Generation of AM Waves
The circuit that generates the AM waves is called as amplitude modulator and in this post we will discuss two such modulator circuits namely :
- Square Law Modulator
- Switching Modulator
Both of these circuits use a non-linear elements such as a diode for their implementation . Both these modulators are low power modulator circuits .
Square Law Modulator
Generation of AM Waves using the square law modulator could be understood in a better way by observing the square law modulator circuit shown in fig.1 .
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It consists of the following :
- A non-linear device
- A bandpass filter
- A carrier source and modulating signal
The modulating signal and carrier are connected in series with each other and their sum V1(t) is applied at the input of the non-linear device, such as diode, transistor etc.
The input output relation for non-linear device is as under :
where a and b are constants.
Now, substituting the expression (1) in (2), we get
The five terms in the expression for V2(t) are as under :
Term 1: ax(t) : Modulating Signal
Term 2 : a Ec cos (2π fct ) : Carrier Signal
Term 3 : b x2 (t) : Squared modulating Signal
Term 4 : 2 b x(t) cos ( 2π fct ) : AM wave with only sidebands
Term 5 : b Ec2 cos2 (2π fct ) : Squared Carrier
Out of these five terms, terms 2 and 4 are useful whereas the remaining terms are not useful .
Let us club terms 2, 4 and 1, 3, 5 as follows to get ,
The LC tuned circuit acts as a bandpass filter . Its frequency responce is shown in fig 2 which shows that the circuit is tuned to frequency fc and its bandwidth is equal to 2fm . This bandpass filter eliminates the unuseful terms from the equation of v2(t) .
Hence the output voltage vo(t) contains only the useful terms .
Comparing this with the expression for standard AM wave i.e.
We find that the expression for Vo(t) of equation (3) represents an AM wave with m = (2b/a) .
Hence, the square law modulator produces an AM wave .
Generation of AM Waves using the switching modulator could be understood in a better way by observing the switching modulator diagram. The switching modulator using a diode has been shown in fig 3(a) .
Fig 3 (a) Fig 3(b)
This diode is assumed to be operating as a switch .
The modulating signal x(t) and the sinusoidal carrier signal c(t) are connected in series with each other. Therefore, the input voltage to the diode is given by :
The amplitude of carrier is much larger than that of x(t) and c(t) decides the status of the diode (ON or OFF ) .
Working Operation and Analysis
Let us assume that the diode acts as an ideal switch . Hence, it acts as a closed switch when it is forward biased in the positive half cycle of the carrier and offers zero impedance . Whereas it acts as an open switch when it is reverse biased in the negative half cycle of the carrier and offers an infinite impedance .
Therefore, the output voltage v2(t) = v1(t) in the positive half cycle of c(t) and v2(t) = 0 in the negative half cycle of c(t) .
Hence , v2(t) = v1(t) for c(t) > 0
v2(t) = 0 for c(t) < 0
In other words , the load voltage v2(t) varies periodically between the values v1(t) and zero at the rate equal to carrier frequency fc .
We can express v2(t) mathematically as under :
where, gp(t) is a periodic pulse train of duty cycle equal to one half cycle period i.e. T0 /2 (where T0 = 1/fc) .
Let us express gp(t) with the help of Fourier series as under :
Substituting gp(t) into equation (4), we get
The odd harmonics in this expression are unwanted, and therefore, are assumed to be eliminated .
In this expression, the first and the fourth terms are unwanted terms whereas the second and third terms together represents the AM wave .
Clubing the second and third terms together , we obtain
This is the required expression for the AM wave with m=[4/πEc] . The unwanted terms can be eliminated using a band-pass filter (BPF) .