Download aic lec 15 feedback v01

28 June 2020
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1441 ‫ ذو القعدة‬7
Analog IC Design
Lecture 15
Negative Feedback
Dr. Hesham A. Omran
Integrated Circuits Laboratory (ICL)
Electronics and Electrical Communications Eng. (EECE) Dept.
Faculty of Engineering
Ain Shams University
Outline
❑
❑
❑
❑
❑
❑
❑
Recapping previous key results
General feedback system
Loop gain
Why negative feedback?
Stability of feedback system
Root locus and Bode plot
Phase and gain margin
15: Negative Feedback
2
Outline
❑
❑
❑
❑
❑
❑
❑
Recapping previous key results
General feedback system
Loop gain
Why negative feedback?
Stability of feedback system
Root locus and Bode plot
Phase and gain margin
15: Negative Feedback
3
MOSFET in Saturation
❑ The channel is pinched off if the difference between the gate and drain voltages is not
sufficient to create an inversion layer
𝑉𝐺𝐷 ≤ 𝑉𝑇𝐻 𝑂𝑅 𝑉𝐷𝑆 ≥ 𝑉𝑜𝑣
❑ Square-law (long channel MOS)
VSG > |VTH|
𝐼𝐷 =
𝜇𝑛 𝐶𝑜𝑥 𝑊
2
𝐿
2 1 + 𝜆𝑉
⋅ 𝑉𝑜𝑣
𝐷𝑆
VSD > |Vov|
𝑉𝑆𝐵 ↑ ⇒ 𝑉𝑇𝐻 ↑
B
VSB
VGS>VTH
VDG < |VTH|
G
VGD<VTH
S
D
VGD < VTH
VDS > Vov
p+
n+
n+
p-sub
15: Negative Feedback
VDS>Vov
VGS > VTH
4
Regions of Operation Summary
OFF
(Subthreshold)
𝑉𝐺𝑆 < 𝑉𝑇𝐻
Triode
𝑉𝐷𝑆 < 𝑉𝑜𝑣
Or
𝑉𝐺𝐷 > 𝑉𝑇𝐻
2
𝑊
𝑉𝐷𝑆
𝐼𝐷 = 𝜇𝐶𝑜𝑥
𝑉 𝑉 −
𝐿 𝑜𝑣 𝐷𝑆
2
15: Negative Feedback
ON
𝑉𝐺𝑆 > 𝑉𝑇𝐻
Pinch-Off
(Saturation)
𝑉𝐷𝑆 ≥ 𝑉𝑜𝑣
Or
𝑉𝐺𝐷 ≤ 𝑉𝑇𝐻
𝜇𝐶𝑜𝑥 𝑊 2
𝐼𝐷 =
𝑉 1 + 𝜆𝑉𝐷𝑆
2 𝐿 𝑜𝑣
5
High Frequency Small Signal Model
𝑔𝑚 =
𝜕𝐼𝐷
𝜕𝑉𝐺𝑆
= 𝜇𝐶𝑜𝑥
𝑊
𝑉
𝐿 𝑜𝑣
𝑔𝑚𝑏 = 𝜂𝑔𝑚
𝑟𝑜 =
1
𝜕𝐼𝐷 /𝜕𝑉𝐷𝑆
=
𝑉𝐴
𝐼𝐷
=
=
𝜇𝐶𝑜𝑥
𝑊
𝐿
⋅ 2𝐼𝐷 =
2𝐼𝐷
𝑉𝑜𝑣
𝜂 ≈ 0.1 − 0.25
1
𝜆𝐼𝐷
𝑉𝐴 ∝ 𝐿 ↔ 𝜆 ∝
𝐶𝑔𝑏 ≈ 0
1
𝐿
𝐶𝑔𝑠 ≫ 𝐶𝑔𝑑
𝑉𝐷𝑆 ↑ 𝑉𝐴 ↑
𝐶𝑠𝑏 > 𝐶𝑑𝑏
Cgd
G
Cgb
D
Cgs
Csb
gmvgs
gmbvbs
ro
Cdb
S
B
15: Negative Feedback
6
Rin/out Shortcuts Summary
𝑟𝑜 1 + 𝑔𝑚 + 𝑔𝑚𝑏 𝑅𝑆
H.I.N.
