Generation Of AM waves

 In this section we will discuss the devices and methods used for the generation of standard AM wave. The that generates the AM wave is called as amplitude modulator and we  will discuss two modulator circuits;

1. Square law modulator
2. Switching modulator

Both of them use a non-linear element such as diode for their Implementation. A non-linear device is the device, with Non-linear relation between its current and voltage. both these modulator are low power modulator circuits.

Square law modulator

A square law modulator consist of elements given below;

• A non-linear device
• A band pass filter 
• A carrier source and a 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.

Thus,             v₁ (t)= x(t) + E_c cos(2πf_c t)

The input output relation for non linear device is 

                      v₂ (t) = av₁ (t) + bv₁²(t)

where a and b are constants, substituting the expression for v₂ (t) we get,

v₂ (t) = a[x(t) + E_c cos (2πf_ct)]+ b[x(t) + E_c cos (2πf_ct)]²

v₂ (t) = ax(t) + aE_c cos (2πf_ct)+ bx²(t) + 2bx(t)E_c cos (2πf_ct) + bE_c²cos² (2πf_ct)

ax(t) =modulating signal

aE_c cos (2πf_ct)=carrier signal

bx²(t)=squared modulated signal

2bx(t)E_c cos (2πf_ct) =AM waves with only sidebands

 bE_c²cos² (2πf_ct)= squared carrier

Out of these there are some useful term and also some unuseful term are present.

so the unuseful term is  ax(t) +  bx²(t) + bE_c²cos² (2πf_ct) and rest of are useful to us.

The LC tuned circuit acts as a bandpass filter. Its frequency response 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 v₂ (t).

Hence the output voltage v₀(t)  contains only the useful terms.

v₀(t) = aE_c cos (2πf_ct)+ 2bx(t)E_c cos (2πf_ct) 

Therefore,    v₀(t) = aE_c [1 + 2bx(t)/a] cos(2πf_ct)

Comparing with the expression for standard AM wave i.e. s(t) = E_c[1 + mx(t)] cos(2πf_ct), we found that the expression for v₀(t) an AM wave with m = (2b/a) , Hence the square law modulator produces AM wave.

 Switching Modulator

The switching modulator using a diode and resistor is given below;

The modulating signal x(t) and the sinusoidal carrier signal c(t) are connected in  series with each other. Therefore the input voltage is given by;

v₁(t) = c(t) + x(t) = E_c cos (2πf_ct) + x(t)

The amplitude of carrier is much larger than that of x(t) and c(t) decides the status of the diode.

Working operation and Analysis

We assume that the diode acts as a ideal switch. It act as a closed switch when it is forward biased in the positive half cycle of the carrier and offers zero impedance. whereas D 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),

Therefore, v₂(t) ={ v₁(t)       for  c(t)>0,

                              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 f_c. The approximate transfer characteristics of the diode load resistor combination has been shown in the figure.

v₂(t) =  v₁(t) . g(t)  = [x (t) + E_c cos (2πf_ct)] g(t)

Where g(t) is a periodic pulse train of duty cycle equal to one half cycle period i.e.,T₀ /2.

Let us express g(t) with the help of Fourier series under :

         g(t)  =  ½ + 2/π  ∑_n=1 (-1)^n-1/2n-1 Cos[2πf_c t(2n – 1)]

         g(t) = 1/2 + 2/π Cos(2πf_ct) + odd harmonic components

Therefore, v₂(t) =  [x (t) + E_c cos (2πf_ct)] [ 1/2 + 2/π Cos(2πf_ct) + odd harmonic components]

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

Hence v₂(t) = 1/2 + 1/2E_c cos (2πf_ct) + 2/π x (t)Cos(2πf_ct) + 2/π E_c cos² (2πf_ct)

In this expression, the first and fourth terms are unwanted terms are unwanted terms whereas the second and third terms together represent the AM wave.