EE314 - RF Circuit Design...EE314 - RF Circuit Design Mailing list: Make sure to subscribe to...

EE314 - RF Circuit Design


Professor Tom Lee

Winter 2002

Recent Announcements:

● 3/12/02 (1:33am): Phase Noise Slides● 3/05/02 (3:45pm): EE314 Winter 2002 Final Exam Wednesday,

March 20, 2002 8:30-11:30 a.m. location TBD● 2/22/02 (12:01am): Netlists for 1st Early Bird must be submitted by

11:59pm on Monday, Feb 25th.● 2/20/02 (3:21pm): A pre-IPO startup in Santa Clara is looking for part-

time help now and during the summer. Candidates would help in product validation, and potentially in some light design work. Please contact Mr. Vince Guel, [email protected], for more information and an appointment.

● 2/19/02 (5:30pm): Noise Lecture Notes are now available. The first half of the notes are what Tom covered in class. The second half is the work of Jung-Suk Goo. His information is the latest on noise theory.

● 2/13/02 (1:16am): HW5 FAQ is on the web● 2/08/02 (5:00pm): HW#5 and a handout

on Noise Figure Measurement are available. ● 2/01/02 (7:38pm): Wednesday's (2/6/02) lecture will be taped ahead

on Tuesday, Feb 5th, from 1:15-2:30 in McCullough 115. Feel free to go to the taping so Dr. Lee has an audience or come to class Wednesday to watch the video.

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Wednesday from 5:45-6:35pm in Skilling 193.

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Final Exam

● Final exam information: ❍ Date: March 20th (Wednesday)❍ Time: 8:30 am ~ 11:30 am❍ Place: Terman Auditorium❍ Map to Terman Auditorium (Terman is at Panama Mall and

Morris Way) ● All on campus students and online/local students whose company site

is local should come to campus for final exam.

● Sample final exam from previous year (Handout 16)

Midterm Design Project

● Midterm Design Project (Handout #12) ● Midterm FAQ ● EE314 Project Models ● Models Handout (Handout #13) ● Older (but still useful) Models Handout (Handout #14)

❍ Getting Started with HSPICE at Stanford (Concise version)❍ Getting Started with HSPICE at Stanford (Long, could be old)❍ Complete HSpice Manual❍ A highly annotated HSPICE input example file❍ Measuring S parameters with Spice ❍ Measure Noise Figure with Spice

● Project Submission Procedure● Submission Diagram (The astute will notice that this is upside down)

❍ Single-Ended Header❍ Differential Header


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Measurement Decks● Scattering Parameters/Pwr Measurement Deck● Noise Figure Measurement Deck● IP3 Measurement Deck

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Instructor

Professor Tom H. Lee

[email protected]

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or by appointment

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Phase Noise in Oscillators

Ali Hajimiri

Stanford University, Stanford, CA 94305

http://smirc.stanford.edu/papers/Orals98s-ali.pdf Email: [email protected]

Outline

Introduction and Definitions

Time-Variant Phase Noise Model

Upconversion of 1/f Noise

Cyclostationary Noise Sources

Measurement Results

Conclusion

Substrate and Supply Noise


Frequency Instability: Time Domain

σ2

∆T

Known As Clock Jitter.

Oscilloscope

Oscillator


D Q

CLK

Flip-Flop

tsetup thold

Oscillator

Data

Clock

Timing Jitter in Digital Applications


Ideal

Actual

Frequency Instability: Frequency Domain

f

f

f0

Spectrum Analyzer

Oscillator

Known As Phase Noise.


Desired SignalStrong Adjacent Channel

Desired Signal

RF

LO

IF

The desired signal is buried under the phase noise of an adjacent strong channel.

Phase Noise in RF Applications

f

f

f

fRF

fLO

fIF

RF

LO

IF


Units of Phase Noise

log(∆ω)=log(ω-ω0)

L ∆ω{ }1

f 3------

1

f 2------

Measured in dB below carrier per unit bandwidth.

(-20dB/dec)

(-30dB/dec)

Sv ω( )

ω

∆ω

ω0

dBc Hz⁄[ ]

ω1 f 3⁄

dBc


Internal noise sources set a fundamental limit for phase noise.

i n2

∆f------

MOSBJT

1f---

Low frequency noise can be an important contributor to the system noise.

Thermal and 1/f Noise

log(f)


Isubstrate

Vsupply

Isupply



Outline





Measurement Results

Conclusion



Oscillator with Input Noise Sources

i n1 t( )

i n2 t( )

i nM t( )

V out t( ) A t( ) f ω0t φ t( )+[ ]⋅=

t

Vout

Oscillator

Non-ideal waveform

Ideal waveform

V out t( ) A ω0t φ0+( )cos⋅=

Noise current

v n1 t( )

v n2 t( )

v nN t( )

sources.

Noise voltagesources.


C Li(t)δ t τ–( )

t

i(t)

t

Vout

t

Vout

Oscillators Are Time-Variant Systems

τ

Impulse injected at the peak of amplitude.

∆V

∆V

Even for an ideal LC oscillator, the phase response is Time Variant.

Impulse injected at zero crossing.τ

τ


Once Introduced, phase error persists indefinitely.

Non-linearity quenches amplitude changes over time.

θ

∆θ

V

dVdt--------

LimitCycle

a

b

∆V

Active Device

C Lδ t τ–( )

t

i(t)

τ

G -G(A)

Amplitude Restoring Mechanism

i(t)


∆V ∆qCnode---------------=

∆φ Γ ω0t( )∆V

Vswing--------------- Γ ω0t( )

∆qqswing--------------= =

CR

i(t)

V(t)

V(t)

∆φ

∆V

Vin

Vout

V+V-

+V1

-V1

Impulse Response of a Relaxation Oscillator

∆q qswing«


2ns 3ns 4ns

1.0

3.0

5.0

2ns 3ns 4nsTime

1.0

3.0

5.0

Nod

e V

olta

ge (

Vol

t)N

ode

Vol

tage

(V

olt)

∆V ∆qCnode---------------=

∆φ Γ ω0t( )∆V

Vswing--------------- Γ ω0t( )

∆qqswing--------------= =

i t( )

1 2 3 4 5δ t τ–( )

tτ

i(t)

Impulse Response of a Ring Oscillator

∆q qswing«


Phase Impulse Response

φ t( )hφ t τ,( )

0 t

i(t)

τ 0 τ

hφ t τ,( )Γ ωoτ( )

qmax-------------------u t τ–( )=

t

i t( )

The unit impulse response is:

Γ x( ) is a dimensionless function periodic in 2π, describing how much

phase change results from applying an impulse at time: t T x2π------=

The phase impulse response of an arbitrary oscillator is a time varying step.


t

t

t

t

V out t( ) V out t( )

Γ ω0t( ) Γ ω0t( )

LC Oscillator Ring Oscillator

Impulse Sensitivity Function (ISF)

The ISF quantifies the sensitivity of every point in the waveform to perturbations.

Waveform

ISF


φ t( ) hφ t τ,( )i τ( )dτ∞–

∞

∫1

qmax---------------- Γ ω0τ( )i τ( )dτ

∞–

t

∫= =

φ t( )i t( )hφ t τ,( )

Γ ω0τ( )

qmax-------------------u t τ–( )=

Superposition Integral:

Phase Response to an Arbitrary Source

Γ ω0t( )

∞–t

∫ ω0t φ t( )+[ ]cosi t( )

qmax----------------

φ t( )ψ t( ) V t( )

IdealIntegration

PhaseModulation

Equivalent representation:


i t( ) φ t( ) V t( )hφ t τ,( ) ω0t φ t( )+[ ]cos

LTV system Nonlinear system

Phase Noise Due to White Noise

i n2

∆f---------

L ∆ω{ }Γrms

2

qmax2

----------------in

2 ∆f⁄

2∆ω2-------------------⋅=

For a white input noise current with the spectral density of

The phase noise sideband power below carrier at an offset of ∆ω is:

Γrms is the rms value of the ISF.


