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

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

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

Suppressed Carrier AM Spectrum • Similar spectrum as AM (without carrier) • But the signal undergoes phase reversal • For a tone: 𝑥𝐶 𝑡 = 𝐴𝐶 𝐴𝑚 𝑐𝑜𝑠𝜔𝑚 𝑡 cos 𝜔𝐶 𝑡 𝐴𝐶 𝐴𝑚 𝐴𝐶 𝐴𝑚 = cos 𝜔𝐶 − 𝜔𝑚 𝑡 + cos 𝜔𝐶 + 𝜔𝑚 𝑡 2 2

1/4ACAm

Amplitude

Amplitude

½AC 1/4mACAm

fC

fC+fm

f fC-fm

fC

fC+fm

fC-fm

f

• An envelope detector can no longer be used 27

Single-Sideband AM • Upper/lower sidebands of DSB carry same information • Cutting either sideband leads to spectrum efficiency |XC(f)| x(t)

Balanced Modulator

Sideband Filter

Sideband Filter SSB

f

ACcoswCt

-fC

fC

0

fC+W

• In practice, we need quadrature carrier/input

x(t)

Ac/2coswCt -90º

+ S - xC(t)

HQ(f)

X^(t)

28

Frequency Modulation • Info carried in frequency

x(t)

𝑥𝐶 𝑡 = 𝐴𝐶 𝑐𝑜𝑠(𝜔𝐶 𝑡 + 2𝜋𝑓Δ න 𝑥 𝜏 𝑑𝜏)

• fD<fC: frequency deviation • Constant envelope • Instantaneous frequency: 𝑓 𝑡 = 𝑓𝐶 + 𝑓Δ 𝑥(𝑡)

t AM

t

FM

• FM & PM are essentially the same

t

29

Narrowband FM • If the frequency (phase) deviation small:

• Then:

𝜙(𝑡) ≪ 1

𝑥𝐶 𝑡 = 𝐴𝐶 𝑐𝑜𝑠(𝜔𝐶 𝑡 + 𝜙(𝑡)) ≈ 𝐴𝐶 [𝑐𝑜𝑠𝜔𝐶 𝑡 − 𝜙(𝑡)𝑠𝑖𝑛𝜔𝐶 𝑡]

• For a tone: ½AC

) A( t

Acb/4

fC-fm

f(t)

fC

fm

fm

AC

fC+fm

f

bAC/2

-Acb/4 30

FM Spectrum • In general, FM spectrum expressed by Bessel series 𝐴 𝛽 = ( )𝑓 • For a tone, with: 𝑓 𝑚

𝑚

𝛥

∞

• We have:

𝑥𝐶 𝑡 = 𝐴𝐶 ෍ 𝐽𝑛 𝛽 cos(𝜔𝐶 𝑡 +𝑛𝜔𝑚 )𝑡 𝑛=−∞

• FM bandwidth is infinite in theory J0(b) AC

J1(b) J2(b)

fC-fm

fC

-J3(b)

f(t)

J3(b) fC+fm

J2(b)

Evenorder

Oddorder

f

-J1(b) 31

Modern Radios Modulation • Use both amplitude & frequency modulation • Created digitally • I/Q inputs are typically uncorrelated Digital Modulator

DAC I

Lowpass Filter Ac/2coswCt -90º

Q

+

S - xC(t)

DAC 32

RF Networks • Available power • Matching networks • Transformers • Parallel-series conversion

• Scattering parameters

33

Available Power Antenna

• The complex power delivered:

1 1 𝑍𝐼𝑁 𝑣𝑠 2 ∗ 𝑃 = 𝑉𝐼𝑁 𝐼𝐼𝑁 = 2 2 𝑍𝑠 + 𝑍𝐼𝑁 2

Zs vs

+ vIN -

ZIN

gmvIN

ZO

ZL

Unilateral Amplifier

• The power maximized if:

𝑍𝐼𝑁 = 𝑍𝑠 ∗

• The available average power is:

𝑣𝑠 2 1 𝑃𝑎 = = 𝑃𝑠 8𝑅𝑠 2

• The available power is only a function of source • Power (conjugate) matching not always beneficial 34

Available Power Gain • By definition, available power gain is: • In the unilateral amplifier:

𝐺𝑎 =

𝑃𝑎,𝑂𝑈𝑇 𝑃𝑎,𝐼𝑁

𝑔𝑚 𝑍𝑜 𝑣𝐼𝑁 2 2 𝑅𝑠 𝑍𝐼𝑁 8𝑅𝑂 𝐺𝑎 = = 𝑔𝑚 𝑍𝑜 2 2 𝑣𝑠 𝑅𝑂 𝑍𝑠 + 𝑍𝐼𝑁 8𝑅𝑠

• When matched at input, Ga is maximized • For an arbitrary 2-port: Ys 𝑌21 𝐺𝑎 = 𝑌𝑠 + 𝑌11

2

𝑅𝑒[𝑌𝑠 ] 𝑅𝑒[𝑌𝑂𝑈𝑇 ]

vs

[Y]

YIN

YL

YOUT 35

Matching Networks • Need to transform the amplifier input impedance • Noise or conjugate matching Matching Netwrok

Source

ZOPT

Amplifier

ZIN

• Amplifier impedance may be different • External components need a well-defined often 50W impedance Matching Netwrok

50W

Receiver

36

Wideband Transformers • In an ideal transformer: 𝑣1 (𝑡) 𝑛1 = 𝑣2 (𝑡) 𝑛2

,

B

i1

i2

+ v1 -

𝑖1 (𝑡) 𝑛2 =− 𝑖2 (𝑡) 𝑛1

+ v2 -

• No energy storing: 𝑣 𝑡 𝑖 𝑡 + 𝑣 𝑡 𝑖 𝑡 = 0 • In practice realized by coupled inductors 1

1

2

2

• Lossy, not as wideband

• Transforms impedances 𝑛1 𝑍𝐼𝑁 (𝑗𝜔) = 𝑛2

2

𝑍𝐿 (𝑗𝜔)

ZL

ZIN n1 : n2

37

Parallel-Series Conversion • For a narrow band, we may approximate transformer • The two circuits equivalent if: 𝑅𝑠 = 𝑅𝑝

𝑋𝑝 2 𝑅𝑝 2 + 𝑋𝑝 2

& 𝑋𝑠 = 𝑋𝑝

𝑅𝑝 2 𝑅𝑝 2 + 𝑋𝑝 2

𝑅𝑠 2 + 𝑋𝑠 2 𝑅𝑠 2 + 𝑋𝑠 2 𝑅𝑝 = & 𝑋𝑝 = 𝑅𝑠 𝑋𝑠 Xs Rs

≡

Xp

Rp

Up

𝑅𝑠 𝑅𝑝 = 𝑋𝑠 𝑋𝑝 Down

38

Example of Amplifier Matching Design • Assume an amplifier with resistive input Ls C L

RIN

Matching Network

𝑅𝑠 = 𝑅𝐼𝑁

+ vIN -

C

Rs

Amplifier

(𝐿𝜔0 )2 𝑅𝐼𝑁 2 + (𝐿𝜔0 )2

𝐿=

𝑅𝐼𝑁 𝑅𝑠 𝜔0 𝑅𝐼𝑁 − 𝑅𝑠

𝐿𝑠 = 𝐿

𝑅𝐼𝑁 2 𝑅𝐼𝑁 2 + (𝐿𝜔0 )2

=

𝑅𝑠 (𝑅𝐼𝑁 − 𝑅𝑠 ) 𝜔0

1 1 𝐶= = 𝐿𝑠 𝜔0 2 𝜔0 𝑅𝑠 (𝑅𝐼𝑁 − 𝑅𝑠 ) 39

Other Features of Matching Circuit • In addition to impedance conversion, 𝐴𝑎 =

𝑅𝐼𝑁 ≈𝑄>1 𝑅𝑠

Available Voltage Gain

• It provides voltage (current) gain • Some filtering

Frequency, GHz

• Works for a narrow band 40

Scattering Parameters • Z or Y parameters cannot be extracted at microwave frequencies V

1

+

V1-

• More convenient to

N-Port

measure incident reflected power

VN+

VN-

• Scattering parameters defined accordingly 𝑉1 − 𝑆11 − 𝑆 𝑉2 = 21 ⋯ ⋯ − 𝑆𝑁1 𝑉𝑁

𝑆12 𝑆21 ⋯ 𝑆𝑁2

⋯ 𝑆1𝑁 𝑉1 + ⋯ 𝑆2𝑁 𝑉2 + ⋯ ⋯ ⋯ ⋯ 𝑆𝑁𝑁 𝑉𝑁 +

• Not unique to microwave design 41

S and Z Matrixes • By definition:

𝑉 + = 𝑍0 𝐼 + 𝑉 − = −𝑍0 𝐼 −

• The voltage/current:

𝑉 = 𝑉+ + 𝑉− 1 𝐼 = 𝐼 + + 𝐼 − = (𝑉 + − 𝑉 − ) 𝑍0

(Z0: characteristics impedance)

• Thus, for the N-port: 𝑉 = 𝑉 + + 𝑉 − = 𝑍 𝐼 = 𝑍ҧ 𝑉 + − 𝑍ҧ 𝑉 − 𝑉− =

𝑍ҧ + 𝑈

−1 ( 𝑍ҧ

− 𝑈 ) 𝑉+

• Z (or Y) and S matrixes are uniquely related [𝑆] =

𝑍ҧ + 𝑈

−1

( 𝑍ҧ − 𝑈 )

Normalized Z

42

S Parameters Example • Consider the lumped circuit below Z0

jB

Z0

• S11 is input reflection with matched load 𝑌0 − 𝑌𝑖𝑛 𝑌0 − (𝑌0 + 𝑗𝐵) −𝑗𝐵 = = 𝑌0 + 𝑌𝑖𝑛 𝑌0 + (𝑌0 + 𝑗𝐵) 2𝑌0 + 𝑗𝐵 For S21 we set V2+ = 0 𝑆11 =

• • For as shunt impedance: 𝑉 = 𝑉 + 𝑉 = 𝑉 = 𝑉 + 𝑉 = 𝑉 • Thus: 𝑉 +𝑆 𝑉 =𝑉 2𝑌 𝑆 = 𝑆 = 1 + 𝑆 = • Or: 2𝑌 + 𝑗𝐵 1

1

+

21

11 1 12

+

2

1

+

1

−

2

2

+

2

−

2

−

−

11

0

0

43

Introduction to RF Filters • Introduction to RF filters • Passive filters • Quadrature signals and Hilbert transform • Polyphase filters

44

Ideal Filters • Extensively used to remove unwanted signals or noise • Ideal filters are non-causal • Impulse response exists for t < 0 • Cannot be realized with finite number of elements • Practical filters are not infinitely sharp h(t) Ð H(f)

2B

|H(f)|

1 B

td

t

1/B

45

Practical Filters • Approximate a certain polynomial • Butterworth (maximally flat), …

• The polynomial must be PR (positive, real e.g. for a 1port), and may be realized by an RLCM circuit jw

RS

√3/2

RL

-1

-0.5

s -√3/2

•

3rd-order transfer function: 𝐻(𝑠) =

1 (1 + 𝑠)(1 + 𝑠 + 𝑠 2 ) 46

RF SAW Filters • The finite Q of inductors limit the filter sharpness • Inductor or capacitor variation may be unacceptable too

• SAW filters: Surface acoustic wave: • Extremely high-Q • Not tunable • But low-loss and reliable

47

Hilbert Transform • Quadrature signals widely used in RF HQ(f) +j

−𝑗 𝑓 > 0 𝐻𝑄 𝑓 = ቊ +𝑗 𝑓 < 0 1 ℎ𝑄 𝑡 = 𝜋𝑡

0

f

-j

• Since the system is not casual, cannot be realized • However, approximated over a frequency range • Hilbert transform on cosine leads to sine 𝑥 𝑡 = 𝐴 cos(𝜔0 𝑡 + 𝜑) 𝑥 ^ 𝑡 = 𝐴 sin 𝜔0 𝑡 + 𝜑 = 𝐴𝑐𝑜𝑠(𝜔0 𝑡 + 𝜑 − 90°)

48

Single-Sideband (SSB) Receivers • Similar to SSB transmitter: coswCt

coswCt

I

I +

x(t)

S

±

+

xC(t) xC(t)

Q

Hilbert

S

±

Q

sinwCt

x(t)

Hilbert

sinwCt

Single-Sideband Transmitter

Single-Sideband Receiver

• Upper and lower sideband signals: 1/2

fm

-fC *

fC

Cosine

0

j/2

fC

-fC

1/2

Sine

-j/2

-j/2

0

fm

j/2

-j/2

1/2 1

I 0

After Addition 1

1/2

1 0

1

1/2 After Subtraction

Q

0 -1/2

j/2

After Multiplication & Lowpass Filtering Q

fC

Receiver Input

After Hilbert

1/2

I

fm

-fC

1/2

1/2

-1/2

0

49

Polyphase Networks • Although real, respond to quadrature sequences differently

• Creates a null at positive (or negative) frequencies 𝐻𝑝𝑝 (𝑓) ≠ 𝐻𝑝𝑝 (−𝑓) 90 2 180

3 1

0

Positive sequence

4

V1

270

V2 V3

270 2 180

HPP(f)

V4

3 1

0

f 0 Negative freq. (or sequence)

Positive freq. (or sequence)

Negative sequence

4 90 50

PPF Frequency Transformation • Use a linear frequency transformation:

𝜔 ↔ 𝜔 − 𝜔0

• A capacitor thus changes to: 𝑗𝐶𝜔 ↔ 𝑗𝐶𝜔 − 𝑗𝐶𝜔0

Complex Resistor

• The complex resistor created using quadrature input C V1

V2 = -j× V1

Cw0 51

Passive PPF Realization • A 1st-order PFF may be realized by an RC circuit • Depending on the sequence, rejects at ±1/RC • Cascade of stages to extend the range R |HPP(w)|

C

3dB

w 0 -1/RC

1/RC

52

Polyphase Filters in SSB Receivers • To realize Hilbert at a narrow frequency range, a polyphase filter may be used coswCt

coswCt

I

I+

+

xC(t)

IS

±

Q

xC(t)

Polyphase Filter

x(t)

Q+

Hilbert Q-

sinwCt

sinwCt

Single-Sideband Receiver Using Hilbert Transform

Single-Sideband Receiver Using Polyphase Filter

• Upper and lower sidebands downconvert with different phase sequence

• One is rejected by the filter 53

Noise • Types of noise • Thermal noise • Passive reciprocal circuits noise • MOSFET noise • Cyclostationary noise

• Amplifier noise representation • Noise figure and noise optimization • Sensitivity • Phase noise 54

Thermal Noise • Einstein in 1906 predicted thermal noise caused by Brownian motion

• Measured by Johnson, proved by Nyquist in 1928 • From equipartition law, PSD is: 𝑣 = 2𝐾𝑇𝑅 2

• The actual thermal noise PSD not flat

𝑆𝑣 (𝑓) =

2𝑅ℎ𝑓 𝑒 ℎ𝑓/𝐾𝑇 − 1

Sv(f)

∞

2 2(𝜋𝐾𝑇) 𝜎𝑣 2 = න 𝑆𝑣 (𝑓)𝑑𝑓 = 𝑅 3ℎ −∞

2KTR

0

1e4

f, GHz

• For all practical purposes however, it is assumed white 55

Maxwell-Boltzmann Statistics • From thermodynamics: E = mgh

Gas h

Gas atom density µ exp(-E/KT)

• For a capacitor, the energy is: • Thus, the probability density is:

1 2 𝐸 = 𝐶𝑣 2

𝑃∝𝑒

−𝐸 (𝐾𝑇 )

=𝑒

−𝐶𝑣 2 ( 2𝐾𝑇 ) 56

Proof of Thermal Noise PSD R

𝑃𝑉 𝑣 = 𝐾0 𝑒

emf

• For an RC circuit, PDF is: −𝐶𝑣 2 ( 2𝐾𝑇 )

∞

• Constant K0 found by: •

The 2nd moment:

C

+ v(t) -

න 𝑃𝑉 𝑣 𝑑𝑣 = 1 −∞

∞

2

𝑣 2 = න 𝑣 𝑃𝑉 𝑣 𝑑𝑣 = −∞

∞ −𝐶𝑣 2 𝐶 𝐾𝑇 2 ( 2𝐾𝑇 ) න 𝑣 𝑒 𝑑𝑣 = 2𝜋𝐾𝑇 −∞ 𝐶

• Thus in the RC circuit we have: ∞

𝑣 2 = 𝑆𝑒𝑚𝑓 (0) න 𝐻(𝑓) −∞

• Leading to:

2

∞

𝑆𝑒𝑚𝑓 (0) 𝐾𝑇 𝑑𝑓 𝑑𝑓 = 2𝑆𝑒𝑚𝑓 (0) න = 2 = 2𝑅𝐶 1 + (2𝜋𝑅𝐶𝑓) 𝐶 0

𝑆𝑒𝑚𝑓 0 = 2𝐾𝑇𝑅 57

Thermal Noise Model • Often only positive frequencies considered • Modeled by Norton or Thevenin equivalent circuit R

Si(f) = 4KT/R

R

Sv(f) = 4KTR

• Noise source must be dealt with statistically 58

Noise Bandwidth • Noise w/ PSD of h/2 passed thru filter:

𝑆𝑦 (𝑓) =

∞ 2 𝑦 = 𝜂 න 𝐻(𝑓) 2 𝑑𝑓

• Noise variance is:

𝜂 𝐻(𝑓) 2 2

0

• For a simple RC circuit:

𝑆𝑦 (𝑓) =

4𝐾𝑇𝑅 𝑓 1 + ( )2 𝐵

R

Sv(f) = 4KTR

R

C

𝐾𝑇 𝐶

• Integration yields: • Effective bandwidth is 1/4RC • For an nth-order Butterworth filter: 𝑦2 =

C

+ v(t) -

𝐵𝑁 =

𝜋𝐵

𝜋 2𝑛𝑠𝑖𝑛(2𝑛)

59

Noise of Passive Reciprocal Circuits • In RLCM circuit, given reciprocity & power conservation: 𝑁

𝑅𝑒 𝑍 𝑓

𝐼(𝑓) 2 = ෍ 𝑘=1

𝑉𝑘 (𝑓) 2 𝑅𝑘

𝑁

𝐻𝑘 (𝑓) 2 𝐼(𝑓) 2 =෍ 𝑅𝑘 𝑘=1

LCM

I

Rk

LCM

V = HkI

⁞

RN

⁞ R1

Hk(f)

Reciprocity

v(t)

Vk = Hk I

Z(s) Rk

LCM

I

⁞

𝑁

Hk(f)

2

• Thus: 𝑅𝑒 𝑍 𝑓 = ෍ 𝐻 𝑅(𝑓) 2𝐾𝑇 𝑆 𝑓 = ෍ 𝐻 (𝑓) = 2𝐾𝑇𝑅𝑒 𝑍 𝑓 • Output noise PSD: 𝑅 𝑘

𝑘=1

𝑘

𝑁

𝑣

𝑘=1

𝑘

𝑘

2

60

Nyquist Theorem & Equipartition law • Equipartition law assigns ½KT degree of freedom • From spectral density: 𝑅 𝜏 = 𝐾𝑇 𝑧 𝜏 + 𝑧 −𝜏 = 𝐾𝑇𝑧 𝜏 𝜏 > 0 𝑣

• In an arbitrary 1-port: 𝑠𝑍2 (𝑗𝜔) 1 = 𝜔→∞ 1 + 𝑗𝐶𝜔𝑍2 (𝑗𝜔) 𝐶

Lossless Reciprocal

R

C

lim 𝑗𝜔𝑍(𝑗𝜔) = 𝑙𝑖𝑚

𝜔→∞

• From initial value theorem: 𝑧 0 = 𝑙𝑖𝑚 𝑗𝜔𝑍(𝑗𝜔) = 𝐶1

+ v(t) -

Z2(s)

+

𝜔→∞

• Average capacitor voltage: • Energy stored on capacitor:

𝐾𝑇 = 𝐾𝑇𝑧 0 = 𝐶 1 2 1 2 1 𝐶𝑣 = 𝐶𝑣 = 𝐾𝑇 2 2 2

𝑣 2 = 𝑅𝑣 0

• Consistent with equipartition law

61

MOSFET Thermal Noise

8 1 + 𝜂 + 𝜂2 𝑆𝑖 = 𝐾𝑇𝑔𝐷𝑆 3 1+𝜂 𝑉𝐷𝑆 𝜂 =1− 𝑉𝐷𝑆,𝑆𝐴𝑇

Noise, A2/Hz

• In strong inversion, the channel acts like a resistor • For long channel devices: 28nm, L=100nm

• g > 2/3 for short channel

• Modelled same as resistors

VDS , V

Si(f) = 4KTggm Sv(f) = 4KTg/gm

62

MOSFET 1/f Noise • Caused by trapping/releasing of electrons near surface • Very long time constant EC --------Long t

n+

Traps

n+

-

Trap time constant

𝑐𝑡 𝜏𝑡 𝑆𝑖 (𝑓) = 1 + 2𝜋𝑓 2 𝜏𝑡 2

• Noise PSD Lorentzian: • Large # of traps with uniform tt leads to 1/f shape • Modelled as a voltage source at gate: 𝑆 (𝑓) = 𝑊𝐿𝐶𝐾 𝑓1 b: 0.7~1.2 • Bigger device, more averaging, less noise 𝑣

𝑂𝑋

2

𝑏

63

Noise of Periodically Varying Devices • In mixers or oscillators operating points change over time

• In a time varying conductance, G(t), If: • Change in the operating points is slow enough so that the dissipative device stays in thermal equilibrium • The fluctuations caused by noise is not large

• We can say:

𝑆𝑖 (𝑓, 𝑡) = 4𝐾𝑇𝐺(𝑡)

• Once viewed in a spectrum analyzer: 𝑆𝑖 (𝑓) =

1 න 𝑆 (𝑓, 𝑡)𝑑𝑡 𝑇 𝑇 𝑖

64

Cyclostationary Processes & Noise • Statistical properties of stationary process are time independent

• A periodically varying process (v(t)) is cyclostationary (in wide sense) if: 𝐸 𝑣 𝑡 + 𝑚𝑇

=𝐸 𝑣 𝑡

= 𝜂(𝑡)

𝑅𝑣 𝑡1 + 𝑚𝑇, 𝑡2 + 𝑚𝑇 = 𝑅𝑣 𝑡1 , 𝑡2

• Often treated as if stationary, e.g.:

• In general:

1 𝑇 𝑅𝑣 𝜏 = න 𝑅𝑣 𝑡 + 𝜏, 𝑡 𝑑𝑡 𝑇 0

Periodic, deterministic 𝑛𝑐𝑦𝑐𝑙𝑜 (𝑡) = 𝑛 𝑡 𝑤(𝑡)

Stationary

65

Cyclostationary Noise Spectrum • From autocorrelation definition: 𝑅𝑛𝑐𝑦𝑐𝑙𝑜 𝑡 + 𝜏, 𝑡 = 𝐸 𝑛 𝑡 + 𝜏 𝑤 𝑡 + 𝜏 𝑛∗ 𝑡 𝑤 ∗ 𝑡

= 𝑅𝑛 (𝜏)𝑤 𝑡 + 𝜏 𝑤 ∗ 𝑡

∞

• Since w(t) is periodic:

𝑤 𝑡 = ෍ 𝑤𝑘 𝑒 𝑗𝜔0𝑡 𝑘=−∞

∞

∞

• Thus:

𝑅𝑛𝑐𝑦𝑐𝑙𝑜 𝑡 + 𝜏, 𝑡 = 𝑅𝑛 (𝜏) ෍ ෍ 𝑤𝑘 𝑤 ∗ 𝑙 𝑒 𝑗𝑘𝜔0𝜏 𝑒 𝑗𝜔0 (𝑘−𝑙)𝑡

• Or:

1 𝑅𝑛𝑐𝑦𝑐𝑙𝑜 𝜏 = න 𝑅𝑛𝑐𝑦𝑐𝑙𝑜 𝑡 + 𝜏, 𝑡 𝑑𝑡 = 𝑅𝑛 (𝜏) ෍ 𝑤𝑘 2 𝑒 𝑗𝑘𝜔0𝜏 𝑇 𝑇

𝑘=−∞ 𝑙=−∞ ∞

∞

• PSD:

∞

𝑘=−∞

𝑆𝑛𝑐𝑦𝑐𝑙𝑜 𝑓 = 𝑆𝑛 𝑓 ∗ ෍ 𝑤𝑘 2 𝛿 𝑓 − 𝑘𝑓0 = ෍ 𝑤𝑘 2 𝑆𝑛 𝑓 − 𝑘𝑓0 𝑘=−∞

𝑘=−∞

Sncyclo |w0|2

Sn …

-kf0

|wk|2

…

0

kf0

f

66

Cyclostationary Noise Example ∞

• If white:

𝑆𝑛𝑐𝑦𝑐𝑙𝑜 𝑓 = 𝑆𝑛 ෍

𝑤𝑘 2 = 𝑆𝑛

𝑘=−∞

1 𝑇 න 𝑤(𝑡) 2 𝑑𝑡 𝑇 0

Parseval’s energy

• In the time varying conductance example: 𝑆𝑛 = 4𝐾𝑇

,

𝑤 𝑡 =

𝐺(𝑡)

4𝐾𝑇 𝑇 𝑆𝑛𝑐𝑦𝑐𝑙𝑜 𝑓 = න 𝐺(𝑡)𝑑𝑡 𝑇 0

,

• A switched resistor as another example: 4𝐾𝑇 𝑆𝑖𝑜𝑛 𝑓, 𝑡 = 𝑆𝑊(𝑡) 2 𝑅 1 𝑇 2𝐾𝑇 𝑆𝑖𝑜𝑛 𝑓 = න 𝑆𝑖𝑜𝑛 𝑓 𝑑𝑡 = 𝑇 0 𝑅 in

SW(t)

SW(t)

in R

ion ion t 67

Two-Port Noise Model • Customary to represent a 2-port w/ input noise sources vn Zs vns vs

Source

in

Noise-Less Two-Port

Noisy Two-Port

• Current source obtained by applying open circuit • Voltage source obtained by applying short circuit • Both needed to support an arbitrary source impedance

68

Example of FET input Noise Rg

vn

Cgd

Cgs Zs

in

+ vgs -

gmvgs

ro

idn ion

• Ignore cgd & ro • Applying short/open yields: 2 𝑖 4𝐾𝑇𝛾 𝑑𝑛 𝑣𝑛 2 = = 𝑔𝑚 2 𝑔𝑚 2

𝑖𝑛 =

𝑖𝑑𝑛 2 𝑔𝑚 𝑗𝜔𝐶𝑔𝑠

4𝐾𝑇𝛾 (𝜔𝐶𝑔𝑠 )2 = (𝜔𝐶𝑔𝑠 )2 𝑣𝑛 2 2 = 𝑔 𝑚 69

Noise Figure • Available input noise is always KT:

