# active circuits lecture 9: active inductors, negative...     Post on 07-Aug-2020

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• Active Circuits Lecture 9: Active Inductors, Negative

Resistance and LC-Ladder Filters ELC 302 – Spring 2013

Dr. Mohamed M. Aboudina maboudina@gmail.com

Department of Electronics and Communications Engineering Faculty of Engineering – Cairo University

mailto:maboudina@gmail.com

• © Mohamed M. Aboudina, 2013

Other Alternatives

Vin Vout R sL

1/sC

𝐻 𝑠 =

1 𝐿𝐶

𝑠2 + 𝑠 𝑅 𝐿 +

1 𝐿𝐶

Replace this RLC filter with an

Active Opamp RC filters

Use Active Circuits to Emulate “L” ≡ Replace L with a

circuit

Vin Vout xR xsL

x/sC

𝑥 = 1/𝑠

1

𝑠2𝐶 = −

1

𝜔2𝐶 ≡ FDNR

Frequency-Dependent Negative Resistance

Leapfrog Filters

Bruton’s transformation

Vin Vout

R/s L

1/s2C

• © Mohamed M. Aboudina, 2013

Emulation of RLC Filters using Opamps

• Active Inductors – Inductor Emulations.

• FDNR: Frequency Dependent Negative Resistance

• (LC Ladder Filters) Leapfrog Filter Architectures.

• GIC (General Impedance Converter)

GII (General Impedance Inverter)

Inductor Emulation Using Two-port Network

𝑇 = 𝐴 =

𝑘 0

0 𝑘

𝑓 𝑠

𝑍𝑖1 = 𝑉1

𝐼1 = 𝐺𝑎𝑍𝐿 𝑠 while 𝐺𝑎 =

𝑎11

𝑎22 = 𝑓(𝑠)

𝐴 = 0 𝑎12

𝑎21 0

• Gyrator

Floating inductor

General Impedance Inverter

Gyrator – Positive Impedance Inverter

𝑍𝑖𝑛 = 1

𝑔1𝑔2 . 1

𝑍

If 𝑍 = 1

𝑠𝐶 ⇒ 𝑍𝑖𝑛 = 𝑠

𝐶

𝑔1𝑔2 = 𝑠𝐿𝑒𝑞

𝑍𝑖𝑛

𝐿𝑒𝑞 = 𝐶

𝑔1𝑔2

Good For Grounded Inductors

𝑔 𝑔

𝐿𝑒𝑞 = 𝐶

𝑔1𝑔2

• Gyrator Example General Impedance Inverter

• 𝑉𝑥 = 2𝑉1

• 𝐼𝑎 = (𝑉1−𝑉𝑥)/𝑅

• 𝐼𝑏 = 𝑉1−𝑉2

𝑅

• 𝐼𝑐 = 𝑉𝑥−𝑉2

𝑅

• 𝐼1 = 𝐼𝑎 + 𝐼𝑏

• 𝑉𝑦 = 𝑉2 − 𝑅𝐼𝑐

• 𝐼𝑑 = 𝑉2−𝑉𝑦

𝑅

• 𝐼2 = 𝐼𝑑 − 𝐼𝑏

• 𝐴 = 0 −𝑅

− 1

𝑅 0

Gyrator Example

R R

R

R

R

R

R

I1

I2 V1

V2

Vx VyV2V2

V1

Ia Ib

Ic

Id

Gyrator Resistance = R

• General Impedance Converters

Antoniou GIC

𝐿𝑒𝑞 = 𝐶𝑅 2

• General Impedance Inverter

• 𝑍𝑖𝑛 = 𝑉1

𝐼1 = 𝑠𝐶𝑅2

• 𝐿𝑒𝑞 = 𝐶𝑅 2

Riordan Gyrator

• Example

For Gyration resistance =1𝑘Ω

General Impedance Inverter

Filter Example

• Antoniou GIC

• Note: Inductance emulation is optimum in case of no floating inductors i.e., LC high-pass filters

General Impedance Converters

Antoniou GIC

𝑓 𝑠 = 𝑠𝐶𝑅 = 𝑘𝑠

𝑘𝑠 𝑠𝑘𝑅𝐿 𝑅𝐿

𝒇 𝒔 = 𝒌𝒔 𝒇 𝒔 = 𝒌𝒔

𝑅𝐿 𝑠𝑘𝑅𝐿

• Antoniou GIC General Impedance Converters Antoniou GIC

𝒇 𝒔 = 𝒌𝒔 𝒇 𝒔 = 𝒌𝒔

𝑍𝐿 𝑠𝑍𝐿

• 3rd Order LPF

6th Order BPF

Example

• Bruton’s transformation

• FDNR

• Bruton’s inductor simulation based on FDNR

• Most suitable for LC LPF with minimum cap realization

FDNR : Frequency-Dependent Negative Resistance

𝑓 𝑠 = 1

𝑠𝐶1𝑅4 ⇒ 𝑍𝑖𝑛 =

1

𝑠2𝐶1𝐶5𝑅4

𝐶 = 𝐶1𝐶5𝑅4