RF Integrated Circuits and Systems Hooman Darabi March 2020
1
Circuit
System
Course Syllabus • RF IC Components • RF Communication Systems • RF networks • Noise • Distortion • Low-noise amplifiers • Mixers • Oscillators • Power amplifiers • Transceiver architectures 2
Teaching Guidelines • Slides only used as a guideline, best to use whiteboard only • Likely not possible to cover all slides for basic RF course • Skip the following sections: ̶
Chapter 3: Slides 41-43 ̶
Chapter 5: Slides 56-57, 61, 64-67 ̶
Chapter 7: Slides 108-111, 115-118 ̶
Chapter 9: Slides 144-151, 157-163
• About 7-8 slides per lecture may be covered in 20-lecture quarter-based curricula
• No material for PLLs (chapter 10). It may be covered in a separate course. 3
RF IC Components • Transistors • High frequency model good for RF (several GHz)
• Resistors • LC circuits • Capacitors • Inductors • Differential, Transformers
• Transmission lines 4
LC Circuits • Widely used in amplifiers, oscillators … • The energy moves between L and C • In lossy circuit however, energy decays eventually
+
iL(t)
C vc(t)
L
+ C
-
Ideal: lossless
R
vc(t) -
Practical 5
Lossless LC Circuit: Ideal Oscillator 𝜕 2 𝑣𝐶 1 + 𝑣 =0 𝜕𝑡 2 𝐿𝐶 𝐶
• Solve the differential equation: 𝑣 𝑡 = 𝑉 𝑐𝑜𝑠𝜔 𝑡 • The capacitor voltage: 𝐶
• And its energy:
0
0
1 𝑊𝐶 𝑡 = 𝐶𝑉0 2 𝑐𝑜𝑠𝜔0 𝑡 2 2 1
𝐿𝐶
V0 : Initial voltage ½CV02 𝑊𝐶 𝑡 =
1 𝐶𝑉0 2 𝑐𝑜𝑠𝜔0 𝑡 2 2
1 𝑊𝐿 𝑡 = 𝐶𝑉0 2 𝑠𝑖𝑛𝜔0 𝑡 2 2
p/2
p
w0t
6
Lossy LC Circuit • The new differential equation: • The Laplace-domain roots are:
𝜕 2 𝑣𝐶 1 𝜕𝑣𝐶 1 + + 𝑣 =0 𝜕𝑡 2 𝑅𝐶 𝜕𝑡 𝐿𝐶 𝐶
𝜔0 1 𝑠1,2 = − ± 𝑗𝜔0 1 − 2 = −𝛼 ± 𝑗𝜔𝑑 2𝑄 4𝑄 𝜔0 =
Where:
• Thus:
𝑖𝐿 𝑡 =
jwd
w0
1
𝐿𝐶 𝑅 𝑄= = 𝑅𝐶𝜔0 𝐿𝜔0
𝑣𝐶 𝑡 = 𝑉0
jw
f
s
-a
𝜔0 −𝛼𝑡 𝑒 cos(𝜔0 𝑡 + 𝜙) 𝜔𝑑
𝑉0 −𝛼𝑡 𝑒 sin𝜔0 𝑡 𝐿𝜔𝑑
Assume Q >> 1 7
Energy Balance in Low-Loss LC Circuit • The total energy is: 𝑊𝑇 (𝑡) = 𝑊𝑐 𝑡 + 𝑊𝐿 𝑡 ≈
1 𝐶𝑉0 2 𝑒 −2𝛼𝑡 2
Capacitor Energy
½CV02
e-2at
p/2
p
• Energy still moves between L and C, but also decays 8
Quality Factor • The power dissipated is:
𝑝(𝑡) = −
𝑑𝑊𝑇 𝜔0 = 2𝛼𝑊𝑇 = 𝑊 𝑑𝑡 𝑄 𝑇
e-2at
WT
Decay Rate
(½CV02)(w0/Q)
p/2
Q p
w0t
• A more physical definition of Q: 𝑊𝑇 𝑇𝑜𝑡𝑎𝑙 𝐸𝑛𝑒𝑛𝑟𝑔𝑦 𝑆𝑡𝑜𝑟𝑒𝑑 𝑄 = 𝜔0 = 𝜔0 𝑝 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃𝑜𝑤𝑒𝑟 𝐷𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑒𝑑 9
More Practical Q Definition • Practical inductors/capacitors have series loss 1 𝑟 − 𝑗𝐿𝜔 1 1 = = + • Since: 1 𝑟 + 𝑗𝐿𝜔 𝑟 + (𝐿𝜔 ) 𝑟(𝑄 + 1) 0
0
2
0
2
2
𝑗𝐿𝜔0 (1 +
𝑄2
)
L
@ r
• In general:
𝑄=
L
R=rQ2
• The circuits below are equivalent (at one frequency)
𝑅 𝐿𝜔0 = 𝐿𝜔0 𝑟
1 1 1 ≈ + 𝑄 𝑄𝐿 𝑄𝐶
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Practical Oscillators • Active circuit to balance the power dissipated I
+ L
C
R
I(t)
vc(t) -
I(t)
Active Circuit
VDD +I0
V -I0
• Power produced:
V0 vc(t)
1 2𝑉0 𝐼0 𝑝 = − න (𝑉0 𝑐𝑜𝑠𝜔0 𝑡)𝐼(𝑡)𝑑𝑡 = 𝑇 𝑇 𝜋
• According to definition:
•
+I0
I(t)
