rf amplifier

1

RF Small Signal AmplifierRF Small Signal Amplifier

2

Low Frequency AmplifierLow Frequency Amplifier

• Transistor is an voltage controlled current source• Device capacitances are negligible • High Zin , rO are desirable for high voltage gain• Amplifier gain drops as frequency increases due to internal and

external capacitances →→→→ Low pass type amplifier Transistor model

inZ+

-

m ing vinv LR

SR

Sv LV+

−Or

)||()( OLmSinS

inL rRgv

ZZZV ⋅−⋅⋅+

=

iL

3

Tuned AmplifierTuned Amplifier

• Gain over a narrow frequency range centered about some high frequency

• If feedback by Cgd is negligible

gsC

+

-

m ing vinv LR

SR

Sv LV+

−Or

gdCLC

( ) ( || ) at 1/(2 )L in m L OV v g R r f LCπ= ⋅ − ⋅ =

inyouty

4

Tuned AmplifierTuned Amplifier

• When the internal capacitances cannot be ignored, yin and yout depend on the load and the source impedances respectively (Miller Effects)

• Additional capacitive loading at the output by Ceq=Cgd[1+gmRs] • Resonance frequency shifts downward due to Ceq→ difficult to

control the resonance frequency→ need relatively large C• If yL is inductive (below resonance frequency), yin shows negative

resistance →may oscillate• Cgd loads the output tank, decreases gain, detunes the resonance, and

most importantly causes instability– Minimizing feedback due to Cgd→ Cascode topology– Bilateral design using S parameters

(1 ) if j ( )(1 / ) if j

out gd m S eq gd gs m

in gs gd m L in gd L

y j C g R j C C C gy j C j C g y j C C y

ω ω ωω ω ω ω

≈ + = + <<

≈ + + = <<

5

S Parameters of FETS Parameters of FET

ECP for RF CMOS with simple substrate model when body is groundedIn general, the substrate model is more complicated

gsC jm ig e vωτ−

iv+

−

dsr

gdC

iR

gRgL

sR

sL

Intrinsics

dR dL

dsCsubC

subR

6

Bias Independent Parameters of FETBias Independent Parameters of FET

• Bias independence: assumption for convenience• Lg, Ls, Ld: parasitic inductances mainly due to electrodes. several

tens of pH, usually ignored for a few GHz application, but important for mm application

• Rg: due to gate poly resistance, reduce the power gain, increase device noise

• Rs: due to ohmic resistance, reduce gm, effective gme = gm/(1+gmRs)• Rd:due to ohmic resistance, affects the device power gain

7

Bias Dependent Parameters of FETBias Dependent Parameters of FETgs: effective charging path resistance for C , / 3, channel resistance at DC

For simple channel model(short channel device)assume uniform sheet charge, and velocity saturated channel

( ),

i c c

d s s gs

R R R

i eWv n v

≈

= −

m

( )

,

/ / 1/ ( : charging time)

3more accurately, (transit time),g 1/4

: parasitic, cause nonunilaterality

n s gs

n s d sgs m s

gs gs gs gs

dm gs s

n

s

ngd

gd

dds

ds

Q qWLn vQ n i nC qWL g qWvv v v v

ig C v LQ

L Lv

QCv

irv

τ τ

τ

=

∂ ∂ ∂ ∂= = = =∂ ∂ ∂ ∂

∂= = =∂

⇒ = ∝

∂=∂

∂=∂

1

cause ouput power loss

, less dependent on Vgs, but sensitive to Vds, cause output power loss and cross talk sub subR C