∞
At low
frequencies ONLY
15: Negative Feedback
1
𝑅𝐷
1+
𝑔𝑚 + 𝑔𝑚𝑏
𝑟𝑜
L.I.N.
7
Summary of Basic Topologies
CS
CG
CD (SF)
RD
RD
RD
vout
vout
vin
vin
vin
RS
RS
Voltage & current amplifier
Voltage amplifier
Current buffer
𝑅𝑆 ||
RS
Voltage buffer
Current amplifier
1
𝑅𝐷
1+
𝑔𝑚 + 𝑔𝑚𝑏
𝑟𝑜
Rin
∞
Rout
𝑅𝐷 ||𝑟𝑜 1 + 𝑔𝑚 + 𝑔𝑚𝑏 𝑅𝑆
𝑅𝐷 ||𝑟𝑜
Gm
−𝒈𝒎
𝟏 + 𝒈𝒎 + 𝒈𝒎𝒃 𝑹𝑺
𝒈𝒎 + 𝒈𝒎𝒃
15: Negative Feedback
vout
∞
𝑅𝑆 ||
1
𝑅𝐷
1+
𝑔𝑚 + 𝑔𝑚𝑏
𝑟𝑜
𝒈𝒎
𝟏 + 𝑹𝑫 /𝒓𝒐
8
Differential Amplifier
Pseudo Diff Amp
𝑨𝒗𝒅
Diff Pair (w/ ideal CS)
−𝑔𝑚 𝑅𝐷
−𝑔𝑚 𝑅𝐷
−𝑔𝑚 𝑅𝐷
𝑨𝒗𝑪𝑴
−𝑔𝑚 𝑅𝐷
0
−𝑔𝑚 𝑅𝐷
1 + 2 𝑔𝑚 + 𝑔𝑚𝑏 𝑅𝑆𝑆
𝑨𝒗𝒅 /𝑨𝒗𝑪𝑴
1
∞
2 𝑔𝑚 + 𝑔𝑚𝑏 𝑅𝑆𝑆
≫1
𝐴𝑣𝐶𝑀2𝑑
𝑣𝑜𝑑
Δ𝑅𝐷
Δ𝑔𝑚 𝑅𝐷
=
≈
+
𝑣𝑖𝐶𝑀 2𝑅𝑆𝑆 2𝑔𝑚1,2 𝑅𝑆𝑆
𝐶𝑀𝑅𝑅 =
15: Negative Feedback
Diff Pair (w/ 𝑹𝑺𝑺 )
𝐴𝑣𝑑
𝐴𝑣𝐶𝑀2𝑑
9
Op-Amp
❑ An op-amp is simply a high gain differential amplifier
▪ The gain can be increased by using cascodes and multi-stage amplification
❑ The diff amp is a key block in many analog and RF circuits
▪ DEEP understanding of diff amp is ESSENTIAL
M3
Vin+
Vin-
Vout
Vout
Vin+
M1
VB
15: Negative Feedback
M4
M2
Vin-
M5
10
Op-Amp vs OTA
❑ In short, an OTA is an op-amp without an output stage (buffer)
❑ Some designers just use op-amp name and symbol for both
Rout
OTA
LOW
HIGH
Rout
vin
Model
Op-amp
iin
Rin
Avvin
iout
vout
iout
vin
iin
Rin
Gmvin
vout
Rout
Diff input, SE output
Fully diff
15: Negative Feedback
11
∗
V-star 𝑽
❑ V-star 𝑉 ∗ is inspired by 𝑉𝑜𝑣 but calculated from actual simulation data
2𝐼𝐷
2𝐼𝐷
2
∗
𝑔𝑚 = ∗ ↔ 𝑉 =
=
𝑉
𝑔𝑚 𝑔𝑚 /𝐼𝐷
❑ Figures-of-merit in terms of 𝑉 ∗
2𝐼𝐷 1
2
𝑔𝑚 𝑟𝑜 = ∗ ⋅
= ∗
𝑉 𝜆𝐼𝐷 𝜆𝑉
𝑔𝑚
1 2𝐼𝐷 1
𝑓𝑇 =
=
⋅ ∗ ⋅
2𝜋𝐶𝑔𝑔 2𝜋 𝑉 𝐶𝑔𝑔
𝑔𝑚
2
= ∗
𝐼𝐷
𝑉
❑ The boundary between weak and strong inversion (𝑛 = 1.2 → 1.5)
𝑉𝑜𝑣 𝑆𝐼 = 𝑉 ∗ 𝑊𝐼 = 2𝑛𝑉𝑇 ≈ 60 → 80𝑚𝑉
15: Negative Feedback
12
Outline
❑
❑
❑
❑
❑
❑
❑
Recapping previous key results
General feedback system
Loop gain
Why negative feedback?
