---
title: "State model of parallel RLC with current source"
subject: "Electrical Circuit Theory"
chapter: 4
key: "semester_3/electrical_circuit_theory:state-model-parallel-rlc-current-source"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/state-model-parallel-rlc-current-source
site: BCT Engineering
---

# State model of parallel RLC with current source

Electrical Circuit Theory · Chapter 4: Transient analysis in RLC circuit by Laplace Transform

- Asked in: 2070 Asar (old course), 8 marks
- Video: [How to do State Space Representation of Electrical Systems | Control Systems](https://www.youtube.com/watch?v=h30pvc-aS6c) — KnowHow
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/state-model-parallel-rlc-current-source
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-4

## Question

Obtain the state model of the network shown in following figure.
[Figure: current source i(t) (arrow up) in parallel with a resistor R, an inductor L and a capacitor C]

## Short answer

State variables: capacitor voltage v and inductor current i_L. State model: d/dt [v; i_L] = [[–1/(RC), –1/C]; [1/L, 0]] [v; i_L] + [[1/C]; [0]] i(t).

## Answer

**Circuit.** Current source i(t), resistor R, inductor L and capacitor C all in parallel across one pair of nodes. Choose the capacitor voltage v (= v_C) and inductor current i_L as the state variables (one for each energy-storing element), with i(t) as the input.

#### Equations
- Inductor: L di_L/dt = v
- KCL at the top node: i(t) = v/R + i_L + C dv/dt

So

```text
dv/dt   = −(1/RC) v − (1/C) i_L + (1/C) i(t)
di_L/dt = (1/L) v
```

#### State model
```text
| v'  |   | −1/(RC)  −1/C | | v   |   | 1/C |
|     | = |               | |     | + |     | i(t)
| iL' |   |   1/L      0  | | iL  |   |  0  |
```
x = [v, i_L]ᵀ, ẋ = A x + B u with

A = [[−1/RC, −1/C], [1/L, 0]], B = [[1/C], [0]], u = i(t)

#### Output equations
- Voltage across the network: y₁ = v = [1 0] x
- Resistor current: i_R = v/R = [1/R 0] x
- Capacitor current: i_C = i(t) − v/R − i_L = [−1/R −1] x + [1] u

Check: the characteristic equation det(sI − A) = s² + s/(RC) + 1/(LC) = 0 is the standard parallel RLC equation, giving damping α = 1/(2RC) and ω₀ = 1/√(LC).

## Check yourself

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

1. For the parallel RLC with a current source, which state variables and input are used?
   A. Capacitor voltage v and inductor current i_L; input i(t)
   B. Inductor voltage and capacitor current; input v
   C. Resistor current and capacitor voltage; input v
   D. Only the capacitor voltage; input i(t)
   Answer: A. One state per energy-storing element: v for C and i_L for L, with the source i(t) as input. (Key point: Choice of states)

2. Which KCL at the top node gives the dv/dt equation?
   A. i(t) = v/R - i_L + C dv/dt
   B. i(t) = v/R + i_L + C dv/dt
   C. i(t) = R v + i_L + C dv/dt
   D. i(t) = v/R + L di_L/dt + C dv/dt
   Answer: B. The source current splits into the resistor, inductor and capacitor currents, so C dv/dt = i(t) - v/R - i_L. (Key point: KCL equation)

3. What is A for x = [v, i_L]?
   A. [[-1/RC, 1/C],[1/L, 0]]
   B. [[-1/RC, -1/C],[0, 1/L]]
   C. [[-1/RC, -1/C],[1/L, 0]]
   D. [[1/RC, -1/C],[1/L, 0]]
   Answer: C. dv/dt = -(1/RC)v - (1/C)i_L + (1/C)i and di_L/dt = (1/L)v, giving this A. (Key point: A matrix)

4. What does det(sI - A) = 0 give?
   A. s^2 + sR/L + 1/(LC) = 0, with alpha = R/(2L)
   B. s^2 + s/(RC) - 1/(LC) = 0
   C. s^2 + RC s + LC = 0
   D. s^2 + s/(RC) + 1/(LC) = 0, with alpha = 1/(2RC)
   Answer: D. This is the standard parallel RLC equation, with damping alpha = 1/(2RC) and omega0 = 1/sqrt(LC). (Key point: Characteristic equation)

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