−

8

ffTT of FETof FET

• Assume unilateral, ignore Cgd• Short circuit current gain, h21=ig/id

21

21

21

T

h drops by 6dB/octave

1 ,2

f is a figure of merit for switching speed

T

m gsd m

g gs gs gs

mT T T

gs

g vi ghi C v C

gh fCω ω

ω ω

ω π ω=

= ≈ =

⇒

= ⇒ = =

gsC mgiv

+

−

gdC

gi di

9

ffmaxmax of FETof FET

• Assume unilateral• Gp : Power gain under matched condition

• To improve fmax, high fT , high Rds, low Rg

gsC mgiv+

−

gi di

gR

dsRgZ

LZgv+

−dv

+

−

dsg

T

p

g

dsT

g

ds

gs

m

gsggsgsmd

g

dsgd

g

Lgd

ggddgLp

RRff

G

RR

ff

RR

Cg

CjIvvgIRRII

ZZII

IVIVPPG

/4

ffat 1

41

41

)/,2/(

/)Re()Re(/

)Re(/)Re(/

max

max

22

22

**

=

==

==

==←

==

==

ω

10

Accurate Accurate ffTT and and fmaxfmax

• Nonunilateral• Includes all parasitic resistances• For CMOS, junction capacitance should be merged to Cgs, Cgd• Substrate parasitics are not included

2

max

)1(23

154

)1()(14

)(]/)(1][[2

smgs

gd

gs

gd

sm

gsmim

mds

T

dsgdmdsdsgdgs

mT

RgCC

CC

RgRRg

RggR

ff

RRCgRRRCCgf

+

++

+

++

=

+++++=

π

11

Low Frequency Approximation of S11, S22Low Frequency Approximation of S11, S22

011

0

022

0

1/

( ||1/ )

in

in

in g in

out

out

out o eq

Z ZSZ Z

Z R j CZ ZSZ Z

Z r j C

ω

ω

−=+

= +

−=+

=

freq (50.00MHz to 10.00GHz)

S(1

,1)

m5

m6


S(2

,2)

m7m8

Series R-C

Parallel R-CL18 CMOS S-parameter (2.15GHz & 5.25GHz) for Finger=32 (width=160um)

Lg, Ls, Ld, Rd, Rs, Ri, ττττ are ignored

12

Low Frequency Approximation of S21,S12Low Frequency Approximation of S21,S120

210

0

012 0

0

2 when are ignored1

(1 )

when Z 1/1

mg

in

in gs gd m

gdgs

gd

g ZS Rj C Z

C C C g Zj C Z

S j Cj C Z

ω

ωω

ω

−=+

= + +

= <<+

-4 -3 -2 -1 0 1 2 3 4-5 5


S(2

,1)

m1m2

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15-0.20 0.20


S(1

,2)

m3

m4

13

RF Small Signal AmplifierRF Small Signal Amplifier

• Small signal amplifier– not voltage amplifier but power amplifier– conjugate impedance matching– usually CE(Common Emitter) and CS(Common Source structure)– characteristics of devices are given by S parameters

• Classification– narrow band amplifier (about 10% BW of carrier frequency)

• Lossless L-section matching– medium band amplifier(20-30% BW)

• double resonance matching, multi-section matching technique– broad band amplifier(more than 50% BW)

• feedback, balanced, traveling wave amplifier

14

Small Signal High Frequency AmplifierSmall Signal High Frequency Amplifier

• ZSO, ZLO: usually 50Ω

• Recall!

oin

oinin ZZ

ZZ+−=Γ

InputMatchingNetwork

transistor[S]

OutputMatchingNetwork

ZZZZsosososoZZZZLoLoLoLo

VsoVsoVsoVso

(VSWR)in (VSWR)out

aΓbΓ

( )L LZΓ( )out outZΓ( )S SZΓ ( )in inZΓ

15

Small Signal High Frequency AmplifierSmall Signal High Frequency Amplifier

• For Re(Zi)>0, always |Γi|≤1• VSWR

• All networks are specified by S parameters and reflection coefficient Γ instead of impedance– practically Γ , S parameters are based on the same normalizing impedance 50 Ω– merely an alias of impedance or admittance– Γin is equivalent to Zin, Γout is equivalent to Zout

• Amplifier Specification– Linear spec :Gain, Bandwidth, VSWR, Noise Figure– Nonlinear spec: P1dB, IIP3 etc.

