Balanced modulator output is DSB-SC

A balanced modulator uses two identical AM modulators driven with opposite-polarity messages and the same carrier. Their outputs are subtracted, so the carrier cancels and only the sidebands remain, giving DSB-SC.

Block diagram

          +----------+  s1(t)
  m(t) -->|   AM     |-------+
          | modulator|       |  +
          +----------+       v
               ^           ( Σ )----> s(t) = DSB-SC
   Ac cos wct--+             ^  -
               v             |
          +----------+       |
 -m(t) -->|   AM     |-------+
          | modulator|  s2(t)
          +----------+

Proof

Both modulators are identical with amplitude sensitivity $k_a$:

$$\begin{aligned} s_1(t) &= A_c[1 + k_am(t)]\cos\omega_ct \\ s_2(t) &= A_c[1 - k_am(t)]\cos\omega_ct \end{aligned}$$

Subtracting:

$$\begin{aligned} s(t) &= s_1(t) - s_2(t) \\ &= A_c\cos\omega_ct + A_ck_am(t)\cos\omega_ct - A_c\cos\omega_ct + A_ck_am(t)\cos\omega_ct \\ &= 2k_aA_c\,m(t)\cos\omega_ct \end{aligned}$$

This is the product of the message and the carrier: a DSB-SC wave. The carrier term cancels exactly when the two modulators are perfectly matched (balanced). In practice each modulator can be a square-law or switching modulator; with square-law devices ($v_o = a_1v_i + a_2v_i^2$) the output is $4a_2A_cm(t)\cos\omega_ct$ after a band-pass filter.

In frequency domain: $S(f) = k_aA_c[M(f-f_c) + M(f+f_c)]$, i.e. only USB and LSB, no impulse at $f_c$.

DSB-SC wave for a sinusoidal message

For $m(t) = A_m\cos\omega_mt$: $s(t) = 2k_aA_cA_m\cos\omega_mt\cos\omega_ct$.

 m(t):  __          __
       /  \        /  \
 -----/----\------/----\---
            \____/

 s(t): /\/\/\      /\/\/\
      /\/\/\/\    /\/\/\/\
 ----+--------X--+--------X--
      \/\/\/\/    \/\/\/\/
       \/\/\/      \/\/\/
            ^ phase reversal at
              each zero of m(t)

The envelope is $|m(t)|$, and the carrier phase reverses by 180° at each zero crossing of the message.