Stability of feedback system
Root locus and Bode plot
Phase and gain margin
15: Negative Feedback
13
General Feedback System
Vin
15: Negative Feedback
+
_+
Verr
Feedforward
(Actuator)
AOL
Vfb
Feedback
(Sensor)
β
Vout
14
General Feedback System
❑ Error signal = 𝑉𝑒𝑟𝑟 = 𝑉𝑖𝑛 − 𝑉𝑓𝑏
❑ Open loop (OL) gain = 𝐴𝑂𝐿 =
❑ Feedback factor = 𝛽 =
𝑉𝑜𝑢𝑡
𝑉𝑒𝑟𝑟
𝑉𝑓𝑏
𝑉𝑜𝑢𝑡
❑ Closed loop (CL) gain = 𝐴𝐶𝐿 =
Vin
15: Negative Feedback
≫1
+
_+
𝑉𝑜𝑢𝑡
𝑉𝑖𝑛
Verr
Feedforward
(Actuator)
AOL
Vfb
Feedback
(Sensor)
β
Vout
15
Closed-loop Gain
𝑉𝑜𝑢𝑡 = 𝐴𝑂𝐿 𝑉𝑖𝑛 − 𝑉𝑓𝑏 = 𝐴𝑂𝐿 𝑉𝑖𝑛 − 𝛽𝑉𝑜𝑢𝑡
𝑉𝑜𝑢𝑡
𝐴𝑂𝐿
𝐴𝑂𝐿
𝐴𝐶𝐿 =
=
=
𝑉𝑖𝑛
1 + 𝛽𝐴𝑂𝐿 1 + 𝐿𝐺
❑ Loop gain = 𝐿𝐺 = 𝛽𝐴𝑂𝐿 ≫ 1
1
𝐴𝐶𝐿 ≈
𝛽
❑ Closed-loop gain is independent of open-loop gain!
Vin
15: Negative Feedback
+
_+
Verr
Feedforward
(Actuator)
AOL
Vfb
Feedback
(Sensor)
β
Vout
16
Error Signal
𝑉𝑒𝑟𝑟 = 𝑉𝑖𝑛 − 𝑉𝑓𝑏 = 𝑉𝑖𝑛 − 𝛽𝑉𝑜𝑢𝑡 = 𝑉𝑖𝑛 − 𝛽𝐴𝑂𝐿 𝑉𝑒𝑟𝑟
𝑉𝑖𝑛
𝑉𝑖𝑛
𝑉𝑒𝑟𝑟 =
=
1 + 𝛽𝐴𝑂𝐿 1 + 𝐿𝐺
❑ Loop gain = 𝐿𝐺 = 𝛽𝐴𝑂𝐿 ≫ 1
𝑉𝑖𝑛
𝑉𝑒𝑟𝑟 =
→0
1 + 𝐿𝐺
❑ Negative feedback loop works to minimize the error signal
Vin
15: Negative Feedback
+
_+
Verr
Feedforward
(Actuator)
AOL
Vfb
Feedback
(Sensor)
β
Vout
17
Feedback Example
❑ Op-amp functions: (1) subtraction and (2) amplification
❑ The network 𝑅1 and 𝑅2 functions: (1) sensing the output voltage and (2) providing a
𝑅2
feedback factor 𝛽 =
𝑅1 +𝑅2
𝐴𝐶𝐿
Vin
+
𝑉𝑜𝑢𝑡
𝐴𝑂𝐿
=
=
=
𝑉𝑖𝑛
1 + 𝛽𝐴𝑂𝐿 1 +
_+
Verr
Vfb
AOL
β
𝐴𝑂𝐿
𝑅1 + 𝑅2
𝑅1
≈
=1+
𝑅2
𝑅2
𝑅2
⋅ 𝐴𝑂𝐿
𝑅1 + 𝑅2
Vout
Vin
Verr AOL
Vfb
Vout
R1
β
R2
15: Negative Feedback
18
Outline
❑
❑
❑
❑
❑
❑
❑
Recapping previous key results
General feedback system
Loop gain
Why negative feedback?