ISF is a periodic function:

cn nω0t θn+( )cos

c0

c1 ω0t θ1+( )cos

Σ ω0t φ t( )+[ ]cos

i t( )qmax-------------

φ t( )

∞–t

∫

∞–t

∫

∞–t

∫

V t( )

ISF Decomposition

Phase can be written as:

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

t

∫ cn i τ( ) nω0τ( )cos dτ∞–

t

∫n 1=

∞

∑+=

Γ ωot( ) c0 cn nω0τ θn+( )cosn 1=

∞

∑+=


c0c1 c2 c3

2ω0ω0 3ω0ω

Sφ ω( )

i n2

∆f------- ω( )1f--- Noise

Sv ω( )

2ω0ω0 3ω0 ω

ω

PM∆ω

∆ω ∆ω ∆ω

∆ω

Noise Contributions from nωo

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

t

∫ cn i τ( ) nωτ( )cos dτ∞–

t

∫n 1=

∞

∑+=


log(ω)

Sφ f( )

1f 2-----

L ∆ω( )1f 3-----

1f 2-----

Amplifier Noise Floor

PSD of φ(t) PSD of V(t)

Power Spectrum of Phase Noise

c0

c1

c2

c3

ω1f---

ω 1f 3----

Noise components around integer multiples of the oscillation frequency havethe strongest effect on phase noise, and their effect is weighted by the Fouriercoefficients of the ISF, cn.

log(ω−ω0)


Outline





Measurement Results

Conclusion



c012π------ Γ x( )dx

0

2π

∫=

t

t

V out t( )

Γ ωt( )

Symmetric rise and fall time

t

t

V out t( )

Γ ωt( )

Asymmetric rise and fall time

Effect of Symmetry

The dc value of the ISF is governed by rise and fall time symmetry, and

controls the contribution of low frequency noise to the phase noise.


1/f 3 Corner of Phase Noise Spectrum

ω1 f

3⁄ω1 f⁄

c0Γrms--------------

2

=

The 1/f3 corner of phase noise is NOT the same as 1/f corner of device noise

log(ω-ωo)

L ∆ω( )

1f 2-----

c0

c1

c2

c3

ω1f---

ω 1f 3----

By designing for a symmetric waveform, the performancedegradation due to low frequency noise can be minimized.


6.4µm0.8µm----------------

6.4µm0.8µm----------------

3.2µm0.8µm----------------

3.2µm0.8µm----------------

i1 t( ) i2 t( )

Ring Oscillator with an Asymmetric Stage

The effect of asymmetry can be seen by comparing the effect of

low frequency injection into symmetric and asymmetric nodes.


0.8GHz 0.9GHz 1.0GHz 1.1GHzFrequency

-100

-50

0

Pow

er b

elow

Car

rier

(dB

c)

Injection into Asymmetric Node

0.8GHz 0.9GHz 1.0GHz 1.1GHzFrequency

-100

-50

0

Injection into Symmetric Node

-40dBc -52dBc

Low Frequency Upconversion


0.5 1.0 1.5 2.0 2.5Wp to Wn ratio

-80.0

-70.0

-60.0

-50.0

-40.0

-30.0

Sid

eban

d po

wer

bel

ow c

arrie

r (d

Bc)

Sidebands Due to Low Frequency Injection

Effect of Rise and Fall Time Symmetry

WN/L

WP/L

fo=1GHzfoff=50MHz

i t( )

1 2 3 4 5Analytical ExpressionSimulation


25µm1µm---------------

25µm1µm---------------

150µm1µm------------------

150µm1µm------------------

1mA

i injection t( )

Gnd

Effect of Differential Symmetry

The noise sources on each of the differential nodes are not fully correlated.

It is the symmetry of the half circuit that matters.


600MHz 700MHz 800MHz 900MHzFrequency (Hz)

-80

-60

-40

-20

0

Pow

er (

dB)

Low Frequency Current Injection into Differential Ring

-46dBc

Effect of Differential Symmetry

Differential symmetry does not automatically eliminate the low frequency upconversion.


L

C

Vdd

A Symmetric LC Oscillator

Possible to Adjust Symmetry Properties of the Waveform

WN/L

WP/L

WN/L

WP/L

Adjust ratiosfor symmetry


Outline





Measurement Results

Conclusion



0

5

10

15

20

Col

lect

or V

olta

ge (

Vol

ts)

0 5ns 10ns 15nsTime

0

1mA

2mA

3mA

Col

lect

or C

urre

nt

Time Varying Current in Colpitts Oscillator

10kΩ

40pF

200pF500µA

200nH

Gnd

Gnd

Vcc


i n t( ) i n0 t( ) α ω0t( )=

i n0 t( )

α ω0t( )

Cyclostationary Properties, Time Domain

φ t( ) in τ( )Γ ω0τ( )

qmax------------------dτ

∞–

t

∫=

in0 τ( )α ω0τ( )Γ ω0τ( )

qmax-------------------------------------dτ

∞–

t

∫=

Γeff x( ) Γ x( ) α x( )⋅=

Effective ISF:

A cyclostationary source can be modeled as stationary with a new ISF.

Noise Modulating Function (NMF)


0.0 2.0 4.0 6.0x (radians)

-1.5

-0.5

0.5

1.5

Colpitts Oscillator

Γeff x( ) Γ x( )α x( )=

Γ x( )

α x( )

Effective ISF

ISFNMF α x( )

Γ x( )

Γeff x( )

Gnd

Vcc


0.0 2.0 4.0 6.0x (radians)

-1.0

-0.5

0.0

0.5

1.0

5 Stage Ring Oscillator

Γeff x( ) Γ x( )α x( )=

Γ x( )

α x( )

Effective ISF

ISFNMF

1 2 3 4 5


Outline





Measurement Results

Conclusion



i t( )i t( ) i t( ) i t( )i t( )

Fully Correlated Sources

1 2 3 4 5

Γk ωot( ) Γ ωot2πkN

---------+ =Similar stages:

ej2π5

------

ej8π5

------

ej4π5

------

ej6π5

------Γk ω0τ( )k 1=

N

∑ Nc0≈

Superposition: φtotal t( ) φ t( )k 1=

N

∑ i τ( )qmax----------- Γk ω0τ( )k 1=

N

∑ dτ∞–

t

∫Nc0qmax----------- i τ( )dτ

∞–

t

∫≈= =


φtotal t( )Nc0qmax----------- i τ( )dτ

∞–

t

∫≈

i t( )i t( ) i t( ) i t( )i t( )

1 2 3 4 5

Fully Correlated Sources

Only the low frequency portion of substrate and supply noise is important, provided:

1. Stages and loadings are the same

2. The noise sources are identical

This is good news since c0 can be significantly reduced by adjusting the symmetry.


Outline





Measurement Results

Conclusion



c0c1 c2 c3

2ω0ω0 3ω0ω

Sφ ω( )

i n2

∆f------- ω( )1f--- Noise

Sv ω( )

2ω0ω0 3ω0 ω

ω

PM∆ω

∆ω ∆ω ∆ω

∆ω

Noise Contributions from nωo

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

t

∫ cn i τ( ) nωτ( )cos dτ∞–

t

∫n 1=

∞

∑+=

∆ω


10kHz 100kHz 1MHzOffset from Multiple Integer of Carrier (fm)

-50.0

-40.0

-30.0

-20.0

-10.0

Sid

eban

d P

ower

bel

ow C

arrie

r (d

Bc)

Injection at Integer Multiples of f0

f=fmf=f0+fmf=2f0+fmf=3f0+fmf=4f0+fm

1 2 3 4 5

i t( ) I n f 0 f m+( )sin=

fm dBc

f0


1µA 10µΑ 100µΑ 1mAInjected Current (amperes)

-80.0

-60.0

-40.0

-20.0

Sid

eban

d P

ower

bel

ow C

arrie

r (d

Bc)

f=fmf=f0+fmf=2f0+fmf=3f0+fm

Sideband Power vs. Injection Current

i t( ) I n f 0 f m+( )sin=

1 2 3 4 5

fm dBc

f0


100kHz 1MHzFrequency of Injection (Hz)

-55

-45

-35

Sid

eban

d P

ower

bel

ow C

arrie

r (d

Bc)

