Frequency Response - ITESO

Dr. J.E. Rayas Sánchez1

Frequency Response

(Part 2)


The FET High-Frequency Model

GD

S

gmvgs rοvgs

Cgd

Cgs

)(2 tGSm VVKg −=

DS

Ao I

Vr =

λ/1=AV

(input capacitance)gdgsiss CCC +=

(output capacitance)gddsoss CCC +=

(reverse capacitance)gdrss CC =


The FET Unity-Gain Frequency (fT)

It is the frequency at which the magnitude of the short-circuit current gain of the Common Source configurationbecomes unity

)(2 gdgs

mT CC

gf+

=π

20 MHz ≤ fT ≤ 100 MHz for JFETs

100 MHz ≤ fT ≤ 2 GHz for MOSFETs

5 GHz ≤ fT ≤ 15 GHz for GaAs MESFETs

10 GHz ≤ fT ≤ 200 GHz for SiGe MOSFETs


The BJT High-Frequency Hybrid π Model

T

Cm V

Ig =C

Ao I

Vr =mg

r βπ =

rπgmvπ rο

B C

vπ

rx

Cπ

Cµ

Eπrhr iex −= rehrr /πµ =

(Collector-Base capacitance)cbob CCC ==µ

(Base-Emitter capacitance)ebCC =π


Typical Behaviour of Cπ


Capacitancia de Difusión (repaso)

DT

d IV

C τ=

1IID = 12 IIID >=

)( jd CC >>ebCC =π


Capacitancia de la Región de Desértica (repaso)

( ) 00

12 ψ

ε

AD

DAj NN

NqNAC+

=

0

0

1ψ

D

jj V

CC

−=

cbobj CCCC === 00µ


Typical Behaviour of β

dB

20

40

60

20 log β o

fβfT

β

0

Frequency (Hz - Log Scale)109108107106105

f

- 3 dB

- 20 dB/decade

fT Unity-Gain Bandwidth


The BJT Unity-Gain Bandwidth (fT)

It is the frequency at which the magnitude of theshort-circuit current gain of the common emitterconfiguration becomes unity