Stability of feedback system
Root locus and Bode plot
Phase and gain margin
15: Negative Feedback
19
Loop Gain
❑ Deactivate the input → Break the loop → Apply a test source → Calculate the gain around
the loop
❑ Loop gain = 𝐿𝐺 = −
𝑉𝑜𝑢𝑡
𝑉𝑡
= 𝛽𝐴𝑂𝐿
❑ A.k.a. loop transmission, return ratio …
❑ But biasing/loading changes when we break the loop!
▪ Make sure dc biasing is properly set
▪ Add a dummy load
Vin = 0
+
+
Verr
Vout
AOL
_
Vfb
15: Negative Feedback
β
Vt
20
Loop Gain
❑ Modern circuit simulators can compute the loop gain without explicitly breaking the loop
▪ Use stability (STB) analysis
▪ Insert a 0V dc voltage source or iprobe in the loop
• Polarity matters for Eldo, but not for Spectre
▪ Loop gain = 𝐿𝐺 = 𝛽𝐴𝑂𝐿 is calculated by the simulator
Vin
+
+
Verr
AOL
Vout
_
V=0
Vfb
15: Negative Feedback
β
21
Outline
❑
❑
❑
❑
❑
❑
❑
Recapping previous key results
General feedback system
Loop gain
Why negative feedback?
Stability of feedback system
Root locus and Bode plot
Phase and gain margin
15: Negative Feedback
22
Why Negative Feedback?
❑ We use a very high gain amplifier 𝐴𝑂𝐿 , but end up with a much smaller gain 𝐴𝐶𝐿
𝐴𝑂𝐿
1
=
≈
1+𝛽𝐴𝑂𝐿
𝛽
❑ We can design high gain amplifiers, but we really do not need all that gain
❑ High gain is the balance that we use to buy other useful properties
❑ Negative feedback useful properties
1. Gain Desensitization → Stable, linear, and accurate gain
2. Bandwidth Extension
3. Modification of I/O Impedances
15: Negative Feedback
23
Gain Desensitization
𝐴𝐶𝐿
𝐴𝑂𝐿
1
=
=
1 + 𝛽𝐴𝑂𝐿 𝛽
1
1
𝛽𝐴𝑂𝐿
1
1
1
1
≈
1−
=
1−
𝛽
𝛽𝐴𝑂𝐿
𝛽
𝐿𝐺
+1
❑ Static gain error
𝐴𝐶𝐿,𝑖𝑑𝑒𝑎𝑙 − 𝐴𝐶𝐿,𝑎𝑐𝑡𝑢𝑎𝑙
1
1
𝜖𝑠 =
≈
=
𝐴𝐶𝐿,𝑖𝑑𝑒𝑎𝑙
𝛽𝐴𝑂𝐿 𝐿𝐺
❑ 𝐴𝑂𝐿 varies due to PVT, load, and input signal variations
❑ 𝐴𝐶𝐿 almost independent of 𝐴𝑂𝐿 (if 𝐿𝐺 ≫ 1)