Sin

Sin

a

a

ΓΓ−Γ−Γ

=Γ−Γ+

=11

1 *

in(VSWR)Lout

Lout

b

b

ΓΓ−Γ−Γ

=Γ−Γ+

=11

1 *

out(VSWR)

16

Input, Output Reflection CoefficientsInput, Output Reflection Coefficients

• If S12 ≠0, Γin(Γout )is a function of ΓL (ΓS) • Instability of Γin, Γout• Usually, high speed active devices have magnitudes of S11, S22

close to 1. • If ΓL( ΓS) is also highly reflective and certain phase condition is

satisfied, magnitude of Γin ( Γout) may become larger than 1 →amplifier oscillates

L

L

L

Lin S

SSSSS

Γ−∆Γ−=

Γ−Γ+=Γ

22

11

22

211211 11

S

S

S

Sout S

SSSSS

Γ−∆Γ−=

Γ−Γ+=Γ

11

22

11

211222 11

21122211 SSSS −=∆

17

Simplified Amplifier NetworkSimplified Amplifier Network

• Transistor S parameters are given • |ΓS|≤1 for all passive ZS, |ΓL|≤1 for all passive ZL

• Amplifier design is simply to choose source and load impedance ZS, ZL to achieve a desired power gain avoiding oscillation

Transistor[S]

)( LL ZΓ)( outout ZΓ)( inin ZΓ)( SS ZΓ

ZZZZSSSS

VsVsVsVs

ZZZZLLLL

18

Amplifier Gain in Terms of S ParametersAmplifier Gain in Terms of S Parameters

• Power– Pavs : power available from source, function of source impedance– Pin : power delivered to transistor– Pavn: power available from transistor output– PL:power delivered to load, function of load impedance

• Transducer power gain GT =PL/Pavs (function of ΓS , ΓL)• Available power gain GA=Pavn/Pavs (function of ΓS)• Operating power gain GP= PL/Pin (function of ΓL)• Measurements

– usual power measurement setup gives GT since signal source indicates Pavs, power meter reads PL

– VNA |S21|2 corresponds to GT for ΓS= ΓL=0

SS

2S

avs Zoffunction ,)8Re(Z

VP =

19

Derivation of Transducer Power GainDerivation of Transducer Power Gain

[ ]S

a1

b1ZS

ZL

a2

b2

)( SS ZΓ ( )in inZΓ ( )L LZΓ( )out outZΓ

20

Transducer Power Gain GTransducer Power Gain GTT

2 2 221

211 22 12 21

2 22

212 222

2 22

212 211

(1 | | ) | | (1 | | )| (1 )(1 ) |

1 | | 1 | | | ||1 | |1 |

1 | | 1 | | | ||1 | |1 |

S LT

S L S L

S L

in S L

S L

S L out

SGS S S S

SS

SS

− Γ − Γ=− Γ − Γ − Γ Γ

− Γ − Γ=− Γ Γ − Γ

− Γ − Γ=− Γ − Γ Γ

21

Available Power Gain GAvailable Power Gain GA, A, Operating Power Gain GOperating Power Gain GTT

*

*

22

212 211

22

212 222

1 | | 1| | ||1 | 1 | |

Function of source impedanceConjugately matched output

Useful for LNA design

1 1 | || | |1 | | |1 |

Function of load impe

L out

S in

SA T

S out

A

LP T

in L

P

G G SS

G

G G SS

G

Γ =Γ

Γ =Γ

− Γ= =− Γ − Γ

− Γ= =− Γ − Γ

danceConjugately matched input

Useful for low input VSWR design

22

ProblemsProblems

• An RF amplifier has the following s-parameters: S11=0.3 ∠ -70°, S21=3.5∠ 85 °, S12=0.2 ∠ -10 °, S22=0.4 ∠ -45 °. The system is shown below. Assuming reference impedance (used for measuring s-parameters) Zo=50Ohm, find:

• (1) Find Γs, ΓL, Γin, Γout

• (a) GT, GA, GP.• (b) PL, PA, Pinc

Amplifier

2221

1211

SSSS ZL=73Ω

40Ω

23

Stability Stability

• Unconditional Stability– for any |ΓS|, |ΓL|≤1 ⇒ |Γin|, |Γout|≤1

• Simple measure of stability →Roulette Stability Factor K

• if K<1 and |∆|<1, potentially unstable– for some |ΓS|, |ΓL|≤1 ⇒ |Γin|, |Γout|≥1– stability depends on ZS and ZL

