Chapter 7 Problem Solutions 7.1 a.

( )( )

( )( )

( )

0 1

1 1

1 1

1/( )

1/

11

i

V s sCT s

V s sC R

T ssR C

= =+⎡ ⎤⎣ ⎦

=+

b.

�T �

1

fH � 159 Hzf

( )( )3 61 1

1 1 159 Hz2 2 10 10H Hf f

R Cπ π −= = ⇒ =

c.

( ) ( )01 1

11iV s V s

sR C= ⋅

+

For a step function ( ) 1iV s

s=

( )

( )( )( )( )

( )

( ) 1 1

1 20

1 1 1 1

1 1 1 2

1 1

1 1 1 1 2

1 1

2 1 1 1 1

1 10

1 1

1 1/

0

1 11 11

1

1 and 1

11

1 11

1 t R C

K KV ss sR C s sR CK sR C K s

s sR C

K s K R C Ks sR C

K K R C KR CV s

s sR C

s sR C

v t e−

= ⋅ = ++ ++ +

=+

+ +=

+= − =

−= ++

= −+

= −

7.2 a.

( ) ( )( ) ( )

( )

0 2

2 2

2 2

2 2

1/

1

i

V s RT sV s R sC

sR CT ssR C

= =+ ⎡ ⎤⎣ ⎦

=+

b.

�T �

1

fL � 1.59 Hz

( )( )4 62 2

1 1 1.59 Hz2 2 10 10 10L Lf f

R Cπ π −= = ⇒ =

×

c.

( ) ( )

( )

( )

( ) 2 2

2 20

2 2

2 20

2 2

2 2/

0

11

111

i

i

t R C

sR CV s V ssR C

V ss

R CV ssR C s

R Cv t e−

= ⋅+

=

= =+ +

=

7.3 a.

( ) ( )( )

0

1

1 1

11

1 1

PP

iP S

P S

PP P

PP P P

PP

RV s sCT sV s

R RsC sC

RsC RR

sC sR CRsC

= =⎛ ⎞

+ +⎜ ⎟⎝ ⎠

⋅= =

++

Then

( )( )

( ) ( )

1 1

1

11/ 1

P

P S P PS

P

P PP S S P P

S S

P SP P PP

P S P S S S P S S P

RT sR R sR C

sCR

R CR R sR R CC sC

sR RR R CT s CR R R R C s R R C R R

=⎛ ⎞

+ + +⎜ ⎟⎝ ⎠

=+ + + +

⎛ ⎞⎡ ⎤⎛ ⎞= × + ⋅ + + ⋅⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟+ + + +⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠

b.

( ) ( ) ( )

( ) ( )

( )( ) ( )

113 11

6 4 6

8

8

10 10 10 11/ 1 5 10 1010 10 20 10 2 10 10

1 112 1 5 10

0.02

1 12 11 5 10

0.02

T s ss

ss

s j

T jj

ω

ωω

ω

−−

− −

−

−

⎛ ⎞⎡ ⎤⎛ ⎞ ⎜ ⎟⎢ ⎥= × + ⋅ + + × ⋅⎜ ⎟ ⎜ ⎟+ × ⋅⎝ ⎠ ⎢ ⎥⎣ ⎦⎝ ⎠

≅ ⋅+ + ×

=

= ⋅⎡ ⎤

+ × −⎢ ⎥⎣ ⎦

( ) ( )( )( )

( )( ) ( )( )

( )

4 6

8

1 1For 502 10 10

1 12 11 50 5 10

50 0.02

1 1 1 12 1 2 2

LS R SR R C

T jj

T jj

ω

ω

ω

−

−

= = =+ ×

= ⋅⎡ ⎤

+ × −⎢ ⎥⎣ ⎦

≈ ⋅ ⇒ = ⋅−

For

( ) ( )( )( )

( )( ) ( )( )

( ) ( )

( )

73 11

7 87

1 1 2 105 10 10

1 12 11 2 10 5 10

2 10 0.02

1 1 1 12 1 2 2

1In each case, 2

HS P P

P

P S

R R C

T j

j

T j T jj

RT jR R

ω

ω

ω ω

ω

−

−

= = = ××

= ⋅⎡ ⎤⎢ ⎥+ × × −

×⎢ ⎥⎣ ⎦

≈ ⋅ ⇒ = ⋅+

= ⋅+

c.