▪ Independent of PVT: stable and robust
▪ Independent of load: stable and robust
▪ Independent of input level: linear
15: Negative Feedback
24
Gain Desensitization
❑ In IC design, we cannot control absolute values due to PVT, load, and input signal variations
❑ But we can precisely control ratios of MATCHED components
𝐴𝐶𝐿
𝑌
𝐴𝑂𝐿
= =
=
𝑋 1 + 𝛽𝐴𝑂𝐿 1 +
❑ 𝑅1 = 3𝑅 𝑎𝑛𝑑 𝑅2 = 𝑅 ⇒ 𝐴𝐶𝐿
𝐴𝑂𝐿
𝑅1 + 𝑅2
𝑅1
≈
=1+
𝑅2
𝑅2
𝑅2
⋅ 𝐴𝑂𝐿
𝑅1 + 𝑅2
= 4 ⇒ Stable, linear, and accurate
Vin
Verr AOL
Vfb
Vout
R1
β
R2
15: Negative Feedback
[Razavi, 2017]
25
Bandwidth Extension
❑ Assume the op-amp (OL) is a first order system
𝐴𝑂𝐿 𝑠 =
𝐴𝑂𝐿𝑜
𝑠
1+
𝜔𝑝,𝑂𝐿
❑ The CL transfer function is also a first order system
𝐴𝐶𝐿
𝐴𝑂𝐿𝑜
𝐴𝑂𝐿 𝑠
𝐴𝐶𝐿𝑜
1 + 𝛽𝐴𝑂𝐿𝑜
𝑠 =
=
=
𝑠
𝑠
1 + 𝛽𝐴𝑂𝐿 𝑠
1+
1+
𝜔𝑝,𝐶𝐿
1 + 𝛽𝐴𝑂𝐿𝑜 𝜔𝑝,𝑂𝐿
▪ But the pole is at a much higher frequency
𝜔𝑝,𝐶𝐿 = 1 + 𝐿𝐺𝑜 𝜔𝑃,𝑂𝐿
❑ CL DC gain reduced by 1 + 𝐿𝐺𝑜
❑ CL bandwidth extended by 1 + 𝐿𝐺𝑜
❑ GBW (and UGF) remains constant
15: Negative Feedback
26
Bandwidth Extension
❑ CL DC gain reduced by 1 + 𝐿𝐺𝑜
❑ CL bandwidth extended by 1 + 𝐿𝐺𝑜
❑ GBW (and UGF) remains constant
𝐴𝐶𝐿
𝐴𝑂𝐿𝑜
𝐴𝐶𝐿𝑜
1 + 𝐿𝐺𝑜
𝑠 =
=
𝑠
𝑠
1+
1+
𝜔𝑝,𝐶𝐿
1 + 𝐿𝐺𝑜 𝜔𝑝,𝑂𝐿
𝐴𝑂𝐿
𝐴𝑂𝐿𝑜
1 + 𝛽𝐴𝑂𝐿𝑜
𝜔𝑝,𝑂𝐿
𝜔𝑢
𝜔𝑝,𝑂𝐿 1 + 𝛽𝐴𝑂𝐿𝑜
15: Negative Feedback
27
Modification of I/O Impedances
❑ Shunt sensing/mixing → R decreases
❑ Series sensing/mixing → R increases
𝑨𝑶𝑳
𝑨𝑶𝑳
𝜷
𝜷
𝑨𝑶𝑳
𝑨𝑶𝑳
𝜷
15: Negative Feedback
𝜷
[Razavi, 2017]
28
The Price We Pay
❑ Negative feedback useful properties
1. Gain Desensitization → Stable, linear, and accurate gain
2. Bandwidth Extension
3. Modification of I/O Impedances
❑ The price we pay to buy these useful properties
1. Gain reduction
2. The risk of instability
15: Negative Feedback
29
Outline
❑
❑
❑
❑
❑
❑
❑
Recapping previous key results
General feedback system
Loop gain
Why negative feedback?
