# Explain the Generation of AM Waves using Square Law Modulator and Switching Modulator

## 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 .

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 V_{1}(t) is applied at the input of the non-linear device, such as diode, transistor etc.

Thus,

…………………………(1)

The input output relation for non-linear device is as under :

…………………………….(2)

where a and b are constants.

Now, substituting the expression (1) in (2), we get

Or,

Or,

The five terms in the expression for V_{2}(t) are as under :

Term 1: ax(t) : Modulating Signal

Term 2 : a E_{c }cos (2π f_{c}t ) : Carrier Signal

Term 3 : b x^{2 }(t) : Squared modulating Signal

Term 4 : 2 b x(t) cos ( 2π f_{c}t ) : AM wave with only sidebands

Term 5 : b E_{c}^{2 }cos^{2 }(2π f_{c}t ) : 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 f_{c }and its bandwidth is equal to 2f_{m }. This bandpass filter eliminates the unuseful terms from the equation of v_{2}(t) .

Fig 2

Hence the output voltage v_{o}(t) contains only the useful terms .

Or,

Therefore ,

………………………….(3)

Comparing this with the expression for standard AM wave i.e.

,

We find that the expression for V_{o}(t) of equation (3) represents an AM wave with m = (2b/a) .

Hence, the square law modulator produces an AM wave .

## Switching Modulator

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 v_{2}(t) = v_{1}(t) in the positive half cycle of c(t) and v_{2}(t) = 0 in the negative half cycle of c(t) .

Hence , v_{2}(t) = v_{1}(t) for c(t) > 0

v_{2}(t) = 0 for c(t) < 0

In other words , the load voltage v_{2}(t) varies periodically between the values v_{1}(t) and zero at the rate equal to carrier frequency f_{c }.

We can express v_{2}(t) mathematically as under :

…………………………(4)

where, g_{p}(t) is a periodic pulse train of duty cycle equal to one half cycle period i.e. T_{0 }/2 (where T_{0 }= 1/f_{c}) .

Fig.4

Let us express g_{p}(t) with the help of Fourier series as under :

…………………………….(5)

……………………………(6)

Substituting g_{p}(t) into equation (4), we get

Therefore,

………………………..(7)

The odd harmonics in this expression are unwanted, and therefore, are assumed to be eliminated .

Hence,

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/πE_{c}] . The unwanted terms can be eliminated using a band-pass filter (BPF) .