basic concepts in rfic designs

RFIC Design and Testing for Wireless Communications

A PragaTI (TI India Technical University) CourseJuly 18, 21, 22, 2008

Lecture 6: Basic Concepts – Linearity, noise figure, dynamic range

By

Vishwani D AgrawalVishwani D. AgrawalFa Foster Dai

200 Broun Hall, Auburn UniversityAuburn, AL 36849-5201, USA

1

RFIC Design and Testing for Wireless Communications

TopicsMonday, July 21, 2008

9:00 – 10:30 Introduction – Semiconductor history, RF characteristics

11:00 – 12:30 Basic Concepts – Linearity, noise figure, dynamic range2:00 – 3:30 RF front-end design – LNA, mixer4:00 – 5:30 Frequency synthesizer design I (PLL)

T d J l 22 2008Tuesday, July 22, 2008

9:00 – 10:30 Frequency synthesizer design II (VCO)

11:00 – 12:30 RFIC design for wireless communications2:00 – 3:30 Analog and mixed signal testing

Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008 2

Units for Microwave and RFIC DesignPeak-to peak voltage: VppRoot-mean-square voltage:

22pp

rms

VV =

Power in Watt :VVPwatt pprms

22

==

22

Power in dBm : RR

Pwatt8

==

⎟⎠⎞

⎜⎝⎛=

mWmWPwattPdBm 1

][ log10 10

On a 50Ohm load, 0dBm=1mW=224mVrms=632mVpp

⎠⎝ mW1

Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008

Noise Figure

• In RF design, most of the front-end receiver blocks are characterized in terms of “noise figure” rather than input referred noisereferred noise.

• Noise factor F is defined as

NNNNNNSSNR addedoaddedosourceototalooiiin 1F )()()()( ++

F

NNNGNNSSNR sourceosourceosourceoi

o

oo

ii

out

in

10log10NF Figure, Noise

1F)(

)(

)(

)()(

)(

)(

=

+======

• Noise figure measures how much the SNR degrades as the signal passes through a system.

• For a noiseless system, SNRin = SNRout, namely, F=1, y , , y, ,NF=0dB, regardless of the gain. This is because both the input signal and the input noise are amplified (or attenuated) by the same factor and no additional noise is introduced.


Thermal NoiseTh l i (J h i ) d t d th l ti f• Thermal noise (Johnson noise) – due to random thermal motion of electrons and is generated by resistors, base and emitter resistance rb,rE,and rc. of bipolar devices, and channel resistance of MOSFETs. Thermal noise is a white noise with Gaussian amplitude distribution.

• Thermal noise floor: KHzdBm

mWkT 0290at /174

1log10 −=⎟

⎠⎞

⎜⎝⎛

2nbV

Qrb

M 2IR2

nV+

_ +

re

Q MnI

n _

2neV

+_

fkTRVn Δ= 42fgkTI mn Δ⎟

⎠⎞

⎜⎝⎛=

3242

>2/3 for submirocn MOS


>2/3 for submirocn MOS

Shot Noise• Shot noise (Schottky noise) – due to the particle-like nature of

charge carriers. Only the time-average flow of electrons and holes appears as constant current. Any fluctuation in the number of charge carriers produces a random noise current at that instant.Shot noise is a Gaussian white process associated with the transferShot noise is a Gaussian white process associated with the transfer of charge across an energy barrier (e.g., a p-n junction). This random process is called shot noise and is expressed in amperes per root hertz.

2,bnI

Q 2,cnIfqII n Δ= 22

Bbn qIi 2= Ccn qIi 2=


Flicker noise• Flicker noise (1/f noise) – found in all active devices. In bipolar

transistors, it is caused by traps associated with contamination and crystal defects in the emitter-base depletion layer. These traps capture and release carriers in a random fashion with noise energyand release carriers in a random fashion with noise energy concentrated in low frequency. K depends on processing and may vary by order of magnitude.

250 ,2 ~.aff

IKI

aC

n ≈Δ=

• In MOSFETs, 1/f noise arises from random trapping of charge at the oxide-silicon interfaces. Represented as a voltage source in series with the gate, the noise spectral density is given by

fWLCoxKVn

12 =


Noise Power Spectral Density

z)

-135

e Po

wer

(dB

m/H

z10dB/Dec

Total Noise

-150

-145

-140N

oise

Thermal and Shot Noise

Flicker Noise

-160

-155

0.1 1 10 100 1000Frequency (kHz)

The 1/f corner frequencyThe 1/f corner frequency can be significantly higher for MOSFET