Stability of feedback system
Root locus and Bode plot
Phase and gain margin
15: Negative Feedback
30
Stability of Feedback System
𝐴𝐶𝐿
𝑉𝑜𝑢𝑡 𝑠
𝐴𝑂𝐿 𝑠
𝑠 =
=
𝑉𝑖𝑛 𝑠
1 + 𝛽𝐴𝑂𝐿 𝑠
❑ Barkhausen’s Oscillation Criteria
𝛽𝐴𝑂𝐿 𝑠 = 1
∠𝛽𝐴𝑂𝐿 𝑠 = −180
Vin
+
+
_
Verr
Vfb
15: Negative Feedback
AOL
Vout
β
31
Stable vs Unstable System: Pole-Zero Plot
Laplace domain
1
𝑠−𝑎
15: Negative Feedback
Time domain
𝑒 𝑎𝑡
[Razavi, 2017]
32
Stable vs Unstable System: Bode Plot
❑ Gain crossover frequency (GX): @ 𝛽𝐴𝑂𝐿 𝑠 = 1
▪ Same as 𝜔𝑢
❑ Phase crossover frequency (PX): @ ∠𝛽𝐴𝑂𝐿 𝑠 = −180
❑ For a stable system: GX < PX
𝟐𝟎 log 𝜷𝑨𝑶𝑳 𝝎
𝟐𝟎 log 𝜷𝑨𝑶𝑳 𝝎
GX
𝟎 𝒅𝑩
∠𝜷𝑨𝑶𝑳 𝝎
−𝟏𝟖𝟎𝒐
15: Negative Feedback
PX
𝝎
𝝎
𝟎 𝒅𝑩
∠𝜷𝑨𝑶𝑳 𝝎
GX
𝝎
PX
𝝎
−𝟏𝟖𝟎𝒐
33
Effect of Feedback Factor 𝜷
❑ We assume 𝛽 is independent of frequency
▪ ∠𝛽𝐴𝑂𝐿 is independent of 𝛽 → PX is independent of 𝛽
❑ Increasing 𝛽 shifts mag up → GX increases → bad for stability
❑ Worst-case stability corresponds to 𝛽 = 1 → 𝛽𝐴𝑂𝐿 = 𝐴𝑂𝐿
▪ Unity-gain feedback → buffer → smallest CL gain
𝟐𝟎 log 𝜷𝑨𝑶𝑳 𝝎
𝟎 𝒅𝑩
∠𝜷𝑨𝑶𝑳 𝝎
𝝎
PX
𝝎
−𝟏𝟖𝟎𝒐
15: Negative Feedback
34
Outline
❑
❑
❑
❑
❑
❑
❑
Recapping previous key results
General feedback system
Loop gain
Why negative feedback?
Stability of feedback system
Root locus and Bode plot
Phase and gain margin
15: Negative Feedback
35
Single-Pole System: Root Locus
❑ Note that 𝛽 does not affect poles of 𝐿𝐺 (assuming 𝛽 is real!)
▪ But it affects poles of 𝐴𝐶𝐿 : Roots of the characteristic equation 1 + 𝛽𝐴𝑂𝐿
❑ The locus exists on real axis to the left of an odd number of poles and zeros.
❑ The locus starts at the open-loop poles and ends at the open-loop zeros or at infinity.
❑ For first-order system, pole always in LHP: Unconditionally stable
jω
β↑
β=0
σ
15: Negative Feedback
36
Single-Pole System: Bode Plot
𝐴𝐶𝐿
𝐴𝑂𝐿𝑜
𝐴𝑂𝐿 𝑠
𝐴𝐶𝐿𝑜
1 + 𝛽𝐴𝑂𝐿𝑜
𝑠 =
=
=
𝑠
𝑠
1 + 𝛽𝐴𝑂𝐿 𝑠
1+
1+
𝜔𝑝,𝐶𝐿
1 + 𝛽𝐴𝑂𝐿𝑜 𝜔𝑝,𝑂𝐿
𝟐𝟎 log 𝜷𝑨𝑶𝑳 𝝎
∠𝜷𝑨𝑶𝑳 𝝎
15: Negative Feedback
[Razavi, 2017]
37
Two-Pole System: Root Locus
❑ Poles always in LHP: Unconditionally stable
𝐴𝐶𝐿𝑜
𝐴𝐶𝐿𝑜
𝐴𝐶𝐿 𝑠 =
=
2
1 𝑠
𝑠
𝑠
𝑠2
1+
+
1 + 2𝜁
+
𝑄 𝜔𝑜 𝜔𝑜2
𝜔𝑜 𝜔𝑜2
▪ 𝜁 > 1 (𝑄 < 0.5): Overdamped (real and distinct CL poles)
▪ 𝜁 = 1 (𝑄 = 0.5): Critical damped (real and equal CL poles)
▪ 𝜁 < 1 (𝑄 > 0.5): Underdamped (complex conjugate CL poles)
15: Negative Feedback
