To solve this problem, we need to determine the maximum modulated frequency that can be detected by a diode detector circuit used for amplitude modulation (AM) detection. The diode detector, also known as an envelope detector, demodulates the AM wave to recover the original modulating signal. For proper detection, the circuit must be able to follow the variations in the envelope of the AM wave. The key parameter here is the time constant of the RC circuit formed by the load resistance (R) and the parallel capacitor (C).

The condition for good detection is that the time constant (τ = R × C) should be much less than the time period of the modulating signal, but greater than the time period of the carrier wave to avoid ripple. However, for the maximum frequency that can be detected without distortion, we use the formula derived from the requirement that the capacitor should discharge rapidly enough to follow the envelope. The maximum modulating frequency (f_max) is given by:

$f_{\max }=\frac{1}{2\pi \mu RC}$

where:

• μ is the modulation index (given as 60%, so μ = 0.6),
• R is the load resistance (100 kΩ = 100 × 10³ Ω),
• C is the capacitance (250 pF = 250 × 10^-12 F).

This formula ensures that the discharge rate of the capacitor is sufficient to track the fastest changes in the modulated envelope. Let's compute step by step:

Step 1: Convert units to SI units
R = 100 kΩ = 100 × 10³ Ω = 10⁵ Ω
C = 250 pF = 250 × 10^-12 F = 2.5 × 10^-10 F
μ = 60% = 0.6

Step 2: Plug into the formula
$f_{\max }=\frac{1}{2\pi \times 0.6\times {10}^5\times 2.5\times {10}^{-10}}$

Step 3: Calculate the denominator
First, compute R × C = 10⁵ × 2.5 × 10^-10 = 2.5 × 10^-5
Then, μ × R × C = 0.6 × 2.5 × 10^-5 = 1.5 × 10^-5
Then, 2π × μ × R × C = 2 × 3.1416 × 1.5 × 10^-5 ≈ 2 × 3.1416 × 0.000015 = 0.000094248
So, denominator ≈ 9.4248 × 10^-5

Step 4: Compute f_max
$f_{\max }=\frac{1}{9.4248\times {10}^{-5}}\approx 10608\text{Hz}$

Step 5: Convert to kHz
10608 Hz = 10.608 kHz ≈ 10.62 kHz (rounded to two decimal places as in the options).

Thus, the maximum modulated frequency that can be detected is 10.62 kHz.

Related Topics

Amplitude Modulation (AM): A technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. In AM, the amplitude of the carrier wave is varied in proportion to the waveform being transmitted.

Envelope Detector: A simple electronic circuit used to demodulate AM signals. It consists of a diode, resistor, and capacitor. The capacitor charges to the peak voltage and discharges through the resistor, following the envelope of the AM wave.

RC Time Constant: The product of resistance and capacitance (τ = R × C) which characterizes the rate of charge/discharge of the capacitor. It is crucial in determining the frequency response of the detector.

Formulae

Maximum detectable modulating frequency for a diode detector: $f_{\max }=\frac{1}{2\pi \mu RC}$

Where μ is modulation index, R is load resistance, C is parallel capacitance.