Analog Communication - AM Modulators

In this chapter, let us discuss about the modulators, which generate ampptude modulated wave. The following two modulators generate AM wave.

Square law modulator

Switching modulator

Square Law Modulator

Following is the block diagram of the square law modulator

Let the modulating and carrier signals be denoted as $mleft ( t ight )$ and $Acosleft ( 2pi f_ct ight )$ respectively. These two signals are appped as inputs to the summer (adder) block. This summer block produces an output, which is the addition of the modulating and the carrier signal. Mathematically, we can write it as

$$V_1t=mleft ( t ight )+A_ccosleft ( 2 pi f_ct ight )$$

This signal $V_1t$ is appped as an input to a nonpnear device pke diode. The characteristics of the diode are closely related to square law.

$V_2t=k_1V_1left ( t ight )+k_2V_1^2left ( t ight )$(Equation 1)

Where, $k_1$ and $k_2$ are constants.

Substitute $V_1left (t ight )$ in Equation 1

$$V_2left (t ight ) = k_1left [ mleft ( t ight ) + A_c cos left ( 2 pi f_ct ight ) ight ] + k_2left [ mleft ( t ight ) + A_c cosleft ( 2 pi f_ct ight ) ight ]^2$$

$Rightarrow V_2left (t ight ) = k_1 mleft ( t ight ) +k_1 A_c cos left ( 2 pi f_ct ight ) +k_2 m^2left ( t ight ) +$

$ k_2A_c^2 cos^2left ( 2 pi f_ct ight )+2k_2mleft ( t ight )A_c cosleft ( 2 pi f_ct ight )$

$Rightarrow V_2left (t ight ) = k_1 mleft ( t ight ) +k_2 m^2left ( t ight ) +k_2 A^2_c cos^2 left ( 2 pi f_ct ight ) +$

$k_1A_cleft [ 1+left ( frac{2k_2}{k_1} ight )mleft ( t ight ) ight ] cosleft ( 2 pi f_ct ight )$

The last term of the above equation represents the desired AM wave and the first three terms of the above equation are unwanted. So, with the help of band pass filter, we can pass only AM wave and epminate the first three terms.

Therefore, the output of square law modulator is

$$sleft ( t ight )=k_1A_cleft [1+left ( frac{2k_2}{k_1} ight ) mleft ( t ight ) ight ] cosleft ( 2 pi f_ct ight )$$

The standard equation of AM wave is

$$sleft ( t ight )=A_cleft [ 1+k_amleft ( t ight ) ight ] cos left (2 pi f_ct ight )$$

Where, $K_a$ is the ampptude sensitivity

By comparing the output of the square law modulator with the standard equation of AM wave, we will get the scapng factor as $k_1$ and the ampptude sensitivity $k_a$ as $frac{2k_2}{k1}$.

Switching Modulator

Following is the block diagram of switching modulator.

Switching modulator is similar to the square law modulator. The only difference is that in the square law modulator, the diode is operated in a non-pnear mode, whereas, in the switching modulator, the diode has to operate as an ideal switch.

Let the modulating and carrier signals be denoted as $mleft ( t ight )$ and $cleft ( t ight )= A_c cosleft ( 2pi f_ct ight )$ respectively. These two signals are appped as inputs to the summer (adder) block. Summer block produces an output, which is the addition of modulating and carrier signals. Mathematically, we can write it as

$$V_1left ( t ight )=mleft ( t ight )+cleft ( t ight )= mleft ( t ight )+A_c cosleft ( 2 pi f_ct ight )$$

This signal $V_1left ( t ight )$ is appped as an input of diode. Assume, the magnitude of the modulating signal is very small when compared to the ampptude of carrier signal $A_c$. So, the diode’s ON and OFF action is controlled by carrier signal $cleft ( t ight )$. This means, the diode will be forward biased when $cleft ( t ight )> 0$ and it will be reverse biased when $cleft ( t ight )< 0$.

Therefore, the output of the diode is

$$V_2 left ( t ight )=left{egin{matrix} V_1left ( t ight )& if &cleft ( t ight )>0 \ 0& if & cleft ( t ight )<0 end{matrix} ight.$$

We can approximate this as

$V_2left ( t ight ) = V_1left ( t ight )xleft ( t ight )$(Equation 2)

Where, $xleft ( t ight )$ is a periodic pulse train with time period $T=frac{1}{f_c}$

The Fourier series representation of this periodic pulse train is

$$xleft ( t ight )=frac{1}{2}+frac{2}{pi }sum_{n=1}^{infty}frac{left ( -1 ight )^n-1}{2n-1} cosleft (2 pi left ( 2n-1 ight ) f_ct ight )$$

$$Rightarrow xleft ( t ight )=frac{1}{2}+frac{2}{pi} cosleft ( 2 pi f_ct ight )-frac{2}{3pi } cosleft ( 6 pi f_ct ight ) +....$$

Substitute, $V_1left ( t ight )$ and $xleft ( t ight )$ values in Equation 2.

$V_2left ( t ight )=left [ mleft ( t ight )+A_c cosleft ( 2 pi f_ct ight ) ight ] left [ frac{1}{2} + frac{2}{pi} cos left ( 2 pi f_ct ight )-frac{2}{3pi} cosleft ( 6 pi f_ct ight )+..... ight ]$

$V_2left ( t ight )=frac{mleft ( t ight )}{2}+frac{A_c}{2} cosleft ( 2 pi f_ct ight )+frac{2mleft ( t ight )}{pi} cosleft ( 2 pi f_ct ight ) +frac{2A_c}{pi} cos^2left ( 2 pi f_ct ight )-$

$frac{2mleft ( t ight )}{3pi} cosleft ( 6 pi f_ct ight )-frac{2A_c}{3pi}cos left ( 2 pi f_ct ight ) cosleft ( 6 pi f_ct ight )+..... $

$V_2left ( t ight )=frac{A_c}{2}left ( 1+left ( frac{4}{pi A_c} ight )mleft ( t ight ) ight ) cosleft ( 2 pi f_ct ight ) + frac{mleft ( t ight )}{2}+frac{2A_c}{pi} cos^2left ( 2 pi f_ct ight )-$

$frac{2mleft ( t ight )}{3 pi} cosleft ( 6 pi f_ct ight )-frac{2A_c}{3pi} cosleft ( 2 pi f_ct ight ) cosleft ( 6 pi f_ct ight )+.....$

The 1^st term of the above equation represents the desired AM wave and the remaining terms are unwanted terms. Thus, with the help of band pass filter, we can pass only AM wave and epminate the remaining terms.

Therefore, the output of switching modulator is

$$sleft ( t ight )=frac{A_c}{2}left ( 1+left ( frac{4}{pi A_c} ight ) mleft ( t ight ) ight ) cosleft ( 2 pi f_ct ight )$$

We know the standard equation of AM wave is

$$sleft ( t ight )=A_cleft [ 1+k_amleft ( t ight ) ight ] cosleft ( 2 pi f_ct ight )$$

Where, $k_a$ is the ampptude sensitivity.

By comparing the output of the switching modulator with the standard equation of AM wave, we will get the scapng factor as 0.5 and ampptude sensitivity $k_a$ as $frac{4}{pi A_c}$ .