Symmetric, n1

Symmetric, n4

Asymmetric, n1

Asymmetric, n4

Symmetric vs. Asymmetric Ring Oscillator

i1 t( )

1 2 3 4 5

i4 t( )

fm dBc

f0


102 103 104 105 106

Offset from the Carrier (Hz)

-130

-80

-30

Sid

eban

d be

low

Car

rier

per

Hz

(dB

c/H

z)

f0=232MHz, 2µm Technology

L ∆f{ } 10 0.84 ∆f 2⁄( )log=

L 500kHz{ } 114.7dBcHz----------–=

ω 1f 3-----

75kHz=

L 500kHz{ } 114.5dBcHz----------–=

ω 1f 3-----

80kHz=

Predicted:

Measured:

5-Stage Single-Ended Ring Oscillator


102 103 104 105 106-130

-80

-30

Sid

eban

d P

ower

bel

ow C

arrie

r pe

r H

z (d

Bc/

Hz)

f0=115MHz, 2µm Process

Offset from the Carrier (Hz)

L ∆f{ } 10 0.152 ∆f 2⁄( )log=

L 500kHz{ } 122.1dBcHz----------–=

ω 1f 3-----

43kHz=

L 500kHz{ } 122.5dBcHz----------–=

ω 1f 3-----

45kHz=

Predicted:

Measured:

11-Stage Single-Ended Ring Oscillator


W/LNtail

W/LNinv

W/LPtail

W/LPinv

Nbias

Pbias

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Symmetry Voltage (VPbias+VNbias) [V]

200kHz

400kHz

600kHz

800kHz

1.0MHz

1.2MHz

1/f3

Cor

ner

Fre

quen

cy

9-Stage Current Starved Single-Ended VCO

Vsym=VPbias+VNbias

f0=600MHz, 0.25µm Process


2 4 6 8 10 12 14 16Tail Current (mA)

-126

-124

-122

-120

-118

-116

-114

-112

Pha

se N

oise

at 6

00K

Hz

offs

et (

dBc/

Hz)

f0=1.8GHz, 0.25µm Process (VDD =3V)

Measurement

Complementary Cross-Coupled LC Oscillator

C

L

Vdd

bias

Gnd

Itail

Γ2rms=0.5

Simulated ISF


Complementary Cross-Coupled VCO

1.5

2

2.5

3

24

68

1012

1416

x 10−3

−126

−124

−122

−120

−118

−116

−114

−112

f0=1.8GHz, 0.25µm Process

Pha

se n

oise

bel

ow c

arrie

r at

600

kHz

offs

et

Vdd Itail (mA)

-121dBc/Hz@600kHz

f0=1.8GHz

P=6mW

CL

Vdd

bias

Gnd

Itail


Complementary vs. NMOS-Only VCO

Vdd

C

bias

Gnd

L/2 L/2

Itail

CL

Vdd

bias

Gnd

Itail

11.5

22.5

3

2 4 6 8 10 12 14 16

x 10−3

−126

−124

−122

−120

−118

−116

−114

−112

f0=1.8GHz, 0.25µm Process

VddItail (mA)

Pha

se n

oise

bel

ow c

arrie

r at

600

kHz

offs

et

NMOS-Only

Complementary


Die Photo of the Complementary Oscillator

700µm x 800µm

L L

C

Active

Bypass Bypass

Driver

Pad limited

0.25µm Process


Conclusion and Contributions

A new general model for phase noise is introduced, which:

● is useful both as an analysis and a design tool,

● is valid for arbitrary sources of noise and interference,

● is independent of the topology of the oscillator,

● predicts the effect of symmetry on the upconversion of 1/f noise,

● incorporates cyclostationary noise sources naturally,

● shows agreement among theory, simulation and measurements.

● reduces to previously existing models as special cases,

● predicts the effect of correlation on phase noise,


AcknowledgmentsAdvisor: Prof. Thomas H. Lee.

Stanford faculty: Prof. K. Saraswat and Prof. J. McVittie.

Tom Lee group: Arvin, Dave, Derek, Hamid, Hirad, Kevin, Mar, Mohan, Raf, Ramin and Tam.

Committee: Prof. Bruce Lusignan, Prof. Bruce Wooley and Prof. Mark Horowitz.

Brains : Ken, Masoud, Mehrdad, Sotirios and Stefanos, Tom Lee group, H. Swain, D. Leeson.

Other groups: Wooley group, Horowitz group, SNF.

Companies: Rockwell Semiconductors, Texas Instrument and Lucent Technologies.

DEDICATED TO ALL MY TEACHERS.

Family: Tabassom, Mom, Dad and Emad.

Friends: Adrian, Ali (x3), Amir (x2), Amirmasoud, Arash, Ardavan, Arvin, Azam, Azita,Babak(x2), Bijan(x2), David(x2), Derek, Eric, Faranak, Farid, Fati, Hamid (x3),Hirad, Hossein, Jalil, Joe, Kambiz, Kati, Kevin, Kiarash, Koohyar, Manohar,Mar, Maryam, Masoud, Matt, Mehdi(x3), Mehrdad, Mina, Mohan, Nogol(x2),Patrick, Philip, Raf, Ramin(x2), Rasoul, Reza(x2), Sha, Shahram, Shideh,

Real power: Ann Guerra.

Sotirios, Stefanos, Tamara, Tayebe, Tina, Yasmin, Younes.

T. Lee, Paul G. Allen Center for Integrated Systems

Narrowband CMOS RF Low-Noise Amplifiers


Prof. Thomas H. LeeStanford University

[email protected]://www-smirc.stanford.edu



Outline

❏❏❏❏ A brief review of classic two-port noise optimization❏❏❏❏ Conditions for minimum noise figure❏❏❏❏ The fundamental importance of correlations

❏❏❏❏ MOSFET noise models in the short-channel regime❏❏❏❏ Equivalent two-port noise generators❏❏❏❏ Second-order noise sources

❏❏❏❏ Power constrained noise optimization❏❏❏❏ Experimental results on devices and circuits❏❏❏❏ Summary and conclusions



Classic Two-Port Noise Optimization

❏❏❏❏ Consider noise in an arbitrary (but linear) system:

❏❏❏❏ Thermal noise of source represented by iS❏❏❏❏ Source admittance is YS

Noisy 2-PortYSiS




❏❏❏❏ The noisy two-port may be modeled as follows:

❏❏❏❏ In general, the external noise sources will be partially corre-lated❏❏❏❏ Correlations arise because an internal noise source may contrib-

ute to both in and en in general

❏❏❏❏ Correlations have strong implications for noise performance

NoiselessYsis

+-

Two-Portin

en




❏❏❏❏ Noise factor, F, is defined as the ratio of the total output noise power divided by that part of the output noise power due to the input source, when source is at 290K

❏❏❏❏ Therefore:

❏❏❏❏ Let noise current in be expressed as sum of two terms

❏❏❏❏ First term, iu, is fully uncorrelated with noise voltage en. Other term, ic, is fully correlated with en.

Fis2 in Ysen+

2+

is2

----------------------------------------------------=




❏❏❏❏ Since ic is correlated with en, we may write one as propor-tional to the other:

❏❏❏❏ Note that Yc has the dimensions of an admittance

❏❏❏❏ Correlation admittance is a mathematical construct, and is not what one measures with an impedance meter

❏❏❏❏ Re-write F as

ic Ycen=

F 1in YSen+

2

iS2

-----------------------------------+ 1iu2 Yc YS+

2en2

+

iS2

--------------------------------------------------+= =




❏❏❏❏ Next, define effective noise resistances (conductances):

❏❏❏❏ Also:❏❏❏❏ Yc = Gc + jBc❏❏❏❏ Ys = Gs + jBs

❏❏❏❏ Finally obtain:

Rnen2

4kT∆f------------------- Gu

iu2

4kT∆f------------------- GS

iS2

4kT∆f-------------------≡,≡,≡

F 1GuGs---------

RnGs-------- Gs Gc+( )

2 Bs Bc+( )2

++ +=




❏❏❏❏ Minimum F occurs when Bs = -Bc = Bopt and

❏❏❏❏ Minimum F is

❏❏❏❏ In general,

❏❏❏❏ Thus, contours of constant noise figure are circles centered about (Gopt, Bopt) in the admittance or Smith plane

GsGuRn--------- Gc

2+ Gopt= =

Fmin 1 2RnGuRn--------- Gc

2+ Gc

2++=

F FminRnGs-------- Gs Gopt–( )

2 Bs Bopt–( )2

++=




❏❏❏❏ Source admittance for optimum noise match does not gener-ally have any relation to the conditions for optimum power gain❏❏❏❏ Possible to have great noise figure and little or no gain

❏❏❏❏ Possible to have great noise figure and a poor impedance match

❏❏❏❏ Classical noise optimization also does not consider power consumption directly❏❏❏❏ Modified approach required to balance all parameters of practi-

cal interest




❏❏❏❏ Classical noise optimization starts with a fixed device and its characteristics and yields the source impedance that min-imizes noise figure

❏❏❏❏ In IC’s, the device geometry is a controllable parameter, so how does one go about selecting a particular device size? How do power and matching constraints influence the choice? What about gain?

