---
title: "Block diagram model of R-L-C-L-R circuit"
subject: "Control Systems"
chapter: 2
key: "semester_3/control_systems:bd-r1l1-c-l2r2"
source: https://www.bctengineer.com/q/semester_3/control_systems/bd-r1l1-c-l2r2
site: BCT Engineering
---

# Block diagram model of R-L-C-L-R circuit

Control Systems · Chapter 2: Component Modeling

- Asked in: 2075 Asoj, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/bd-r1l1-c-l2r2
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2

## Question

Develop block diagram model for the circuit shown in figure below. [Figure: input Vi; series R1 and L1 to a node; capacitor C from that node to ground; series L2 to the output node; R2 from output node to ground; Vo across R2]

## Short answer

Define V1 across C, I1 in R1–L1, I2 in L2–R2. Block diagram: I1=(Vi-V1)/(R1+sL1), V1=(I1-I2)/(sC), I2=V1/(R2+sL2), Vo=R2 I2. Reduction yields Vo/Vi = R2 / [CL1L2 s^3 + C(L1R2+L2R1)s^2 + (CR1R2+L1+L2)s + R1+R2].

## Answer

Let $V_1$ = voltage across $C$, $I_1$ = current in $R_1$–$L_1$, $I_2$ = current in $L_2$–$R_2$.

#### Element equations (Laplace)
$$
\begin{aligned}
I_1(s) &= \frac{V_i(s) - V_1(s)}{R_1 + sL_1} \\
V_1(s) &= \frac{1}{sC}\left[I_1(s) - I_2(s)\right] \\
I_2(s) &= \frac{V_1(s)}{R_2 + sL_2} \\
V_o(s) &= R_2\,I_2(s)
\end{aligned}
$$

#### Block diagram
Here $Z_1 = R_1 + sL_1$ and $Z_2 = R_2 + sL_2$.
```text
 Vi─►(Σ)►[1/Z1]─►(Σ)►[1/sC]─┬─►[1/Z2]─┬─►[R2]─►Vo
      ▲-          ▲-        │          │
      │           └── I2 ───┼──────────┘
      └── V1 ───────────────┘
```

#### Reduction
Inner loop (between $I_1$ and $V_1$, feedback via $\frac{1}{R_2+sL_2}$):
$$
\frac{V_1}{I_1} = \frac{\frac{1}{sC}}{1 + \frac{1}{sC(R_2+sL_2)}} = \frac{R_2 + sL_2}{sC(R_2+sL_2) + 1}
$$
Outer loop with forward $\frac{1}{R_1+sL_1}$ and unity feedback of $V_1$, then multiply by $\frac{R_2}{R_2+sL_2}$:
$$
\frac{V_o(s)}{V_i(s)} = \frac{R_2}{(R_1 + sL_1) + (R_2 + sL_2) + sC(R_1 + sL_1)(R_2 + sL_2)}
$$
Expanded:
$$
\frac{V_o(s)}{V_i(s)} = \frac{R_2}{CL_1L_2s^3 + C(L_1R_2 + L_2R_1)s^2 + (CR_1R_2 + L_1 + L_2)s + R_1 + R_2}
$$

## Check yourself

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

1. Which equation gives I2?
   A. I2 = Vo/R1
   B. I2 = V1 sC
   C. I2 = V1/R2
   D. I2 = V1/(R2 + sL2)
   Answer: D. I2 flows through L2 and R2 in series with V1 across them. (Key point: Element equation)

2. The capacitor block gives V1 = (I1 - I2)/(sC). This means:
   A. I2 is fed back negatively to the second summer
   B. Vo is fed to the input
   C. I2 is added
   D. V1 is not fed back
   Answer: A. C carries the difference I1 - I2, so I2 is subtracted. (Key point: Feedback structure)

3. At s = 0 the transfer function Vo/Vi equals:
   A. R1/(R1+R2)
   B. 0
   C. R2/(R1 + R2)
   D. 1
   Answer: C. At DC the inductors are shorts and C is open, giving a divider R2/(R1+R2). (Key point: DC gain)

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