– you should find stable ZS and ZL

• if -1 < K<0 , unstable for almost values of ZS and ZL

2 2 211 22

11 22 12 2112 21

1 | | | | | | 1, | | 1, where 2 | |

S SK S S S SS S

− − + ∆= > ∆ < ∆ = −

24

Simultaneous Conjugate MatchingSimultaneous Conjugate Matching

• Only if unconditionally stable, simultaneous conjugate matching yields maximum gain– ΓS

* =Γin(ΓL), ΓL* =Γout(ΓS)

– Solution ΓMS, ΓML are little bit complicated. CAD will help you.

• Under simultaneous conjugate matching condition – GTmax = GPmax = GAmax

221Tmax

12

| | ( 1)| |SG K KS

= − −

* 111

22

* 222

11

1

1

Ls

L

sL

s

SS

SS

− ∆ΓΓ = Γ =− Γ

− ∆ΓΓ = Γ =− Γ

25

Stability CirclesStability Circles

• Potentially Unstable(Conditional Stability)– find stable ZS and ZL using stability circle

• Input (Source) Stability Circle– locus of ΓS on Smith chart producing |Γout|=1– if |S11|<1, ZS in the region including origin(ZS=Z0) is stable source impedance

• Output (Load) Stability Circle– locus of ΓL on Smith chart producing |Γin|=1– if |S22|<1, ZL in the region including origin(ZS=Z0) is stable load impedance

• You can easily draw stability circle using CAD

26

Load Stability Load Stability Circle(LSCCircle(LSC))

( )22

22

211222

22

**1122

22

211211 1

1

DSSS

DSDSS

SSSS

L

L

L

−=

−

−−Γ⇒

=Γ−Γ+

LLL RC =−Γ⇒ Load stability circle

Center of circle Radius of circle

Re

Im

RLCL

0

From

(2)

plane LΓ

27

Source Stability Source Stability Circle(SSCCircle(SSC))

( )22

11

211222

11

**2211

22

211222 1

1

DSSS

DSDSS

SSSS

s

s

s

−=

−

−−Γ⇒

=Γ−Γ+

sss RC =−Γ⇒ Source stability circle

Center of circle Radius of circle

Re

Im

RsCs

0

From

(3)

planeSΓ

28

Stability RegionsStability Regions

• The source and load stability circles only indicate the value of Γs and ΓL where |Γ2 | = 1 and |Γ1 | = 1. We need more information to show the stability regions for Γ s andΓL.

• For example for LSC, when ΓL =0, |Γ1 | = |S11|.• Let the LSC does not encircle S11=0 point. If |S11| < 1 then

ΓL =0 is a stable point, else if |S11| > 1 then ΓL=0 is an unstable point.

LSC

|S11|<1

StableRegion LSC

|S11|>1

29

Stability Region Cont...Stability Region Cont...

• Let the LSC encircles S11=0 point. Similarly if |S11| < 1 then ΓL =0 is an stable point, else if |S11| > 1 then ΓL=0 is an unstable point.

• This argument can also be applied for SSC.

LSC

|S11|<1

LSC

|S11|>1

StableRegion

30

Summary for Stability RegionSummary for Stability Region

• For both Source and Load reflection coefficients (Γs and ΓL ) :

LSC or SSC

|S11| or |S22| <1

LSC or SSC

|S11| or |S22| >1

LSC or SSC

|S11| or |S22| <1

LSC or SSC

|S11| or |S22| >1

31

Unconditionally Stable AmplifierUnconditionally Stable Amplifier

• There are times when the amplifier is stable for all passive source and load impedance.

• In this case the amplifier is said to be unconditionally stable.