❏❏❏❏ To answer these questions, let’s now consider some amplifi-er topologies❏❏❏❏ By understanding their properties, perhaps we can converge on

a practical circuit



MOSFET Two-Port Noise Parameters

❏❏❏❏ Consider only drain and induced gate current noise. Then, the following two-port parameters apply:

Parameter Expression

Gc Ý 0

Bc

Rn

Gu

ωCgs 1 α cδ5γ--------+

γgd0

gm2

------------------γα----- 1

gm-----------⋅=

δω2Cgs2 1 c 2–

5gd0----------------------------------------------------------




❏❏❏❏

❏❏❏❏ Bopt is inductive, except for frequency behavior. Difficult to provide this behavior over a large bandwidth.

❏❏❏❏

❏❏❏❏

❏❏❏❏ Note that Fmin = 0dB if gate and drain noise were fully cor-related. The mere presence of noise sources does not neces-sarily imply nonzero NF.

Bopt Bc– ω– Cgs 1 α cδ5γ--------+

= =

GoptGuRn----------- Gc

2+ αωCgs

δ5γ-------- 1 c 2–

= =

Fmin 1 2Rn Gopt Gc++ 125

-------- ωωT----------- γδ 1 c 2–

+≈=




❏❏❏❏ Let’s now compile a short table of Fmin values:

❏❏❏❏ Numbers pessimistically assume that hot electron effects tri-ple the mean-square noise densities.

❏❏❏❏ Even with such effects, achievable noise figures are very good.

❏❏❏❏ Question: How can these values be approached in practice?

gm/ωCgs

Fmin (dB)

20 0.515 0.610 0.95 1.6



Second-Order Noise Sources

❏❏❏❏ Practical NF values are affected by series gate resistance and epi noise.

❏❏❏❏ F is increased by Rg/Rs, so just 10Ω by itself sets a lower NF bound of 0.8dB in a 50Ω system.❏❏❏❏ Must use multi-fingered devices (Rfinger= RSHWfinger/3L).

❏❏❏❏ Cannot use planar spiral inductors in gate circuit if best NF is to be achieved (NF typically > 2-3dB).

❏❏❏❏ Thermal noise of substrate (epi) resistance modulates the back gate, giving rise to additional drain current noise:

ind2

∆f---------- 4kT γgd0 gmb

2 Repi+ 4kTgd0 γ

gmb2 Repi

gd0-----------------------------+

= =



Second-Order Noise Sources

❏❏❏❏ Effect of epi noise is equivalent to an increase in γ:

❏❏❏❏ One may compute that, typically, epi noise increases γ by ~10%, an amount smaller than the uncertainty in γ itself.

❏❏❏❏ Epi noise also contributes to equivalent input current noise, but this is fully correlated with the drain noise.❏❏❏❏ Again, fundamental NF limits are set by the uncorrelated gate

and drain noise components.

γeff γgmb

2 Repigd0

-----------------------------+=



Narrowband LNA

❏❏❏❏ Choose inductive source degeneration to produce desired real part:

❏❏❏❏ Equation assumes a cascode stack with equal-sized devices

❏❏❏❏ Choose sum of gate and source degenerating inductances ei-ther to resonate with Cgs or to provide a susceptance equal to Bopt.

❏❏❏❏ First choice maximizes gain, second choice minimizes NF. Differ-ence is small because Bopt Ý ωCgs.

❏❏❏❏ Note that classic noise optimization says nothing about pow-er dissipation, nor anything about how to select device width.

LSRS 1 2 Cgd Cgs⁄( )+[ ]⋅

ωT-------------------------------------------------------------------≈



Power-Constrained Noise Optimization

❏❏❏❏ Good approximation: Select device width roughly equal to (500µm-GHz)/f0 (for a 50Ω system).

❏❏❏❏ Adjust bias to obtain desired power dissipation.❏❏❏❏ Keep VDS–VDSAT as small as practical to minimize hot-electron

effects (say, under half a volt or so).

❏❏❏❏ For equal-sized cascoding and main devices, continue to se-lect source degeneration inductance according to:

❏❏❏❏ Add gate inductance to bring input to resonance.❏❏❏❏ Noise factor bound is 1 + 2.4(γ/α)(ω/ωT), so scaling contin-

ues to help directly.

LSRS 1 2 Cgd Cgs⁄( )+[ ]⋅

ωT-------------------------------------------------------------------≈



LNA Input Stage: Some Observations

❏❏❏❏ As noted, overall stage transconductance is independent of device gm if resonant frequency and current density are held constant.❏❏❏❏ Theoretically, may use arbitrarily narrow devices and still ob-

tain the desired transconductance.

❏❏❏❏ If drain current noise were the only noise source, narrower devices would lead to monotonically decreasing noise.

❏❏❏❏ Since gain is fixed, noise figure approaches 0dB as device narrows. Power dissipation would also approach zero.

❏❏❏❏ Absurd conclusion of zero dB NF, zero power dissipation and nonzero gain should make one suspect that something is missing from the foregoing.



Experimental Results: Devices

❏❏❏❏ For 0.5µm technology (drawn), NFmin Ý 1.0dB @ 2mA, 1GHz.❏❏❏❏ NFmin decreases to Ý 0.7dB @ high ID.

❏❏❏❏ NFmin increases to Ý 1.3dB @ 2GHz @ high ID.

❏❏❏❏ NFmin still below 2dB @ 400µA, 1GHz.

❏❏❏❏ These values apply to a single device without regard for in-put impedance.❏❏❏❏ Practical NFmin values are perhaps 0.5dB to 1dB higher.

❏❏❏❏ Contrary to expectations, no increase in NFmin is observed in these devices as VDS increases in saturation.

❏❏❏❏ Drain engineering possibly responsible (G. Klimovitch et al., 1997).



Experimental Results: Circuits

❏❏❏❏ Single-ended versions consume half the power for a given NF than differential versions, but:❏❏❏❏ No rejection of common-mode noise.

❏❏❏❏ Very sensitive to parasitics, particularly inductances in the source lead of the main transistor.

❏❏❏❏ Differential versions are relatively insensitive to hard-to-model and hard-to-control parasitics.❏❏❏❏ Attractive for high-volume production.❏❏❏❏ Common-mode rejection highly desirable for mixed-signal envi-

ronments.



Experimental Results: Circuits

❏❏❏❏ Series gate inductance provided by bondwires to avoid inev-itable NF degradation associated with spiral inductors.❏❏❏❏ Difficult to obtain accurate value without trimming, but repeat-

ability with automated die attach and bonding equipment is very good.

❏❏❏❏ Input Q is generally 3-5, so LNA is somewhat forgiving of mod-erate element value variation.

❏❏❏❏ Measured and simulated NF agree to within 0.2dB.❏❏❏❏ S11 < –15dB.

❏❏❏❏ Receiver IIP3 > –16dBm (measurement confounded by lin-earity limitation of subsequent receiver stages).❏❏❏❏ IIP3 > –6dBm for LNA itself (simulated).



Summary and Conclusions

❏❏❏❏ CMOS devices are capable of excellent noise performance in the low-GHz frequency range.❏❏❏❏ Noise performance will continue to improve, despite fears that

hot-electron effects will nullify the benefits of scaling.