( ) ( )( ) ( )( )

( )( ) ( )

3 74 7

43

10 k , 0.1 F

1 1 1 11/ 1 5 10 102 2 1 2 10 10

1 12 1 11 5 10

2 2 10

S P S PR R C C

T s ss

s j

T j

j

μ

ω

ω

ωω

−

−

−−

= = Ω = =

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥= ⋅ + ⋅ + + ×⎜ ⎟⎢ ⎥⎜ × ⎟⎣ ⎦⎝ ⎠

=

= ⋅⎡ ⎤⎢ ⎥+ + × −

×⎢ ⎥⎣ ⎦

( )( )( )

( )( ) ( )( )

( ) ( )

4 7

43

1For 5002 10 10

1 12 11.5 500 5 10

500 2 10

1 1 0.2982 1.5 0.75

T j

j

T jj

ω

ω

ω

−

−−

= =×

= ⋅⎡ ⎤⎢ ⎥+ × −

×⎢ ⎥⎣ ⎦

= ⋅ ⇒ =−

( )( )

( ) ( )( ) ( )( )

( ) ( )

( )

33 7

3 43 3

1For 2 105 10 10

1 11/ 1.5 2 10 5 102 2 10 2 10

1 1 0.2982 1.5 0.75

1In each case, 2

P

P S

T j j

T jj

RT jR R

ω

ω

ω

ω

−

−−

= = ××

⎧ ⎫⎛ ⎞⎡ ⎤⎪ ⎪⎜ ⎟⎢ ⎥= ⋅ + × × −⎨ ⎬⎜ ⎟× ×⎢ ⎥⎪ ⎪⎣ ⎦⎝ ⎠⎩ ⎭

= ⋅ ⇒ =+

< ⋅+

7.4 Circuit (a):

( )( )

( ) ( )

0 2 2

1 122

1 11

2 1 1 1 12 2

1 2 1 2 1 1 1 2 1 2 121 1

1/11/

1 1 or

11

i

o

i

V R RTR sCV

RRR sCsC

R sR C sR CVR RR R sR R C R V R R sR R CRsR C

= = =++

+

+ += = = ⋅

+ + + +++

Low frequency:

2

1 2

20 210 20 3

o

i

V RV R R

= = =+ +

High frequency:

( )( )

( ) ( ) ( )

4 61 1 1 1

1

3 62 1 1 2 22

2

1

110 10 10 0.10 1.59 2

110 20 10 10 10 0.0667 2.39 2

o

i

VV

R C f Hz

R R C f Hz

τπτ

τ τπτ

−

−

=

= = × = ⇒ = =

= = × × × ⇒ = ⇒ = =

�T �

1.0

0.67

1.59 2.39 f Circuit (b):

( )

22

2 2 2

212 1

2 22

2

1 2 1 2 2

11

11

11

o

i

RRsCV sR CT

RV RR R sR CsC

RR R s R R C

+= = =

++ +

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠

Low frequency:

( ) ( )

2

1 2

3 61 2 2

20 220 10 3

10 20 10 10 10 0.0667

1 2.39 2

o

i

V RV R R

R R C

f Hz

τ

πτ

−

= = =+ +

= = × × × =

= =

�T �

0.67

2.39 f 7.5 a.

( ) [ ]( )

3 6

3 12

30 10 10 10 10 0.40 s

30 10 10 50 10 0.375 sS i P S S

P i P P P

r R R C r

r R R C r μ

−

−

= + = + × × × ⇒ =

= = ⎡ ⎤× × × ⇒ =⎣ ⎦

b.

( )

( )6

1 1 0.398 Hz2 2 0.4

1 1 424 kHz2 2 0.375 10

L LS

H HP

f fr

f fr

π π

π π −

= = ⇒ =

= = ⇒ =×

At midband. SC → short, PC → open

( )( ) ( )30 10 7.5

o i i P

i P

V I R R

T s R R T s k

=

= = ⇒ = Ω

c.

�T �

7.5 k�

fL fH 7.6 (a)

( ) ( )( ) ( )

( )

( ) ( ) ( )

2 2 22

max

2

10

1 1

1 1 11 2 1 21 2

11 1 1At

2 21 1

120log 6 2

Phase 2 tan 2 2 tan 1 2 45 Phase 90

dB dB

T Tj f ff

T

f T

T T dB

f

π τ π τπ τ

πτ

π τ− −

= ⇒ = =+ ++

=

= ⇒ = =+

⎛ ⎞= ⇒ ≅ −⎜ ⎟⎝ ⎠

= = − = − ° ⇒ = − °

(b) ( )( )

Slope 2 6 / 12 / 40 /Phase 2 90 Phase 180

dB oct dB oct dB decade= − = − = −= − ° ⇒ = − °

7.7 (a)

( ) ( )( )

( )

( )

34

4

2 2

10

20 1 2000 120 2000

5 10 2.5 10

1 1 1 12000 20 2000

2.5 10

1 120 2000

jT j

j j

jj

j j j j

T

ωω

ω ω

ωωω

ω ω ω ωω

ω

ω ω

−−

−

−=

⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞− × ⎜ ⎟ ×⎝ ⎠= =⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞+ + + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