• Assuming |S11| > 1 and |S22| < 1, the stability region would look like this:

LSC

|S11|>1

ΓL can occupy any point in the Smith chart

SSC

|S22|<1

Γs can occupy any point in the Smith chart

32

Problem 2Problem 2

• Use the s-parameters of the amplifier in Problem 1, draw the load and source stability circles and find the stability region.

SSC LSC

33

Summary for Stability CheckSummary for Stability Check

Set frequency range

Get S-parameters withinfrequency range

K factor > 1and |∆| < 1 ?

Amplifier Unconditionally Stable

Yes

Draw SSC and LSC

No

Find |S11| and |S22|

Circles intersectSmith Chart ?

Amplifier is conditionallystable, find stability regions

Yes

No Amplifier isnot stable

Start

End

34

Stabilization MethodsStabilization Methods

• |Γin | > 1 and |Γout | > 1 can be written in terms of input and output impedances:

• This implies that Re[Zin] < 0 or Re[Zout] < 0.• Thus one way to stabilize an amplifier is to add a series

resistance or shunt conductance to the port. This should made the real part of the impedance become positive.

1 and 1in o out oin out

in o out o

Z Z Z ZZ Z Z Z

− −Γ = > Γ = >+ +

35

Stabilization Methods Cont...Stabilization Methods Cont...

2 - port Network

Z1

SourceNetwork

LoadNetwork

Z1+R1’

2221

1211

SSSS

R1’ R2’

Z2 Z1+R1’

2 - port Network

Y1

SourceNetwork

LoadNetwork

Y1+G1’

2221

1211

SSSS

G1’ G2’

Y2 Y2+G2’

36

Example Example -- SS--parameters measurement and stability parameters measurement and stability analysisanalysis

DCDC1

DC

S_ParamSP1

Step=1.0 MHzStop=1.0 GHzStart=50.0 MHz

S-PARAMETERS

CCc2C=470.0 pF

CCc1C=470.0 pFTerm

Term1

Z=50 OhmNum=1

TermTerm2

Z=50 OhmNum=2

LLb2

R=L=330.0 nH

LLb1

R=L=330.0 nH

LLc

R=L=330.0 nH

RRb1R=10.0 kOhm

RRb2R=4.7 kOhm

CCeC=470.0 pF

RReR=100 Ohm

pb_phl_BFR92A_19921214Q1

V_DCSRC1Vdc=5.0 V

37

Example Cont...Example Cont...

m1freq=600.0MHzK=0.956

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.0 1.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.8

1.2

freq, GHz

K

m1

D

Plotting K and ∆ versus frequency(from 50MHz to 1.0GHz):

This is the frequencywe are interested in

Amplifier isconditionallystable

38

Example Example -- Viewing SViewing S1111 and Sand S2222 at f=600MHzat f=600MHz

freq600.0MHz

S(1,1)0.263 / -114.092

S(2,2)0.491 / -20.095

39

Example Cont...Example Cont...

indep(SSC) (0.000 to 51.000)

SS

C

indep(LSC) (0.000 to 51.000)

LS

C

Since |S11| < 1 @ 600MHz Since |S22| < 1 @ 600MHz

40

ExampleExample

• The S-parameters for a BJT at a particular bias point and f=750MHz are:

• S11 = 0.76<38o

• S21 = 2.35<33o

• S12 = 0.04<-52o

• S22 = 0.66<-42o

• Check the transistor stability, plot the Source and Load Stability Circles and determine the stability regions.

41

Transistor Power GainTransistor Power Gain

• Used as a figure of merit for transistor• Independent of source and load impedance• Classification

– K>1, MAG(Maximum Available Gain)

– K<1, MSG(Maximum Stable Gain)

– Unilateral Power Gain U :MAG obtained using neutralization

||||MSG

12

21

SS=

)1(||||MAG 2

12

21 −−= kkSS

)/Re(2|/|2|1/|

12211221

21221

SSSSkSSU−

−=

K

MAG(6dB/octave)MSG

3dB/oct

f [log]

[dB]

42

Narrow Band Amplifier DesignNarrow Band Amplifier Design

• Unilateral Design– assume S12=0– approximate design– use unilateral transducer power gain GTU=GT|S12=0