❏❏❏❏ Inductively-degenerated LNA architecture simultaneously provides near-optimum gain and NF.❏❏❏❏ Proper device width is important, also.

❏❏❏❏ At under 10mW dissipation, practical single-ended amplifi-er noise figures of ~1.5dB at 1GHz are achievable with 0.5µm technology.

❏❏❏❏ Short-channel effects improve linearity, so dynamic range per power will improve with scaling.

❏❏❏❏ Epi and gate resistance noise effects are minor, or can be made so.

1

ABabcdfghiejkl Stanford UniversityCenter for Integrated SystemsIntegrated Circuits Lab

High Frequency Noise inCMOS Low Noise Amplifiers

Jung-Suk Goo

Center for Integrated Systems, Stanford [email protected]

7


Fundamentals of Noise (Continue)(Noise Sources in MOSFETs)

✧ Shot Noise❖ Dominant in subthreshold region

✧ Generation-Recombination Noise❖ Low-frequency noise

✧ Flicker (1/f ) Noise❖ Low-frequency noise❖ Important in mixers (up-conversion error)

✧ Thermal Noise❖ White (frequency independent) noise❖ Ultimate performance limit of amplifiers

Sin =

4 k TR

,

i2n�f

9


✤ Minimum Noise Factor (Fmin) : Best achievable noise performance

✤ Optimum Source Admittance (Yopt) : Source admittance yielding NFmin

✤ Equivalent Noise Resistance (Rn) : Sensitivity of NF when Ys differs from Yopt

Fundamentals of Noise (Continue)(4 Noise Parameters)

−10

1

−1−0.500.510

5

10

15

20

13


A B

D

CE

F

Two-Port Theory (Continue)(How to Use in Analysis ?)

CA = A1CA2Ay1

+CA1 CY = CY1 +CY2CZ = CZ1 +CZ2

Cascade ( A) Series ( Z) Parallel ( Y)

❃ H. Hillbrand and P. H. Russer (IEEE Trans. on Circuits and Systems, Apr. 1976)

15


Rs

Rg

Rd

Rsub

S

G D

BSource Drain

Gate

Sin

Sig

Sid

MOSFET Noise Modeling(High Frequency Noise Sources)

Parasitics Intrinsic

16


MOSFET Noise Modeling (Continue)(Classical van der Ziel Model)

i2d

i2g

=

i2d

4 k T �f gd0

� =

i2g

4 k T �f

17


Typical Approach

Proposed Approach

MOSFET Noise Modeling (Continue)(Modeling Approach)

❖ Deembed all parasitic components (Rg, Rs, Rd, and Rsub) usinglumped equivalent circuit.

❖ Derive intrinsic noise model based on dc transport physics.

❖ Deembed components from Rg, Rs, and Rd only.

❖ Find contribution mechanism from Rsub using simulation.

❖ Identify root cause of excess intrinsic noise based onsimulation.

19


105

1010

10−21

10−20

10−19

Frequency [Hz].

Dra

inC

urre

ntN

oise

[A2

/Hz]

I

II

III

IV

Cdepl

Rsub

Vsub

gmbVdepl

MOSFET Noise Modeling (Continue)(Drain Current Noise Spectrum)

105

1010

10−21

10−20

10−19

Frequency [Hz].

Dra

inC

urre

ntN

oise

[A2

/Hz]

1=f Noise

Channel Thermal Noise

Commonly Assumed Proposed

Sid;sub =

4kTRsubg2mb

1 + (!RsubCdepl)2

20


MOSFET Noise Modeling (Continue)(Reported Clues)

❖ Low frequency measurement studies reported very large γ.

❖ A frequency dependence is observed in Rn at a few GHz.

❖ A simulation study suggested a frequency dependence.

❃ J. J. Ou (Symp. VLSI Tech., 1999)❃ S. Donati (IEDM, 1998)

22


106

108

1010

10−24

10−23

10−22

10−21

Frequency [Hz].

Dra

inC

urre

ntN

oise

[A2

/Hz]

VDS=2.0V

0.4V

0.2V

0.0V

106

108

1010

10−24

10−23

Frequency [Hz].

Dra

inC

urre

ntN

oise

[A2

/Hz]

Lg=0.5�m

1.0�m

2.0�m

MOSFET Noise Modeling (Continue)(Substrated Induced Drain Current Noise)

✤ Match the description of excess noise.

✤ May exaggerate γ at low f. Sid;sub =

4kTRsubg2mb

1 + (!RsubCdepl)2

Lines : Simulation

Circles : Model

30


Rs

i2dp

i2Rs

ro i2n

MOSFET Noise Modeling (Continue)(Relation to BSIM4 Approach)

Physical BSIM4Simulation Implementation

✤ High ac resistance ro nearsource junction causesexcess noise.

✤ Capture excess noise byadjusting source sideextra resistance Rs.

31


❖ Parasitic noise sources from Rs, Rd, and Rg can be easilydeembedded.

❖ Substrate induced noise results in a frequency dependence indrain noise spectrum and exaggerate γ at low f.

❖ Substrate doping better be deep and high.

❖ The non-local transport behavior causes high local ac resistancenear the source junction and subsequently generates excessdrain current noise.

❖ Broader channel charge profile may improve γ : buried channel,JFET, MESFET, etc.

MOSFET Noise Modeling (Continue)(Summary)

33


Integrated LNA Design(Implications of CMOS Circuits)

❖ Basically higher γ and δ increase Fmin.

❖ Improvement of gm offsets impact of γ and δ.

❖ Thus, shorter channel length yields lower NFmin (low enough).

❖ Too small. |Γopt| is almost 1. ❖ Much larger than HEMTs, due to γ.

Achieving good NF of actual circuits is a major challenge!

Fmin

Yopt Rn

F = Fmin +(Ys � Yopt)2 Rn

Gs

34


Integrated LNA Design (Continue)(Noise Optimization)

State of the Art

Proposed Approach

❖ Exploit freedom in tailoring device geometries.

❖ Small-signal circuit is simplified.

❖ Constant γ and δ are assumed.

❖ Relied heavily on mathematical derivations. -> less intuitive

❖ Directly use measured s-parameters and noise parameters.

❖ Based on the four noise parameter concept. -> highly intuitive

❖ Applicable to any IC technologies.

35


Integrated LNA Design (Continue)(Basic Idea)

✧ Measure data extensively from one device size.

✧ Use the minimum channel length.

✧ Understand scaling effect of each parameters.❖ Fmin is not scaled by W.❖ Yopt is scaled by W.❖ Rn is scaled by (1/W).❖ Y-parameters are scaled by W.


Gs

36


Lg

Ls

ZC ZB ZA

Off-ChipMatching

Vs

Rs

CmTm

LgCp

Ls

Integrated LNA Design (Continue)(Tuned LNA Architecture)

Basic Architecture Conjugate Power Match

Controls noise performance. Why ?

37


10−1

100

101

0

50

100

150

200

Source Inductance [nH].

Noi

seR

esis

tanc

e[

]

M1 = 100/0.24

M2 = 50/0.24

VDS1= 2.5VVGS1= 0.8Vf = 4GHz

Rn;LNA

10−1

100

101

0

1

2

3

4

5


Noi

seF

igur

e[d

B]

M1 = 100/0.24

M2 = 50/0.24

VDS1= 2.5V

VGS1= 0.8V

f = 4GHz

NFLNA

NFmin;LNA

Integrated LNA Design (Continue)(Power Matched Design)

❃ M2 contribution is excluded.


Gs

38



M1 = 100/0.24

M2 = 50/0.24

VDS1= 2.5V

VGS1= 0.8V

f = 4GHz

Zconj;bestZopt;best

10−1

100

101

0

1

2

3

4

5


Noi

seF

igur

e[d

B]

M1 = 100/0.24

M2 = 50/0.24

VDS1= 2.5V

VGS1= 0.8V

f = 4GHz

NFLNA

NFmin;LNA



Gs

39


0.5 1 1.5 2 2.510

−2

10−1

100

101

102

Gate Bias ofM1 [V].

