A carrier wave of peak voltage 14 V is used to transmit a message. What should be the peak voltage of the modulating signal in order to have a modulation index of 70%?
9.8 V
Understanding Amplitude Modulation Voltage Calculation
This question involves the concept of amplitude modulation (AM), specifically calculating the peak voltage of the modulating signal required to achieve a specific modulation index, given the peak voltage of the carrier wave.
What is Amplitude Modulation?
Amplitude modulation is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. In AM, the amplitude (strength) of the carrier wave is varied in proportion to the instantaneous amplitude of the message signal (also known as the modulating signal).
Key Components in AM
- Carrier Wave: A high-frequency wave that carries the information. It has a constant amplitude and frequency before modulation.
- Modulating Signal (Message Signal): The information signal (e.g., audio signal) that we want to transmit.
- Modulated Wave: The resulting wave after the carrier wave's amplitude has been varied according to the modulating signal.
Modulation Index
The modulation index (\(\mu\)), also known as the modulation depth, is a measure of the extent to which the amplitude of the carrier wave is varied by the modulating signal. It is a crucial parameter in AM because it indicates how efficiently the carrier is being used to transmit the message signal and affects the quality of the received signal.
For sinusoidal modulation, the modulation index is defined as the ratio of the peak voltage of the modulating signal (\(V_m\)) to the peak voltage of the carrier wave (\(V_c\)):
\(\mu = \frac{V_m}{V_c}\)
The modulation index is often expressed as a percentage.
Calculating the Modulating Signal Voltage
We are given the following information:
- Peak voltage of the carrier wave, \(V_c = 14\) V
- Desired modulation index, \(\mu = 70\%\)
First, convert the modulation index from a percentage to a decimal:
\(\mu = 70\% = \frac{70}{100} = 0.70\)
We use the formula for the modulation index and rearrange it to solve for the peak voltage of the modulating signal (\(V_m\)):
\(\mu = \frac{V_m}{V_c}\)
\(V_m = \mu \times V_c\)
Now, substitute the given values into the rearranged formula:
\(V_m = 0.70 \times 14 \text{ V}\)
Performing the multiplication:
\(V_m = 9.8 \text{ V}\)
Therefore, the peak voltage of the modulating signal should be 9.8 V to achieve a modulation index of 70% when the carrier wave peak voltage is 14 V.
Summary of Calculation
| Parameter | Symbol | Value |
|---|---|---|
| Carrier Peak Voltage | \(V_c\) | 14 V |
| Modulation Index (decimal) | \(\mu\) | 0.70 |
| Modulating Signal Peak Voltage | \(V_m\) | ? |
Formula: \(V_m = \mu \times V_c\)
Calculation: \(V_m = 0.70 \times 14 \text{ V} = 9.8 \text{ V}\)
Revision Table: Amplitude Modulation Formulas
| Concept | Formula | Notes |
|---|---|---|
| Modulation Index | \(\mu = \frac{V_m}{V_c}\) | For sinusoidal modulation |
| Modulating Signal Voltage | \(V_m = \mu \times V_c\) | Rearranging the index formula |
| Carrier Voltage | \(V_c = \frac{V_m}{\mu}\) | Rearranging the index formula |
| Maximum Modulated Voltage | \(V_{max} = V_c + V_m\) | Occurs when signals add constructively |
| Minimum Modulated Voltage | \(V_{min} = V_c - V_m\) | Occurs when signals subtract |
| Modulation Index (from peak/min voltages) | \(\mu = \frac{V_{max} - V_{min}}{V_{max} + V_{min}}\) | Alternative formula |
Additional Information: Significance of Modulation Index
The value of the modulation index is important for the proper functioning of an AM system.
- \(\mu < 1\) (Undermodulation): The carrier amplitude never drops to zero. The original message signal can be recovered easily at the receiver. This is the desired range for standard AM broadcasting.
- \(\mu = 1\) (100% Modulation): The carrier amplitude varies from zero to twice the unmodulated carrier amplitude. This provides the maximum signal power efficiency for standard AM, but it requires careful design to avoid distortion. \(V_{min} = 0\).
- \(\mu > 1\) (Overmodulation): The carrier amplitude drops to zero and stays at zero for a certain period. This causes distortion in the transmitted signal, as information is lost when the amplitude is zero. \(V_{min} < 0\), which is not physically possible for amplitude, indicating clipping. Overmodulation generates extra sidebands and makes signal recovery difficult and distorted.
In this problem, a modulation index of 70% (\(\mu = 0.7\)) falls within the desired undermodulation range.
Important Questions from Communication Systems
The wavelength of radiation emitted when He+ makes a transition from the state n = 3 to the state n = 2 will be:
(Take Rydberg constant R = 1.097 × 10⁷ m⁻¹)
Match List - I with List - II
List-I List-II (A) Range (I) Range of frequencies over which communication system works (B) Band width (II) The largest distance between transmitter and receiver (C) Attenuation (III) Loss of strength of a signal during propagation (D) Transducer (IV) A device that receives an input in electrical form or provides an output in electrical form Choose the correct answer from the options given below:
Match List - I with List - II
List-I List-II (A) Range (I) Range of frequencies over which communication system works (B) Band width (II) The largest distance between transmitter and receiver (C) Attenuation (III) Loss of strength of a signal during propagation (D) Transducer (IV) A device that receives an input in electrical form or provides an output in electrical form Choose the correct answer from the options given below:
A carrier wave of peak voltage 14 V is used to transmit a message. What should be the peak voltage of the modulating signal in order to have a modulation index of 70%?
Which of the following frequency would be suitable for beyond the horizon communication using sky waves?
