High Frequency Cs Amplifier

8/11/2019 High Frequency Cs Amplifier

1/31

NTUEE Electronics L.H. Lu 8-1

CHAPTER 8 FREQUENCY RESPONSE

Chapter Outline8.1 Low-Frequency Response of the CS and CE Amplifiers

8.2 Internal Capacitive Effects and the High-Frequency Model

8.3 High-Frequency Response of the CS and CE Amplifiers

8.4 Tools for the Analysis of the High-Frequency Response of Amplifiers8.5 A Closer Look at the High-Frequency Response*

8.6 High-Frequency Response of the CG and Cascode Amplifiers*


2/31


Frequency response of amplifiersMidband:

The frequency range of interest for amplifiers

Large capacitors can be treated as short circuit and small capacitors can be treated as open circuit

Gain is constant and can be obtained by small-signal analysis

Low-frequency band: Gain drops at frequencies lower thanfL Large capacitors can no longer be treated as short circuit

The gain roll-off is mainly due to coupling and by-pass capacitors

High-frequency band:

Gain drops at frequencies higher thanfH Small capacitors can no longer treated as open circuit

The gain roll-off is mainly due to parasitic capacitances of the MOSFETs and BJTs


3/31


4/31

Determining the lower 3-dB frequency Coupling and by-pass capacitors result in a high-pass frequency response with three poles

If the poles are sufficiently separated

Bode plot can be used to evaluate the response for simplicity

The lower 3-dB frequency is the highest-frequency pole

P2 is typically the highest-frequency pole due to small resistance of 1/gm If the poles are located closely

The lower 3-dB frequency has to be evaluated by the transfer function which is more complicated

Determining the pole frequency by inspection

Reduce Vsig to zero

Consider each capacitor separately(treat the other capacitors as short circuit)

Find the total resistance between the terminals

Selecting values for the coupling and by-pass capacitors

These capacitors are typically required for discrete

amplifier designs CSis first determined to satisfy neededfL CC1 and CC2 are chosen such that poles are 5 to 10

times lower thanfL



5/31

The CE amplifierSmall-signal analysis

Considering the effect of each capacitor separately

Considering only CC1:


sigB

LCBmM

sigBC

P

sigBC

sigB

LCBm

sig

o

LCmo

C

sigB

Bsig

RrR

RRrRgA

RrRC

RrRCs

s

RrR

RRrRg

V

V

RRVgV

sCRrR

rRVV

||

)||)(||(

)||(

1

)||(

1||

)||)(||(

)||(

1||

||

1

1

1

1


6/31

Considering only CE:

Considering only CC2:


1

||

1

1

||1)1(||

)||(

)||(

1)1(||

1

2

sigB

eE

P

sigB

eE

esigB

LC

sigB

B

sig

o

LCbo

E

esigB

sigB

Bsigb

RRrC

RRrC

s

s

rRR

RR

RR

R

V

V

RRIV

sCrRR

RR

RVI

)(

1

)(

1||

)||)(||(

1

||

||

2

3

2

2

LCC

P

LCC

sigB

LCBm

sig

o

L

L

C

C

Cmo

sigB

Bsig

RRC

RRCs

s

RrR

RRrRg

V

V

R

RsCR

RVgV

RrR

rRVV


7/31

Determining the lower 3-dB frequency Coupling and by-pass capacitors result in a high-pass frequency response with three poles

The lower 3-dB frequency is simply the highest-frequency pole if the poles are sufficiently separated

The highest-frequency pole is typically P2 due to the small resistance ofRE An approximation of the lower 3-dB frequency is given by

Selecting values for the coupling and by-pass capacitors

These capacitors are typically required for discrete amplifier designs CEis first determined to satisfy neededfL CC1 and CC2 are chosen such that poles are 5 to 10 times lower thanfL


221 PPP

M

sig

o

s

s

s

s

s

sA

V

V

2211

111

2

1

CCEECC

LRCRCRC

f


8/31

8.2 Internal Capacitive Effects and the High-Frequency Model

The MOSFET deviceThere are basically two types of internal capacitance in the MOSFET