𝑁𝑖 =

4𝐾𝑇𝑅𝑠 = 𝐾𝑇 4𝑅𝑠

𝑆𝑖 /𝐾𝑇 𝑁𝑜 𝑁𝑖 = = 𝑆𝑜 /𝑁𝑜 𝐺𝑎 𝐾𝑇 𝐾𝑇

• By definition, the noise figure is: • Could model 2-port with its noise sources 𝐹=

Rs

vns

vn

in

Noise-Less

ZIN

• Thus:

(𝑣𝑛 + 𝑍𝑠 𝑖𝑛 ) 2 + 4𝐾𝑇𝑅𝑠 𝑁𝑖 = 4𝑅𝑠

(𝑣𝑛 + 𝑍𝑠 𝑖𝑛 ) 2 + 4𝐾𝑇𝑅𝑠 (𝑣𝑛 + 𝑍𝑠 𝑖𝑛 ) 2 𝐹= =1+ 4𝐾𝑇𝑅𝑠 4𝐾𝑇𝑅𝑠

70

Noise Figure Optimization • Noise figure a function of source impedance • Compares amplifier noise to a reference (𝑣𝑛 + 𝑅𝑠 𝑖𝑛 ) 2 + 4𝐾𝑇𝑅𝑠 (𝑣𝑛 + 𝑅𝑠 𝑖𝑛 ) 2 𝐹= =1+ 4𝐾𝑇𝑅𝑠 4𝐾𝑇𝑅𝑠

• An optimum expected when Rs swept F

Fmin

Rs Rsopt

71

Minimum Noise Figure • Systematically optimize NF for a given amplifier 𝑣 =𝑣 +𝑣 =𝑍 𝑖 +𝑣 • We can write: • Assign nonphysical resistances: 𝑣 = 4𝐾𝑇𝑅 & 𝑖 = 4𝐾𝑇𝐺 𝑛

𝑢

𝑐

2

𝑢

𝑐 𝑛

𝑢

𝑛

2

𝑢

𝑛

𝑅𝑢 + 𝐺𝑛 [(𝑅𝑠 + 𝑅𝑐 )2 +(𝑋𝑠 + 𝑋𝑐 )2 ] 𝐹 = 1+ 𝑅𝑠

• Rearranging NF equation: • Taking derivative to minimize: 𝑅𝑠𝑜𝑝𝑡 =

𝑅𝑢 + 𝑅𝑐 2 𝐺𝑛

&

𝑋𝑠𝑜𝑝𝑡 = −𝑋𝑐

• Noise figure for the optimum impedance is 𝐹𝑚𝑖𝑛 = 1 + 2𝐺𝑛 (𝑅𝑐 +

𝑅𝑢 + 𝑅𝑐 2 ) 𝐺𝑛

• Both amplifier & source need to be optimized 72

Noise Circles • Away from optimum point, noise figure contour is family of circles

𝐺𝑛 2 𝐹 = 𝐹𝑚𝑖𝑛 + 𝑍𝑠 − 𝑍𝑠𝑜𝑝𝑡 𝑅𝑠

• Remains circle on Smith chart, given bilinear relation between impedance and reflection coefficient Xs

F↑ Xsopt

Fmin

Rsopt

Rs 73

FET Minimum Noise Figure • Knowing single FET input noise: 𝑍𝑠𝑜𝑝𝑡 =

−1 𝑗𝐶𝑔𝑠 𝜔

&

𝐹𝑚𝑖𝑛 = 1 Ls = 1/Cgsw2

vn

vn

Zs in

• Noise figure in general is:

Cgs

𝛾 𝐹 =1+ [(𝑅𝑠 𝐶𝑔𝑠 𝜔)2 +(1 − 𝑋𝑠 𝐶𝑔𝑠 𝜔)2 ] 𝑔𝑚 𝑅𝑠

• At optimum condition, source resonates with Cgs • Infinite gain from source to input, not practical 74

Source Follower Minimum Noise Figure • Only cgd ignored • in comprises noise of M1/2 • ZL includes ro and gmb • Noise figure is: 1 + 𝑗𝜔𝐶𝑔𝑠1 𝑍𝑇𝐻 𝐹 =1+ 𝑔𝑚1 + 𝑗𝜔𝐶𝑔𝑠1

2

𝛾(𝑔𝑚1 + 𝑔𝑚2 ) 𝑅𝑒[𝑍𝑇𝐻 ]

1

ZTH

in

M1 +vgsM2

ZL

ZL=CL||ro1||ro2||1/gmb1

• Thus: 𝑗𝐶 𝜔 • At the optimum source impedance, gain is one 𝑍𝑠𝑜𝑝𝑡 =

𝑔𝑠1

𝑔𝑚1 + 𝑗𝜔𝐶𝑔𝑠 𝑉𝑂 = =1 𝑉𝑇𝐻 𝑔𝑚1 + 𝑗𝜔𝐶𝑔𝑠 + 𝑍𝐿 (1 + 𝑗𝜔𝐶𝑔𝑠1 𝑍𝑇𝐻 ൯ 75

Passive Lossy Circuit Noise Figure • Noise figure of passive lossy circuit is equal to its loss Rs vs

ZOUT(s)

• From Nyquist theorem:

𝑆𝑣𝑂𝑈𝑇 𝑓 = 2𝐾𝑇𝑅𝑒 𝒁𝑂𝑈𝑇 = 4𝐾𝑇𝑅𝑂𝑈𝑇

• Available input noise is:

4𝐾𝑇𝑅𝑂𝑈𝑇 1 4𝑅𝑂𝑈𝑇 𝑁𝑖 = = 𝐺𝑎 𝐺𝑎

• Thus, the noise figure is:

𝐹=

1 = 𝐿𝑜𝑠𝑠 𝐺𝑎

76

Noise of Cascade of Stages Zs vs

F1, Ga1

F2, Ga2

Stage 1

Stage 2

• Available input noise of a given stage: • Total available input noise is:

• Hence, the noise figure is:

𝑁𝑖 = 𝐹 − 1 𝐾𝑇

(𝐹2 −1)𝐾𝑇 𝑁𝑖 = 𝐾𝑇 + 𝐹1 − 1 𝐾𝑇 + 𝐺𝑎1 𝐹2 − 1 𝐹 = 𝐹1 + 𝐺𝑎1

• No need for stages to be matched, only Ga is

available power gain w/ respect to previous stage 77

Example of Cascade of Stages • Consider an amplifier preceded by 2 passive stages

Input

L1

L2

F3

• May represent antenna switch, filter, & the receiver • Total noise figure is: 𝐹 = 𝐿1 +

𝐿2 − 1 𝐹3 − 1 + = 𝐿1 𝐿2 𝐹3 1 1 𝐿1 𝐿1 𝐿2

• Or in dB: 𝑁𝐹 = 𝐿𝑜𝑠𝑠1 + 𝐿𝑜𝑠𝑠2 + 𝑁𝐹3 78

Impact of Feedback on Noise • Noiseless FB does not change input noise • Accordingly, min NF not affected vn

Zs

in

vs/Zs

vs

ZIN

gvin

+ vo -

fvo

Zs

Shunt Feedback Norton Equivalent

Zs

vn/Zs

Shunt FB Example

+ vin -

in

+ vin -

ZIN

Norton Equivalent

Both vn & in brought out

79

Sensitivity • From noise figure definition: 𝑆𝑖 𝑆𝑁𝑅𝑖 𝑁𝑖 𝐹= = 𝑆𝑁𝑅𝑜 𝑆𝑁𝑅𝑜

• Thus, in dB: 𝑁𝐹 = 10𝑙𝑜𝑔𝑆𝑖 − 10log (𝐾𝑇𝐵) − 10𝑙𝑜𝑔𝑆𝑁𝑅𝑜

• The minimum signal to establish a certain SNR: 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = −174 + 𝑁𝐹 + 10𝑙𝑜𝑔𝐵 + 𝑆𝑁𝑅

• Bandwidth and SNR set by standard, modem • Linearly dependent on noise figure 80

Phase Noise • Mixers to downconvert the RF signal • Reduces power of subsequent stages XLO (f) Downconverted Output

RF Input

Phase Noise

LO xLO (t)=Accos(2pfCt+fn(t)) fC

f

• LO used to downconvert is noisy 𝑥𝐿𝑂 𝑡 = 𝐴𝐶 cos(2𝜋𝑓𝐶 𝑡 + 𝜙𝑛 𝑡 ) ≈ 𝐴𝐶 [cos2𝜋𝑓𝐶 𝑡 − 𝜙𝑛 𝑡 𝑠𝑖𝑛2𝜋𝑓𝐶 𝑡]

• Phase noise defined as noise per Hz, relative to carrier 81

Distortion • Blockers in wireless systems • Input intercept point • Dynamic range • Compression • Reciprocal mixing • Harmonic mixing

82

Cellular Systems and Blockers • Handsets communicate to

Base Station

base stations

• Blockers inevitable in wireless environment

• Receiver must maintain good NF despite blockers TX0

Receiver

Desired

Blockers

TX1

Interferer

TX1

TX2

TX2

Interferer

Desired 83

FDD Radios • In full duplex radios, RX and TX operate concurrently • TX acts as a blocker • Puts more stringent requirements on both RX and TX TX Leakage

Other Blocker RX Desired

TR Switch

Receiver

fRX

fTX

Receiver

Transmitter

FDD

Transmitter

TDD 84

Nonlinear Systems • In a nonlinear amplifier: y = a1x+a2x2+a3x3

x

𝑎2 𝐴2 3𝑎3 𝐴3 𝑎2 𝐴2 𝑎3 𝐴3 𝑦= + 𝑎1 𝐴 + 𝑐𝑜𝑠𝜔𝑡 + 𝑐𝑜𝑠2𝜔𝑡 + 𝑐𝑜𝑠3𝜔𝑡 2 4 2 4

• Harmonic distortion however not a concern in NB radios NB RF System

f

f

2f

3f

85

Two-Tone Blockers

f1 f2

f1 f2

2f2-f1

2f1-f2

• Apply a 2-tone blocker to a NB radio:

• Now, close-by terms exist 𝑦 = 𝑎1 𝐴1 𝑐𝑜𝑠 𝜔1 𝑡 + 𝐴2 𝑐𝑜𝑠 𝜔2 𝑡

3𝑎3 𝐴1 2 𝐴2 3𝑎3 𝐴2 2 𝐴1 + cos 2𝜔1 − 𝜔2 𝑡 + cos 2𝜔2 − 𝜔1 𝑡 + ⋯ 4 4

IM3 products

86

3rd-Order Input Intercept Point • IIP3 defined as the point the 2 curves meet Never reach due to compression

a1A DP3

• Thus:

f1 f2

𝐴𝐼𝐼𝑃3 =

2f2-f1

2f1-f2

3/4a3A3

20logA IIP3

4 𝑎1 3 𝑎3

𝑎1 𝐴 = 2𝐼𝐼𝑃3 𝑖𝑛 𝑑𝐵𝑚 − 40𝑙𝑜𝑔𝐴 = 2𝐼𝐼𝑃3 − 2𝑃 3𝑎3 𝐴3 4 Δ𝑃3 𝐼𝐼𝑃3 = 𝑃 + 2 Δ𝑃3 = 20𝑙𝑜𝑔

• In dB:

• Therefore, in an otherwise ideal receiver: 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = 3𝑃𝐵 − 2𝐼𝐼𝑃3 + 𝑆𝑁𝑅

87

FET IIP3 • Consider a short channel MOSFET • The I-V characteristics is: 1 𝑊 (𝑉𝐺𝑆 − 𝑉𝑇𝐻 )2 𝐼𝐷 = 𝜇0 𝐶𝑂𝑋 2 𝐿 1 + ( 𝜇0 + 𝜃)(𝑉𝐺𝑆 − 𝑉𝑇𝐻 ) 2𝑣𝑠𝑎𝑡 𝐿 𝜃′

• Use Taylor series to expand around Veff: 𝜕𝑦 1 𝜕2𝑦 𝑎1 = ቤ ,𝑎 = อ 𝜕𝑥 𝑥=0 2 2 𝜕𝑥 2

𝑥=0

1 𝜕3𝑦 , 𝑎3 = อ 6 𝜕𝑥 3

,⋯ 𝑥=0

• Hence, if the input nonlinearity is dominant: 𝐴𝐼𝐼𝑃3 = (1 + 𝜃 ′ 𝑉𝑒𝑓𝑓 )

2𝑉𝑒𝑓𝑓 4 (𝑉𝑒𝑓𝑓 2 + )≈ 3 𝜃′

8 𝑉𝑒𝑓𝑓 3 𝜃′ 88

Cascade of Nonlinear Stages • For multiple stages, expand

y = a1x+a2x2+a3x3

y = b1x+b2x2+b3x3

using Taylor series

• We have: 𝑦 = 𝑏1 (𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 ) + 𝑏2 (𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 )2 + 𝑏3 (𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 )3

• Thus:

𝑦′ 0

= 𝑎1 𝑏1 ,

𝑦 ′′ 0 𝑦 ′′′ 0 2 = 𝑎2 𝑏1 + 𝑎1 𝑏2 , = 𝑎3 𝑏1 + 𝑏3 𝑎1 3 + 2𝑎1 𝑎2 𝑏2 2 6

• In a dominantly 3rd-order system: 1 1 𝑎 = + • Or: 𝐴 𝐴 𝐴 1

𝐼𝐼𝑃3,𝑇

2

𝐼𝐼𝑃3,1

2

𝐴𝐼𝐼𝑃3,𝑇 ≈

2

𝐼𝐼𝑃3,2

4 𝑎1 𝑏1 3 𝑎3 𝑏1 + 𝑏3 𝑎1 3

2

• The 2nd stage is expected to dominate

89

2nd-Order Input Intercept Point • IIP2 defined same as IIP3

20logA IIP2

• In a system with 2nd-order nonlinearity

• Hence:

𝑎2 𝐴2 𝑎2 𝐴2 𝑦 = 𝑎1 𝐴𝑐𝑜𝑠𝜔𝑡 + + 𝑐𝑜𝑠2𝜔𝑡 2 2 2𝑎1 𝐼𝐼𝑃2 = 𝑎2

• If several stages cascaded:

1 𝐴𝐼𝐼𝑃2,𝑇

=

1 𝐴𝐼𝐼𝑃2,1

+

𝑎1 𝐴𝐼𝐼𝑃2,2 90

Impact of Feedback on Nonlinearity • Feedback expected to

y = a1x+a2x2+a3x3

improve linearity

• Use Taylor series to expand

+

S -

𝑦 = 𝑎1 (𝑥 − 𝛽𝑦) +𝑎2 (𝑥 − 𝛽𝑦)2 + 𝑎3 (𝑥 − 𝛽𝑦)3 b

• We will have: 2 𝑎1 𝑎2 𝑎 1 + 𝑎 𝛽 − 2𝛽𝑎 3 1 2 2+ 3 𝑦= 𝑥+ 𝑥 𝑥 1 + 𝑎1 𝛽 (1 + 𝑎1 𝛽)2 (1 + 𝑎1 𝛽)5

• Hence:

𝐼𝐼𝑃2,𝑓𝑏 = (1 + 𝑎1 𝛽) × 𝐼𝐼𝑃2 3 𝐼𝐼𝑃3,𝑓𝑏 ≈ (1 + 𝑎1 𝛽)2 × 𝐼𝐼𝑃3

• 2nd-order amplifier w/ fb becomes 3rd-order nonlinear 91

Dynamic Range • Spurious free dynamic range the most common • Maximum input to have IM3 in noise floor Pmax

Noise Floor

• Hence:

2𝐼𝐼𝑃3 + 𝑁𝑜𝑖𝑠𝑒 𝐹𝑙𝑜𝑜𝑟 2𝐼𝐼𝑃3 + −174 + 𝑁𝐹 + 10𝑙𝑜𝑔𝐵𝑊 𝑃𝑚𝑎𝑥 = = 3 3

• Minimum input at sensitivity • Thus:

2(𝐼𝐼𝑃3 − 𝑁𝑜𝑖𝑠𝑒 𝐹𝑙𝑜𝑜𝑟) 𝑆𝐹𝐷𝑅 = 𝑃𝑚𝑎𝑥 − 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑡𝑦 = − 𝑆𝑁𝑅 3

92

Compression • A single blocker, if large enough reduces the gain y 1dB

x

1dB Compression

• We have:

3𝑎3 𝐴3 𝑦 = 𝑎1 𝐴 + 𝑐𝑜𝑠𝜔𝑡 + ⋯ 4

• The point that gain drops by 1dB is 1dB compression 4 𝑎1 4 𝑎1 𝐴1𝑑𝐵 = 1 − 0.89 = 0.33 3 𝑎3 3 𝑎3

93

Desensitization • Receiver gain reduction leads to more noise • Called desensitization Less slope AB AD AB

• Since: 𝑦 = 𝑎1 𝐴𝐷 𝑐𝑜𝑠𝜔𝐷 𝑡 + 3𝑎3 𝐴𝐷 𝑐𝑜𝑠𝜔𝐷 𝑡(𝐴𝐵 𝑐𝑜𝑠𝜔𝐵 𝑡)2 + ⋯ ≈ (𝑎1 +

• We have:

𝐷𝑒𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑧𝑒𝑑 𝐺𝑎𝑖𝑛 3𝑎3 2 =1+ 𝐴 𝐿𝑖𝑛𝑒𝑎𝑟 𝐺𝑎𝑖𝑛 2𝑎1 𝐵

3𝑎3 2 𝐴 )𝐴𝐷 𝑐𝑜𝑠𝜔𝐷 𝑡 2 𝐵

94

Reciprocal Mixing • LO phase noise SNR

reciprocally mixes with blocker

BW

AB

• Sets a noise floor AD

XLO (f) PN (dBc/Hz)

DfB

• Based on phase noise definition: 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = 𝑃𝐵 + 𝑃𝑁 + 10𝑙𝑜𝑔𝐵𝑊 + 𝑆𝑁𝑅

fC

f Df

• In an otherwise ideal receiver: 𝑁𝐹𝐵 = 174 + 𝑃𝐵 + 𝑃𝑁 95

Harmonic Mixing • Actual mixers are hard switching

IIPk

Hard Switching

⅕

…

5fLO

3fLO

fLO

(nfLO ± fIF)/k

fLO +fIF

⅓

• That along with LNA nonlinearity causes issues • A blocker at (nfLO±fIF)/k apears at IF 96

Image and Half-IF Blockers

Image Blocker

fLO +fIF

fLO -fIF/2

fLO

fLO -fIF/2

fLO +fIF

fLO

fLO -fIF

• Image blocker: n = k = 1 • Half-IF blocker: n = k = 2

Half-IF Blockers

• Half-IF blocker closer, harder to filter • But may be reduced by a differential design 97

Low-Noise Amplifiers • Role of matching • RF tuned amplifiers • Feedback LNAs • Practical degenerated CS LNA • Feedforward LNAs

98

Role of Matching Network 𝑅𝑒[𝑍𝑇𝐻 ] 𝑅𝑠 𝑆𝑣𝑇𝐻 = 4𝐾𝑇𝑅𝑒[𝑍𝑇𝐻 ] 𝑣𝑇𝐻 2 = 𝑣𝑠 2

• From Nyquist theorem:

≡

vs

is=vs/Rs

Rs

Rs

+ v(t) -

ZTH

|H(f)|2=RsRe[ZTH]

ZTH

≡

vTH

v2TH=4KTRe[ZTH]

• Matching won’t affect available noise & power 4𝐾𝑇𝑅𝑒[𝑍𝑇𝐻 ] 𝑁𝑎 = = 𝐾𝑇 4𝑅𝑒[𝑍𝑇𝐻 ]

&

𝑣𝑇𝐻 2 𝑉𝑠 2 𝑃𝑎 = = 4𝑅𝑒[𝑍𝑇𝐻 ] 4𝑅𝑠

99

Practical LNA Interface • Matching may include all the input parasitics • A simple Thevenin equivalent captures the input Rs ZTH

Rs

vs

≡

Lossless

Package Matching Network

Bondwire Amplifier

Pad+ESD

vs

v2TH=4KTRe[ZTH]

vTH

100

Noise vs. Power Matching • Optimum noise and matching require different values of ZTH

• Main challenge of LNA design is to satisfy both • Feedback: to alter input impedance, preserve noise • Feedforward: to cancel the noise

ZTH

Noisy Amplifier

vTH ZIN

ZTH = Zsopt ZTH = ZIN* 101

RF Tuned Amplifiers • Traditional analog amplifiers either too noisy, or low gain

• Many practical LNAs use tuned load VDD

VDD

VDD

VDD

L

RL

CL

Active Load

Complementary

Resistive Load

CL

Tuned Load

102

LNA and Filtering • Tuned load & matching has the added advantage of filtering far-out blockers

• But very narrowband Blockers Bandpass Filters Amplifier NB Response

Desired

VDD

L

CL

103

CS and CG Stages • Traditional CS and CG amplifiers commonly used • But may incorporate matching & inductive load • Noise figure not so good however VDD

VDD

L

L

CL

CL

CS Stage

CG Stage

104

CS Amplifier Noise • 3dB barrier as matching made of noisy resistor • RP may represent matching loss ZTH

≡

L

Cgs

CL

Rp

2

• General noise figure:

𝐹 =1+

𝑍𝑇𝐻 𝑍𝑇𝐻 + 1+ 𝑅𝑝 𝑅𝑒[𝑍𝑇𝐻 ] 𝑅𝑝

• Once matched:

𝐹 =2+

4 1 𝛾+ >2 𝑔𝑚 𝑅𝑝 𝑔𝑚 𝑅𝐿

2

1 𝑔𝑚 𝑅𝐿 𝑔𝑚 𝑅𝑒[𝑍𝑇𝐻 ] 𝛾+

105

CS LNA Noise vs. S11 • Noise traded for matching:

1 𝛾 + 2 4 𝑔𝑚 𝑅𝐿 𝐹= + 1 + 𝑆11 (1 + 𝑆11 )2 𝑔𝑚 𝑅𝑒[𝑍𝑇𝐻 ]

4 3.5

NF, dB

3 2.5 2

gmRe[ZTH ]= 10

1.5 1 0.5

|S11|<-10 gmRe[ZTH ]= 20 -2 6 .0 -2 0 .0 -1 6 .5 -1 4 .0 -1 2 .0 -1 0 .5 -9 .1 -8 .0 -6 .9 -6 .0 -5 .2 -4 .4 -3 .7 -3 .1 -2 .5 -1 .9 -1 .4 -0 .9 -0 .4 0 .0

0 |S11|, dB

• Reasonable noise figure for S11 = -10dB

106

CG Amplifier Noise

1 𝑔𝑚 + 𝑔𝑚𝑏

• General NF: • Once matched:

CL

1/(gm+gmb) Cgs

𝑍𝐼𝑁 =

L

gmro>>1

• Matching thru a noisy device • Same as CS, 3dB barrier • Though gmb helps • Lower power than CS • Input impedance:

𝑔𝑚 1 + (𝑔𝑚 + 𝑔𝑚𝑏 )𝑍𝑇𝐻 2 𝐹 =1+ (𝛾 + ) 𝑔𝑚 + 𝑔𝑚𝑏 𝑅𝑒 𝑍𝑇𝐻 𝑔𝑚 + 𝑔𝑚𝑏 (𝑔𝑚 +𝑔𝑚𝑏 )𝑅𝐿 𝑔𝑚 4 𝑔𝑚 𝐹 =1+𝛾 + ≈1+𝛾 𝑔𝑚 + 𝑔𝑚𝑏 (𝑔𝑚 +𝑔𝑚𝑏 )𝑅𝐿 𝑔𝑚 + 𝑔𝑚𝑏 1

107

Shunt FB Amplifier • Shunt feedback lowers amplifier input impedance • If noiseless however, won’t affect min noise figure • Replace FB w/ its 2-port equivalent ZF

+ v1 -

ZTH

vo

+ v1 -

ZF

fv1

ZF

fv2

≡

≡

ZL

+ v2 -

ZF

+ v2 -

f = -1/ZF 108

Shunt FB LNA Noise Analysis 4KTRe[ZTH] ZTH

4KTRe[YF]

• Input impedance:

𝑍𝐼𝑁 =

ZF||ZL

ZF

(gm-1/ZF)vgs

+ vgs -

fvo

vo

4KT(ggm+Re[YF]+Re[YL])

𝑍𝐹 𝑍𝐹 +𝑍𝐿 = 1 1 + (𝑔𝑚 − 𝑍 )(𝑍𝐹 ||𝑍𝐿 ) 1 + 𝑔𝑚 𝑍𝐿 𝐹

• And the noise figure: 2

𝑍𝐹 +𝑍𝑇𝐻 4𝐾𝑇𝛾𝑔𝑚 + 4𝐾𝑇𝑅𝑒 𝑌𝐿 4𝐾𝑇𝑅𝑒 𝑌𝐹 𝑍𝐹 +𝑍𝑇𝐻 𝐹 =1+ + ( 𝑔𝑚 𝑍𝐹 + 1 4𝐾𝑇𝑅𝑒 𝑍𝑇𝐻 4𝐾𝑇𝑅𝑒 𝑍𝑇𝐻 𝑔𝑚 𝑍𝐹 + 1

• If matched, ignoring FB noise:

2

+ 𝑍𝑇𝐻 2 )

2

𝑍𝐹 +𝑍𝑇𝐻 𝛾𝑔𝑚 + 𝑅𝑒[𝑌𝐿 ] 𝐹 = 1+ 𝑔𝑚 𝑍𝐹 + 1 𝑅𝑒 𝑍𝐼𝑁

109

Resistive Shunt FB LNA • If feedback & load resistors large: • Thus: 𝑍

𝑇𝐻

∗

𝑍𝐹 +𝑍𝑇𝐻 𝑔𝑚 𝑍𝐹 + 1

2

1 ≈ 𝑔𝑚 2

𝑍𝐹 +𝑍𝐿 1 𝑍𝐹 = 𝑍𝐼𝑁 = ≈ (1 + ) 1 + 𝑔𝑚 𝑍𝐿 𝑔𝑚 𝑍𝐿

• Ignoring cgd, noise figure is: 𝑅𝑒[𝑌𝐿 ] 𝛾 𝛾 𝑔𝑚 𝐹 = 1+ ≈1+ =1+ 𝑅 𝑍 𝑍 1 + 𝑅𝐹 𝑅𝑒 1 + 𝑍𝐹 𝑅𝑒 1 + 𝑍𝐹 𝐿 𝐿 𝐿