BJT Model with Noise Sourcesrb cb’

vbb

4kT

Cμ

Cπ gmvb’erπicnibn

ibf

2qIB

4kTrb

ro

2qIC

ee1/f noise Base

Shot Noise

CollectorShot Noise

ca) cb)rbb

vrb

4kTrb

b’ icn

2qICibn

a)rbb b’ icn

c

2qIi

b)

4kTrb

baseshot noise

collectorshot noise

e2qIB2qIB

q Cbn

4kTr

e2qIB2qIB

2qICibn


shot noise rb

CMOS Model with Noise Sources

Input-referred noise

vgs CGS

CGD

gmvgs r

vo

IndkTV 82

=ro

kT2 8

mgV

3n =

⎟⎠⎞

⎜⎝⎛= mgkTI

324

2

nd mgZkTI 2

in

2

n 38

=

• Gate resistance can be added to the noise model with gate resistivity ρ

LWRGATE ρ

31

=


Linear vs. Nonlinear Systems

• A system is linear if for any inputs x1(t) and x2(t), x1(t) y2(t), x2(t) y2(t) and for all values of constants a and b, it satisfies

a x1(t)+bx2(t) ay1(t)+by2(t)a x1(t)+bx2(t) ay1(t)+by2(t)

A t i li if it d t ti f• A system is nonlinear if it does not satisfy the superposition law.


Effects of Nonlinearity

• Harmonic DistortionHarmonic Distortion• Gain Compression• DesensitizationDesensitization• Intermodulation

• For simplicity, we limit our analysis to memoryless time invariant system Thusmemoryless, time invariant system. Thus,

...)()()()( 33

221 +++= txtxtxty ααα (3.1)


Effects of Nonlinearity -- HarmonicsIf a single tone signal is applied to a nonlinear system, the output generally exhibits fundamental and harmonic frequencies with respect to the input frequency. In Eq. (3.1), if

tAtAtAty ωαωαωα coscoscos)( 3322 ++

frequencies with respect to the input frequency. In Eq. (3.1), if x(t) = Acosωt, then

ttA

tAtA

tAtAtAty

ωωα

ωαωα

ωαωαωα

)3coscos3(4

)2cos1(2

cos

coscoscos)(3

32

21

321

++++=

++=

tA

tAtA

AAω

αω

αω

αα

α 3cos4

2cos2

cos)4

3(

2

33

22

33

1

22 ++++=

Observations:1. even order harmonics result from αj with even j and vanish if the system has odd symmetry, i.e., differential circuits.


2. For large A, the nth harmonic grows approximately in proportion to An.

Effects of Nonlinearity -- Gain Compression

tA

tAtA

AAty ωα

ωα

ωα

αα 3cos

42cos

2cos)

43

(2

)(3

32

23

31

22 ++++= (3.2)

• Under small-signal assumption, the system is

4242

g ynormally linear and harmonics are negligible. Thus, α1A dominates small-signal gain = α1.

• For large signal nonlinearity becomes evident• For large signal, nonlinearity becomes evident. large-signal gain = . The gain varies when input level changes.

4/3 331 Aαα +

• If α3 < 0, the output is a “compressive” or “saturating” function of the input the gain is compressed when A increases


compressed when A increases.

Output of Bipolar Differential Pair

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

T

idCEEF

id

od

VV

RIVV

2tanhα( ) K+−+−= 753

3157

152

31tanh xxxxx


Effects of Nonlinearity – 1dB Compression Point (d

BV)

11 13

log20α

=−dB dBA

ut V

olta

ge

1dB11

1

31311

380801450

43

αα

αα

==

+ −−

dB

dBdB

A

AA

dBA1

Out

pu

20logAin

331 3808.0145.0

αα−dBA

VdB pp 8/

l102

dBA −1 20logAin

• 1-dB compression point is defined as the input signal level that causes small-signal gain to drop 1 dB It’s a measure of the maximum input

mWdBm pp

150log10

×Ω=

small signal gain to drop 1 dB. It s a measure of the maximum input range.

•1-dB compression point occurs around -20 to -25 dBm (63.2 to 35 6 V i 50 Ω t ) i t i l f d d RF lifi


35.6mVpp in a 50-Ω system) in typical frond-end RF amplifiers.

