ELC 3060: Active Circuits Introduction & Filter Transfer Function (Poles/Zeroes)
ELC 3060: Active Circuits Introduction & Filter Transfer Function (Poles/Zeroes)
ELC 3060: Active Circuits Introduction & Filter Transfer Function (Poles/Zeroes)
Spring 2022
Topic
Introduction & Filter Transfer Function (Poles/Zeros)
Single Opamp First and Second Order Filters
Two and Three Opamp-RC Filters
MOS-C Filters + Filter Performance
Filter Approximation & Frequency Transformation
Higher-Order Filters (Cascaded and Leapfrog)
Active Inductors & Negative Resistors
Introduction to Gm-C Filters
Linearized Gm Circuits
Gm-C Filter Implementation
Introduction to Switched-Capacitors (SC)
Circuit Implementation of SC Filters
3
This Course
Information
• Instructor:
– Dr. Mohamed Mobarak
• Teaching Assistant:
– Eng. Ahmed Abdelraouf
• Email: mohamedmobarak@eng.cu.edu.eg
• References:
– Course notes / Sheets
– T.L.Deliyannis, Y.Sun, J.K.Fidler, “Continuous-Time Active Filter Design”
(Chapters 2-6, 8-11)
– B. Razavi, “Fundamentals of Microelectronics” (Chapter 14)
– T. Carosune/D. Johns/K. Martin “Analog Integrated Circuit Design”
(Chapter 10)
4
Introduction
Why Filters?
Biomedical Military
Wireless Communication Hard disk drive
Hearing aid mm-Wave
Video Space
Sensors Imaging
Sigma delta converters Wireless Communication
Loop filter in PLL
• Band-Stop or Band-Reject
Filter (BSF):
o Notch is a specific case
13
Filters
Summary of Filter Classifications
14
Filters
Continuous-Time Filters
𝑁 𝑠 𝑎 𝑛 ς𝑛𝑖=1(𝑠−𝑧𝑖 )
• 𝐻 𝑠 = = ς𝑚
𝐷 𝑠 𝑖=1(𝑠−𝑝𝑖 )
• Since all coefficients of N(s) are real, zeroes of H(s) can be real or
complex conjugate.
• Similarly for D(s), Poles of H(s) can be real or complex conjugate.
For stability purposes, all real part of poles must be negative (< 0).
16
Filters
Stability
A B
• 𝐻 𝑠 = 𝑠 − 𝑠𝑧 = 𝑠 + 𝜎𝑧
𝑠 + 𝜎𝑧
– 𝑠𝑧 : Is called a Zero.
– Minimum distance exists at 𝜔 = 0 →
Minimum 𝐻 𝑗𝜔 happens at 𝜔 = 0. −𝜎𝑧 𝜎
𝐻 𝑗𝜔
– Distance increases as 𝜔 increases →
𝐻 𝑗𝜔 increases.
2𝐴𝑚𝑖𝑛
– Phase at DC = 0 𝐴𝑚𝑖𝑛
– Phase gradually increases till it reaches
90𝑜 at 𝜔 = ∞. 𝜔
∠𝐻 𝑗𝜔
– @𝜔 = 𝜎𝑧 → ∠𝐻 𝑗𝜔 = +45𝑜
90𝑜
45𝑜
• What if 𝑠𝑧 is at the RHP?
𝜔
23
Poles and Zeroes
Examples (First Order Filters)
• Amplitude Equalizers:
– The amplitude equalizer has an amplitude response that does not
belong to any of the filter responses considered above.
– It is used to compensate for the distortion of the frequency spectrum
that the signal suffers when passing through a system (cable, wire,
optical fiber … etc.).
– Its amplitude response is therefore drawn as complementary to the
signal spectrum.
25
𝜔0
• 𝐻1 𝑠 = (ideal or lossless integrator)
𝑠
𝑘
• 𝐻2 𝑠 = 𝑠 (1st order filter or lossy integrator)
(1+ )
𝜔0
𝑠
(1+𝜔 )
• 𝐻3 𝑠 = 𝑠
𝑧
(1+𝜔 )
𝑝
– Left-half plane zero.
𝑠
(1−𝜔 )
• 𝐻4 𝑠 = 𝑠
𝑧
(1+ )
𝜔𝑝
– Right-half plane zero
26
• Ideal op amp is a special case of ideal differential amplifier with infinite gain,
infinite Rid and zero Ro .
v
v = o and lim vid = 0
id A A→
𝐻1 𝑠 : Lossless Integrator
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☺ Independent Tuning
29
☺ Parasitic Insensitive
☺ Independent Tuning
☺ Small Parameter Sensitivity
☺ Small Component Spread
☺ No Opamp input Swing (virtual ground)
30
Parasitic Capacitance
𝐾
𝐶𝑝
𝐶 + 𝐶𝑝 𝑅
𝐻 𝑠 =
1
𝑠+
𝐶 + 𝐶𝑝 𝑅
Sensitivity
dP dC P=Parameter
SC =
P
P C C=Component
Δ𝑃/𝑃 𝐶 𝜕𝑃
• 𝑆𝐶𝑃 = =
Δ𝐶/𝐶 𝑃 𝜕𝐶
Example: Sensitivity
0 = 1 / (R1C1 )
d 0 −1
=
dR1 R12C1
d 0 dR1
=−
0 R1
0
S R1 = −1
37
𝑘𝑦 𝑦 𝑦
• 𝑆𝑥 = 𝑆𝑘𝑥 = 𝑆𝑥
1/𝑦 𝑦 𝑦
• 𝑆𝑥 = 𝑆1/𝑥 = −𝑆𝑥
𝑦𝑛 𝑦 𝑦 1 𝑦
• 𝑆𝑥 = 𝑛𝑆𝑥 & 𝑆𝑥 𝑛 = 𝑆
𝑛 𝑥
𝑦 𝑦
• 𝑆𝑥 = 𝑆𝑥2 × 𝑆𝑥𝑥2
𝑛ח
𝑖=1 𝑦𝑖 𝑦
• 𝑆𝑥 = σ𝑛𝑖=1 𝑆𝑥 𝑖
38
Analog IC Design
Design Kit
▪ In range of pF
▪ Accuracy +/-15%
43
Analog IC Design
Integrated Capacitors: MIM (Metal-Insulator-Metal)
44
Analog IC Design
Integrated Capacitors: MOM (Metal-Oxide-Metal)
45
Analog IC Design
Integrated Inductors