𝛾+

• < 3dB noise figure possible • Input impedance varies w/ load

C∞

VDD

RF CL

110

Cascode CSLNA • Cgd creates shunt feedback • Load may be expressed as: 1

1 𝑍𝐿 = || 𝑗(𝐶𝐹 + 𝐶𝑝 𝜔൯ 𝑔𝑚

CF

• Input impedance is: 𝑍𝐼𝑁 =

CP

−1 = 𝑓 𝑔𝑚 + 𝑓 𝑍𝐿 ||𝑍𝐹

𝑅𝑒[𝑍𝐼𝑁 ] =

𝐶𝑝 𝑔𝑚 1 + 𝐶 + 𝑗𝜔𝐶 𝐹

𝑔𝑚 − 𝑗𝜔𝐶𝐹

𝐶𝑝 𝑔𝑚 (2 + 𝐶 )

𝐶𝑝 2+𝐶

𝑔𝑚 + 𝑗𝜔𝐶𝐹

𝑔𝑚

𝐹

2 ≈

𝐹

f = -jwCF

𝐹

𝛾(𝑔 + 𝑅𝑒 𝑌 ) 𝐹 = 1+ • The noise figure is: 𝑔 + 𝑗𝜔𝐶 𝑅𝑒 𝑍 • Cgd creates noiseless real part 𝑚

𝑚

𝐿

𝐹

2

𝐼𝑁

=1+

𝛾

𝐶𝑝 1 + 2𝐶

𝐹 111

Series FB LNA • Series feedback increases LNA input impedance ZTH

Cgs

gmvgs

io

ZTH

≡

1/gmb

io Zs

Zs

≡

io

Cgs

ZTH Zs

Series FB

• Input impedance is: 1 𝑍𝐼𝑁 = + 𝑍𝑠 𝑗𝜔𝐶𝑔𝑠

𝑔𝑚 𝑍𝑠 1 𝑔𝑚 𝑍𝑠 1+ = + 𝑍𝑠 + 1 + 𝑗𝜔𝐶𝑔𝑠 𝑍𝑠 𝑗𝜔𝐶𝑔𝑠 𝑗𝜔𝐶𝑔𝑠

fio=Zsio

Zs

112

Series FB LNA Noise • Zs must be noise less ideally: Inductive •

𝑍𝑠 = 𝑟 + 𝑗𝑋 = 𝑗𝜔𝐿𝑠

1 − 𝜔𝐶𝑔𝑠 𝑋 + 𝑗𝜔𝐶𝑔𝑠 𝑍𝑇𝐻 2 4𝐾𝑇𝛾𝑔𝑚 𝑔𝑚 General noise expression: 𝐹 = 1 + 4𝐾𝑇𝑅𝑒 𝑍𝑇𝐻 𝑔𝑚 𝑋 1 ∗ 𝑍𝑇𝐻 = 𝑍𝐼𝑁 = − − 𝑗𝑋 𝜔𝐶𝑔𝑠 𝑗𝜔𝐶𝑔𝑠

• Since:

𝑍𝐼𝑁 =

• Once matched:

𝑔𝑚 𝐿𝑠 1 + + 𝑗𝜔𝐿𝑠 𝐶𝑔𝑠 𝑗𝜔𝐶𝑔𝑠 𝐹 = 1 + 𝛾𝜔𝐶𝑔𝑠 𝑋 = 1 + 𝛾𝐶𝑔𝑠 𝐿𝑠 𝜔2 𝑖𝑜 1 = 𝑣𝑇𝐻 𝑗𝑋

FB adds a degree of freedom • And the gain: • Noise improved if X lowered • But more gain (worse linearity), narrow matching

113

Practical CS LNA • Source inductor can be realized thru bondwire • May be implemented differentially • Often limited by matching practicality and loss L

Cp

Ls

CL

Ls 114

Feedforward LNA + vIN -

vs

+ vo -

avIN

RIN rmiIN

• Noisy resistor to match • But cancel its noise

Rs

iIN

• Requires to measure voltage and current

• General noise expression: 𝑅𝐼𝑁 𝛼𝑅𝑠 + 𝑟𝑚 𝐹 = 1+ 𝑅𝑠 𝛼𝑅𝐼𝑁 − 𝑟𝑚

2

𝛼𝑅𝑠 + 𝑟𝑚 =1+ 𝛼𝑅𝑠 − 𝑟𝑚

No cancellation

3 2

2 1

NF, dB Gain/a Perfect cancellation

0

If matched

-2

-1 rm/aRs

0 115

Noise Cancelling LNA Circuit • CG branch measures current • Also provides matching • CS branch measures voltage • CG noise entirely cancelled • Very wideband

R2

R1

+ vo M1

Rs

vs

+ vIN -

M2

• Optimum noise (once CG noise cancelled): 𝛾

𝑅1 + 𝑅2 𝛾 𝐹 =1+ + ≈1+ 𝑔𝑚2 𝑅𝑠 𝑅𝑠 𝑔𝑚2 𝑅2 2 𝑔𝑚2 𝑅𝑠 116

LNA Power Optimization • Target S11 = 1/3 (or ~-10dB) 𝐹 = 1.5 +

4.5 1 4.5 𝛾+ ≈ 1.5 + 𝑔𝑚 𝑅𝑝 𝑔𝑚 𝑅𝐿 𝑔𝑚 𝑅𝑝

• For a given NF, current lowered if RP increased

• Higher RP leads to worse

𝑅𝑝 𝑅𝐼𝑁 = 2 Rs

CL

RIN

C

L

CIN

Rp

linearity, narrower band

Gain = RP/2Rs

• Ultimately limited by matching network loss 117

Impact of Cgd & Degeneration • Cgd creates noiseless real part • Helps reduce matching gain

𝐶𝑝 2+𝐶

1 𝐹 𝑌𝐼𝑁 = 𝑗𝜔𝐶𝐼𝑁 + + 𝑗𝜔𝐶𝐹 + 𝜔𝐶𝐹 2 𝑅𝑃 𝑔𝑚 Rp

CIN

CF

CF

(2+CP/CF)/gm

C

L CIN

Rp CIN

• Inductive degeneration helps further

Rp

CF

gm/(2+CP/CF)(wCF)-2

≡

CP

118

Mixers • Mixers fundamentals • Ring mixers • Gilbert analog multiplier • Active mixers • Active mixer noise analysis

• Passive mixers

119

Mixer Operation: System Perspective • Frequency shift realized by:

𝑥(𝑡)𝑒 ±𝑗2𝜋𝑓0 𝑡 ↔ 𝑋(𝑓 ∓ 𝑓0 ൯ e-j2pf0t

f -fC

0

fC 1

-f0

0

f -fC-f0

0 fC-f0

• However, 𝒆𝒋𝟐𝝅𝒇𝟎𝒕 is not real

120

Frequency Shift Practical Realization • In practice, sine and cosine

cos(2pf0t)

I

RF Signal

are used:

I - jQ sin(2pf0t)

Q

e-j2pf0t

• Sidebands distinguished by keeping track of phase cos(2pf0t)

f -fC

0

fC 1/2

1/2 1/2

1/2

1/2

-f0

f0

fC+f0

sin(2pf0t)

f -fC

0

f 0

fC-f0

-fC-f0

-fC+f0

1/2

0

fC +j/2

-j/2

+j/2

f0

-j/2

-f0

0

fC+f0

f fC-f0

-fC+f0

-fC-f0

+j/2

0 -j/2

121

Mixers as Nonlinear Circuits • Mixing caused by either nonlinearity or time variance • For example in a 2nd-order system: y = a1x + a2x2 + a3x3 y

RF ∑

LO

x

IF

≡

IF

RF LO

• A LPF extracts the desired signal in case of receiver 𝑦 = ⋯ + 𝑎2 𝐴1 cos 𝜔1 𝑡 + 𝐴2 cos 𝜔2 𝑡

2 + ⋯ = ⋯ + 𝑎 𝐴 𝐴 cos 𝜔 ∓𝜔 2 1 2 1 2

𝑡+⋯

• Practical mixers mostly based on time variance • Linear with respect to the input 122

Diode-Ring Mixer • Diode-ring mixers been around before transistors D1

D3

>VON

LO-

Output

D4

LO

D2 +1 LO+

P(t) -1

𝐴𝑐𝑜𝑠𝜔𝑅𝐹 𝑡

Input

• Input voltage effectively multiplied by: 𝑃 𝑡 =

4 1 1 (sin𝜔𝐿𝑂 𝑡 − sin3𝜔𝐿𝑂 𝑡 + sin5𝜔𝐿𝑂 𝑡 − ⋯) 𝜋 3 5

• Desired output (once filtered):

2 𝐴𝑐𝑜𝑠 𝜔𝑅𝐹 − 𝜔𝐿𝑂 𝑡 𝜋

123

FET Multiplier

Input

Output

+ LO

LO -

• FETs are better switches, more linear • Commutates the AC voltage • Requires a high impedance buffer to minimize loss

124

Gilbert Analog Multiplier (1+y)IE/2

(1+y)IE/2

VX+

Q5

Io

Q4

Q2

Q1

(1-y)IE/2

(1-x)IB/2

(1+x)IB/2

Q3

Q6

IE

VX-

(1-y)IE/2 Simplified Gilbert Multiplier

• KVL: • Thus: •

•

Generation of Differential Current

𝑉𝐵𝐸1 − 𝑉𝐵𝐸2 = 𝑉𝐵𝐸3 − 𝑉𝐵𝐸4 = 𝑉𝐵𝐸5 − 𝑉𝐵𝐸6

𝐼𝐸1 𝐼𝐸3 𝐼𝐸5 1 + 𝑥 = = = 𝐼𝐸2 𝐼𝐸4 𝐼𝐸6 1 − 𝑥 𝐼𝐸 𝐼𝐸1 + 𝐼𝐸2 = (1 − 𝑦) 2 KCL: 𝐼𝐸 𝐼𝐸3 + 𝐼𝐸4 = (1 + 𝑦) 2 𝐼𝑜 Normalized output: 𝑧 = 𝐼 = 𝑥 × 𝑦 𝐸

125

Active Mixers • Same circuit, but apply a large signal to LO port Io »2VT VX+

Q2

Q1

VX-

(1-y)IE/2

Q3

Q4

VX

VX+

(1+y)IE/2

• Removes dependence on LO port

Io

IEE

Io VX+

Q1

Q2

VX-

2VT -IEE

Io

VX

−𝑉𝑋 𝐼𝑜 = 𝐼𝐸𝐸 tanh 2𝑉𝑇

IEE |VX|>>2VT t

• Normalized output:

𝐼𝑜 4 2 𝑧 = = 𝑦 × sin𝜔𝐿𝑂 𝑡 = 𝑦[@(𝜔𝑅𝐹 −𝜔𝐿𝑂 )] 𝐼𝐸 𝜋 𝜋

126

FET Active Mixers • Commutate DC and AC current • Comprises a V-I, switches, and an I-V VDD

M3

-

M5

M4

M6

LO+

M1

I0-gmvin/2

+vin/2

I0+gmvin/2

LO-

M2

-vin/2

Switching

LO+

vo

V-I

+

I-V

RL

127

Active Mixer 1/f Noise • No frequency translation in load, minimal 1/f noise • Input stage only contributes through leakage • For switches • Noise linear, time-varying: Superposition • Fast switching • Tail current ideal: Only matters at very high frequencies ion

M3

M5

M4

I

M6

AsinwLOt

vn

-AsinwLOt

AsinwLOt

Iod

I

«A

Vid

-I I 128

Switches 1/f Noise • Noise slowly advances or retards switching points Time vn vn Dt 2I

Time Dt

Train of Impulses Time

T/2

• Output noise:

vn

𝑖𝑜𝑛 =

2 𝑣𝑛 𝐼 × 2𝐼 × Δ𝑡 = 4𝐼 = 𝑣 𝑇 𝑆 × 𝑇 𝜋𝐴 𝑛

129

Input Stage 1/f Noise • Switches offset modulates zero-crossings

Time Dt=VOS/S

• Output noise:

gmivni

𝑖𝑜𝑛 = 4𝐼

𝑉𝑂𝑆 𝑉𝑂𝑆 𝐼 𝑉𝑂𝑆 = 4𝑔𝑚 𝑣𝑛𝑖 = 𝑣𝑛𝑖 𝑆×𝑇 𝑆 × 𝑇 𝜋𝐴 𝑉𝑒𝑓𝑓

• For typical design, switches 1/f noise dominant Switches Input Stage

𝐼 𝑣 𝜋𝐴 𝑛 𝐼 𝑉𝑂𝑆 𝑖𝑜𝑛 = 𝑣𝑛𝑖 𝜋𝐴 𝑉𝑒𝑓𝑓

𝑖𝑜𝑛 =

130

Active Mixer Switches White Noise • We have:

𝑔𝑚1 𝑔𝑚2 𝑖𝑂𝑈𝑇 = 2 𝑣 𝑔𝑚1 + 𝑔𝑚2 𝐿𝑂

iOUT

LO+

• Thus:

M1

M2 in1

𝜕𝑖𝑂𝑈𝑇 𝑔𝑚1 (𝑡)𝑔𝑚2 (𝑡) 𝐺𝑚 𝑡 = =2 𝑣𝐿𝑂 𝑔𝑚1 𝑡 + 𝑔𝑚2 (𝑡)

LO-

in2 I

• Switches only contribute noise at zero-crossings • Since vLO ≈ 0 at zero-crossings, we can show: 𝑔𝑚1 (𝑡)𝑔𝑚2 (𝑡) 𝑆𝑖𝑂𝑈𝑇 = 16𝐾𝑇𝛾 = 8𝐾𝑇𝛾𝐺𝑚 𝑡 𝑔𝑚1 𝑡 + 𝑔𝑚2 (𝑡) 131

