CNU EE 8.1-1

Microelectronic Circuits II

Ch 8 : Frequency Response

8.1 Low-Frequency Response of CS & CE Amplifier

CNU EE 8.1-2

Introduction- gain is constant independent of the frequency of the input signal à infinite bandwidth à Not true,- midband : gain remains almost constant over a wide frequency range

coupling & bypass capacitors à short, internal capacitance à open- lower freq. band : coupling & bypass capacitors no longer have low impedance,

fL : lower end of mid-frequency, reactance 1/jwC- high freq. band : internal capacitance of transistor no longer have high impedance,

fH : upper end of mid-frequency , reactance 1/jwC- Amplifier bandwidth = fH – fL- Extension of bandwidth (i.e., increase fH) à source degeneration resistances

Magnitude of the gain of a discrete BJT or MOS amplifier vs. frequency

CNU EE 8.1-3

Appendix E : STC (Single Time Constant) circuits

- single-time-constant (STC) circuits : one reactive component (L or C) & one resistance R- An STC circuit : time constant t = L/R or CR- Evaluating the time constant

1) reduce the excitation to zero : voltage source à short, current source à open2) find equivalent resistance Req seen by the one reactive component : Req=R4||[R3+(R2||R1)]3) time constant is either L/Req or CReq t =C{R4||[R3+(R2||R1)]}

CNU EE 8.1-4

Classification of STC circuits- Two categories : low-pass (LP) & high-pass (HP) types- low-pass circuits pass dc (i.e., signals w/ zero freq.) and attenuate high freq. w/ zero transmission at w=- At w = 0 , capacitor à open circuit (1/jwC = ), inductor à short circuit (jwL = 0 )- At w = , capacitor à short circuit (1/jwC = 0 ), inductor à open circuit (jwL = )

¥

¥¥

¥

low-pass type high-pass type

CNU EE 8.1-5

Low-Pass circuits- STC low-pass circuit transfer function

K: gain at w=0, w0=1/t t :time constant- Magnitude response, phase response

- Magnitude axis (dB) : 20log|T(jw)/K|- freq. axis (rad/sec):logarithmic scale(w/w0)- low-frequency asymptote |T(jw)| ~ 1~0dB- w/w0 >> 1 : if w doubles, |T(jw)| is halvedà slope : -6dB/octave, -20dB/decade

- two straight-line asymptotes of the magnitude – response curve meet at the Corner freq. or break freq. w0

- w =w0 , gain drops by a factor of relative to the dc gain à corner freq. = 3-dB frequency

( ) ( )00 1)(

1)(

www

w jKjT

sKsT

+=

+=¥

¥

phase response

( )( )0

1

20

tan)(1

)( wwwfww

w --=+

=KjT

magnitude response

www 0)( KjT »

2)( 0 KjT =w

2

CNU EE 8.1-6

High-Pass circuits- STC high-pass circuit transfer function

K: gain at w= , w0=1/t t :time constant- Magnitude response, phase response

- Magnitude axis (dB) : 20log|T(jw)/K|- freq. axis (rad/sec):logarithmic scale(w/w0)- high-frequency asymptote |T(jw)| ~ 1~0dB- w/w0 << 1: if w doubles, |T(jw)| is doubledà slope : +6dB/octave, +20dB/decade

- two straight-line asymptotes of the magnitude – response curve meet at the Corner freq. or break freq. w0

- w =w0 , gain drops by a factor of relative to the gain at high frequencyà corner freq. = 3-dB frequency

( )www

w 00 1)()(

jKjT

sKssT

-=

+=

phase response

( )( )wwwf

www 0

1

20

tan)(1

)( -=+

=KjT

magnitude response

0

)(www KjT »