Gate capacitance effect: the gate electrode forms a parallel-plate capacitor with gate oxide in the middle

Junction capacitance effect: the source/body and drain/body are pn-junctions at reverse bias

The gate capacitive effect

MOSFET in triode region:

MOSFET in saturation region:

MOSFET in cutoff region:

Overlap capacitance:

The junction capacitance

Junction capacitance includes components from the bottom side and from the side walls The simplified expression are given by


ovoxgdgs CWLCCC 2

1

ovoxgs CWLCC 3

2

ovgdgs CCC

ovgd CC

oxgb WLCC

oxovov CWLC

0

0

/1 VV

CC

DB

dbdb

0

0

/1 VV

CC

SB

sbsb


9/31

The high-frequency MOSFET model

Simplified high-frequency MOSFET model

Source and body terminals are shorted

Cgdplays an important role in the amplifierfrequency response

Cdb is neglected to simplify the analysis


oDB

dbdb

oSB

sbsb

oxovgd

oxovoxgs

DAo

mmb

Doxnovoxnm

VV

CC

VV

CC

CWLC

CWLWLCC

IVr

gg

ILWCVLWCg

/||1

/||1

3

2

/||

)/(2||)/(

0

0


10/31

The unity-gain frequency (fT) The frequency at which the current gain becomes unity

Is typically used as an indicator to evaluate the high-frequency capability

Smaller parasitic capacitances Cgs and Cgdare desirable for higher unity-gain frequency

The unity-gain frequency can also be expressed as

The unity-gain frequency is strongly influenced by the channel length

Higher unity-gain frequency can be achieved for a given MOSFET by increasing the bias current orthe overdrive voltage


gdgs

mT

gdgs

m

i

o

gdgs

igs

gsmgsgdgsmo

CC

gf

CCs

g

I

I

CCs

IV

VgVsCVgI

2

1

)(

)(

243)/(

21

22

3)/(2

2

1

LV

CCVLWCf

WLC

I

LCC

ILWCf

VgVsCVgI

OVn

gdgs

OVoxnT

ox

Dn

gdgs

Doxn

T

gsmgsgdgsmo


11/31

The BJT DeviceHigh-frequency hybrid- model:

The base-charging or diffusion capacitance Cde:

The base-emitter junction capacitance Cje:

The collector-base junction capacitance C:

The cutoff (unity-gain) frequency:


T

CFmFdeV

IgC

02 jje CC

m

c

CB

V

V

CC

0

0

1

T

CmT

mfe

m

b

c

fe

bb

mc

VCC

I

CC

gf

rCCsrCCs

rgh

CCsr

sCg

I

I

h

sCsCr

ICCrIV

VsCgI

)(2

1

2

1

)(1)(1

)(/1

/1)||||(

)(

0


12/31

8.3 High-Frequency Response of the CS and CE Amplifiers

The common-source amplifierMidband gain:

Frequency response:

The common-source amplifier has one zero and two poles at higher frequencies

The amplifier gain falls off at frequencies beyond midband The amplifier bandwidth is defined by the 3-dB frequency which is typically evaluated by the dominant

pole (the lowest-frequency pole) in the transfer function


)( LmsigG

GM Rg

RR

RA

L

ogsmogsgd

ogsgdgsgs

G

gs

sig

gssig

R

VVgVVsC

VVsCVsCRV

RVV

)(

)(

gsgd

Gsig

GsigL

gsgdLm

G

L

sig

L

Gsig

Gsig

m

gd

Gsig

GLm

sig

o

CCRR

RRRsCCRg

R

R

R

R

RR

RRs

g

C

sRR

RR

g

V

V

211

1


13/31

Simplified analysis technique Assuming the gain is nearly constant (-gmR

L)

Find the equivalent capacitance of Cgdat the input (with identicalIgd)

Miller effect

Neglect the small currentIgdat the output

The dominant pole is normally determined by Ceq The frequency response of the common-source amplifier is approximated by a STC