Switches Noise Spectral Density • Noise is cyclostationary 𝑖 (𝑡) = 𝑃(𝑡) × 𝑣 (𝑡) • Spectral density of each switch is 4KTg, sampled by 𝑜𝑛

𝑃 𝑡 =

• Hence:

𝑛

𝐺𝑚 (𝑡)

4𝐾𝑇𝛾 8𝐾𝑇𝛾 𝜕𝑖𝑜 8𝐾𝑇𝛾 𝜕𝑖𝑜 /𝜕𝑡 𝑆𝑖𝑜𝑛 = න 𝐺𝑚 (𝑡)𝑑𝑡 = න 𝑑𝑡 = න 𝑑𝑡 𝑇 𝑇 𝜕𝑣 𝑇 𝜕𝑣 /𝜕𝑡 𝐿𝑂 𝐿𝑂 𝑇 𝑇/2 𝑇/2

• Output current is a square-wave toggling between ±I • Thus (at t = 0 for instance):

𝜕𝑖𝑜 ≅ 2𝐼𝛿(𝑡) 𝜕𝑡

• Which leads to: 𝑆𝑖𝑜𝑛 =

8𝐾𝑇𝛾 2𝐼𝛿(𝑡) 4𝐼 𝐼 න 𝑑𝑡 = 4𝐾𝑇𝛾 = 4𝐾𝑇𝛾 𝑇 𝑆×𝑇 𝜋𝐴 𝑇/2 𝜕𝑣𝐿𝑂 /𝜕𝑡 132

Active Mixer Noise Figure • The input stage noise multiplies by ±1: 𝑆𝑖𝑜𝑛 = ±1 2 𝑆𝑖𝑛,𝑔𝑚 = 𝑆𝑖𝑛,𝑔𝑚 = 4𝐾𝑇𝛾𝑔𝑚

• Output noise current spectral density is: 𝑆𝑖𝑜𝑛 = 2 × 4𝐾𝑇𝛾𝑔𝑚 + 4 × 4𝐾𝑇𝛾

• And the noise figure:

𝐼 4𝐾𝑇 +2× 𝜋𝐴 𝑅𝐿

𝜋2 2𝛾 𝐼 1 𝐹 = 1 + 4 (𝛾 + + ) 𝑔𝑚 𝑅𝑆 𝑔𝑚 𝜋𝐴 𝑔𝑚 𝑅𝐿

• Or assuming short channel input device: 2𝑉𝑒𝑓𝑓 𝜋 2 𝑉𝑒𝑓𝑓 𝐹≈ 𝛾(1 + ) 4 𝑅𝑆 𝐼 𝜋𝐴 133

Passive Current-Mode Mixers • Active mixer nonlinearity & 1/f noise improved if: LO+ I-V M3

M5

M4 LO-

M6

LO+

LO-

I-V

M3

V-I

LO+

ZIN≈0

M4

M5 M6 V-I LO+ Active Mixer

Passive Mixer

• Only commutates the AC current • Rail-to-rail LO for good switching: Lower frequency

134

Passive Mixer Noise • Gm stage modelled by a current source • Noise PSD:

𝑖𝑂𝑈𝑇

• Current gain:

2

4𝐾𝑇𝑅𝑆𝑊 [ 𝑆𝑊1 𝑡 2 + 𝑆𝑊2 (𝑡) 2 ] 4𝐾𝑇𝑅𝑆𝑊 = = 𝑅𝑠 + 𝑅𝑆𝑊 2 𝑅𝑠 + 𝑅𝑆𝑊 2

𝑖𝑂𝑈𝑇 𝑅𝑠 = 𝑖𝑠 𝑅𝑠 + 𝑅𝑆𝑊

SW1

• Input referred noise: 4𝐾𝑇𝑅𝑆𝑊

• Switch noise negligible • Gm noise dealt with like LNAs

iOUT

𝑅𝑠 2 is

Ideal TIA

Rs

SW2 135

Oscillators • Linear oscillators • Leeson’s equation • Nonlinear oscillators • Amplitude and phase noise • Bank’s general result • Practical LC oscillators

136

Linear Oscillator • Consists of a linear negative resistor • Only an abstraction VIN

VOUT gmVIN

ingm(t)

Energy-Restoring Circuit (transconductance)

Lp

Cp

Rp

VOSC inRp(t)

iSU(t)

ZTANK

∑

ZTANK

VOSC(t)

gm

Start-up current

• Recast as a FB system Hs =

iSU(t) + inRp(t) + ingm(t)

𝑍𝑇𝐴𝑁𝐾 (𝑠) = 1 + 𝑔𝑚 𝑍𝑇𝐴𝑁𝐾 (𝑠)

1 𝑠𝐶

𝑝

𝑠2 + 𝑠

1 + 𝑔𝑚 𝑅𝑝 1 + 𝑅𝑝 𝐶𝑝 𝐿𝑝 𝐶𝑝

137

Steady Oscillation in Linear Oscillator • Marginally stable if gm = -1/Rp • Oscillation amplitude depends on initial conditions iSU(t) = qSUδ(t) ∑

ZTANK

VOSC(t)

gm

VOSC(t)

VOSC(t)

qSU/Cp

gm > -1/Rp

qSU/Cp VOSC(t)

qSU/Cp

gm < -1/Rp

gm = -1/Rp

138

Phase Noise of Linear Oscillators • In steady oscillation, tank impedance: 1 𝐻 𝑗(𝜔0 +Δ𝜔) 2 ≈ 2Δ𝜔𝐶𝑝

• Near carrier:

2

𝜔𝐿𝑝 𝐻 𝑗𝜔 2 = 1 − 𝜔 2 𝐿𝑝 𝐶𝑝

𝜔0 2 𝐿 𝑝 = 2Δ𝜔

2

𝑅𝑝2 1 𝜔0 2 = 4 𝑄2 Δ𝜔

vn(ω)= in(ω)|H(ω)|2 in(ω)

in

Vn

ZTANK

∑

gm=-1/Rp

ω0

ω0

H(ω)

vn Lp in

H(ω) =(1/jωCp) || jωLp

Cp

Rp

1/gm Energy-restoring circuit model as negative resistance, gm = -1/Rp

139

2

SSB Phase Noise Definition • By definition, phase noise is noise normalized to carrier

𝑖ഥ𝑛2 ℒ Δ𝜔 = 2

𝐻 𝑗(𝜔0 +Δ𝜔) 𝐴2 /4

2

𝑘𝑇𝐹𝑅𝑝 1 𝜔0 2 = 𝐴2 𝑄2 Δ𝜔

• Known as Leeson’s equation (F is unknown) • A factor of 2 accounts for PM noise (vs. AM) Carrier Power A2/4

Carrier Power A2/4 Ratio between carrier power and singlesideband (L{Δω})

Sidebands due to noise (vn{ω})

-ω0-Δω -ω0

-ω0+Δω

0

ω0-Δω

ω0

ω0+Δω

140

General Phase Noise Characteristics L{Δω} [dBc/Hz]

• Three regions for general phase noise curve -30dB/decade (flicker noise region) -20dB/decade (thermal noise region) Flat region (buffer/instrument noise)

Flicker noise corner

Carrier Offset, Δω 𝑃𝑡𝑎𝑛𝑘 (𝑡) 𝐴2 /(2𝑅𝑝 ) 𝜂= = 𝑃𝐷𝐶 𝑉𝐷𝐷 𝐼𝐷𝐷

• Efficiency defined as: • Common to assign a figure of merit 𝜔0 2 𝛥𝜔

𝜂𝑄2 𝐹𝑂𝑀 = = (𝑃𝐷𝐶 /1𝑚𝑊) ℒ 𝛥𝜔 𝐹

2 𝑘𝑇

10−3 141

Nonlinear Oscillator • Linear oscillator is not practical

vosc(t) Lp

inr(t)

DC Characteristic of the Active Resistor

gnr(t) Rp=100Ω

Cp

• Negative resistance always smaller than tank loss

Start-up Behavior of the Nonlinear LC Oscillator

• Oscillation amplitude set by nonlinearity -1/Rp -1/Rp

During start-up gnr(t)<-1/Rp

gnr(t)<-1/Rp for only a portion of the oscillation

142

Steady Oscillation in Practical Oscillator • As a thought experiment, let us drive oscillator externally vDriven(t) itank(t) Lp

|Pnr|>Ptank Growing Oscillation

Cp

inr(t) Rp=100Ω

gnr(t) = dinr(t)/dvnr(t)

GEFF=-1/Rp |Pnr|<Ptank Decaying Oscillation

143

Power Conservation • Energy balance:

𝑇

𝑇

𝐴2 න 𝑣𝑜𝑠𝑐 (𝑡)𝑖𝑛𝑟 (𝑡)𝑑𝑡 = − න 𝑣𝑜𝑠𝑐 (𝑡)𝑖𝑅𝑝 𝑡 𝑑𝑡 = − 𝑇 2𝑅 𝑃 0 0

𝑣 𝑡 = 𝐴𝑐𝑜𝑠𝜔 𝑡 • Output is nearly sinusoidal: • Express negative resistance as: 𝑔 𝑡 = 𝑑𝑖 𝑡 /𝑑𝑣 𝑡 • We can rearrange: 𝑑𝑣 𝑡 𝑜𝑠𝑐 𝑛𝑟

0

𝑛𝑟

𝑛𝑟

𝑛𝑟

𝑇

𝑇 𝐴 𝐴 න 𝑣𝑜𝑠𝑐 (𝑡)𝑖𝑛𝑟 (𝑡)𝑑𝑡 = − න ( 𝑠𝑖𝑛𝜔0 𝑡)𝑑𝑖𝑛𝑟 (𝑡) = − න ( 𝑠𝑖𝑛𝜔0 𝑡)𝑔𝑛𝑟 𝑡 (−𝐴𝜔0 𝑠𝑖𝑛𝜔0 𝑡)𝑑𝑡 0 0 𝜔0 0 𝜔0

• Hence:

𝑇

𝑇 𝐴2 2 2 𝐴 න 𝑔𝑛𝑟 𝑡 𝑠𝑖𝑛 𝜔0 𝑡𝑑𝑡 = − 𝑇 2𝑅 𝑃 0 𝑇

1 −1 න 𝑔 𝑡 (1 − 𝑐𝑜𝑠2𝜔0 𝑡)𝑑𝑡 = 𝑇 0 𝑛𝑟 𝑅𝑃

• Fourier series basic definition leads to: 1 𝐺𝐸𝐹𝐹 = 𝐺𝑛𝑟 0 − 𝐺𝑛𝑟 2 = − 𝑅𝑝

Sets amplitude 144

Power Conservation: General Approach • Energy balance: • In Fourier domain:

𝑇

𝑇

න 𝑣𝑜𝑠𝑐 (𝑡)𝑖𝑛𝑟 (𝑡)𝑑𝑡 = − න 𝑣𝑜𝑠𝑐 (𝑡)𝑖𝑅𝑝 𝑡 𝑑𝑡 0

0 ∞

∞

෍ 𝑉𝑜𝑠𝑐 𝑘 𝐼𝑛𝑟 −𝑘 = − ෍ 𝑉𝑜𝑠𝑐 𝑘 𝐼𝑅𝑝 −𝑘 𝑘=−∞

𝑘=−∞

• Output is near sinusoid • Express the negative resistance

𝐼𝑛𝑟 −1 + 𝐼𝑛𝑟 1 = −

2𝑉𝑜𝑠𝑐 1 𝐴 =− 𝑅𝑝 𝑅𝑝

𝑔𝑛𝑟 𝑡 = 𝑑𝑖𝑛𝑟 𝑡 /𝑑𝑣𝑛𝑟 𝑡

𝑖𝑛𝑟 𝑡 = 𝐼𝑛𝑟(𝐷𝐶) + න 𝑔𝑛𝑟 𝑡

In Fourier domain:

• Thus:

𝐺𝐸𝐹𝐹 = 𝐺𝑛𝑟 0 −

𝐼𝑛𝑟 𝑘 =

𝐴 2𝑘

𝑑𝑣𝑛𝑟 𝑡 𝑑𝑡

𝑑𝑡

𝐺𝑛𝑟 𝑘 − 1 − 𝐺𝑛𝑟 𝑘 + 1

1 1 1 𝐺𝑛𝑟 −2 − 𝐺𝑛𝑟 2 = − 2 2 𝑅𝑝

Sets amplitude 145

Phase & Frequency Noise • Noise may be decomposed to AM & PM sidebands A

G*A

A

A

Fundamental

Fundamental

ΔAejθ 0.5ΔAejθ

Fundamental modulated by single-sideband

G*0.5ΔAejθ

-0.5ΔAe-jθ

PM sidebands

0.5ΔAe-jθ

Limiter with infinitely sharp transition

Fundamental modulated by PM-sidebands

-G*0.5ΔAe-jθ

PM sidebands

0.5ΔAejθ

AM sidebands

AM sidebands

PM-sidebands preserved at output

AM-sidebands attenuated at output

• AM noise eliminated once passed thru a limiter 𝑣𝑜𝑠𝑐 𝑡 = 𝐴 (1 + 𝑚 𝑡 ) 𝑐𝑜𝑠 𝜔0 𝑡 + 𝜙 𝑡

≈ 𝐴𝑐𝑜𝑠𝜔0 𝑡 + 𝑚 𝑡 𝐴𝑐𝑜𝑠𝜔0 𝑡 − 𝜙 𝑡 𝐴𝑠𝑖𝑛𝜔0 𝑡 146

Cyclostationary Noise in VCOs • VCO cyclostationary noise decomposed to AM & PM 𝑣𝑜𝑠𝑐 𝑡 ≈ 𝐴 cos 𝜔0 𝑡 + 𝑚 𝑡 𝐴 cos 𝜔0 𝑡 − 𝜙 𝑡 𝐴 sin 𝜔0 𝑡

