AM modulation index; square law modulator

Modulation index of AM

The modulation index $\mu$ (or $m_a$) of an AM wave is the ratio of the message amplitude to the carrier amplitude; it shows how deeply the carrier amplitude is varied.

$$\mu = \frac{A_m}{A_c} = \frac{A_{max} - A_{min}}{A_{max} + A_{min}}$$

For no envelope distortion, $0 < \mu \le 1$. Multiplied by 100 it is the percentage modulation.

Square law modulator for DSB-AM

A non-linear device (diode or FET in its square-law region) is fed with the sum of message and carrier; its square term produces the product $m(t)\cos\omega_ct$, and a band-pass filter selects the AM wave.

 m(t) -->(+)--> v1 --> Non-linear --> v2 --> BPF --> s(t)
          ^            device              (fc,2W)
 Ac cos wc t

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

Device: $v_2(t) = a_1v_1(t) + a_2v_1^2(t)$

$$\begin{aligned} v_2(t) = {} & a_1m(t) + a_1A_c\cos\omega_ct + a_2m^2(t) \\ & + 2a_2A_cm(t)\cos\omega_ct + a_2A_c^2\cos^2\omega_ct \end{aligned}$$

The BPF centred at $f_c$ with bandwidth $2W$ keeps only the carrier and the product term:

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

This is DSB-AM with $k_a = 2a_2/a_1$.

Unwanted outputs removed: $a_1m(t)$ (0–$W$), $a_2m^2(t)$ (0–$2W$), and $a_2A_c^2\cos^2\omega_ct$ (DC and $2f_c$). For the filter to separate them, $f_c > 3W$.

The circuit is simple, but it is a low-level modulator and suffers distortion if the device is not purely square law.