Question

The plot of modulation index versus carrier amplitude yields a

• A. Horizontal line
• B. Circle
• C. Hyperbola
• D. Parabola

Hint:

Modulation index \(m = V_m / V_c\).
If \(V_m\) is held constant, then \(m\) is inversely proportional to \(V_c\).
The graph of \(y = k/x\) (where k is a constant) is a rectangular hyperbola.

Answer 1

The Correct Option is C

The modulation index (\(m\)) for Amplitude Modulation is defined as \(m = \frac{V_m}{V_c}\), where \(V_m\) is the amplitude of the modulating signal (message) and \(V_c\) is the amplitude of the carrier signal. The question asks for the plot of modulation index (\(m\)) versus carrier amplitude (\(V_c\)). Let \(m\) be on the y-axis and \(V_c\) be on the x-axis. So, \(y = m\) and \(x = V_c\). The relationship is \(y = \frac{V_m}{x}\). If we assume that the amplitude of the modulating signal \(V_m\) is constant, then this equation is of the form \(yx = V_m\) (constant), or \(y = \frac{\text{constant}}{x}\). This is the equation of a rectangular hyperbola. As \(V_c\) (carrier amplitude) increases, the modulation index \(m\) decreases for a fixed \(V_m\). As \(V_c\) decreases, \(m\) increases. \[ \boxed{\text{Hyperbola}} \]