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

This is a direct-recall style question rather than a multi-step calculation, so here is why option (1) holds up against the others.
Reasoning: a sum of sinusoids is periodic only when every individual frequency is an integer multiple of one common base frequency, and that base (the fundamental) must be the greatest such value, i.e. the GCD of \( 100, 300, 500 \).
Checking: \( 100 = 1 \times 100 \), \( 300 = 3 \times 100 \), \( 500 = 5 \times 100 \), so \( 100 \) rad/s divides all three exactly and is the largest number that does so.
Values like \( 300 \), \( 500 \), or \( 1500 \) either fail to divide one of the terms evenly or are larger common multiples rather than the greatest common divisor. \[ \boxed{\omega_{\text{fundamental}} = 100 \text{ rad/s}} \]

Answer 2

An alternative way to see this is to find the period of each individual sinusoid separately and then find the smallest interval over which all three periods line up simultaneously.

1. Option 100 rad/s: The individual periods are \( T_1 = \dfrac{2\pi}{100} \), \( T_2 = \dfrac{2\pi}{300} = \dfrac{T_1}{3} \), \( T_3 = \dfrac{2\pi}{500} = \dfrac{T_1}{5} \). Since \( T_1 \) is an exact integer multiple of both \( T_2 \) and \( T_3 \), the whole signal repeats every \( T_1 = \dfrac{2\pi}{100} \), meaning its angular frequency is \( 100 \) rad/s.
2. Option 300 rad/s: This corresponds to a period of \( \dfrac{2\pi}{300} \), which is shorter than \( T_1 \); the \( 100 \) rad/s term does not complete a full cycle in this span, so the composite signal does not repeat this quickly.
3. Option 500 rad/s: Same issue in a more extreme form, its period is even shorter and neither the \( 100 \) rad/s nor the \( 300 \) rad/s term completes a whole cycle within it.
4. Option 1500 rad/s: This period is shorter still than the true repeat interval of the signal; it is related to the signal's structure only as a multiple, not as the actual repetition rate.

The true fundamental is the frequency belonging to the longest period that still lets every component finish whole cycles, and that is achieved only at one of the four choices.

So the correct answer is 100 rad/s.