])([)( gsLmgsgdogsgdgd VRgVsCVVsCI

gsmgd VRgsC )1(

gseqgd VsCI

gdLmeq CRgC )1(

H

M

gsgdLmsig

Lm

Gsig

G

sig

o

s

A

CCRgRs

RgRR

R

V

V

1]})1[({1

1])1[( gsgdLmsigH CCRgR

)( Lm

sigG

GM Rg

RR

RA


14/31

The common-emitter amplifier

Simplified analysis for frequency response

Miller effect: replacing Cwith Ceq The response is approximated by a STC


1]}||][)1({[

||

1

sigBLmH

Lm

sigBsigB

BM

H

M

sig

o

RRrCRgC

RgRRr

rRR

RA

A

V

V


15/31

8.4 Useful Tools for the Analysis of the High-Frequency Response of Amplifiers

Determining the upper 3-dB frequencyGeneral transfer function of the amplifier

The upper 3-dB frequency

dominant-pole response

One of the poles is of much lower frequency than any of the other poles and zeros A dominant pole exists if the lowest-frequency pole is at least 4 away from the nearest pole or zero


)/1()/1)(/1(

)/1()/1)(/1()()(

21

21

PnPP

ZmZZMHM

sss

sssAsFAsA

22

21

22

21

2

2

2

1

2

22

21

2

2

2

2

1

4

2

2

2

1

2

22

21

4

22

21

2

21

221

2

21

221

22

21

21

112

11

1

111

111

11111

11111

)/1)(/1(

)/1)(/1(

2

1)(

)/1)(/1()/1)(/1()()(

ZZPP

H

PP

H

ZZ

H

PP

H

PP

H

ZZ

H

ZZ

H

PHPH

ZHZHHH

PP

ZZMHM

jF

ss

ssAsFAsA

1

22

21

22

21

112

11

1

P

ZZPP

H


16/31

Open-circuit time constant method to evaluate amplifier bandwidthGeneral transfer function of the amplifier

Open-circuit time constant (exact solution):

Dominant pole approximation:

Millers Theorem

A technique to replace the bridging capacitance

The equivalent input and output impedances are:


PnPP

b

111

11

1

n

i

iiRCb1

1

i

ii

PH

PPnPP

RCb

b

11

1111

1

1

121

1

n

n

m

m

PnPP

ZmZZH

sbsbsb

sasasa

sss

ssssF

221

221

21

21

1

1

)/1()/1)(/1(

)/1()/1)(/1()(

K

ZZ

K

ZZ

/11

1

2

1

12 KVV


17/31

Time-constant method*A technique used to determine the coefficients of the transfer function from the circuit

Determining b1:

Set all independent sources zero

R0ii: the equivalent resistance in parallel with Ciby treating the other capacitors as open circuit

Determining b2:

Set all independent sources zeroR jii: the equivalent resistance in parallel with Ciby treating Cj short and the other capacitors open


0333

0222

01111 RCRCRCb

n

n

m

m

PnPP

ZmZZH

sbsbsb

sasasa

sss

ssssF

221

221

21

21

1

1

)/1()/1)(/1(

)/1()/1)(/1()(

2333

0222

1333

0111

1222

01112 RCRCRCRCRCRCb


18/31

8.5 A Closer Look at the High-Frequency Response*

The equivalent circuit of a CS amplifier*

Analysis using Millers Theorem*

Bridging capacitance Cgdis replaced by C1 and C2

Transfer function and H:


Lm

gs

o

gd

Lm

gdgd

Lmgdgd

Lm

gs

o

RgV

VK

CRg

CKCC

RgCKCC

RgV

VK

)1

1()1(

)1()1(

12

1

)/1)(/1(

1

21 PPsigG

GLm

sig

o

RR

RRg

V

V

LgdLLLP

sigLmgdgssiginP

RCCRC

RRgCCRC

)(

)]1([

2

1

122

21 /1/1

1P

PP

H


19/31

Analysis using open-circuit time constants*

Open-circuit time constants:


LC

LLmsiggd

siggs

RR

RRgRR

RR

L

)1(

LLLLmsiggdsiggsCLgdgdgsgsH

H

LLLLmsiggdsiggsCLgdgdgsgsH

RCRRgRCRCRCRCRC

RCRRgRCRCRCRCRC

L

L

])1([

11

])1([


20/31

Exact analysis*Transfer function of the amplifier:

Step 1: define nodal voltages

Step 2: find branch currents

Step 3: KCL equations

Step 4: transfer function by solving the linear equations

Poles and zero:


oLogsmogsgd

ogsgdgsgssiggssig

VsCRVVgVVsCVVsCVsCRVV

/)()(/)(

LsiggdLgsgdLLgdLsigLmgdgs

mgd

sigG

Gm

sig

o

LsiggdLgsgdLLgdLsigLmgdgs

mgdLm

sig

o

RRCCCCCsRCCRRgCCs

gCs

RR

RRg

V

V

RRCCCCCsRCCRRgCCs

gCsRg

V

V

])[(})()]1({[1)/(1

])[(})()]1({[1

)]/(1[

2

2

gdmZ

sigLgdLgsgdL

LgdLsigLmgdgs

P

LgdLsigLmgdgs

P

PPPPPPPPP

Cg

RRCCCCC

RCCRRgCC

RCCRRgCC

ssssss

sD

/

])[(

)()]1([

)()]1([1

111

111111)(

1

2

1

21

2

121

2

2121


21/31

CS amplifier with a small source resistance*The case where source resistance is zero

The transfer function of the amplifier:

Midband gain:

The transfer function has one pole and one zero

Midband gain and zero are virtually unchanged

The lowest-frequency pole no longer exists

Gain-bandwidth product:

Gain rolls off beyondfH(20 dB/decade)

The gain becomes 0 dB atft:


sigsig

sigsig

VV

RR

0

])[(1)/(1)(LgdL

mgdm

sig

o

RCCsgCsRg

VV

gdmZ

LgdLP

Cg

RCC

/])/[(1

1

1

LmM RgA

LgdL

mHMt

RCC

gfAf

)(2||


22/31

8.6 High-Frequency Response of the CG and Cascode Amplifiers*

High-frequency response of the CG amplifier*The frequency response by neglecting ro:

Midband gain:

Poles:

The upper 3-dB frequency H P2Typically higher than the CS amplifier

The frequency response including ro:

Open-circuit time constant method


)/1)(/1( 21 PP

M

sig

o A

V

V

)1/(1

sigmLm

sigm

LmM RgRg

RgRgA

LLgd

P

msiggs

P

RCC

gRC

)(

1

)]/1(||[

1

2

1

])(/[1

||

||

)1/()(

gdLgdgsgsH

Logd

insiggs

sigomsigoo

omLoin

RCCRC

RRR

RRR

RrgRrR

rgRrR


23/31

High-frequency response of the cascode amplifier*Open-circuit time constant method:

Capacitance Cgs1 sees a resistanceRsig Capacitance Cgd1 sees a resistanceRgd1

Capacitance (Cdb1 + Cgs2) sees a resistanceRd1

Capacitance (CL + Cgd2) sees a resistanceRL ||Ro

Effective time constant

The upper 3-dB frequency:

In the case of a largeRsig:

The first term dominates especially if the Miller multiplier is large (typically with largeRd1 andRL)

A smallRL (to the order of ro) is needed for extended bandwidth

The midband gain drops as the value ofRL decreases

A trade-off exists between gain and bandwidth


12212 oomooo rrgrrR

)||)(()(])1[(212111111 oLgdLdgsdbdsigdmgdsiggsH

RRCCRCCRRRgCRC

]/)[(|||| 2221211 omLooinod rgRrrRrR

1111 )1( dsigdmgd RRRgR

1221111111 )})(||()()]1([{

2

1

2

1 gdLoLgsdbgdddmgdgssigH

H CCRRCCCRRgCCRf


24/31

In the case of a smallR

sig:The first term becomes negligible and the third term dominates

A largeRL (to the order ofA0ro) can be used to boost the amplifier gain

In the case of zeroRsig:

Summary of CS and cascode amplifiers


2

2

2

1

))(||(

1

2

1

gdL

mt

gdLoL

H

CC

gf

CCRRf


25/31

High-frequency response of the bipolar cascode amplifierThe similar analysis technique can be applied for bipolar cascode amplifier.