Effects of Nonlinearity – Desensitization (Blocking)• Desensitization -- small signal experiences a vanishingly small gain when co-exists with a large signal, even if the small signal itself does not drive the system into nonlinear range.A l i t t i t x(t) = A cosω t+ A cosω t t E (3 1) h

,cos23

43)( 1

2213

31311 L+⎟

⎠⎞

⎜⎝⎛ ++= tAAAAty ωααα

Applying two-tone inputs x(t) = A1cosω1t+ A2cosω2t to Eq.(3.1), we have

gain of desired 24 ⎠⎝

,cos23)( 11

2231 L+⎟⎠⎞

⎜⎝⎛ += tAAty ωααFor A1<< A2, it reduces to

signal

2 ⎠⎝

Observations:• Weak signal’s gain decreases as a function of A2 if α3 < 0. For sufficiently largeWeak signal s gain decreases as a function of A2 if α3 < 0. For sufficiently large A2, the gain drops to zero the weak signal is “blocked” by the strong signal. (Why cannot we see stars during day?)• Many RF receivers must be able to withstand blocking signals 60 to 70 dB

t th th t d i l


greater than the wanted signals.

Effects of Nonlinearity – Intermodulation

• Harmonic distortion is due to self-mixing of a single-t i l It b d b l filt itone signal. It can be suppressed by low-pass filtering the higher order harmonics.

• However there is another type of nonlinearity• However, there is another type of nonlinearity --intermodulation (IM) distortion, which is normally determined by a “two tone test”.

• When two signals with different frequencies applied to a nonlinear system, the output in general exhibits some components that are not harmonics of the input frequencies. This phenomenon arises from cross-mixing (multiplication) of the two signals


mixing (multiplication) of the two signals.

Effects of Nonlinearity – IntermodulationAA)(

DC Term++=

+=

AAty

AtAtx

)(21)(

coscos)(2

22

12

2211

α

ωω

1st Order Terms

⎥⎤

⎢⎡

+⎥⎦⎤

⎢⎣⎡ ++

tAAAA

tAAAA

)2(3

cos)2(43

22

122

211311 ωαα

2nd Order [ ]++

+⎥⎦⎤

⎢⎣⎡ ++

tAtA

tAAAA

2cos2cos21

cos)2(4

22

212

12

222

212321

ωωα

ωαα

Terms[ ][ ]++

+−++

tAtA

ttAA

3cos3cos41

)cos()cos(2

23

213

13

2121212

ωωα

ωωωωα

3rd Order terms

[ ][ ][ ] ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−++

+−++

ttAA

ttAA

)2cos()2cos(

)2cos()2cos(43

4

12122

21

212122

13

ωωωω

ωωωωα


[ ] ⎭⎩ 121221

Intermodulation – Why do we care about IM3 mostly?

In wireless communication system such as cellular handsets with narrow-band operating frequencies (i.e., a few tens of MHz), only the IM3 spurious signals (2w1 - w2) and (2w2 - w1) fall within the filter passband.


Intermodulation -- Third Order Intercept Point (IP3)

• Two-tone test: A1=A2=A and A is sufficiently small so that higher-order nonlinear terms are negligible and the gain is relatively constant and equal to α1.constant and equal to α1.

• As A increases, the fundamentals increases in proportion to A, whereas IM3 products increases in proportion to A³.

⎤⎡⎤⎡

+=

+=

Aty

AtAtx

)(

coscos)(2

2

21

α

ωω 3133

13

231 and

34

49 IIPOIPIIPA α

αα

αα ==→>>

dBA dB 69145.01

[ ] [ ]+++++

+⎥⎦⎤

⎢⎣⎡ ++⎥⎦

⎤⎢⎣⎡ +

ttAttA

tAAtAA

)cos()cos(2cos2cos1

cos49cos

49

22

22

3112

31

ωωωωαωωα

ωααωααdB

AIP

dB 6.93/43

1 −≈=

[ ] [ ]

[ ] [ ][ ] ⎭

⎬⎫

⎩⎨⎧

−+++−++

++

+−++++

tttt

AttA

ttAttA

)2cos()2cos()2cos()2cos(

433cos3cos

41

)cos()cos(2cos2cos2

2121

21213321

33

21212212

ωωωωωωωω

αωωα

ωωωωαωωα


[ ] ⎭⎩ ++ tt )2cos()2cos(44 2121 ωωωω

Intermodulation – IP2 vs. IP3

20

40 IP2

FundamentalA1α A1α

A1=A2=A2ND Order IM Product:

2

12 α

α=IPA

20

0

20 Fundamental

22 Aα

1

22 Aα

-60

-40

-20IP3

1

1

121 ωω − 1ω 2ω 21 ωω +

Freq

3RD Order IM Product 4 α

-80 3

-100

3rd Order IM Product

2nd Order IM Product

1

3 Order IM Product3

13 3

4αα

=IPA

PΔ

A1α A1α

3

-60

-120 IIP3

-40 -20 0 20 40

IM Product2

Freq

334

3 Aα 334

3 Aα


0212 ωω − 1ω 2ω

Freq122 ωω −

Calculate IIP3 without Extrapolation

1 AA A 1

PΔ

A1α A1α

33 A 3

20logAi

33 A

P

Freq

334Aα 3

343 Aα

20logAin

P/2

212 ωω − 1ω 2ωFreq

122 ωω −

][2

][][3 dBmPdBPdBmIIP in+Δ

=


Relationship Between 1-dB Compression and IP3

1-dB compression point with single tone applied:

3

13 3

4αα

=IPA3

11 145.0

αα

=−dBA dBAA

dB

IP 66.904.31

3 ==

1-dB compression point with two tones applied:

dBAIP 4142553

23

1

3 αα

dBA dB

IP 4.1425.522.0

3

11

3 ===

αα


Determine IIP3 and 1-dB Compression Point from Measurement

• An amplifier operates at 2 GHz with a gain of 10dB. Two-tone test with equal power applied at the input, one is at 2.01 GHz. At the output, four tones are observed at 1.99, 2.0, 2.01, and 2.02GHz. The power levels of the tones are -70,-20,-20, and -70dBm. Determine the IIP3 and 1-dB compression point for this amplifier.

• Solution: 1.99 and 2.02 GHz are the IP3 tones.

( ) [ ] [ ]dBP

dBmPPGPIIP

66146695

57020211020

213 311 −=+−+−−=−+−=

OIP3)log(20 1 A

dBmPdB 66.1466.951 −=−−=

334

3log20 AP

][2

][][3 dBmPdBPdBmIIP in+Δ

=

IIP320logAin

4

P/2


2

Intermodulation of Cascade Nonlinear Stages

...)()()()(

...)()()()(3

132

12112

33

2211

+++=

+++=

tytytyty

txtxtxty

βββ

ααα

L

)(t)(t)(t )(2 ty)(1 ty)(tx

A A A1,3IPA 2,3IPA 3,3IPA

Linearity is more important for back-end stages.

K+++≈ 23,3

21

21

22,3

21

21,3

23

11

IPIPIPIP AAAAβαα


,,,

Noise Figure of Cascade Stages

( ) NFNFNF 111 −−−( )GGG

NFGG

NFG

NFNFNFmppp

m

ppptot

)1(.....2111

3

1

21

1.....

1111

−

++++−+=

NFtot – total equivalent Noise FigureNFm – Noise Figure of mth stage

G A il bl i f th tGpm – Available power gain of mth stage

Noise figure is more important for front-end stages.


Sensitivity

• Sensitivity -- defined as the minimum signal level that the system can detect with acceptable SNR.

OUT

RSsig

OUT

in

SNRPP

SNRSNRNF

/==

• The overall signal power is distributed across the channel bandwidth, B, integrating over the bandwidth to obtain total mean square power

BSNRNFPP OUTRStotsig •••=,

here P is the so rce resistance noise po er

BSNRNFPPdBdBHzdBmRSdBmin log10min/min, +++=


where PRS is the source resistance noise power.

Maximum Input Power

OIP3)log(20 1 A

2,

3outIMout

inIIP

PPPP

−+=

20logA

334

3log20 AP

23

2,, inIMininIMin

in

PPPPP

−=

−+=

2 PP +IIP320logAin

P/2 32 ,3 inIMIIP

in

PPP

+=

where PIM,out denotes output-referred power of IM3 products, Pout=Pin+G, PIM,OUT= PIM,in+G. The input level for which the IM products become equal to the noise floor F is thus given by

3log101742

32 33 BNFdBmPFPP IIPIIP

in++−

=+

=


Dynamic Range• Dynamic Range (DR) -- defined as the ratio of the maximum to

minimum input levels that the circuit provides a reasonable signal quality.

• Spurious Free Dynamic Range (SFDR) determine the upper• Spurious-Free Dynamic Range (SFDR) -- determine the upper end of dynamic range on the intermodulation behavior and the lower end on sensitivity.

• The upper end of the dynamic range is defined as the e uppe e d o t e dy a c a ge s de ed as t emaximum input power in a two tone test for which the 3rd IM products do not exceed the noise floor F=-174dBm+NF+10logB.

• The SFDR is thus given by

min3

min3

,max, 3)(2

)(3

2SNR

FPSNRF

FPPPSFDR IIPIIP

miminin −−

=+−⎟⎠

⎞⎜⎝

⎛ +=−=

• Example: NF=9dB, PIIP3=-15dBm, B=100kHz, SNRmin=12dB SFDR=(-15-(-174+9+50))/1.5-12=54.7dB.


basic concepts in rfic designs

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