LECTURE NOTES for Radio Frequency Integrated Circuits and Systems 2nd Edition by Hooman Darabi

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

10

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

12

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

16

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

19

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

21

RF Communication Systems • Analog linear and nonlinear modulation • Communication systems • AM and FM

22

Communication Systems • Almost all radios communicate at a carrier frequency • Accommodate more users • More efficient • Antenna size, … Transmitter

Channel

Receiver

0

fc Noise, Interference, ... 23

Amplitude Modulation • Traditionally used in AM radio 𝑥𝐶 𝑡 = 𝐴(𝑡)𝑐𝑜𝑠(𝜔𝐶 𝑡) = 𝐴𝐶 [1 + 𝜇𝑥 𝑡 ]𝑐𝑜𝑠(𝜔𝐶 𝑡)

• m is modulation index and typically less than one • For the BB data: < 𝑥(𝑡) >≤ 1 2

1

x(t) t

-1 xC(t)

AC(1+m) x(t)

xC(t)

AC(1-m) t

AM Waveforms

Simplified Peak Detector

24

AM Spectrum • Applying Fourier transform leads to spectrum |XC(f)| Carrier Upper sideband

Lower sideband -fC

0

fC-W

fC

f fC+W

• The power of modulates signal is: < 𝑥𝐶 (𝑡)2 >=

1 2 1 1 𝐴𝑐 1 + 𝜇 2 < 𝑥 𝑡 2 > = 𝐴𝑐 2 + 2 × 𝐴𝑐 2 𝜇 2 < 𝑥 𝑡 2 > 2 2 4

• Half the power wasted on the unsuppressed carrier 25

Suppressed Carrier AM • Carrier suppressed using a balanced modulator 1/2x(t)

AM Modulator

Ac(1+.5x(t))coswCt

+ S ACcoswCt -1/2x(t)

AM Modulator

- x(t)A cosw t C

C

Ac(1-.5x(t))coswCt

• Known as double-sideband suppressed carrier AM (DSB) 𝑥𝐶 𝑡 = 𝐴𝐶 𝑥 𝑡 cos(𝜔𝐶 𝑡) 26