---
title: "RLC circuit current"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:app-rlc-circuit-current"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/app-rlc-circuit-current
site: BCT Engineering
---

# RLC circuit current

Mathematics I · Chapter 4: Ordinary Differential Equations and their Applications

- Asked in: 2072 Chaitra
- Video: [RLC Circuits - Differential Equation Application](https://www.youtube.com/watch?v=8mEYrHq6RA8) — Mu Prime Math
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/app-rlc-circuit-current
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

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.

## 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.

## Answer

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 → ∞.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which equation governs the charge?
   A. L q'' + R q' + C q = E
   B. L q' + R q + q/C = E
   C. L q'' - R q' + q/C = E
   D. L q'' + R q' + q/C = E
   Answer: D. Voltage drops sum to E. (Key point: Governing equation)

2. For q'' + 4q' + 5q = 5 the auxiliary roots are:
   A. -2 ± i
   B. -2, -2
   C. -4 ± i
   D. 2 ± i
   Answer: A. m = (-4 ± sqrt(16 - 20))/2. (Key point: Roots)

3. From q(0) = 0 and q'(0) = 0, A and B are:
   A. A = 0, B = 0
   B. A = -1, B = -2
   C. A = 1, B = 2
   D. A = -1, B = 2
   Answer: B. 1 + A = 0 and -2A + B = 0. (Key point: Initial conditions)

4. The current is:
   A. 5 e^(-2t) cos t
   B. 5 e^(2t) sin t
   C. e^(-2t) sin t
   D. 5 e^(-2t) sin t
   Answer: D. i = dq/dt; underdamped since 6400 < 8000. (Key point: Current)

## Related in Mathematics I

- [RL circuit current as a function of time](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aapp-rl-circuit-current) — Mathematics I, chapter 4, asked 4×

## Related in other subjects

- [Parallel RLC with exponential current source, Laplace](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aparallel-rlc-exponential-current-laplace) — Electrical Circuit Theory, chapter 4
- [Initial symmetrical current, load short circuit](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Ainitial-current-load-short-circuit) — Power System Analysis II, chapter 3
- [State equations of current-source RLC circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-current-source-1h-1f-2h) — Control Systems, chapter 8
- [Surface runoff with initial loss and phi-index, 100-min storm](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Asurface-runoff-initial-loss-0-6) — Engineering Hydrology, chapter 3