2)( 0 KjT =w

2

¥

CNU EE 8.1-7

Frequency response of STC networks

CNU EE 8.1-8

MOSFET 등가회로

§ Small-signal equivalent circuit models

( )2/

21

tnGSnD VvL

Wki -÷øö

çèæ=

§ saturation mode

T modelp 모델

CNU EE 8.1-9

§ frequency independent analysis- CC1, CC2,: coupling capacitor- CS (mF): bypass capacitor- Effect of capacitances on gain Vo/Vsig- at midband frequencies, CC1, CC2,CS (mF)ànegligibly small impedances, and

assume perfect short circuità midband gain

- at low frequencies, reactance 1/jwC of CC1, CC2,CS (mF) increases & amplifier gain decreases

( )LDomsigG

G

sig

oM RRrg

RRR

VVA ||||

+-=º

Low-Frequency response of CS amplifier

§ three frequency bands- midband : useful band of amplifier- low-frequency band : CC1, CC2,CS- high-frequency band : Cgs, Cgd

CNU EE 8.1-10

Low-frequency response of CS amplifier§ Determining Vo/Vsig at low frequency- current source I : open- voltage source VDD : short- direct small-signal analysis (No ro)- Determining Vg, Id, Io, & Vo

sig

sig1

V V 1G

g

GC

R

R RsC

=+ +

sigsig

1 sig

V V (4.133)1( )

Gg

G

C G

R sR R s

C R R

=+ +

+

1 01 sig

1 (4.134)( )P

C GC R Rw w= =

+

Coupling capacitor CC1 introduces a high-pass STC response w/ a break frequency

§ Drain current Id from amplifier input Vg

S

mP

S

mgm

Sm

gd C

g

Cg

s

sVg

sCg

VI =

+=

+= 211

wCS introduces a frequency-dependent factor& STC high-pass type w/ a break frequency wP2

T model

CNU EE 8.1-11

( )[ ]

( )[ ]LDmsigG

GM

PPPM

PPPLDm

sigG

G

d

o

g

d

sig

g

sig

o

RRgRR

RAs

ss

ss

sA

ss

ss

ssRRg

RRR

IV

VI

VV

VV

||

||

321

321

÷÷ø

öççè

æ

+-÷÷

ø

öççè

æ+÷÷

ø

öççè

æ+÷÷

ø

öççè

æ+

=

÷÷ø

öççè

æ+÷÷

ø

öççè

æ+÷÷

ø

öççè

æ+÷

÷ø

öççè

æ

+-==

=www

www

Low-frequency response of CS amplifier

2

1D

o d

D LC

RI IR R

sC

= -+ +

§ VO from Id by using current-divider rule

2

V (4.137)1( )

D Lo o L d

D L

C D L

R R sI R IR R s

C R R

= = -+ +

+

CC2 introduces a 3rd high-pass factorw/ a 3rd break frequency at

§ Overall low-frequency transfer function of CS amplifier3

2

1 (4.138)( )P

C D LC R Rw =

+

AM : Midband gain with CC1, CC2,CS acting as perfect short circuits

CNU EE 8.1-12

Low-frequency response of CS amplifier§ Determining lower 3-dB frequency, fL- Lower 3-dB frequency, fL = frequency at

which |Vo/Vsig| drops to |AM|/- If the break frequency wP1, wP2, & wP3

are sufficiently separated, their effectappear distinct.

- At each break frequency, gain slope increases by 20dB/decade

- fP1, fP2 & fP3 : low-frequency poles- If the highest-frequency pole, fP2, is

separated from the nearest pole, fP3 by atleast a factor of 4 (two octaves) :

- The highest pole, fP2 ~ Cs because Csinteracts with 1/gm, which is low

§ Determining the pole frequencies by inspection1. Reduce Vsig to zero2. Consider each capacitor separately; that is, assume that the other two capacitors are acting as

perfect short circuits3. For each capacitor, find the total resistance seen between its terminals. This is the resistance that

determines the time constant associated with this capacitor

2L Pf f@

2

CNU EE 8.1-13

BJT 등가회로§ Small-signal equivalent circuit models

T modelp 모델

CNU EE 8.1-14

Low-frequency response of CE amplifier§ frequency independent analysis

- CC1, CC2,CE (mF): short circuit- Cp , Cm (pF range): open circuit- |AM| = constant in the midband