– detailed design procedure (Refer to Gonzales)– not practical, just shows attainable variable range of GT

• Bilateral Design(if S12 is not negligible)– Simultaneous conjugate matching design

• valid for k>1(unconditionally stable), fixed gain GTMAX

– GP or GA design• if unconditionally stable(k>1) and for gain other than GTMAX

• if potentially unstable(k<1)

43

Operating Power Gain DesignOperating Power Gain Design

• Mismatched output, matched input • (VSWR)in=1, (VSWR)out>1, • Constant GP circle on ΓL plane

– locus of ΓL(ZL) that yields constant GP at a frequency – GP is only a function of ΓL(ZL)

• Unconditionally stable case– GPMAX exists at a single point of ΓL(ZL)– Design procedure for a gain less than GPMAX

① Determine GP

② Draw GP circle and select the desired ΓL

③ Matched source impedance is ΓS= Γin*

221 12| / | ( 1)PMAXG S S K K= − −

44

Design Procedure Using GDesign Procedure Using GP P (potentially unstable)(potentially unstable)

① Determine GP(Gp<MSG)② Draw GP circle ③ Draw load stability circle(Γin stability)④ Select ΓL on the GP circle far from stability circle⑤ Matched source impedance is ΓS= Γin

*

⑥ Draw input stability circle(Γout stability)– Check if ΓS placed in the stable region– If stable, design completed– If unstable. go to step 4 and select new ΓL

⑦ If input match is made, GP becomes GT

45

GGP P circle when K<1 circle when K<1

46

Available Power Gain DesignAvailable Power Gain Design

• Mismatched input, matched output • (VSWR)in>1, (VSWR)out=1• Constant GA circle on Γ S plane • Design procedures are equivalent to that of using GP gain except that

Γs replaces ΓL

• If output match is made, GA becomes GT

47

NoiseNoise

• Random variation of current or voltage

• White and Color noise• Thermal noise(white)

– PSD(Power Spectral Density)=kT, – k Boltzman constant ,T Kelvin Temperature – Pn at 290°K, PSD =-174dBm/Hz– Spectrum analyzer with 1MHz resolution bandwidth shows noise floor

-114dBm/MHz

• Shot noise(white)– PSD= q electron charge, I dc current

0)(lim == ∫+

∞→dttVV

Tt

tn

Tn

constant== ∫+

∞→dttVv

Tt

tn

Tn

22 )]([lim 2, nrmsn vv =

qIin 22 =

48

NoiseNoise

• Flicker noise– PSD=Γ/fα, Γ: proportional constant, α≅ 1 Other noise source

• Lorentz noise– PSD=kτ/(1+(ωτ)2)

49

Thermal noise of ResistanceThermal noise of Resistance

• Available thermal noise power Pn=kTB• Equivalent circuit of noisy R

– can deliver the same available noise power to matched load R– Pn = v2

n,rms/4R=i2n,rmsG/4

Noisy R at T

Noiseless R

Vn,rms

G=1/Rin,rms

Equivalent Noise Voltage Source Model

kTBRvv nrmsn 42, ==

Equivalent Noise Current Source Model

kTBGii nrmsn 42, ==

yield same available noise power Pn=kTB

50

Equivalent Noise Temperature TEquivalent Noise Temperature Tee

• Te equivalent noise temperature• Passive network Te

– ambient temperature

• Active network Te– not physical temperature– can be much larger than the ambient temperature

NoisyNetworks

Pa[available noise power]

R at Te

Pa=kTe

Equivalent Thermal Noise Model

51

Noise FactorNoise Factor

• Noise Factor F

• Reference input noise power Ni=kToB, To= 290°K• Noise Figure NF=10logF• Ex: LNA Te=464 °K(not real Temp), F=2.16, NF=4.15dB

