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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

    © Mohamed M. Aboudina, 2013

    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

    © Mohamed M. Aboudina, 2013

  • 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

    © Mohamed M. Aboudina, 2013

  • 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

    © Mohamed M. Aboudina, 2013

  • General Impedance Converters

    Antoniou GIC

    𝐿𝑒𝑞 = 𝐶𝑅 2

    © Mohamed M. Aboudina, 2013

  • General Impedance Inverter

    • 𝑍𝑖𝑛 = 𝑉1

    𝐼1 = 𝑠𝐶𝑅2

    • 𝐿𝑒𝑞 = 𝐶𝑅 2

    Riordan Gyrator

    © Mohamed M. Aboudina, 2013

  • Example

    For Gyration resistance =1𝑘Ω

    General Impedance Inverter

    Filter Example

    © Mohamed M. Aboudina, 2013

  • 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

    © Mohamed M. Aboudina, 2013

  • Antoniou GIC General Impedance Converters Antoniou GIC

    𝒇 𝒔 = 𝒌𝒔 𝒇 𝒔 = 𝒌𝒔

    𝑍𝐿 𝑠𝑍𝐿

    © Mohamed M. Aboudina, 2013

  • 3rd Order LPF

    6th Order BPF

    Example

    © Mohamed M. Aboudina, 2013

  • Bruton’s transformation

    © Mohamed M. Aboudina, 2013

  • 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

    © Mohamed M. Aboudina, 2013

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