---
title: "State equations of RLC network, 2 F capacitor output"
subject: "Control Systems"
chapter: 8
key: "semester_3/control_systems:ss-rlc-network-2f"
source: https://www.bctengineer.com/q/semester_3/control_systems/ss-rlc-network-2f
site: BCT Engineering
---

# State equations of RLC network, 2 F capacitor output

Control Systems · Chapter 8: State Space Analysis

- Asked in: 2078 Kartik, 5 marks; 2068 Baisakh (old course), 8 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/ss-rlc-network-2f
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8

## Question

Write the state space equation for the electrical network shown below. Take voltage across 2F capacitor as output. [Figure: source Vs – series 1 Ω – node A; 1 F capacitor from node A to ground; node A – series 1 H inductor – node B; 1 Ω resistor and 2 F capacitor each from node B to ground; Vo across the 2 F capacitor]

## Short answer

Define states x1 = voltage across 1F capacitor, x2 = inductor current, x3 = voltage across 2F capacitor (output). Then x_dot = [-1 -1 0; 1 0 -1; 0 0.5 -0.5] x + [1;0;0] V_s, y = [0 0 1] x.

## Answer

Choose the energy-storing elements' variables as states: capacitor voltages and inductor current.

- $x_1 = v_1$: voltage across the 1 F capacitor (node A)
- $x_2 = i_L$: current in the 1 H inductor (from A to B)
- $x_3 = v_2$: voltage across the 2 F capacitor (node B) $= V_o$

```text
  Vs  1 ohm   A   1 H      B
 (+)-/\/\/--+--UUUU--+------+---o Vo
  |         |   iL   |      |
  |        === 1F   1 ohm  === 2F
  |         |        |      |
 (-)--------+--------+------+---o
```

#### KCL at node A
Current from the source through 1 Ω $= (V_s - v_1)/1$:

$$
1\cdot\frac{dv_1}{dt} = (V_s - v_1) - i_L \;\Rightarrow\; \dot{x}_1 = -x_1 - x_2 + V_s
$$

#### KVL around the inductor

$$
1\cdot\frac{di_L}{dt} = v_1 - v_2 \;\Rightarrow\; \dot{x}_2 = x_1 - x_3
$$

#### KCL at node B

$$
2\frac{dv_2}{dt} = i_L - \frac{v_2}{1} \;\Rightarrow\; \dot{x}_3 = 0.5x_2 - 0.5x_3
$$

#### State and output equations

$$
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} -1 & -1 & 0 \\ 1 & 0 & -1 \\ 0 & 0.5 & -0.5 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}V_s
$$

$$
V_o = y = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}
$$

**Check:** from this model $\dfrac{V_o(s)}{V_s(s)} = \dfrac{1}{2s^3 + 3s^2 + 4s + 2}$, the same as found directly by nodal analysis with impedances ($1/s$ for 1 F, $s$ for 1 H, $1/(2s)$ for 2 F).

## Check yourself

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

1. How are state variables chosen for an RLC network?
   A. Resistor currents and capacitor charges
   B. Capacitor voltages and inductor currents
   C. Only the source voltage
   D. Resistor voltages and inductor voltages
   Answer: B. Energy-storing elements define the states: capacitor voltage and inductor current. (Key point: Choosing states)

2. How many state variables does this network have?
   A. 2
   B. 4
   C. 3
   D. 5
   Answer: C. Two capacitors (1 F and 2 F) and one inductor (1 H) give three states. (Key point: Number of states)

3. Which equation holds at node A for the 1 F capacitor?
   A. dv1/dt = (Vs - v1) + iL
   B. dv1/dt = v1 - v2
   C. dv1/dt = iL - v1
   D. dv1/dt = (Vs - v1) - iL
   Answer: D. KCL: current through the 1 ohm resistor minus the inductor current flows into the 1 F capacitor. (Key point: Capacitor equation at node A)

4. What is the inductor state equation (L = 1 H)?
   A. d iL/dt = v1 - v2
   B. d iL/dt = v1 + v2
   C. d iL/dt = iL - v2
   D. d iL/dt = Vs - v2
   Answer: A. The voltage across the inductor is the difference of the node voltages A and B. (Key point: Inductor equation)

## Related in Control Systems

- [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
- [State space model of circuit with current source](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-circuit-4ohm-2h-is) — Control Systems, chapter 8
- [State equations of two-source circuit with 2 F](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-two-source-1-2h-2f) — Control Systems, chapter 8
- [Block diagram and Vo/Vi of L-R ladder network](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abd-l1-r1-l2-r2) — Control Systems, chapter 2
- [State model of two-source RLC circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-model-two-source-circuit) — Control Systems, chapter 8
- [State space model of a two-stage RC circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-model-rc-two-stage) — Control Systems, chapter 8

## Related in other subjects

- [Laplace: series RLC with 6e^(-2t) source](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-series-rlc-exponential-source-6e) — Electrical Circuit Theory, chapter 4
- [Node equations and Ybus, three-source network](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-node-equations-three-sources) — Power System Analysis II, chapter 1, asked 2×
- [Classical: inductor current and capacitor voltage, 100 V](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aclassical-inductor-current-capacitor-voltage-100v) — Electrical Circuit Theory, chapter 3
- [Mesh analysis: voltage across 1 Ω resistor](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Amesh-analysis-dependent-source-1ohm) — Electrical Circuit Theory, chapter 1