1 2 𝐶𝑉 𝑊𝑇 𝜋 𝑉0 0 𝑄 = 𝜔0 = 𝜔0 2 = 𝜔0 𝐶 = 𝑅𝐶𝜔0 2𝑉0 𝐼0 𝑝 4 𝐼0 𝜋 4 𝑉0 = 𝑅𝐼0 Thus: 𝜋
-V0
-I0
t
11
Linear Capacitors • Fringe capacitors, cheap and dense • Scales with technology • Almost 6fF/mm2 in 28nm B
A
A A
B
Top View
r
C
B
Cbottom Cbottom Substrate
CSUB/RSUB
Side View
Lumped Model
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MOS Capacitors • More density by using MOS capacitors • However, nonlinear • Either bias properly • Or use as a varactor: provides continuous tuning
• In 28nm as dense as 22/10fF/mm2 for thin/thick VG VCTRL n+
n+
Regular
n-well
p-sub
Accumulation
Accumulation-Mode MOSCAP 13
Tunable Capacitors • For noise reasons, the varactor size must be limited • Use discrete tuning to extend the range S1 S2
… C
2C
2nC
… Sn VCTRL Discrete Tuning Cocept
VCTRL Differential Design
14
Inductors • Ampere’s circuital law describes magnetic field versus current: dL I
ර 𝑯. 𝑑𝑳 = 𝐼
Arbitrary closed path
• The magnetic flux is: 𝜙 = න 𝐁 ⋅ 𝑑𝐒 = 𝜇0 න 𝑯 ⋅ 𝑑𝐒 𝑆
𝑆
• By definition, the inductance is: 𝑆𝑑 ⋅ 𝑯 𝑆 𝜙 𝐿 = = 𝜇0 𝐼 𝑯 ׯ. 𝑑𝐿 15
Integrated Inductors More constructive H fields
• Inevitably, need spirals
d I
√2 d
H
• Single- or multi-turn often using top metal 𝐿𝑇 = 𝐿 + 𝑀+ − 𝑀− W
S
Strong M
DIN Weak M
DOUT
Complex function of geometry
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Inductor Q: Ohmic Loss • Metal resistance a key contributor 𝜇 2𝑙 • For a piece of wire: 𝐿 ≈ 2𝜋 𝑙(𝑙𝑛 + 0.5) 𝑊+𝑡 0
• The resistance is:
𝑟 = 𝑅□
𝑙 𝑊
• At high frequency the current propagates mostly at surface
• The skin depth is:
𝛿=
1 1 𝜔 𝜇 𝜎 2 0 0
• The resistance becomes: 𝑟 = 𝑅□
𝑙 𝑡/𝛿 𝑊 1 − 𝑒 −𝑡/𝛿 17
Inductor Q: Other Loss Mechanisms • More dominant at higher frequencies
Displacement Current
H
RSUB Substrate
Capacitive Loss
CSUB
Induced Current in substrate Substrate
Magnetic Loss 18
Differential Inductors • Most RF circuits, particularly oscillators are differential
S1 S2 P1 P2 Differential Circuit
Differential Circuit
Two Single-Ended Inductors
Differential Inductors
Transformer
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Inductor Lumped Model • Physical model, with good accuracy 5.E-09
CF r
L COX
4.E-09 3.E-09 2.E-09 1.E-09
5.0E+08 1.5E+09 2.5E+09 3.5E+09 4.4E+09 5.4E+09 6.4E+09 7.4E+09 8.4E+09 9.4E+09 1.0E+10 1.1E+10 1.2E+10 1.3E+10 1.4E+10 1.5E+10 1.6E+10 1.7E+10 1.8E+10 1.9E+10 2.0E+10
0.E+00
CSUB/RSUB
Inductance versus frequency 𝐼𝑚[𝑍𝐼𝑁 ] 𝐿 𝜔 = ≅𝐿 𝜔
1 − 𝐿𝐶𝑆𝑈𝐵 𝜔2 (1 − 𝐿𝐶𝑆𝑈𝐵
𝜔 2 )2 +
𝐿
(𝑅 + 𝑟𝐶𝑆𝑈𝐵 )𝜔 𝑆𝑈𝐵
2
20
Integrated Transformers • Similar to differential inductors • An example of 2-turn with center tap:
P1 S1
12mm
S2 P2
5mm
245mm
AP Layer
145mm
270mm C
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RF Communication Systems • Analog linear and nonlinear modulation • Communication systems • AM and FM
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