Active Circuits Lecture 9: Active Inductors, Negative Resistance and LC-Ladder Filters ELC 302 – Spring 2013 Dr. Mohamed M. Aboudina [email protected] Department of Electronics and Communications Engineering Faculty of Engineering – Cairo University © Mohamed M. Aboudina, 2013
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Active Circuits Lecture 9: Active Inductors, Negative
Resistance and LC-Ladder Filters ELC 302 – Spring 2013
Dr. Mohamed M. Aboudina [email protected]
Department of Electronics and Communications Engineering Faculty of Engineering – Cairo University
© Mohamed M. Aboudina, 2013
© Mohamed M. Aboudina, 2013
Other Alternatives
Vin Vout
R sL
1/sC
𝐻 𝑠 =
1𝐿𝐶
𝑠2 + 𝑠𝑅𝐿 +
1𝐿𝐶
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
0𝑘
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
𝑅0
Gyrator Example
RR
R
R
R
R
R
I1
I2V1
V2
Vx
VyV2V2
V1
IaIb
Ic
Id
Gyrator Resistance = R
© Mohamed M. Aboudina, 2013
General Impedance Converters
Antoniou GIC
𝐿𝑒𝑞 = 𝐶𝑅2
© Mohamed M. Aboudina, 2013
General Impedance Inverter
• 𝑍𝑖𝑛 =𝑉1
𝐼1= 𝑠𝐶𝑅2
• 𝐿𝑒𝑞 = 𝐶𝑅2
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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