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Circuit Analysis: AC, Linear Circuits and Transients

The Laplace transform of a voltage in a circuit is V(s) = 10(s + 3)/(s(s + 2)(s + 5)). Which of the following is the voltage v(∞) after a long time?

  1. A0 V
  2. B3 V
  3. C10 V
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Answer: B. 3 V

Method 1 (final-value theorem): v(∞) = lim(s→0) sV(s) = 10 × 3/(2 × 5) = 3 V. The theorem can be used when all the poles of sV(s) are in the left half-plane (the response converges). Method 2 (check with partial fractions): writing V(s) = A/s + B/(s + 2) + C/(s + 5) gives A = 3, B = −5/3 and C = −4/3, so v(t) = 3 − (5/3)e^(−2t) − (4/3)e^(−5t). As t → ∞ the exponential terms vanish, leaving 3 V, and at t = 0 the value is 3 − 5/3 − 4/3 = 0, which agrees with the initial-value theorem lim(s→∞) sV(s) = 0.

  • A. 0 V is the initial value v(0+) = lim(s→∞) sV(s).
  • B. Correct. lim(s→0) sV(s) = 3 V.
  • C. This wrongly takes the numerator coefficient 10 as the final value.
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