)(2 µππ CCgf m

T +=

2 MHz ≤ fT ≤ 100 MHz for general purpose BJTs

1 GHz ≤ fT ≤ 10 GHz for high speed BJTs

1 GHz ≤ fT ≤ 50 GHz for HBTs and HEMTs


Typical Behaviour of fT

fT

log I C

fTmax

ICM


High-Frequency Response of FET Amplifiers

Common Source

gmvgs rο||rLvgs

Cgd

Cgs vo

rS

virL vovi

rS

Sgs rR = )1(/ gs

oS

Sgs

ogsgd v

vrrvvv

R −=−

=

)||)(( LogsmS

gso rrvg

rv

v +−= )]/1)(||(1[ SmLoSgd rgrrrR ++=

)/(1 gdgdgsgsH CRCR +≈ω



gmvgs

rLvgs Cgd

Cgs vo

rS

vi

roCommon Gate

rL vovi

rS

1|| ZrR Sgs =Lo

gs

rvv

Z/1

−= L

o

ogsgsmo r

rvv

vgv

−−+−= )(

oLLm

oLL

rrrgrrrZ/

)/1(1 +

+=m

Lo gZrr 1thenif 1 ≈>>

To calculate Rgd, assume ro >> rS , then vgs = 0 : Logd rrR ||=




Common Drain

gmvgs rL

Cgs

Cgdvo

rS

vi+ vgsrL vo

vi

rS

Logsm

gsgs rvvg

vR

/−=

L

ogsm

S

ogs

rvvg

rvv

−=+

Sgd rR =

LS

gs

LS

Smgso rr

vrr

rgvv

/1/1)1(

+−

≈+

−=

)(1)/(11

SLm

SL

SLmgs rrg

rrrrg

R++

+=++

=



High-Frequency Response of BJT Amplifiers

Common Emitter

rL vovi

rS rπgmvπ rovπ

rx

Cπ

CµrS

vorLvi

)1(/ π

πππ

πµ v

vRRvvvR oo −=−=ππ rrrR xS ||)( +=

)||)(( Lomo rrvgRvv π

π

π +−= )]/1)(||(1[ ππµ RgrrRR mLo ++=

)/(1 µµππω CRCRH +≈



Common Base

rL vo

rS

vi

It can be shown that

Sm

Sx

rgrrrR

++=

1||ππ LSxoL rrrrrR ≈+= )||(||µ




Common Collector

rL vovi

rS

It can be shown that

)||(1)||(||

oLm

oLSx

rrgrrrrrR

+++= ππ )(1

||xSLm

xSL

rrrgrrrrR+++

++= πµ



Comparison of Low-Frequency FET Responses

CD

CG

CS

GCRSCR

DCR

mS g

R 1||21 || GGSS RRR + DL RR +

)1||(m

SSS gRR +21 || GG RR

LD RR +

Lm

S Rg

R +)1||( DR21 || GGSS RRR +

DCSCGCL CRCRCR

DSG

111 ++≈ω


Comparison of Low-Frequency BJT Responses

CC

CB

CE

CCRBCR

ECR

1)||||(|| 21

++

βπ BBS

ERRRrR)||||( 21 πrRRR BBS + )||( oCL rRR +

)]||)(1([|||| 21 SEBB RRrRR βπ ++ mES grRR /1|||| π+ )||( oCL rRR +

)]||([|| LEoC RRrR +1

)||||(|| 21

+++

βπ BBS

ELRRRrRR)]||)(1([|||| 21 LEBBS RRrRRR βπ +++

CCECBCL CRCRCR

CEB

111 ++≈ω


Comparison of High-Frequency FET Responses

gsR

)]/1)(||(1[ SmLoS rgrrr ++

CD

CG

CS

gdR

Sr

mS g

r 1|| Lo rr ||

)(1 SLm

SL

rrgrr++

+Sr



Comparison of High-Frequency BJT Responses

πR

)]/1)(||(1[ ππ RgrrR mLo ++πrrr xS ||)( +

Sm

Sx

rgrrr

++

1||π )||(|| SxoL rrrr +

)||(1)||(||

oLm

oLSx

rrgrrrrr

+++

π


CC

CB

CE

µR

)(1||

xSLm

xSL

rrrgrrrr+++

++π


Miller´s Theorem

V2

Z1

V1

Z2

ZV2

V1

1

2

VVAV =

VAZZ

−=

11 12 −=

V

V

AZAZ

V1

C1

V2

C2

CV2

V1)1(1 VACC −=

−=V

V

AACC 1

2


Miller Effect

Common Source

gmvgs rο||rLvgs

Cgd

Cgs vo

rS

virL vovi

rS

)]/1)(||(1[ SmLoSgd rgrrrR ++=

)]/1)(||(1[ ππµ RgrrRR mLo ++=

rL vovi

rS rπgmvπ rovπ

rx

Cπ

CµrS

vorLvi

Common Emitter


The Cascode Configuration

RC

R3

R2

vs

RS

CS

RE CE

VCC

RLvo

CCR1

CBQ2

Q1


The Cascode Configuration (cont)

vs

RS RC || RLvo

Q2

Q1

R3 || R2


The Cascode Amplifier at High Frequencies

23 |||| RRRr SS =23

23

||)||(

RRRRRvv

S

si +=

rS

vi ro1

gm1vπ 1

Cµ 1

Cπ 1

rx1

vπ 1

rπ 1 rπ 2

gm2vπ 2

ro2

vπ 2Cπ 2 Cµ 2rLvo

11

1'π

π

rrrrvvxS

ii ++

= 11 ||)(' πrrrr xSS +=

vi'gm1vπ 1

Cµ 1

Cπ 1 vπ 1

rπ 2

gm2vπ 2

ro2

vπ 2Cπ 2 Cµ 2rLvo

rS'


The Cascode Amplifier at High Frequencies

22

21|| em

rg

r =π

vi'gm1vπ 1

Cµ 1

Cπ 1

rS'

rπ 2 vπ 2Cπ 2 1/gm2vπ 1

vi'gm1vπ 1

Cµ 1

Cπ 1

rS'

re2Cπ 2vπ 1

vx

'1 SrR =π 22 erR =π )1(''/ 11

11

ππ

πµ v

vrrvvvR x

SS

x −=−=

2111 )'

( emS

x rvgrvv π

π +−=2

221 '2)

'1(' eS

S

eemS rr

rrrgrR +≈++=µ


The Cascode Configuration

RC

R3

R2

vs

RS

CS

RE CE

VCC

RLvo

CCR1

CBQ2

Q1

'1 SrR =π 22 erR =π

21 '2 eS rrR +=µ

)/(1 112211 µµππππω CRCRCRH ++≈

11 ||)(' πrrrr xSS +=

23 |||| RRRr SS =


Frequency Response of a Differential Amplifier

ππ rrRR xS ||)( +=

)]/1)(||(1[ ππµ RgRrRR mCo ++=


vC1

RC

RS

+vd /2vC1 vC2

I

VCC

VEE

RCRC

vdRS

RS


Cascode Differential Amplifier

RS RCvC1+vd /2

vC1 vC2

I

VCC

VEE

RCRC

vdRS

RS

R2

R1

VCC

VEE

])/[(1 21 µµπππω CRCRRH ++≈

erR =2π

erRR += 12 πµ

ππ rrRR xS ||)(1 +=


Hybrid Microwave Integrated Circuits

M. Pozar (1998),Microwave Engineering. Amherst, MA: John Wiley and Sons.


Hybrid Microwave Integrated Circuits (cont)

M. Pozar (1998), Microwave Engineering. Amherst, MA: John Wiley and Sons.


Monolithic Microwave Integrated Circuits

M. Pozar (1998), Microwave Engineering. Amherst, MA: John Wiley and Sons.


Active Device Models at Microwave Frequencies

rπgmvπ rοvπ

rx

Cπ

Cµ

B

E

C

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Frequency Response - ITESO