Square law approximation for DSB-AM

Square law approximation

Any non-linear device (diode, transistor) has an output–input characteristic that can be expanded as a power series:

$$i = a_0 + a_1v + a_2v^2 + a_3v^3 + \dots$$

When the input is small and the device is biased in its non-linear region, terms above the second order are negligible. Then

$$i \approx a_1v + a_2v^2$$

This is the square law approximation. The $v^2$ term multiplies the input components with each other, so it can produce product terms such as $m(t)\cos\omega_ct$ needed for modulation (and for detection).

Square law modulator for DSB-AM

 m(t) -->(+)--> v1 --> Non-linear --> v2 --> BPF --> s(t)
          ^           device (diode)     (fc,2W)  DSB-AM
          |           v2 = a1 v1 + a2 v1^2
 Ac cos wc t

Input:

$$v_1(t) = m(t) + A_c\cos\omega_ct$$

Output:

$$\begin{aligned} v_2(t) &= a_1[m(t) + A_c\cos\omega_ct] + a_2[m(t) + A_c\cos\omega_ct]^2 \\ &= a_1m(t) + a_1A_c\cos\omega_ct + a_2m^2(t) \\ &\quad + 2a_2A_cm(t)\cos\omega_ct + \frac{a_2A_c^2}{2}(1 + \cos2\omega_ct) \end{aligned}$$

A BPF centred at $f_c$ with bandwidth $2W$ passes:

$$s(t) = a_1A_c\left[1 + \frac{2a_2}{a_1}m(t)\right]\cos\omega_ct$$

which is DSB-AM with amplitude sensitivity $k_a = 2a_2/a_1$.

 |V2(f)|
  m, m^2       carrier+sidebands      2fc
 /\____         /\ | /\                |
 0   2W      fc-W  fc  fc+W           2fc
               [ BPF passes ]

Conditions: $f_c > 3W$ (so the $m^2$ band up to $2W$ does not overlap $f_c - W$); $|k_am(t)| \le 1$ to avoid over-modulation; small signals so higher-order terms are negligible, otherwise distortion appears.