AM

PM

𝑛𝐶𝑌𝐶𝐿𝑂 𝑡 = 𝑛 𝑡 𝑤 𝑡 = 𝐴𝑚 𝑡 cos 𝜔0 𝑡 − 𝐴𝜙 𝑡 sin 𝜔0 𝑡

• PM noise extracted:

Sn(ω)

2 𝜙 𝑡 = − 𝑛 𝑡 𝑤 𝑡 sin 𝜔0 𝑡 𝐴

-2ω0

-ω0

|W[-1]-W[1]|2

|W[-2]-W[0]|2 |W[-3]-W[-1]|2

• After straightforward math

2ω0

ω0 |W[0]-W[2]|2

|W[1]-W[3]|2

A2SΦ(ω)

∞

1 𝑆𝜙 Δ𝜔 = 2 ෍ 𝑆𝑛 Δ𝜔 − 𝑘ω0 𝑊 𝑘 − 1 − 𝑊[𝑘 + 1] 2 𝐴 𝑘=−∞

147

Cyclostationary PM Noise: Special Cases • We have:

2 2 𝜙 𝑡 = − 𝑛 𝑡 𝑤 𝑡 sin 𝜔0 𝑡 = − 𝜙෨ 𝑡 𝐴 𝐴

𝑅𝜙෩ 𝑡 + 𝜏, 𝑡 = 𝑅𝑛 𝜏 𝑤(𝑡)𝑤 ∗ (𝑡 + 𝜏) sin 𝜔0 (𝑡 + 𝜏) sin 𝜔0 𝜏 = 𝑅𝑛 𝜏 𝑤(𝑡)2 sin 𝜔0 𝑡2

1 1 𝑅𝜙෩ 𝜏 = න 𝑅𝜙෩ 𝑡 + 𝜏, 𝑡 𝑑𝑡 = 𝑅𝑛 𝜏 න 𝑤(𝑡)2 sin 𝜔0 𝑡2 𝑑𝑡 𝑇 𝑇 𝑇 𝑇

I.

White noise shaped by p(t) = w2(t) If memory-less 𝑆𝑃𝑀 ω0 + Δ𝜔 = 𝑖𝑛2

1 1 1 𝑖𝑛2 𝑃 0 − 𝑃 −2 − 𝑃[2] = 𝑃 0 − 𝑃[2] 2 4 4 2

II. White noise of time-varying device (GM(t)= p(t)) 𝑆𝑃𝑀 ω0 + Δ𝜔 = 𝑘𝑇𝛾 𝐺𝑀 0 − 𝐺𝑀 [2]

III. White stationary noise (p(t) =1) 𝐴2 𝑆𝑛 𝑆𝑃𝑀 ω0 + Δ𝜔 = 𝑆𝜙 Δ𝜔 = 4 2

148

Noise Passing Thru Nonlinearity Gc*A

A SSB

Creation of LSB due to LTV repsonse Case 1: Single-sideband

Case 1

g(t)

A

Gc*A

GPM*USB

USB

LSB

Case 2: PM sidebands

Nonlinearity

GPM*LSB

Bandpass filter @ ω0

Case 2

G*A

A

LSB

PM noise

USB

GAM*LSB

Case 3: AM sidebands

Δi𝑜𝑢𝑡 𝑡 = −𝐴𝜙 𝑡 sin 𝜔0 𝑡 𝑔 𝑡 = − Δi𝑜𝑢𝑡 𝑡 = −𝐴𝜙 𝑡

Carrier and PM sidebands experience same gain, i.e.

Gc =GPM=G[0]-G[2]

GAM*USB

Case 3

𝐴𝜙 𝑡 2𝑗

Carrier and AM sidebands experience different gain, i.e.

Gc = G[0]-G[2] GAM = G[0]+G[2]

∞

෍ G k − 1 − 𝐺[𝑘 + 1] 𝑒 𝑗𝑘𝜔0 𝑡 𝑘=−∞

G 0 − 𝐺 2 sin 𝜔0 𝑡 = G 0 − G 2 Δv𝑖𝑛 𝑡

149

Tank Response to AM/PM Noise inr(t)

vosc(t) PM noise w.r.t. inr(t)

Cp

Lp

Rp

gnr(t)

AM noise w.r.t. inr(t)

ZAM

ZPM

Lp

Cp

Lp

PM sidebands see lossless LC tank

|ZPM|2

Cp

Rp

Cp

Rp||1/GAM

AM sidebands see lossy LC tank

|ZPM|2 Noise TF

|ZAM|2

Noise TF

AM/PM noise see different LTI responses

Lp

inr(t)

vosc(t)

-20dB/decade

|ZAM|2

ω

Carrier Offset [Hz]

ω0

v𝑃𝑀 (𝑠) 1 1 𝜔0 2 𝐿 𝑃 Z𝑃𝑀 𝑠 = = | 𝑠𝐿𝑃 |𝑅𝑃 || ≈ i𝑃𝑀 (𝑠) 𝑠𝐶𝑃 G𝑛𝑟 0 − G𝑛𝑟 2 2Δ𝜔

-Rp

2

𝑅𝑝2 1 𝜔0 2 = 4 𝑄2 Δ𝜔 150

Bank’s General Result • Tank response to noise: 2 𝑅 1 𝜔0 2 𝑝 2 2 2 2 𝑣𝑃𝑀 = 𝑖𝑃𝑀 𝑍𝑃𝑀 𝑗(𝜔0 +Δ𝜔) = 𝑖𝑃𝑀 4 𝑄2 Δ𝜔

• The phase noise is: ℒ Δ𝜔 =

• Since:

2 2 𝑣𝑅𝑝(𝑃𝑀) + 𝑣𝑛𝑟(𝑃𝑀)

𝐴2 /4 2 𝑖𝑅𝑝 𝑃𝑀 =

=

2 2 𝑖𝑅𝑝(𝑃𝑀) + 𝑖𝑛𝑟(𝑃𝑀)

2 𝑖𝑅𝑝

𝐴2 /4 = 𝑘𝑇/𝑅𝑝

𝑍𝑃𝑀 𝑗(𝜔0 +Δ𝜔)

2

Power conservation

2 2𝑘𝑇𝛾 2 𝑖𝑛𝑟 = 𝐺𝑛𝑟 0 − 𝐺𝑛𝑟 [2] = 𝑘𝑇𝛾/𝑅𝑝 𝑃𝑀 2

𝑘𝑇𝑅𝑝 (1 + 𝛾) 1 𝜔0 2 ℒ Δ𝜔 = 𝐴2 𝑄2 Δ𝜔

• We have: • Applicable to almost all oscillators • Validates Leeson’s equation for nonlinear oscillators

151

vout(t)

NMOS LC Oscillator 2Cp

t

2Cp Lp/2

Lp/2

vout(t)

Rp/2

inr(t)

IBIAS/2 Rp/2

t -IBIAS/2

gnr(t)

inr(t)

t

Active Element IBIAS

• Negative resistance:

• Amplitude:

inr(t)

Cp

vout(t) Rp

Lp

𝑑𝑖𝑛𝑟 𝑑𝑖𝑛𝑟 𝑑𝑖𝑛𝑟 −𝐼𝐵𝐼𝐴𝑆 𝑇 𝑑𝑡 𝑑𝑡 𝑔𝑛𝑟 𝑡 = = = = ෍ 𝛿(𝑡 − 𝑘 ) 𝑑𝑣𝑛𝑟 𝑑𝑣𝑛𝑟 −𝐴𝜔0 𝑠𝑖𝑛𝜔0 𝑡 𝐴𝜔0 4 𝑘 𝑑𝑡 −𝐼𝐵𝐼𝐴𝑆 −𝑗𝑘𝜋/2 𝐺𝑛𝑟 𝑘 = 𝑒 𝐴𝜋 −2𝐼𝐵𝐼𝐴𝑆 1 𝐺𝐸𝐹𝐹 = 𝐺𝑛𝑟 0 − 𝐺𝑛𝑟 2 = =− 𝐴𝜋 𝑅𝑝 2 𝐴 = 𝐼𝐵𝐼𝐴𝑆 𝑅𝑝 152 𝜋

Phase Noise in NMOS Oscillator vnr(t) Current [A]

inr(t) inL(t)

0 -0.5IBIAS

inR(t)

-2VDD

0

Voltage [V]

gR(t) IBIAS

inCS(t) gCS

Conductance [S]

gL(t)

0.5IBIAS

2VDD

-gR||gL

-2VDD

0

Voltage [V]

0

2VDD

𝑔𝑅 (𝑡)𝑔𝐿 (𝑡) 𝑔𝑅 𝑡 + 𝑔𝐿 (𝑡)

• Negative resistance is:

𝑔𝑛𝑟 𝑡 = −

• And its noise is:

2 𝑖𝑔𝑛𝑟 = 2𝑘𝑇𝛾

𝑔𝑅 (𝑡)𝑔𝐿 (𝑡) = 2𝑘𝑇𝛾𝑔𝑛𝑟 𝑡 𝑔𝑅 𝑡 + 𝑔𝐿 (𝑡)

• Bank’s result applies 153

Tail Current Noise • Tail current multiplied by ±½, thus: 𝑝 𝑡 = 𝑤 2 𝑡 = 1/4

• The tail current PM noise is then: 2 2 𝑖𝑐𝑠(𝑃𝑀) = 𝑖𝑐𝑠

2 1 1 1 𝑖𝑐𝑠 𝑃 0 − 𝑃 −2 − 𝑃[2] = = (1/4)𝑘𝑇𝛾𝑔𝐶𝑆 2 4 4 8

• And the total phase noise is: ℒ Δ𝜔 =

2 2 2 𝑖𝑅𝑝(𝑃𝑀) + 𝑖𝑔𝑛𝑟(𝑃𝑀) + 𝑖𝑐𝑠(𝑃𝑀)

Where:

𝐴2 /4

𝐹 =1+𝛾+

𝑍𝑃𝑀 𝑗(𝜔0 +Δ𝜔)

2 =

𝑘𝑇𝐹𝑅𝑝 1 𝜔0 2 𝐴2 𝑄2 Δ𝜔

𝛾𝑔𝐶𝑆 𝑅𝑝 4 154

Optimum Operation of NMOS VCO (2/π)IBIASRP

• Best to operate in

𝐴=

2 𝐼 𝑅 = 2𝑉𝐷𝐷 𝜋 𝐵𝐼𝐴𝑆 𝑝

• Using square law: 𝜋𝐴 𝐹 =1+𝛾+𝛾 4𝑉𝑒𝑓𝑓

• FOM is:

Current Limited Regime

Voltage Limited Regime

FOM [dB20]

• If current source ideal:

Amplitude [V]

current mode

Current [A] 2𝑄2

𝐴 𝐹𝑂𝑀 = 1𝑚𝑊 = 𝜋𝐹𝑉𝐷𝐷 𝑘𝑇

𝑉𝑒𝑓𝑓 2 1− 𝑉 𝐷𝐷

𝛾𝜋 𝑉 𝜋 1 + 𝛾 + 2 𝑉𝐷𝐷 − 1

2𝑄2 1𝑚𝑊 𝑘𝑇

𝑒𝑓𝑓

• Best if:

𝑉𝑒𝑓𝑓 =

𝑉𝐷𝐷 2

155

Oscillator Design Guidelines • Tuning range:

𝐶𝑚𝑎𝑥 𝐶𝑚𝑖𝑛 − 1

𝜔𝐻 − 𝜔 𝐿 𝑇𝑅 = =2 𝜔0 𝐶𝑚𝑎𝑥 𝐶𝑚𝑖𝑛 + 1

• Maximize Q and A (amplitude) 𝐴=

2 𝐼 𝑅 = 2𝑉𝐷𝐷 𝜋 𝐵𝐼𝐴𝑆 𝑝

• Set RP based on required phase noise 𝑘𝑇𝐹𝑅𝑝 1 𝜔0 2 ℒ Δ𝜔 = 𝐴2 𝑄2 Δ𝜔

• Set IBIAS, and gm (gm×RP ~ 5-10) • With Q and RP known, set LP = RP/Qw0 • If Cmin too large, a smaller inductance may be needed

156

1/f Noise Mechanisms • For an arbitrary cyclostationary noise: ∞

𝑆𝜙 Δ𝜔 =

1 2 ෍ 𝑆 Δ𝜔 − 𝑘ω 𝑊 𝑘 − 1 − 𝑊[𝑘 + 1] 𝑛 0 𝐴2 𝑘=−∞

• In case of 1/f noise:

Sn(ω) -2ω0

∞

1 𝑆𝜙 Δ𝜔 = 2 ෍ 𝑆𝑛 Δ𝜔 𝑊 −1 − 𝑊[1] 2 𝐴

-ω0

|W[-1]-W[1]|2

|W[-2]-W[0]|2 |W[-3]-W[-1]|2

2ω0

ω0 |W[0]-W[2]|2

|W[1]-W[3]|2

𝑘=−∞

• If w(t) is symmetric:

A2SΦ(ω)

𝑊 −1 = 𝑊[1]

• Flicker noise appears only as AM 157

1/f Noise – Nonlinear Capacitance • We have thus far assumed tank is linear

• If tank is nonlinear: 𝜔0 𝐴 =

inr(t)=2.5tanh(-4vout(t)) [mA] Cyclo-Stationary Flicker Noise Source Si=(4kT105/f)inr(t)

inr(t)