[Sedra/Smith, 2015] & [Razavi, 2017]
38
Two-Pole System: Bode Plot
❑ Phase shift always < 180𝑜
▪ Unconditionally stable
❑ 𝜁 > 1 (𝑄 < 0.5): Overdamped
❑ 𝜁 = 1 (𝑄 = 0.5): Critical damped
❑ 𝜁 < 1 (𝑄 > 0.5): Underdamped
▪ Overshoot in step response
𝟐𝟎 log 𝜷𝑨𝑶𝑳 𝝎
❑ 𝜁 < 1/ 2 = 0.707
▪ 𝑄 > 1/ 2 = 0.707
▪ Peaking in frequency response
▪ Ringing in step response
15: Negative Feedback
∠𝜷𝑨𝑶𝑳 𝝎
𝑨𝑪𝑳 𝝎
[Razavi, 2017]
39
Three-Pole System: Root Locus
❑ Poles cross 𝑗𝜔 axis at a specific value of 𝛽 = 𝛽𝑐𝑟𝑖𝑡
▪ The system becomes unstable
❑ Pole are NOT always in LHP
▪ Conditionally stable: 𝛽 < 𝛽𝑐𝑟𝑖𝑡
jω
β↑
σ
15: Negative Feedback
40
Three-Pole System: Bode Plot
❑ Instability (oscillation) condition can be satisfied
𝟐𝟎 log 𝜷𝑨𝑶𝑳 𝝎
∠𝜷𝑨𝑶𝑳 𝝎
𝑨𝑪𝑳 𝝎
15: Negative Feedback
[Razavi, 2017]
41
Stability Summary
❑ First-order system
▪ Unconditionally stable
❑ Third order system
▪ Conditionally stable: Set 𝛽 < 𝛽𝑐𝑟𝑖𝑡
❑ Second-order system
▪ Unconditionally stable
▪ But may suffer from CL peaking/ringing if close to oscillation condition
▪ How much margin is needed?
15: Negative Feedback
42
Outline
❑
❑
❑
❑
❑
❑
❑
Recapping previous key results
General feedback system
Loop gain
Why negative feedback?
Stability of feedback system
Root locus and Bode plot
Phase and gain margin
15: Negative Feedback
43
Phase and Gain Margin
𝑃𝑀 =
180𝑜
− ∠𝛽𝐴𝑂𝐿 𝐺𝑋
=
180𝑜
−
tan−1
𝜔𝑢
𝜔𝑢
−1
− tan
𝜔𝑝1
𝜔𝑝2
𝐺𝑀 = 0 − 20 𝑙𝑜𝑔 𝛽𝐴𝑂𝐿 𝑃𝑋
𝟐𝟎 log 𝜷𝑨𝑶𝑳 𝝎
GX
𝟎 𝒅𝑩
GM
𝝎
∠𝜷𝑨𝑶𝑳 𝝎
PX
𝝎
−𝟏𝟖𝟎𝒐
PM
15: Negative Feedback
44
Phase Margin (PM)
𝑃𝑀 =
90𝑜
− tan−1
𝜔𝑢
𝜔𝑝2
PM > 0 → stable
But low PM means:
→ frequency domain peaking
→ time domain ringing
𝜷𝑨𝑶𝑳 𝝎
∠𝜷𝑨𝑶𝑳 𝝎
𝑨𝑪𝑳 𝝎
Frequency domain peaking
→ noise amplification
Time domain ringing
→ poor settling time
15: Negative Feedback
[Razavi, 2017]
45
Phase Margin (PM)
𝑃𝑀 =
90𝑜
− tan−1
𝜔𝑢
𝜔𝑝2
PM > 0 → stable
But low PM means:
→ frequency domain peaking
→ time domain ringing
𝜷𝑨𝑶𝑳 𝝎
∠𝜷𝑨𝑶𝑳 𝝎
𝑨𝑪𝑳 𝝎
Frequency domain peaking
→ noise amplification
Time domain ringing
→ poor settling time
15: Negative Feedback
[Razavi, 2017]
46
Phase Margin: Ultimate GBW
❑ If 𝜔𝑝2 = 𝜔𝑢 : PM = 45o
▪ Typically inadequate (peaking/ringing)
❑ Thus 𝜔𝑝2 should be > 𝜔𝑢 → 𝜔𝑝1 ≪ 𝜔𝑢 < 𝜔𝑝2
▪ 𝜔𝑝1 defines OL BW and 𝜔𝑝2 defines ultimate GBW (max CL BW)