Opt

imum

Sou

rce

Indu

ctan

ce[n

H]

W2 =W1 /2

VDS1= 2.5V

f = 4GHz

20mA10mA

5mA

IDS=2mA

0.5 1 1.5 2 2.50

0.5

1

1.5

2

Gate Bias ofM1 [V].

Noi

seF

igur

e[d

B]

W2 =W1 /2

VDS1= 2.5V

f = 4GHz

NFLNA

NFmin

NFmin;LNA



❖ Optimum Ls is bias dependent and linearly scaled by thecurrent specification.

❖ The achievable noise figure is independent of the currentspecification and quite close to the intrinsic NFmin.

40


101

102

103

104

0

0.5

1

1.5

2

2.5

Width ofM2 [�m].

Noi

seF

igur

e[d

B]

M1 = 80/0.24

VDD = 2.5V

IDD = 5mA

f = 4GHz

Overall

M1 only

Integrated LNA Design (Continue)(Cascode Stage Design)

❖ Cascode device adds at least 40% extra noise.

❖ W2 must be large enough.

41


Integrated LNA Design (Continue)(Power Constrained Performance, Zs=Zin=50Ω)

Optimum Impedance (Zopt)

W2 =W1

VDD = 2.5V

f = 4GHz

Zconj=50

Zopt

W2 =W1=2

VDD = 2.0V

f = 800MHz

IDD=2mA

5mA

10mA Zconj=50

Zopt

Matched Cascode Mismatched Cascode


Gs

42


Integrated LNA Design (Continue)(Power Constrained Performance, Zs=Zin=50Ω)

0.5 1 1.50

50

100

150

200

250

300

Gate Bias ofM1 [V].

Noi

seR

esis

tanc

e[

]

W2 =W1=2

VDD = 2.0V

f = 800MHz IDD=2mA

5mA

10mA

0.5 1 1.50

2

4

6

8

Gate Bias ofM1 [V].N

oise

Fig

ure

[dB

]

W2 =W1=2

VDD = 2.0V

f = 800MHz

IDD=2mA

5mA

10mA

NFmin

Noise Resistance(Rn) Noise Figure(NF)


Gs

43


Integrated LNA Design (Continue)(Impact of Pad Capacitance, Zs=Zin=50Ω)

W2 =W1=2

VDD = 2.0V

f = 800MHz

IDD = 3.75mA

Zconj=50

Zopt

0.5 1 1.50

50

100

150

Gate Bias ofM1 [V].N

oise

Res

ista

nce

[

]

W2 =W1=2

VDD = 2.0V

f = 800MHzIDD = 3.75mA

Cp=0pF

20pF

50pF

Optimum Impedance (Zopt) Noise Resistance(Rn)


Gs

44


Integrated LNA Design (Continue)(Impact of Pad Capacitance, Zs=Zin=50Ω)

0.5 1 1.50

1

2

3

4

5

6

Gate Bias ofM1 [V].

Noi

seF

igur

e[d

B]

W2 =W1=2

VDD = 2.0V

f = 800MHz

IDD = 3.75mA Cp=0pF

20pF

50pF

NFmin

0.5 1 1.50

5

10

15

20

25

30

Gate Bias ofM1 [V].

jS21

j

[dB

]

W2 =W1=2

VDD = 2.0V

f = 800MHz

IDD = 3.75mA

Cp=0pF20pF

50pF

Gain (s21)Noise Figure(NF)

45


Integrated LNA Design (Continue)(Implementation , Zs=Zin=50Ω)

✤ 800MHz single-ended

✤ 0.24µm, silicided-poly, 5-metals

✤ W1=90, W2=45 (not optimized)

✤ 5µm-long gate fingers

✤ M5 spiral inductors w/patterned ground shield

✤ M5/M1 pad capacitors

✤ 24-pin LLP package

✤ An off-chip inductor for Lg

46


Integrated LNA Design (Continue)(Performance)

Parameters Measured Value

Frequency 800 MHz

Supply Voltage 2.0 V

Power Consumption 7.5 mW

Noise Figure 0.9�0.2 dB

Available Gain 8.8 dB

s11 -38.1 dB

IIP3 7.1 dBm

Die Area 0.19mm2

Just adds 0.3dB to NFmin

47


Integrated LNA Design (Continue)(Summary)

❖ The tuned LNA provides better noise performancethan the intrinsic MOSFET.

❖ Overall NF is controlled by Ls : Optimal Ls achieves NFclose to intrinsic NFmin with a perfect power match.

❖ For a fixed Zs, simultaneous choice of Vgs and width ofinput stage is most critical in design.

❖ Cascode stage adds at least 40% extra noise.

❖ Pad capacitance provides another design flexibility.

❖ CMOS LNA can be competitive with GaAs andBipolar in low GHz range.

EE314 Handout #9: Winter 2002

Noise Figure Measurement

1999 Thomas H. Lee, rev. February 8, 2002; All rights reserved Page 1 of 23


1.0 Introduction

One of the most important performance metrics for low-level amplifiers is noise figure,

NF

, or noise factor,

F

.

The two terms are used interchangeably in the literature, but we adopt the following arbitrary convention in this text: We will denote noise figure by

NF

, and define it as 10log

F

. We will be somewhat sloppy about using the terms (reflecting common usage), but context should make clear whether or not the decibel version is being discussed.

The definition of noise factor now in use was first formally proposed by Harald Friis

1

of Bell Labs. At its core, the definition involves signal-to-noise ratios (SNRs):

(1)

This definition shows that

F

is the factor by which an amplifier degrades the signal-to-noise ratio of the input signal. As such, it is never smaller than unity. As simple and straightforward as the definition appears to be, numerous subtleties are buried in it, and it will soon be clear that we have provided an incomplete definition. Accurate measurement of noise figure depends on a full appreciation of all of these subtleties, and an understand-ing of how to identify and correct sources of measurement error. As we’ll soon see, auto-mated noise figure instruments do not eliminate the need for a knowledgeable operator. As has been noted, “automated equipment merely lets you produce more wrong answers per unit time.” The purpose of this chapter is to reduce the rate of erroneous answer genera-tion.

2.0 Basic Definitions and Noise Measurement Theory

One important subtlety concerns the temperature at which the measurement of noise figure is made. Specifically, the temperature of the source has a profound effect on the noise fig-ure. Intuitively, this temperature dependence may be understood as follows: The device under test (DUT) generates its own internal noise, independent of the source temperature. If the latter is very low, then the source noise will be correspondingly low, so the noise added by the DUT will have a comparatively greater effect. The measured noise figure will thus be higher than if the source were hotter (and thus noisier). Because of this sensi-tivity, a meaningful comparison of noise figures requires that the measurements be made at a standard temperature. Friis proposed a reference temperature, denoted

T

0

, of 290 kelvins (about 62

°

F or 17

°

C), a temperature which is considerably cooler than the interior of most laboratories. An oft-cited reason for this choice is the approximate equality of this

1. “Noise Figures of Radio Receivers,”

Proc. of the IRE

, July 1944, pp. 419-422.

FSNRiSNRo

≡




temperature with that commonly seen by antennas used in terrestrial wireless communica-tions. However, perhaps a stronger motivation for its selection is simply that

kT

0

is then 4.00 x 10

-21

J, a round number with undeniable appeal in an era of slide rule computation, particularly to an eminently practical gentleman like Friis.

The final statement on standard conditions, made by a committee of the Institute of Radio Engineers (a forerunner of the IEEE), is that the noise figure measurement is to be made with a source whose available noise power is the same as that of an input termination whose temperature is 290K. Recall that available power is defined as the power that

could

be delivered to a (conjugately) matched load. Hence, even if the source does not in fact happen to drive a matched load, the power remains

available

. Available power is precisely what the words imply: a potential power, independent of the actual load. The standards committee accepted Friis’s recommendation for basing noise figure on available power, because this parameter can be related directly to the temperature of a thermal noise gener-ator, such as a resistor. Confusion about this definition is all too common, and can lead to serious errors, as will be made clear later in this chapter.

A second consideration is that determining input and output signal-to-noise ratios is by no means trivial.