+

-

Rs at Ts

Vs

Noisy 2 Port

GA

s

e

i

addedi

TT1

NInput) to ReferredPower Noise (AddedN Power) Noise(Input N

+=+

•Available output noise power Pno=kTSBGA+Pn,added=kTSBGA+KTeBGA

=kBGATS(1+Te/TS)Te: equivalent noise temperature of 2 port network

referred to input

power noiseoutput oo

i

o

i

i

o

i

i

i

Ao NSNRSNR

SS

NN

SS

NGNFcf ,/)( ===

52

NF of Cascaded NetworkNF of Cascaded Network

• NF of 1st stage is important• Gain of 1st stage should be high enough to suppress the

2nd stage noise• N cascaded network

G1F1

G2F2

NiNo

1

21 G

1FFF

,

−+=

−+−=+=+= 222112221121 )1()1(/1 GFNGGFNGNGGNN

NGGNF iio

i

o

G1, G2 available gain of each stageN1, N2 input referred added noise power of each stage

....G1)/G(F1)/G(FFF 213121 +−+−+=

53

CLASSICAL TWOCLASSICAL TWO--PORT NOISE THEORYPORT NOISE THEORY

Noisy 2 portSYis is Noiseless 2 portSY+-

en

in→→→→

( )

( ) ( )

22

2

2 22 2 2

2 2

2 2 2

2 22

, ,

1

, ,4 4 4

1 1

s n s nn c u c c n

s

s u c s n u c s n

s s

n u sn u s

u c s c s nu c s n

s s

i i Y eF i i i i Y e

i

i i Y Y e i Y Y eF

i i

e i iR G GkT f kT f kT f

G G G B B RG Y Y RF

G G

+ += = + =

+ + + + += = +

≡ ≡ ≡∆ ∆ ∆

+ + + ++ + = + = +

54

CLASSICAL TWOCLASSICAL TWO--PORT NOISE THEORYPORT NOISE THEORY

( ) ( )

2

2min

2 2

min

,

1 2 1 2

us c opt s c opt

n

un opt c n c c

n

ns opt s opt

s

GB B B G G GR

GF R G G R G GR

RF F G G B BG

= − = = + =

= + + = + + +

= + − + −

55

Noise Figure of 2 Port NetworkNoise Figure of 2 Port Network

• Noise factor of 2 port network is dependent on the source admittance• Noise 4 parameters are dependent on frequency and bias conditions• Manufacturer provides noise 4 parameters

])()[( 22min optsopts

s

n BBGGGRFF −+−+=

Fmin minimum noise factorYs=Gs+jBs source admittanceYopt=Gopt+jBopt source admittance at FminRn equivalent input noise resistance of 2 port networkNoise 4 parameters : Fmin, Yopt, Rn

in,rms

YsNoisy 2 Port

Network

56

Noise CircleNoise Circle

• Rn, Γopt, Fmin are device parameters and constants if bias and frequency are fixed

• Locus of Γs on the Smith chart for a given noise figure Fi is a circle→Constant Noise Figure Circle

• CAD will automatically draw these family of circles

onn

opts

optsn

ZRr

rFF

/

|1|)||1(

||422

2

min

=

Γ+Γ−

Γ−Γ+=

2min |1|4 opt

n

ii r

FFN Γ+−

= where,i

iioptFi

i

optFi N

N)N|-|Γ(R

NC

+

+=

+Γ

=1

1

1

22

Radius Center

57

LNA DesignLNA Design

• Impossible to achieve maximum gain and minimum noise figure simultaneously

• Compromise between Gain, NF, and VSWR• Potentially unstable bilateral design① Determine GA(<MSG)② Draw source stability circle, GA circle, NF circle on Γs plane③ Select Γs.. close to Γopt and far from source stability circle④ ΓL= Γ*

out, automatically VSWRout=1⑤ Output stability check (|Γin| <1)⑥ Input is always mismatched. Therefore always (VSWR)in>1

58

Output stability circle

Input stability circle

20dB GA circle

2dB noise circle

•S parameters S11=0.641/-171 °S21=5.89/9.6 °S12=0.057/163 °S22=0.572/-95.7 °

•Stabilityk=0.617MSG=20.1dBpotentially unstable

•Noise ParametersNFmin=1.5dBΓopt=0.58/151 °Rn=7.5Ω

rf amplifier

Documents