Q.Derive an expression for modulation index in terms V_max and V_min.

The modulation index (or modulation factor, or depth of modulation) tells us how strongly the message controls the carrier amplitude. For AM it is the ratio of peak modulating voltage to peak carrier voltage:

and it is often quoted as a percentage, $M = (V_m/V_C)\times 100$. For distortion-free modulation $V_m$ must be less than $V_C$, so $m_a$ lies between 0 and 1, with a maximum permitted value of 1.0 (100% modulation). If $V_m > V_C$ then $m_a > 1$, giving over-modulation: the negative peaks are clipped, the envelope no longer matches the message, and distortion known as splattering appears.

When an AM wave is displayed on an oscilloscope, the index is read directly from the largest and smallest envelope amplitudes:

which rearrange to the standard working formula

The same relation holds with antenna currents, $m_a = (I_{max}-I_{min})/(I_{max}+I_{min})$.

When several sinusoidal signals of different frequencies modulate the carrier at once, their indices combine as the root of the sum of squares, $m_t = \sqrt{m_1^2 + m_2^2 + \dots + m_n^2}$, and this total should still not exceed unity. Because it fixes both the quality and the strength of the transmitted signal, the modulation index is one of the most important quantities in the whole study of AM.