Cp

200Ω

1.6nH

vout(t)

Nonlinear capacitance results in flicker noise in output spectrum

1 𝐿 𝐶(𝐴)

• AM-FM conversion: 𝑑𝜔0 1 1 𝑑𝐶(𝐴) 𝐾𝐴𝑀𝐹𝑀 = =− 𝑑𝐴 2 𝐿𝐶(𝐴) 𝐶(𝐴) 𝑑𝐴

• And phase noise is:

2 𝑆Ω Δ𝜔 𝐾𝐴𝑀𝐹𝑀 2 𝑣𝐴𝑀 ℒ Δ𝜔 = = Δ𝜔 2 Δ𝜔 2

158

Effective Nonlinear Capacitance inr

iC ACw0 A vnr

vC

Memory-Less

Capacitor

ර𝑖𝑛𝑟 𝑑𝑣𝑛𝑟 = 0

ර𝑖𝐶 𝑑𝑣𝐶 = 𝜋𝐴2 𝐶𝜔0

• Linear cap: • Nonlinear cap:

𝑣𝐶 = 𝐴𝑐𝑜𝑠𝜔0 𝑡 𝑖𝐶 = −𝐶𝜔0 𝐴𝑠𝑖𝑛𝜔0 𝑡

Thus,

𝑖𝐶 2 𝑣𝐶 2 ( ) +( ) = 1 𝐴𝐶𝜔0 𝐴

‫𝐶𝑣𝑑 𝐶𝑖 ׯ‬ 𝐶𝐸𝐹𝐹 = = 𝐶𝑛𝑟 0 − 𝐶𝑛𝑟 2 𝜋𝐴2 𝜔0 159

Impacts of Harmonics • Output has harmonics • Creates 1/f noise • If Gnr memory-less: • In Fourier domain:

inr(t)

vout(t)

• Thus:

Fundamental Component

Cp

Higher Order Harmonic Components

ර𝑖𝑛𝑟 𝑑𝑣𝑛𝑟 = 0 ∞

∞

෍ 𝑘𝐼𝑛𝑟 [𝑘]𝑉𝑛𝑟 [−𝑘] = 0 𝑘=−∞

• Tank impedance:

Rp

Lp

Or

𝑘 𝑉𝑛𝑟 [𝑘] 2 ෍ =0 𝑍[𝑘]

𝑘=−∞

𝐿 𝑝 𝜔0 2 𝑍[1] ≈ 2Δ𝜔

&

𝑘 𝑍𝑘 ≈ 2 𝐿 𝜔 𝑘 −1 𝑝 0

k>1

∞

𝛥𝜔 1 𝑉𝑛𝑟 [𝑘] 2 2 ≈− ෍ 𝑘 −1 𝜔0 2 𝑉𝑛𝑟 [1] 2 𝑘=2

• In terms of current:

∞

Δ𝜔 1 𝑘 2 I𝑛𝑟 [k] 2 ≈− 2෍ 2 𝜔0 2𝑄 𝑘 − 1 I𝑛𝑟 [1] 2 𝑘=2

160

VCO with 2f0 Resonance • Current source noise eliminated if bypassed • Tank de-Q resolved by creating resonance at 2f0 LC Tank Resonates @ FLO

Vtail High Impedance @ 2FLO

Voutn

VDD

voltage

Voutp

VCS

Parasitic Cap

Vcs Filtered Current Source (Optional)

time

Vtail oscillates @2ω0

161

Alternative VCO with 2f0 Resonance • No need for additional tail inductor

CCM

k

Differential Mode (Resonates @ FLO )

Common Mode (Resonates @ 2FLO )

k

k

LS 2CDM

IDM

2VDM

IDM

ICM

ICM VCM

wDM =

1 LS (1+ k)(CCM +CDM)

wCM =

1 LS (1 − k )CCM

162

Class-C VCO • For NMOS VCO: • If injecting impulses:

vnr(t)

vnr(t)

2 𝐴 𝜋 𝑉𝐷𝐷 𝐴 𝜂= 𝑉𝐷𝐷 𝜂=

inr(t)

VB<VDD IBIAS/2

inr(t)

VDD-VB (offset increases effective VTH)

IBIAS

Ctail

t

VB=VDD

t

-IBIAS/2

• Current source bypassed, tank d-Q not an issue • Good FoM, but modest phase noise

163

Power Amplifiers • Role of matching in PAs • PA classes • Class A PA • Class B PA

164

Power Matching • Unlike LNAs, PAs are not conjugate matched • Otherwise, half the power wasted (< 50% efficiency) 1 1 𝑃𝑂𝑈𝑇 ∝ 𝑅𝐴𝑁𝑇 2 𝑅𝐴𝑁𝑇 + 𝑍𝑂𝑈𝑇

2

• Efficiency is very important

PA

LNA

RANT

ZOUT

50W Maximum Power

Maximum Efficiency 165

Efficiency Definitions • The main goal is to deliver the required power • But, Critical to maintain good efficiency and linearity • Two efficiency definitions: Drain efficiency

𝑃𝑂𝑈𝑇 𝜂= 𝑃𝐷𝐶

Power added efficiency

𝜂=

𝑃𝑂𝑈𝑇 − 𝑃𝐼𝑁 𝑃𝐷𝐶

• Almost equal if PA gain high 166

Role of Matching in PAs • Swing at the device limited to supply (battery) • Also device stress a concern VDD

+33dBm in GSM 28V

2VDD

Matching Network LCM 1:n

50W

50/n2 W PA

Antenna

• Matching network also filters harmonics 167

Matching Network Efficiency • Loss of matching directly affects efficiency • Large matching gain means high loss, low efficiency 1:n

C L

𝑅𝐿 >1 𝑅𝐿𝑃𝐴

RP=Lw0Q

𝑛=

PA

𝑅𝐿 𝑅𝐿𝑃𝐴 𝐿= 𝜔0 𝑅𝐿 − 𝑅𝐿𝑃𝐴

RL

RLPA

PA Matching

• Thus:

𝑅𝑃 = 𝑄𝑅𝐿

𝑅𝐿𝑃𝐴 𝑄𝑅𝐿 = 𝑅𝐿 − 𝑅𝐿𝑃𝐴 𝑛2 − 1

• Relative power loss:

𝑃𝐿𝑜𝑠𝑠 𝑅𝐿𝑃𝐴 𝑛2 − 1 𝑛 = = ≈ 𝑃𝑂𝑈𝑇 𝑅𝐿 𝑄 𝑄 168

Class A PA • Device is always on • Best linearity, but worst efficiency IDS

VDD

IDC L

A

VDS

Lossless Matching RL

• Waveforms: • Efficiency:

VDD 50W

t

𝑉𝐷𝑆 = 𝑉𝐷𝐷 + 𝐴𝑠𝑖𝑛𝜔0 𝑡 𝐴 𝐼𝐷𝑆 = 𝐼𝐷𝐶 + 𝑠𝑖𝑛𝜔0 𝑡 𝑅𝐿 𝐴2 𝐴2 𝑃𝑂𝑈𝑇 𝐴2 2𝑅𝐿 2𝑅𝐿 𝜂= = = = 2 < 50% 𝑃𝐷𝐶 𝐼𝐷𝐶 𝑉𝐷𝐷 𝑉𝐷𝐷 𝑉 2𝑉𝐷𝐷 𝐷𝐷 𝑅𝐿

169

Class A Efficiency at Lower Powers • At lower power, swing reduces, h degrades: 𝐴 𝜂 = • Consider: 2𝑅 𝐼 𝑉 2

𝜂=

𝐴2 2𝑉𝐷𝐷 2

𝐿 𝐷𝐶 𝐷𝐷

• Need to lower both IDC and VDD Switcher

Variable VDD

LCM

170

Differential PAs • Common to use balanced design • Takes advantage of all good differential properties INN

VDD …

…

INP

171

Class B PA • Not a good idea to leave the device on all the time • Bias at threshold instead, less linear VTH

VDD

IDS

IP t

LCM

50W

A

VDS

VDD

RL t

• • •

𝐼𝑃 𝐼𝑃 2𝐼𝑃 𝐼𝐷𝑆 = + 𝑠𝑖𝑛𝜔0 𝑡 − 𝑐𝑜𝑠2𝜔0 𝑡 + ⋯ Waveforms: 𝜋 2 3𝜋 𝐼𝑃 𝐴 To establish swing of A: 2 = 𝑅𝐿 𝐴2 𝑃𝑂𝑈𝑇 𝜋 𝐴 2𝑅𝐿 Efficiency: 𝜂= = = 𝐼𝑃 𝑃𝐷𝐶 𝑉𝐷𝐷 4 𝑉𝐷𝐷 𝜋

172

Class A & B PA Efficiency 𝐴2

• Class A:

𝜂=

• Class B:

𝜋 𝐴 𝜂= 4 𝑉𝐷𝐷

h

2𝑉𝐷𝐷 2

p/4 Class-B

½

Class-A

VDD

A

• Class AB compromises linearity vs. efficiency

173

Push-Pull Class B PA • Pseudo-differential extension • Similar to push-pull amplifiers, but NMOS only • Same efficiency VDD

50W

174

Transceiver Architectures • Conceptual receiver • Super-heterodyne receiver • Zero-IF receiver • Low-IF receiver • Direct-conversion transmitter • PLL-based transmitters • Polar transmitters

175

Conceptual Receiver • If we were to build a receiver: • Needed gain & progressive filtering of blocker

RF Filter

IF Filter

LNA To ADC

In-Band

Out-of-Band

LO

Desired IF

176

Super-Heterodyne Receiver • Most robust receiver IF Filter LNA

RF Filter

RF Filter LO

IF > 22.5MHz

To achieve sufficient IR

Image Blocker

Half-IF Blockers

960M

925M

fLO +fIF

fLO -fIF/2

fLO

fLO -fIF/2

fLO +fIF

fLO

fLO -fIF

20M

• But needs external filters for image & other blockers

GSM Example

177

Zero-IF Receiver • Set IF to zero, relaxed filtering • Signal its own image

-fLO

• Needs quadrature

≈

≈

Downconversion

0

fLO coswLO t

down-conversion

I

Q sinwLO t

178

Image in Zero-IF Receivers 1/2

1/2

1/2 fm -fLO

fm

≈

High Sideband

• The two sidebands have different phase information

*

-fLO

fLO =

j/2

fLO

0 j/2

fLO 0

-fLO -j/2

-j/2

1/2

1/2 fm -fLO

fm -fLO

fLO

0 =

*

≈

Low Sideband

1/2

j/2

j/2

fLO

fLO -fLO

0 -j/2

-j/2

179

Single-Sideband Receivers • Apply Hilbert transform 1/2

1/2

1/2 fm

High-Side

I

1/2 fm

0

0 1/2

1/2

j/2 Hilbert Tranform Q 0

0

-j/2 1/2 fm

I Low-Side

1/2

1/2

1/2 fm

0

0

j/2 Hilbert Tranform Q 0

0 -j/2

-1/2

-1/2

180

Image Cancellation • A conceptual zero-IF receiver then looks like this: coswLOt I +

0

≈

High

+

fLO Q

sinwLOt

+ -

Low

Hilbert 0 Complex Output

• I/Q paths do not double power consumption (apart from LO generation) 181

Zero-IF Receiver Issues • Mismatches limit image rejection • Very critical in high data rate radios: Need calibration Residual Image

SNR 0

• Requires quadrature LO • Also more sensitive to 1/f noise, 2nd-order distortion 182

Modern Zero-IF Receivers • Enhances image rejection, IIP2 digitally • Requires a high SNR ADC coswLO t ADC

I

LNA

DSP (IQ Cal)

DC, IIP2

ADC Q sinwLO t

183

Low-IF Receiver • IF low enough to locate image in band • More relaxed image rejection • Same mechanism as zero-IF for image rejection coswLO t

Image Rejection

+ ± Hilbert sinwLO t

184

Practical Low-IF Receivers • Image rejection done by active/passive PPF coswLO t

PPF

ADC

I

LNA

DSP

ADC Q

sinwLO t

• The higher IF, the better • But more stringent image rejection 185

Alternative Low-IF Topology • Use LPF to perform image rejection digitally • More stringent ADC requirement coswLO t I

ADC

LNA

DSP (IMR)

Lowpass Filter

ADC Q sinwLO t

• Identical to zero-IF receiver, except for IF≠0 186

Direct-Conversion Transmitters • Dual of direct-conversion receiver • Similarly, IQ imbalance, DC offset problematic: Calibrate coswLO t I

DAC PA/PAD

DSP

DAC Q sinwLO t 187

Pulling in Transmitters • Modulated PA output disturbs VCO • Operate VCO at twice frequency, helps IQ generation coswLO t coswmt

fm

coswmt

… fm

sinwmt

PA/PAD

PA/PAD

…

½f0 ( )2

sinwmt

fLO

sinwLO t

To TX

2

fm

fm fLO

2fm 2fm f0

188

Translational Loops • Direct-conversion transmitter too noisy • Stringent far-out noise needed in cellular applications • PLL helps filter the noise

PA/PAD

fVCO

LO

PFD/CP

fIF

Modulator

DAC 189

PLL-Based Transmitter • Use PLL directly • No modulator or LO needed • Sensitive to PLL non-idealities PA/PAD

MMD PFD/CP

REF

DS

Modulator/ Predistortion 190

Polar Transmitter • Extend PLL-based TX to support amplitude modulation • Good AM-PM alignment needed • PM spectrum is very wide: More sensitive I

AM CORDIC

PM

PLL

PA

Q

VDD

191