---
title: "Block diagram of a two-stage RC circuit"
subject: "Control Systems"
chapter: 2
key: "semester_3/control_systems:bd-rc-two-stage"
source: https://www.bctengineer.com/q/semester_3/control_systems/bd-rc-two-stage
site: BCT Engineering
---

# Block diagram of a two-stage RC circuit

Control Systems · Chapter 2: Component Modeling

- Asked in: 2078 Kartik, 4 marks
- Video: [How to draw the block diagram of any electrical circuit(from transfer function)](https://www.youtube.com/watch?v=K9ehi3DG5LA) — Saibal Manna
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/bd-rc-two-stage
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2

## Question

Draw the block diagram of circuit shown below. [Figure: input Vi; series R1 to a node; capacitor C1 from that node to ground; series R2 to the output node; capacitor C2 from output node to ground; Vo across C2; mesh currents i1(t), i2(t)]

## Short answer

Block diagram uses three summing junctions with feedback loops. Overall transfer function is V_o(s)/V_i(s) = 1/[R₁R₂C₁C₂ s² + (R₁C₁ + R₁C₂ + R₂C₂)s + 1], which includes loading term R₁C₂.

## Answer

Write one equation per element so each gives one block, then join the blocks.

#### Equations (Laplace domain)
Let $V_1$ = voltage across $C_1$.
$$
\begin{aligned}
I_1(s) &= \frac{V_i(s) - V_1(s)}{R_1} \\
V_1(s) &= \frac{I_1(s) - I_2(s)}{sC_1} \\
I_2(s) &= \frac{V_1(s) - V_o(s)}{R_2} \\
V_o(s) &= \frac{I_2(s)}{sC_2}
\end{aligned}
$$

#### Block diagram
```text
 Vi─►(Σ)►[1/R1]─►(Σ)►[1/sC1]─┬─►(Σ)►[1/R2]─┬─►[1/sC2]─┬─►Vo
      ▲-          ▲-         │   ▲-        │          │
      │           └── I2 ────┼───┼─────────┘          │
      └── V1 ────────────────┘   └── Vo ──────────────┘
```
- $V_1$ is fed back (negative) to the first summer, $I_2$ to the second and $V_o$ to the third.

#### Resulting transfer function
Reducing the three interacting loops (or by Mason's rule) gives
$$
\frac{V_o(s)}{V_i(s)} = \frac{1}{R_1R_2C_1C_2s^2 + (R_1C_1 + R_1C_2 + R_2C_2)s + 1}
$$
The term $R_1C_2$ shows the **loading** of the second stage on the first; the result is not simply the product $\dfrac{1}{(1+sR_1C_1)(1+sR_2C_2)}$.

## Check yourself

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

1. How is the block diagram of the RC circuit built?
   A. Replace C with a gain
   B. Use Bode plots
   C. Guess a standard form
   D. Write one equation per element, each gives one block, then join them
   Answer: D. Each Laplace equation (output = block x input) becomes a block and a summing point. (Key point: Method)

2. The voltage across C1 is:
   A. V1 = (Vi - Vo)/R1
   B. V1 = (I1 - I2)/(sC1)
   C. V1 = I1 sC1
   D. V1 = (I1 + I2)/(sC1)
   Answer: B. C1 carries the difference of the two mesh currents I1 - I2. (Key point: Capacitor voltage equation)

3. What does the R1C2 term in the denominator show?
   A. The two stages are isolated
   B. The circuit is unstable
   C. R1 is shorted
   D. The second stage loads the first, so the result is not a simple product of two first-order lags
   Answer: D. Because stage two draws current from stage one, the TF is not 1/((1+sR1C1)(1+sR2C2)). (Key point: Loading effect)

## Related in Control Systems

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