DSB-AM Modulation Index for Multitone Signals

[paul_harris77]

Homework Statement

A carrier wave has an rms voltage of 10V. It is modulated by a signal having compenents of frequencies f1 and f2 and the rms voltage of the modulated carrier rises to 11.5V. If the modulation index due to one of the components is 60%, calculate the modulation index for the other component.

The Attempt at a Solution

I have started by working out the peak voltage of the message signal:

V_message = 11.5$$\sqrt{2}$$ - 10$$\sqrt{2}$$ = 1.5$$\sqrt{2}$$ V

Surely then the total effective modulation index, m, is:

m= V_message(peak) / V_carrier peak = 1.5/10 = 0.15 (15%)

Looking on the internet, I found a formula for the effective modulation index as a function of the individual modulation indices:

m_eff = $$\sqrt{m1^2 + m2^2 ...}$$

So: 0.15 = $$\sqrt{0.6^2 + m2^2}$$

m2 = $$\sqrt{0.15^2 - 0.6^2}$$ = 0.34i (imaginary)

I have clearly done something wrong as I am sure the modulation index cannot be imaginary.

Does anyone have any ideas as to what I am doing wrong?

Any replies would be greatly appreciated.

Many thanks

Paul

 

[KataKoniK]

,

I can understand your confusion and frustration with this problem. I believe the issue lies in your calculation of the total effective modulation index. While your calculation is correct for a single modulating component, it does not take into account the presence of multiple components.

To find the total effective modulation index, we must first calculate the individual modulation indices for each component. We can do this using the formula:

m_i = (V_i(peak) - V_carrier(peak)) / V_carrier(peak)

Where V_i(peak) is the peak voltage of the individual component and V_carrier(peak) is the peak voltage of the carrier wave.

In this case, we know the modulation index for one of the components is 60%, so we can calculate the peak voltage for that component using the formula:

V_i(peak) = V_carrier(peak) * m_i + V_carrier(peak)

Substituting in the known values, we get:

V_i(peak) = 10 * 0.6 + 10 = 16V

Now, we can use this peak voltage to calculate the modulation index for the other component:

m2 = (V2(peak) - V_carrier(peak)) / V_carrier(peak) = (11.5 - 10) / 10 = 0.15 (15%)

So, the total effective modulation index is:

m_eff = \sqrt{m1^2 + m2^2} = \sqrt{0.6^2 + 0.15^2} = 0.62 (62%)

I hope this helps clear up any confusion and leads you to the correct solution. Keep up the good work in your scientific endeavors!