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RLC circuit current

Short answer

The differential equation for charge is 20q''+80q'+100q=100 with q(0)=0, i(0)=0. Solving yields q=1-e^(-2t)(cos t+2 sin t). The current is i(t)=q'=5e^(-2t) sin t amperes.

Find a current i(t) in the RLC circuit assuming zero initial current and charge q, if R = 80 ohms, L = 20 Henry, C = 0.01 Farads and E = 100 volts.

For a series RLC circuit with charge q on the capacitor and i = dq/dt, Kirchhoff's voltage law gives

L q'' + R q' + q/C = E

Here L = 20 H, R = 80 Ω, C = 0.01 F (so 1/C = 100), E = 100 V.

Equation for q

20q'' + 80q' + 100q = 100 → q'' + 4q' + 5q = 5

Initial conditions: q(0) = 0 and i(0) = q'(0) = 0.

Solve

Auxiliary equation m² + 4m + 5 = 0 gives m = −2 ± i.

q_c = e^(−2t)(A cos t + B sin t)

Particular solution: q_p = 1 (constant, since 5q = 5).

q = 1 + e^(−2t)(A cos t + B sin t)

  • q(0) = 0: 1 + A = 0, so A = −1.
  • q'(0) = 0: q' = e^(−2t)[(−2A + B) cos t + (−2B − A) sin t], so at t = 0: −2A + B = 0, B = −2.

q = 1 − e^(−2t)(cos t + 2 sin t)

Current

i = dq/dt = e^(−2t)[(2 cos t + 4 sin t) + (sin t − 2 cos t)] = 5e^(−2t) sin t

i(t) = 5 e^(−2t) sin t amperes

(Alternatively, differentiate the circuit equation: L i'' + R i' + i/C = 0 with i(0) = 0, i'(0) = E/L = 5, which gives the same result.) The current is a damped oscillation (underdamped, since R² < 4L/C: 6400 < 8000) that dies out as t → ∞.

✓ Answer checked by Claude Opus 5.5 (high effort)

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