§ three frequency bands- midband : useful band of amplifier- low-frequency band : CC1, CC2,CE- high-frequency band : Cp , Cm

)||||()||(

)||(LCom

sigB

B

sig

oM RRrg

RrRrR

VVA

+-=º

p

p

§ bandwidth or 3-dB bandwidthBW = fH – fL

~ fH when fL << fH§ gain-bandwidth product

GB = |AM|BW: trade-off gain for bandwidth

CNU EE 8.1-15

Low-frequency response of CE amplifier

§ Simple circuit for Low Frequency response- current source I : open- voltage source VCC : short- ignore Cp & Cm : open- neglect rx since rx << rp§ Analysis of the low-frequency response of the CE amplifier

- consider the effect of the three capacitors CC1, CE & CC2 one at a time

T model

CNU EE 8.1-16

Low-frequency response of CE amplifier

])||[(1

11

1

sigBCP

PM

sig

o

RrRC

ssA

VV

+=

+-=

p

w

w

§ Effect of CC1 with CE & CC2 short-circuited

§ [(RB||rp+Rsig] is the resistance seen between the terminals of CC1 when vsig = 0

CC1 introduces a STC high-passtype w/ a break frequency wP1

úúúú

û

ù

êêêê

ë

é

+++

-=

-=++

=

])||[(1

)||()||(

)||(

)||(1)||(

||

1

1

sigBC

LCmsigB

B

sig

o

LCmo

CsigB

Bsig

RrRCs

sRRgRrR

rRVV

RRVgV

sCRrR

rRVV

p

p

p

p

p

pp

p 모델

CNU EE 8.1-17

Low-frequency response of CE amplifier

÷÷ø

öççè

æ+

+=

úû

ùêë

é÷÷ø

öççè

æ+

+++++

-=

÷÷ø

öççè

æ+++

+-=-=

÷÷ø

öççè

æ+++

+=

1||

1

1||

/1)1()||(

)||(

1)1()||(

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

1

2

b

w

bb

b

b

bbb

sigBeE

PsigB

eEesigB

LC

sigB

B

sig

o

sig

CEesigB

LC

sigB

BLCbo

CEesigB

sigB

Bsigb

RRrCRR

rCs

srRR

RRRR

RVV

V

sCrRR

RRRR

RRRIV

sCrRR

RRRVI

§ Effect of CCE with CC1 & CC2 short-circuited & utilizing Thevenin theorem

§ [re+((RB||Rsig)/(b+1))] is the resistance seen between the two terminals of CE when vsig = 0

CE introduces a STChigh-pass type w/ a break frequency wP2

T 모델

CNU EE 8.1-18

Low-frequency response of CE amplifier

)(1

23

3

LCCP

PM

sig

o

RRC

ssA

VV

+=

+-=

w

w

úúúú

û

ù

êêêê

ë

é

+++

-=

++-=

+=

)(1)||(

)||()||(

1)||(||

2

2

LCC

LCmsigB

B

sig

o

L

LC

C

Cmo

sigB

Bsig

RRCs

sRRgRrR

rRVV

RR

sCR

RVgVRrR

rRVV

p

p

pp

pp

§ Effect of CC2 with CE & CC1 short-circuited

§ (RC+RL) is the resistance seen between the terminals of CC2 when vsig = 0

CC2 introduces a STC high-passtype w/ a break frequency wP3

p 모델

CNU EE 8.1-19

Low-frequency response of CE amplifier

§ Overall low-frequency transfer function of CE amplifier

÷÷ø

öççè

æ+÷÷

ø

öççè

æ+÷÷

ø

öççè

æ+

-=321 PPP

Msig

o

ss

ss

ssA

VV

www

§ 3-dB frequency fL is determined by the highest of the three break frequencies the break frequency is caused by the bypass capacitor CE , that is fL ~ fP2