---
title: "Gyrator simulation of inductor"
subject: "Filter Design"
chapter: 8
key: "semester_5/filter_design:gyrator-inductor-simulation"
source: https://www.bctengineer.com/q/semester_5/filter_design/gyrator-inductor-simulation
site: BCT Engineering
---

# Gyrator simulation of inductor

Filter Design · Chapter 8: Design of High-Order Active Filters

- Asked in: 2075 Chaitra, 1+4 marks; 2074 Chaitra, 1+4 marks (asked 2 times)
- Video: [Gyrator Device :  How to design and analyze Gyrator Circuit](https://www.youtube.com/watch?v=Ls_FvD37yHE) — Engineering Prof.
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/gyrator-inductor-simulation
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-8

## Question

What is (ideal) gyrator? How can you simulate inductor using gyrator? Explain with necessary derivation.

## Short answer

An ideal gyrator is a two-port with I1=g V2, I2=-g V1, giving Z_in = r^2 / Z_L. Loading with capacitor C yields Z_in = s (C/g^2) = s r^2 C, simulating an inductor L = C/g^2 = r^2 C. A grounded inductor needs one gyrator; a floating inductor requires two back-to-back.

## Answer

#### Ideal gyrator
An **ideal gyrator** is a lossless two-port that converts a voltage at one port into a proportional current at the other. With gyration conductance $g$ (gyration resistance $r = 1/g$):
$$
I_1 = g V_2, \qquad I_2 = -g V_1
$$
```text
 I1 ->                    <- I2
 o------+  +---------+  +------o
  V1    |--| gyrator |--|    V2
 o------+  |   g     |  +------o
           +---------+
```
It is non-reciprocal and inverts impedance: a load $Z_L$ at port 2 appears at port 1 as $Z_{in} = \dfrac{1}{g^2Z_L} = \dfrac{r^2}{Z_L}$.

#### Simulating an inductor
Terminate port 2 with a capacitor $C$. The load current is $sCV_2$, and it leaves port 2, so $I_2 = -sCV_2$.
$$
\begin{aligned}
-sCV_2 &= -gV_1 \;\Rightarrow\; V_2 = \frac{g}{sC}V_1 \\
I_1 &= gV_2 = \frac{g^2}{sC}V_1 \\
Z_{in} &= \frac{V_1}{I_1} = s\frac{C}{g^2} = s\,r^2C
\end{aligned}
$$
So the input looks like an inductor
$$
L_{eq} = \frac{C}{g^2} = r^2C
$$
```text
 o----+  +---------+  +----+
      |--| gyrator |--|   === C   => L = r^2 C
 o----+  +---------+  +----+
```
**Example:** $r = 10\ \text{k}\Omega$, $C = 0.1\ \mu$F gives $L = 10^8 \times 10^{-7} = 10$ H.

**Practical realization:** a gyrator is built with two voltage-controlled current sources of opposite sign (transconductance $\pm g$), or with op-amps (e.g. the Riordan circuit). Antoniou's GIC can also simulate a grounded inductor directly.

**Grounded vs floating:** one gyrator with a grounded C gives a **grounded** inductor. A **floating** inductor needs two gyrators connected back to back with a grounded capacitor between them, giving $L = r^2C$ between the two outer terminals.

Simulated inductors are small, have high Q and can be integrated, so a passive LC ladder can be converted to an active RC filter while keeping its low sensitivity.

## Check yourself

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

1. Which equations describe an ideal gyrator?
   A. I1 = g V2, I2 = -g V1
   B. V1 = g V2, V2 = g V1
   C. I1 = g V1, I2 = g V2
   D. V1 = g I1
   Answer: A. A gyrator turns a voltage at one port into a proportional current at the other, non-reciprocally. (Key point: Gyrator equations)

2. A load ZL at port 2 appears at port 1 as what?
   A. ZL
   B. 1/(g^2 ZL)
   C. g^2 ZL
   D. g ZL
   Answer: B. Zin = 1/(g^2 ZL) = r^2/ZL, so it inverts impedance. (Key point: Impedance inversion)

3. What is the equivalent inductance when port 2 is terminated with capacitor C?
   A. C g^2
   B. C/g^2 = r^2 C
   C. 1/(g^2 C)
   D. g C
   Answer: B. Zin = sC/g^2, i.e. L = r^2 C. (Key point: Inductor from capacitor)

4. How is a floating inductor simulated?
   A. One gyrator with grounded C
   B. Two gyrators back to back with a grounded capacitor between them
   C. A gyrator with an inductor load
   D. A single op-amp inverter
   Answer: B. One gyrator gives only a grounded inductor; two back-to-back give a floating one. (Key point: Floating inductor)

## Close variants set in other exams

- [Ideal gyrator and floating inductor simulation](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Agyrator-floating-inductor) — Filter Design, chapter 8, asked 3×

## Related in Filter Design

- [GIC simulation of a grounded inductor](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Agic-grounded-inductor-simulation) — Filter Design, chapter 8, asked 7×
- [GIC simulation of floating inductor](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Agic-floating-inductor) — Filter Design, chapter 8