones.

In theory, the same device can work in reverse - if there is an oscillating with certain frequency electric current in the circuit, it will induce the oscillations of an electric current in an antenna, which, in turn, will create electromagnetic waves around this antenna, oscillating with the same frequency.
If these electromagnetic waves are of certain strength, the device will serve as a transmitter.

To make this design work, we have to assure an uninterrupted oscillations of the current in our circuit with its own inherent frequency.
Generally speaking, the oscillations in our circuit are damped because of natural resistance. Let's design a system with some sort of energy support to compensate the damping effect of resistance in a circuit.

Consider a swing at the children's playground. After initial push it will eventually stop because of a friction, unless we give another push. That new push should be synchronized with a movement of a swing and directed along its movement. So, periodic push in some direction at every moment when a swing moves in that direction is a solution.

Let's design an electronic circuit that does similar supporting pushes to electric current.
One of many kinds of circuits that delivers this solution is below.
It utilizes a triode as an ON/OFF switch to turn ON or OFF the current from a battery that feeds a circuit exactly at the moments it's needed to support loss of energy due to circuit resistance.
Assume, there are some oscillations in a circuit that contains inductor L₁ and capacitor C.
These oscillations cause the variable electric current in the inductor L₁, which, in turn, induces the electromotive force in the inductor L₂.

Considering every circuit has certain resistance, the oscillations of electric current in the L₁C circuit, if not connected to the inductor L₂, would be damped and their amplitude would gradually diminish towards zero.

Note, however, that the oscillations of the electric current in the L₁C circuit create a variable magnetic field with a variable magnetic flux going through the inductor L₁ and, since it's paired with inductor L₂, through L₂ as well.

Since this magnetic flux is variable, it induces an electromotive force in inductor L₂ proportional to a speed of change of the magnetic flux.
This variable electromotive force, in turn, causes a grid of a triode to be periodically charged positively or negatively.

When the grid is positive, electrons from the hot cathode go through a grid to the anode and feed the capacitor, thus compensating losses of energy due to resistance of a circuit.
If, however, the grid is negative, there is no current going from a cathode to an anode and capacitor is not receiving any charge from a battery.

To properly calculate the resulting oscillations of the current in the circuit we have to include this extra dosage of electricity supplied by a battery when the triode is open. This significantly complicates the calculations, and we will not go through this.
All we can state is that, with properly evaluated parameters of all components of this circuit, the feed from a battery will be exactly at the time the capacitor needs it, and the oscillations of the current in the circuit will be sustained as long as the battery has energy in it.

The above is just an idea. To practically implement it, we need to go through some calculations, which are beyond the scope of this course.

The above example is based on feeding a capacitor with additional charge. Alternatively, we can design a circuit, when a battery would feed an inductor with extra electromotive power.

Many other ideas and designs for transmitters were developed throughout the time, all have certain pluses and minuses, but the main principle of synchronously turn on the flow of new energy into an oscillator to compensate the losses due to resistance is still at the foundation of them all.

Antenna and LC Circuit: UNIZOR.COM - Physics4Teens - Waves - Radio

Notes to a video lecture on http://www.unizor.com

Antenna and LC Circuit

This lecture is about radio signals and receiving of these signals by a device we call "radio".
We will not talk in this lecture about how to produce and transmit radio signals, only about receiving them.

In this and subsequent lectures we will use the concept of forced oscillation presented in lectures "Forced Oscillation 1", "Forced Oscillation 2" and "Problems 4" of the "Mechanical Oscillations" topic of the "Waves" part of this course.
We strongly recommend to familiarize yourselves with the material presented there prior to study the material of this lecture.

The space around us is filled with radio signals, which are, in essence, oscillations of electromagnetic field.
We don't see them because our eyes are sensitive to certain range of frequencies of oscillations of electromagnetic field called visible spectrum.
Radio frequencies used to carry information to our radio, television, wireless phones and other devices are outside of this visible spectrum and, therefore, are invisible, but they do exist and we live in space filled with oscillations of electromagnetic field.