𝟐𝟎 log 𝜷𝑨𝑶𝑳 𝝎
Frequency domain peaking
→ noise amplification
Time domain ringing
→ poor settling time
GX
𝝎
∠𝜷𝑨𝑶𝑳 𝝎
𝝎
PM
15: Negative Feedback
47
Asymptotic vs Actual UGF
❑ For 𝜔 < 𝜔𝑢 the Bode plot is similar to a 1st order system
❑ For a true 1st order system: GBW = UGF = 𝜔𝑢 = 𝜔𝑢1
❑ But the 2nd pole causes some bending below the asymptote
▪ Actual 𝜔𝑢 slightly < 𝜔𝑢1 : UGF < GBW
15: Negative Feedback
[Jespers & Murmann, 2017]
48
Optimum Phase Margin (Freq Response)
❑ Maximum CL BW without peaking occurs at 𝜁 = 𝑄 = 0.707
▪ Maximally flat response
15: Negative Feedback
49
Optimum Phase Margin (Freq Response)
❑ Maximum CL BW without peaking occurs at 𝜁 = 𝑄 = 0.707
▪ Maximally flat response
▪ 𝜔𝑝2 = 2𝜔𝑢1 and 𝑃𝑀 ≈ 65𝑜
15: Negative Feedback
[Jespers & Murmann, 2017]
50
Optimum Phase Margin (Freq Response)
❑ Maximum CL BW without peaking occurs at 𝜁 = 𝑄 = 0.707
▪ Maximally flat response
▪ 𝜔𝑝2 = 2𝜔𝑢1 and 𝑃𝑀 ≈ 65𝑜
❑ But 𝜁 < 1: Underdamped system
▪ Overshoot exists in transient response
15: Negative Feedback
51
Optimum Phase Margin (Tran Response)
❑ Fastest settling without overshoot occurs at 𝜁 = 1 (𝑄 = 0.5)
▪ Critical damped system
▪ 𝜔𝑝2 = 4𝜔𝑢1 and 𝑃𝑀 ≈ 76𝑜
15: Negative Feedback
[Jespers & Murmann, 2017]
52
Second-Order System Speed-up
❑ First order system corresponds to 𝜔𝑝2 /𝜔𝑢1 → ∞
▪ Too much overdamping
❑ The settling of 2nd order system with optimum PM is faster
▪ Critical damped system is faster than overdamped system
❑ But we must take some extra margin to account for variations
15: Negative Feedback
[Jespers & Murmann, 2017]
53
Thank you!
15: Negative Feedback
54
References
❑ A. Sedra and K. Smith, “Microelectronic Circuits,” Oxford University Press, 7th ed., 2015
❑ B. Razavi, “Design Of Analog CMOS Integrated Circuit,” 2nd ed., McGraw-Hill, 2017.
❑ T. C. Carusone, D. Johns, and K. W. Martin, “Analog Integrated Circuit Design,” 2nd ed.,
Wiley, 2012.
❑ P. Jespers and B. Murmann, Systematic Design of Analog CMOS Circuits Using PreComputed Lookup Tables, Cambridge University Press, 2017.
15: Negative Feedback
55
Bandwidth Extension
❑ Cascade of feedback amplifiers provides the same gain and a much faster response
▪ But power consumption and area doubled
15: Negative Feedback
[Razavi, 2017]
56