2

Since noise figure is an intrinsic property of the DUT alone (assuming lin-earity, as we must if noise figure is to be uniquely definable), and therefore not of how you drive the DUT, it should be possible to devise a measurement that does not involve the use of an explicit signal. To do so, it is helpful to note that the available noise appearing at the output of the DUT results from two contributions. One is the amplified available source noise power (with the source at

T

0

= 290K), which has a value

, (2)

where

B

is the noise (brickwall) bandwidth and

G

av

is the available power gain of the DUT.

The other component of output noise is simply the noise added by the DUT itself. We call this noise contribution

N

a

. The total available output noise power is therefore

. (3)

Now let’s revisit, and revise, the noise figure definition of Eqn. 1:

. (4)

This quantity must be evaluated at 290K, as stated earlier.

2. A third subtlety arises in cases where the system contains frequency-translating elements such as mixers. We defer a discussion of this consideration to Section 8.0.

Nos kT0BGav=

N1 kT0BGav Na+=

FSNRiSNRo

≡Si Ni⁄

So No⁄=




Interpreting all quantities as available powers, the ratio of output signal

S

o

to input signal

S

i

is the available gain,

G

av

. The available input noise power is simply

kT

0

B

, and the avail-able output noise power is

N

1

as defined in Eqn. 3. So we may write

. (5)

The last expression on the right,

, (6)

is the definition officially adopted by the IRE.

3

It initially appears more attractive as a basis for measurement than Eqn. 1 because it contains no terms related to an explicit input or output signal. Using Eqn. 6, measurement of noise figure reduces to the measurement of noise, available gain and bandwidth. Unfortunately, there are still serious practical diffi-culties associated with trying to base a measurement directly on this equation. In particu-lar, it is not easy to measure the product of the effective noise bandwidth and available gain,

BG

av

, with high accuracy. The experimental difficulties are perhaps best appreciated after comparing the various noise measurement methods discussed in Section 6.0.

One of these alternative noise figure evaluation methods, which is implemented in com-mercial instruments such as the HP8970A, cleverly sidesteps the need to measure gain-bandwidth by employing a ratio of noise measurements performed at two different source temperatures. As a general philosophy, it is always advantageous to replace absolute mea-surements with ratiometric ones wherever dimensional considerations permit it. Fortu-nately noise factor is a dimensionless quantity, so a purely ratiometric measurement is possible. Gain-bandwidth product is not dimensionless, so measuring it should not be fun-damentally necessary here.

The basis for the ratiometric technique is that the use of a hot source increases the compo-nent of output noise due to the source, without changing the noise added by the DUT. If the ratio of the source temperatures is accurately known, then measuring the output noise powers under the hot and cold conditions permits us to solve for the noise added by the DUT and, hence, compute the noise figure.

The following plot of output noise power as a function of source temperature illustrates how such a ratiometric measurement solves our problem:

3. See

Proc. of the IRE

, vol. 51, no. 3, March 1963, pp. 434-442.

FSi Ni⁄

So No⁄1

Gav

NoNi

1

Gav

N1Ni

N1

Nos

kT0BGav Na+

kT0BGav= = = = =

FkT0BGav Na+

kT0BGav=




FIGURE 1. Output noise power vs. source temperature

Comparing features of this drawing with Eqn. 6, note that the slope and y-intercept tell us everything we need to compute

F

:

. (7)

Clearly, the need to measure gain-bandwidth has disappeared because two points deter-mine a line. Despite the straightforward nature of this observation, engineers have devised a surprising number of different ways to use noise data from two points to determine noise figure. Just keep in mind that underlying the seeming complexity in what follows is the extremely simple geometric picture of Figure 1.

If we make a noise power measurement at a source temperature,

T

h

, that is above the ref-erence temperature by an amount

T

ex

, then the available output noise power becomes

. (8)

Combining the hot measurement with the one at

T

0

(Eqn. 3), a little algebra allows us to find that the noise factor may be expressed as

. (9)

The ratio

N

2

/

N

1

is often called the “

Y

factor” in the literature (why? because it comes after

X

...). Figure 1 shows a cold temperature equal to the reference temperature,

T

0

, but it should be clear that any temperature other than

T

h

could be used to figure out the slope and intercept of the line. More generally, if the cold temperature

T

c

is not

T

0

, the numerator changes, so that the noise factor is

0TS (K)ThT0

Na

slope = kGavBN1

N2

Tex

Available outputnoise power (W)

–Te

Nos

FkT0BGav Na+

kT0BGav= 1

NakT0BGav

+ 1 y-interceptT0( ) slope( )

+= =

N2 kBGavTh Na+ kBGav T0 Tex+( ) Na+= =

FTex T0⁄

N2N1

1−

=




. (10)

The ratio

T

ex

/

T

0

is a property of the noise source, and is information (almost) supplied by the manufacturer. The qualifier “almost” applies because the manufacturer actually speci-fies a slightly different quantity called the

excess noise ratio

(ENR), which is defined as the ratio of noise powers actually delivered to a 50

Ω

load (or occasionally some other standard impedance level). However the ratio

T

ex

/

T

0

results from a consideration of

avail-able

powers (as does the

Y

factor). The two ratios are equivalent only in the special case where the noise source happens to have an impedance of precisely 50

Ω

. Despite the best efforts of manufacturers, this condition is not perfectly satisfied in practice, so substituting ENR for

T

ex

/

T

0

is one (generally small) potential source of error. Because it is much easier to determine ENR, however, that’s what the NBS (the National Bureau of Standards, now the National Institute for Standards and Technology, NIST) decided to do, and what man-ufacturers consequently measure and report.

In the “old days,” actual hot and cold sources were used, commonly with resistors at 77K (the boiling point of liquid nitrogen) and 373K (the boiling point of water, although the resistor was electrically heated to this temperature, rather than being immersed in an actual water bath). Clearly, the greater the temperature difference, the more accurately we can compute the slope and intercept, for a given magnitude of uncertainty in the power measurement. A limitation on the hot side is the difficulty of accurately determining or controlling the temperature. And the higher the temperature, the more significant the prob-lems of materials properties (e.g., melting).

Nowadays, it is common to use noise diodes

4

(see the chapter on RF diodes) which can produce the noise of an exceptionally hot source (e.g., 10,000K, higher than the melting point of any known metal) while remaining at room temperature. The same diode can pro-vide the cold reference as well, simply by turning it off, causing an internal resistive matching network to provide an available noise power that corresponds to the ambient temperature (RF choke RFC is simply an inductor large enough to be considered an open-circuit at all frequencies of interest):

FIGURE 2. Typical noise diode

4. See, e.g.,

HP Journal

, April 1983, p. 26.

F

TexT0

YTcT0

1− −

Y 1−=

OUTRFC DC block

DC bias current




One drawback is that, unlike true hot and cold resistors, such diodes are not fundamental standards; their hot noise cannot be computed from first principles. Since ENR must be known to great accuracy to be useful, it is usually traceable to a primary noise standard (which is a heated or cooled physical resistor) maintained by national laboratories, such as the NIST. This traceability accounts in part for the relatively high cost of noise diodes.

3.0 Noise Temperature

Noise temperature,

T

e

, is an alternative figure of merit used in place of noise figure in some cases. As seen in Figure 1, noise temperature is (minus) the extrapolated intercept of the noise power curve with the temperature axis. An intuitively appealing meaning of noise temperature can be extracted by translating the noise power curve to the right by a temperature equal to the noise temperature:

FIGURE 3. Noise temperature

The translated curve is that of a noiseless amplifier (because the noise at zero source tem-perature is zero) with the same slope (= available gain-bandwidth product, times

k

) as the original amplifier. As can be seen, this noiseless amplifier produces an available output noise power equal to the available output noise of the original amplifier, if the source is now heated to a temperature

T

0

+

T

e

. The increase in available output noise power due to the hotter source is precisely equal to the available noise (

N

a

) added by the original DUT:

. (11)

Noise temperature is used most often in satellite communications systems for several rea-sons. One is that objects in the sky generally don’t have an effective temperature any-where near 290K, so choosing such a reference temperature has a weaker physical justification. The other is that space communication systems generally have exceptionally low noise figures, and noise temperature is a higher resolution measure of very low noise figure values. The following table compares noise figure, noise factor and noise tempera-ture over a range generally considered very low noise:

0 TS (K)T0

Na

N1

N2

–Te T0 + Te

Available outputnoise power (W)

Na kTeBGav=




TABLE 1. Comparison of noise figure, noise factor and noise temperature

It is sometimes helpful to note that, in the very low noise figure regime (e.g., below about 1dB), the noise figure in dB is approximately the noise temperature divided by 70-75. Stated alternatively, each tenth of a dB corresponds to roughly 7-7.5K.

