Question

For a periodic signal $v(t) = 30\sin(100t) + 10\cos(300t) + 6\sin(500t + \pi/4)$, the fundamental frequency in rad/s is

• A. 100
• B. 300
• C. 500
• D. 1500

Hint:

For periodic signals, the fundamental frequency is the GCD of all individual angular frequencies.

Answer 1

The Correct Option is A

Step 1: Identify angular frequencies.
From the given signal:
\[ \omega_1 = 100,\quad \omega_2 = 300,\quad \omega_3 = 500 \text{ rad/s} \]
Step 2: Determine fundamental angular frequency.
The fundamental frequency is the greatest common divisor (GCD) of all angular frequencies.
Step 3: Compute GCD.
\[ \gcd(100, 300, 500) = 100 \]
Step 4: Final conclusion.
Hence, the fundamental angular frequency of the signal is 100 rad/s.

Answer 2

The signal is \( v(t) = 30\sin(100t) + 10\cos(300t) + 6\sin(500t + \pi/4) \), and each option is tested by checking whether the individual angular frequencies \( 100, 300, 500 \) rad/s could all complete whole numbers of cycles within a period built from that candidate fundamental.

1. Option 100 rad/s: Checking ratios, \( 100/100 = 1 \), \( 300/100 = 3 \), \( 500/100 = 5 \); all three are whole numbers, so a period of \( T = 2\pi/100 \) makes every term repeat exactly, with no smaller common period possible since \( 100 \) is the largest number dividing all three frequencies evenly.
2. Option 300 rad/s: Here \( 100/300 = 1/3 \), which is not an integer, so a period of \( 2\pi/300 \) would not bring the \( 100 \) rad/s term back to its starting phase; this cannot be the fundamental.
3. Option 500 rad/s: Both \( 100/500 = 1/5 \) and \( 300/500 = 3/5 \) are non-integers, so this value fails the same integer-ratio test.
4. Option 1500 rad/s: While \( 1500 \) is a common multiple of all three (in fact the LCM), the fundamental frequency is by definition the largest value that fits, not just any common multiple; \( 1500 \) rad/s corresponds to a much shorter period than necessary and is not the true fundamental.

Only the option equal to the greatest common divisor of the three angular frequencies makes every term complete whole cycles over the shortest possible repeating interval.

So the correct answer is 100 rad/s.