These oscillations are not synchronous because they are produced by many unrelated to each other sources, including our Sun, planet Earth, numerous sources of electromagnetic field around our own devices, including transmitters that intentionally produce oscillating electromagnetic fields and those produced unintentionally because any electric device is a source of electromagnetic field around it.

So, the space around us is filled with chaotic waves of electromagnetic field, and our purpose is to "catch" specific waves that have specific frequency of oscillations, used to transmit specific information.
The circuit that contains an inductor and a capacitor can help, and here is how.

Any oscillations of electromagnetic field will cause the oscillations of electric current in a conductor (like a piece of wire) in this field. This is the Faraday's Law of Induction.
While oscillations of electromagnetic field cannot be felt or observed by us directly, the induced electric current in a conductor is something tangible, we can use it in some devices to measure, observe, feel and use.

The problem is, chaotic oscillations of electromagnetic field with many different frequencies superposed on each other cause exactly similar chaotic oscillations of the electric current induced in the conductor.
Using a circuit with an inductor and a capacitor, we can select a particular oscillations of a particular frequency in this conductor, while suppressing the others.
Out of numerous oscillations with different frequencies superposed on each other we will select a particular frequency and use it to receive specific information.

The principle of selection of a particular frequency of oscillation out of many oscillations of different frequencies superposed on each other is based on a relationship between an oscillating external electromotive force and the own frequency of oscillation in a circuit that includes an inductor and a capacitor (LC circuit).
It is exactly analogous to forced mechanical oscillations and a principle of resonance presented in the lectures mentioned above.

Consider the following circuit that includes antenna, inductor and a variable capacitor (the one with moving plates, so its capacitance can be varied.)

Antenna is a conductor, where a chaotically oscillating electromagnetic field around it produces chaotically oscillating induced electric current.

The oscillations of an electric current in the antenna are transferred to forced oscillations of the current in the inductor of a circuit through some transformer device.

Using this setup we have introduced an external electromotive force into an LC circuit that has its own parameters (inductance of an inductor and capacitance of a capacitor), including its own natural frequency of oscillation
ω₀ = 1/√L·C
(see "Electronic Oscillator" topic of this course's "Waves" part.)

External electromotive force induced by an antenna in the inductor of a circuit is a combination of many oscillations of electric current with many different wave frequencies.
Each one will interact with the circuit's own frequency of oscillations ω₀.

At this point we would like to mention once more that the equations for an electric current in the LC circuit are exactly the same as those for mechanical oscillations, including forced oscillations with periodic external force.
So, the concept of resonance in an LC circuit is exactly the same as that described for mechanical oscillations in lecture "Forces Oscillation 2" of the "Mechanical Oscillations" topic of this part of the course.

It means that, out of many different frequencies of oscillations of the external electromotive force, the oscillations with a frequency equal to the circuit's own natural frequency will cause an increase in value of electric current oscillating with this frequency, while all other external waves with frequencies not equal to the circuit's own frequency will remain as low intensity noise of oscillations of electric current in the LC circuit.

Notice that the capacitor in the circuit presented above is a variable one, that is it's capable to change its capacitance, usually achieved by changing the position of its plates relative to each other, thus changing the common area of opposing plates and, consequently, changing the capacitance C.

Change in capacitance results in a change of a circuit's own frequency of oscillation. That, in turn, allows to select a different external frequency of oscillations out of all waves chaotically received by an antenna, thus performing a tuning to any specific frequency within a range of values specific for a particular technical characteristics of a circuit.

The explanation above is just a principle of selection of electromagnetic oscillations of a specific frequency using an LC circuit. The real devices in radio, TV, phones etc. are much more complex and involve significant enhancements to the process, but it's important to be familiar with this principle, as it lies in the foundation of most electronic devices we use.