To relate noise temperature and noise factor, return again to the official IRE noise figure definition:

. (12)

Substituting Eqn. 11 for

N

a

yields

, (13)

which simplifies to

. (14)

If the noise added by the DUT equals the noise power of the source, the noise figure will be 3dB, corresponding to a noise temperature of 290K. Many LNAs with effective noise temperatures well below 100K (corresponding to noise figures below 1.3dB) are commer-cially available.

The noise temperature may be found indirectly by relating Eqn. 14 to Eqn. 9, or directly from the hot and cold noise measurements of Figure 1. Pursuing the latter strategy, we may write

(15)

and

NF

(dB)

F T

e

(kelvins)

0.5 1.122 35.4

0.6 1.148 43.0

0.7 1.175 50.7

0.8 1.202 58.7

0.9 1.230 66.8

1.0 1.259 75.1

1.1 1.288 83.6

1.2 1.318 92.3

FN1Nos

≡kT0BGav Na+

kT0BGav=

FkT0BGav Na+

kT0BGav

kT0BGav kTeBGav+

kT0BGav= =

F 1Te

290+=

N2 kThBGav Na+ k Te Th+( ) BGav= =




, (16)

so that

. (17)

Solving for

T

e

yields

. (18)

Perhaps the most common reason that noise temperature is used in some contexts is that the quantity

F

– 1 recurs frequently in calculations of cascaded noise figure, as we shall see in Section 4.0. By rearranging Eqn. 14, it’s clear that noise temperature

T

e

is propor-tional to

F

– 1, so its use simplifies such calculations considerably.

Because both noise figure and noise temperature fully convey the information of the other (as implied by Eqn. 14, for example), you may use either. The choice of which to use is made largely on the basis of culture and convenience.

3.1 Spot noise figure

In many cases, one is interested in the noise performance of an amplifier as a function of frequency. In those situations, the measurement bandwidth is restricted to some known value (e.g, 4MHz, as in the HP8970) and the noise figure for that bandwidth is reported at a specific frequency. Since the parameter is thus a noise figure measured in a narrow band centered around a specific spot, it is known as the spot noise figure. The noise figures most often reported in the literature are spot noise figures.

4.0 Friis’ Formula for the Noise Figure of Cascaded Systems

Computing the noise figure of a cascade of systems is often carried out incorrectly. Once again, the problem is a failure to appreciate certain subtleties. One difficulty is that indi-vidual noise figures do not combine in any simple way to yield the overall cascaded noise figure. Another is that each stage may see a different source impedance, and the noise fig-ure must be computed with respect to that impedance. To understand these and other issues in detail, we now derive the correct equation for the cascaded noise figure, called Friis’ formula.

Consider a noisy system that is driven by yet another noisy system:

N1 kTcBGav Na+ k Te Tc+( ) BGav==

YN2N1

k Te Th+( ) BGavk Te Tc+( ) BGav

= =

TeTh YTc−

Y 1−=




FIGURE 4. Cascaded systems

The first stage has a noise factor

F

1

and available power gain

G

1

measured with

R

S

as source resistance. The second stage has an available power gain

G

2

and a noise factor

F

2

when these quantities are measured

with the

output impedance of the previous stage as source resistance

. If there were additional stages, the available gain and noise figure of each one would be determined using the output impedance of the preceding stage as the source resistance. A common error is to use

R

S

as the source impedance for all stages, but this choice is correct only if the output impedances happen to be

R

S

.

The easiest way to derive Friis’ formula is to make use of the concept of noise tempera-ture. Because the available output noise power added by each DUT is

kT

e

BG

av

, the avail-able noise power at the output of the first DUT is

. (19)

The second stage takes this noise, amplifies it, and adds to it another

kT

e

BG

av

of its own:

. (20)

We could just as well regard the overall system as a single amplifier with available gain

G

av

1

G

av

2

, driven by a source

R

s

. Hence, we may also write

, (21)

where

T

e

12

is the overall noise temperature of the cascade. Equating Eqn. 20 and Eqn. 21 yields

. (22)

The overall noise temperature is therefore the noise temperature of the first stage, plus the input-referred noise temperature of the second stage. This formula reflects the understand-ing that the signal boost provided by the first stage diminishes the effect of noise of subse-quent stages. Clearly, Eqn. 22 can be extended to an arbitrary number of stages, yielding one form of Friis’ formula:

. (23)

G1, F1 G2, F2RS

DUT 1 DUT 2

No1 kTsBGav1 kTe1BGav1+ k Ts Te1+( ) BGav1= =

No2 k Ts Te1+( ) BGav1Gav2 kTe2BGav2+=

No2 k Ts Te12+( ) BGav1Gav2=

Te12 Te1Te2

Gav1+=

Te12 Te1Te2

Gav1

Te3Gav1Gav2

...+ + +=




An alternative expression in terms of noise factors is readily derived by using Eqn. 14 to relate noise temperature and noise factor:

. (24)

From inspection of the last two equations, we see that the expression for cascaded noise temperature is somewhat simpler (none of those pesky –1 terms to clutter up the equation). The noise temperature contributed by the

n

th stage can be computed simply by dividing through by the product of the available gains of the (

n – 1

) stages preceding it. For this reason, the noise temperature formulation is frequently favored when considering cas-caded systems.

5.0 Noise Measure

From Friis’ formula, we see that if an amplifier has good noise figure but low gain, sup-pression of noise from subsequent stages is poor. Unfortunately, classical noise optimiza-tion design methods sometimes lead to an “optimum” amplifier design with precisely this combination of characteristics. Because both the noise figure and gain of an amplifier are important in general, another figure of merit known as

noise measure

is sometimes used to guide engineers toward a balanced design. Its formal definition initially seems to combine these two quantities in a puzzling way:

(25)

The rationale for this definition becomes clear when we examine Friis’ formula for the special case of an infinite cascade of identical amplifiers:

, (26)

which ultimately simplifies to

. (27)

Therefore, this definition of noise measure is actually the normalized noise temperature of the infinite cascade:

. (28)

F12 F1F2 1−

Gav1

F3 1−

Gav1Gav2...+ + +=

MF 1−

11

Gav−

≡

Ftot FF 1−Gav

F 1−

Gav2

...+ + +=

Ftot 1F 1−

11

Gav−

+ 1 M+= =

Te tot, Ftot 1−( ) T0 MT0 M⇒Te tot,

T0= = =




We see that, for good noise performance, we want the noise measure to be not much greater than the normalized noise temperature of the device itself.

Just to keep you on your toes, noise measure is defined in some references as

F

tot

, rather than as

F

tot

– 1. Be sure to identify which definition is being used, as the difference can introduce considerable error for low noise systems. Finally, note that this definition of noise measure has no particular relationship to the definition of noise measure for negative resistance devices, such as Gunn and tunnel diodes (see chapter on RF diodes).

6.0 Typical Noise Figure Instrumentation

Having derived multiple expressions for noise figure, we’re now in a position to examine several different methods for carrying out an actual measurement. As usual, we start with a little history, partly for entertainment, but partly because methods that were used long ago tend to be ones that hobbyists can implement economically today.

6.1 The (good?) old days

From Figure 1 we see that measuring noise figure is equivalent to determining the equa-tion of the noise power-vs.-source temperature line. Measuring two points along the line is sufficient, but so is knowing a single point and the line’s slope. The former method is the modern way, but it is worthwhile discussing the latter. Even though it poses nontrivial experimental challenges, the equipment required is within the reach of most RF hobbyists, so a description of this technique merits inclusion here.

Prior to the development of calibrated hot and cold sources, the only noise source avail-able was one at room temperature. With that limitation, one can determine the available output noise only at that one (perhaps i

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