One of the primary uses of Amplitude Modulation is in aiding the transmission of information.
Let us suppose we are in a room full of students, and the students in the back of the classroom cannot hear the teacher. He can increase his volume and speak louder so the students can hear him.
But, if we need to cover a greater distance, we can use electronics to help in this transmission of acoustic information. For example, we can speak into a microphone and that acoustic signal can be converted into an electronic signal. We can use electronics to amplify it and then apply that amplified signal to a transducer (i.e. a loudspeaker) to again produce an acoustic wave that contains information.
Because we have increased the amplitude of the signal at the source, this signal will be transmitted further, before being attenuated below the threshold of hearing.
If we want to cover miles of distance and transmit this information, a good way of doing this is to encode the audio information on top of an electromagnetic wave, we can amplitude modulate a high frequency carrier wave with our voice (the information we try to send) on top of it and broadcast that electromagnetic wave.
Amplitude modulation (AM) is known as the modification of the amplitude of a waveform by variation of a second waveform.
First, let us see how to form an amplitude modulated signal from two cosine waves:
Carrier: c(t) = A cos(ωct)
Message: m(t) = k cos(ωmt)
We refer to the first equation as the carrier signal. Cosine function that varies between +1 and -1 at a carrier frequency ωc. We multiply that by an amplitude A, so that the entire signal varies between +A and –A. For the second cosine function we refer to it as the message signal.
Combining these two functions as presented below, we obtain, as it is called, an amplitude modulated signal:
AM Signal: s(t) = [1 + k cos(ωmt)] A cos(ωct)
We can notice that in a cosine function like c(t) the amplitude A is a constant quantity and it does not vary with time. But in the AM signal we can consider the entire quantity [1 + kcos(ωmt)]A to be the amplitude of the cosine wave with frequency ωc. In this case, the amplitude is now time varying, because of the cosine function ωmt. So the amplitude of the cosine wave varies at frequency ωm, while the signal itself varies at frequency ωc.
Modulation index: k – typically varies between 0 and 1.
Let us suppose we consider k equal to one. This means that the quantity between square brackets will vary between 2 and 0, which means that the amplitude of the modulated signal will vary between 2A and 0 as time changes.
Now, let us look at the following picture.
On the Voltage vs time graph there are plotted both the message waveform and the carrier waveform.
In this particular case there are chosen: 10 KHz frequency for the carrier waveform and 100Hz for the message waveform.
Typically, when forming an AM signal waveform, the carrier frequency, ωc is much greater that the modulating (message) frequency ωm. So, in this case, there is a factor of 100 difference between the message waveform and the carrier waveform.
When the two waveforms presented above are combined together, we get the AM modulated signal:
We can observe that within the obtained waveform we still have the carrier frequency varying at 10 KHz, but the amplitude of that signal is varying now at 100Hz.
If we connect the peaks (maximum values of the modulated waveform) we have, what is called, the envelope of the AM modulated waveform.
Envelope detector circuit
Broadcasting the electromagnetic wave that carries the audio signal, we will receive the signal far away using a receiving antenna attached to an AM radio that will decode (demodulate) the carrier signal and recover the audio information.
In an AM radio signal, the high frequency carrier allows better propagation of the signal, allows smaller receiving antennas and the big difference between the modulating waveform and the carrier waveform, allows the modulated waveform to be easily demodulated.
The carrier carries the message and the message is contained in the envelope of the AM modulated signal. So, to extract the information that we are interested in, we can use an electronic circuit that is called an envelope detector, whose output is the envelope of the input AM waveform.
Let’s take a look at the circuit below.
This is not actually an envelope detector, but rather a portion of it, also known as half-wave rectifier.
If we apply the previously mentioned AM modulated signal to the input of the circuit we obtain the positive half of the waveform.
We notice that any portion that is below zero Volts is eliminated.
Now to extract the envelope from this waveform, we have to modify the circuit slightly.
We can see that the only change has been the addition of the capacitor in parallel with the output resistor.
Now let us take look on how the circuit works. As the AM waveform goes positive, the diode will become forward biased and the output capacitor will be connected directly across the input waveform and will charge up to the input AM value. Then, as the AM voltage decreases with time, eventually the voltage across the capacitor will be larger than the input, the diode becomes reverse biased and connection between the input and the output will be cut (the portion containing the diode will become an open circuit). The voltage on the capacitor can then discharge through the resistor at a rate determined by the time constant (τ) (for more information on this subject, see the StudentZone—April 2018 ADALM1000 SMU Training Topic 4: Transient Response of RC Circuit). This process is repeating, as the waveform increases again, the voltage becomes greater than the voltage on the capacitor, the diode will turn on again and the capacitor charges up.
The result of this behavior is presented below:
You can see what is happening. The orange waveform is tending to follow the peak amplitude (envelope) of the carrier signal.
Let us zoom out to see this more clearly.
In this zoomed out version of the graph we have the rectified AM signal (purple) and the capacitor voltage (orange). You can see that the capacitor voltage is following the audio signal (information) that we are interested in.
So, the output the output if the envelope detector circuit would look like this:
It is the 100 Hz message signal with some 10 KHz variations on it. Both frequencies are in the audio range so we could probably hear the small 10 KHz variation on top of the 100Hz message signal. But typically in AM radio, the small variations would be in the MHz range, far outside the audio band so we will not detect these variations, we will hear only the message signal which is in the audio band.
An important consideration in designing circuits like this is the time constant set by the capacitor and the resistor. If we make the time constant too fast, then instead of tracking the envelope of the signal, we will tend to track the carrier (the capacitor will charge and discharge too fast in between the variations of the carrier signal). In the other extreme, if we make the time constant too large, let’s say infinitely large (the resistor value tending to infinity), when the diode turns off the capacitor could never discharge. The capacitor will retain the maximum value of the AM waveform and the output will become a DC voltage having an AC input voltage (also called a peak detector circuit).