26/31

8.7 High-Frequency Response of the Source and Emitter Followers

The source follower:Low-frequency (midband) gain and output resistance:

High-frequency characteristics: High-frequency zero:

Output becomes 0 at s = sZ = -gm/CgsHigh-frequency zero: Z = gm/CgsfZfT (transistors unity-gain frequency)

The 3-dB frequencyfH:


)/1(||

1/1/1||

||

moo

mL

L

moL

oLM

grR

gR

R

grR

rRA

)(2

1

)/1(||||

1

L

L

CLgsgsgdgdH

moLC

m

Lsig

gs

siggd

RCRCRCf

grRR

Rg

RRR

RR


27/31

The emitter follower:Low-frequency (midband) gain and output resistance:

High-frequency characteristics: High-frequency zero:

Output becomes 0 at s = sZ = -1/CreHigh-frequency zero: Z = 1/CrefZfT (transistors unity-gain frequency)

The 3-dB frequencyfH:


)]1/([||||

1

sigeoLo

eL

LM

RrrRR

rR

RA

)(2

1

1

])1([||

RCRCf

r

R

r

R

RRR

RrRR

H

e

Lsig

Lsig

Lsig


28/31

8.8 High-Frequency Response of Differential Amplifiers

Resistively Loaded MOS Differential Amplifier:Differential-mode operation:

Use differential half-circuit for analysis

Identical to the case for common-source amplifier

Can be approximated by a dominant-pole system

Common-mode operation: Use common-mode half-circuit for analysis

The capacitance CSS is significant

CSS results in a zero at lower frequency

Other capacitance high-frequency poles and zeros


SSSS

Z

SSSS

D

D

SS

DSS

SSD

DD

D

D

SS

Dcm

D

D

SS

Dcm

CR

CsRR

R

R

RsC

RR

RR

R

R

Z

RsA

R

R

R

RA

1

12

1

22)(

2


29/31

Common-mode rejection ratio (CMRR):

The frequency dependence of CMRR can be evaluated

CMRR decreases at higher frequencies due to the pole ofAdand the zero ofAcm



30/31

Active-Loaded MOS Differential Amplifier:Differential-mode operation:

Equivalent capacitances:

Transconductance (Gm):

The pole and zero of Gm(s) are at very high frequencies

Minor pole and zero near unity-gain frequency


xdbgddbgdL

gsgsdbdbgdm

CCCCCC

CCCCCC

4422

43311

T

m

mZ

T

m

mP

fC

gf

fC

gf

2

2

2/2

3

32

3

31

13

1

24

3

1

3

14344

3

13

/1

2/1

2//1

2/

/1

2/2/

2/

mm

mmm

id

om

idmmm

idm

ddo

mm

idm

mm

idmmgmd

mm

idmg

gsC

gsCg

V

IG

VggsC

Vg

III

gsC

Vg

sCg

VggVgI

sCg

VgV


31/31

The transfer function of the amplifier:

The dominant pole is typically

Common-mode operation:

By taking CSS into account for the mid-band common-mode gain:

Contributes to a zero at lower frequency

Common-mode rejection ratio (CMRR): CMRR decreases due to the dominant pole ofAd(s) and the zero ofAcm(s)


oL

PRC

f2

11

oL

m

m

m

m

om

id

o

ooL

ooidm

oL

oidmo

RsC

g

Cs

g

Cs

RgV

V

rrsC

rrVG

RsC

RVGV

1

1

1

21

)||(1

||

1

3

31

42

42

SSm

SSSS

SSSS

SSm

cmRg

CsR

CsR

Rg

A2

22

1

12

1

High Frequency Cs Amplifier

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