×=

⎛ ⎞ ⎛ ⎞+ ⋅ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

�T �

5 � 10�3

2.5 � 10�4

1 20 2000 �

(b)

( )( )( )

( )2

2

10 10 110

1000 11000

0.10 110

11000

j

T jj

T

ω

ωω

ω

ω

⎛ ⎞+⎜ ⎟⎝ ⎠=

⎛ ⎞+⎜ ⎟⎝ ⎠

⎛ ⎞+ ⎜ ⎟⎝ ⎠=

⎛ ⎞+ ⎜ ⎟⎝ ⎠

10

0.1010 1000

�T �

�

7.8 (a)

( ) ( )( ) ( )( )105 10 5

5 55 10 5 500 10 500 1 110 500

10 10500 1 1

10 500

T s

j

j j

ω

ω ω

= = ⋅+ + ⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠= ⋅

⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

�T �

0.02

10 500 �(rod/s) (b) Midband gain = 0.02 (c) 500 rad/sω = (d) 10 rad/sω = 7.9 (a)

( )( )

( )( )

4

3 5

4

3 5

3 5

4

3 5

2 10( )10 10

2 10 110 10 1 1

10 102 10( )

1 110 10

T sS S

S S

T jj j

ωω ω

−

×=+ +

×= ⋅⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

×=⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

�T �

2 � 10�4

103 105 �(rod/s) (b) 42 10−× (c) 310 rad/sω = (d) no low freq 3dB− freq. 7.10 a.

( )( )

0

midband

5.229 65.2 0.5

159

m L iS

m LS

rV g V R V V

r R

rT g Rr R

T

ππ π

π

π

π

⎛ ⎞= − = ⎜ ⎟+⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠=

b. ( )

( )

( )3

1 1 1 5.31 ms Open-circuit2 2 2 30

1 1 0.332 s Short-circuit2 2 480 10

S S C

L S SS L

P PH

r R r C

f r rr f

rf

π

π π π

τ μπ π

= +

= ⇒ = = ⇒ =

= = ⇒ =×

c.

( ) ( )3

3

6

3

5.31 10 0.932 F0.5 5.2 10

0.332 10 55.3 pF6 10

SC C

S

P L L

PL L

L

rC CR

r R CrC CR

π

μτ

−

−

×= = ⇒ =+ + ×

=×= = ⇒ =

×

7.11 Computer Analysis 7.12 Computer Analysis 7.13 a.

( )

( )( )

( )( )

1 2

2

1 2

10 1.5 1.304 k

1.5 12 1.565 V1.5 10

1.565 0.7 0.0759 mA1.30 101 0.17.585 mA100 0.026

0.343 k7.59

7.59 292 mA/V0.026

TH

TH CC

BQ

CQ

m

R R R

RV VR R

I

I

r

g

π

= = = Ω

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠−= =

+=

= = Ω

= =

( )

( )( )

( ) [ ]

( )

1 2

3 6

4

4

1

10 1.5 0.343 101 0.1

1.30 10.44 1.159 k

0.5 1.16 10 0.1 101.659 10 s

1 1 959 Hz2 2 1.66 10

i E

i

S i C

L L

R R R r R

R

r R R Cr

f fr

π β

π π

−

−

−

= + +⎡ ⎤⎣ ⎦

= +⎡ ⎤⎣ ⎦= ⇒ = Ω

= + = + × × ×= ×

= = ⇒ =×

b.

r�V� gmV�

R1��R2Vi��

�

�

V0

RE

RC

RS

Rib

Ib

( )( )

( )( )

( )

( ) ( )

( ) ( )( )( )

0

1

1 2

1 2

1 2

0 0 0

midband midband

10.343 101 0.1 10.44 k

1.30 0.1111.30 10.4

0.5 1.3 10.44

1.6590.111 100 1 0.111

6.691.659 1.659

b C

b E

b iib

i i

ii

S ib

i

ii

C

i i i

V I RR r R

R RI I

R R R

I I

VIR R R R

V

VI

RV V VV V V

π

ββ

β

= −= + += + = Ω

⎛ ⎞= ⎜ ⎟⎜ ⎟+⎝ ⎠⎛ ⎞= =⎜ ⎟+⎝ ⎠

=+

=+

=

= ⇒ = ⇒ =

c.

6.69

fL � 959 Hz

V0Vi

f

� �

7.14

( )( )

( )

( )( )

2

2

1 2 1

11

22

2

0.5 mA 0.5 0.5 0.25 V

0.5 1.5 3.08 V0.2

3.08 0.25 3.33 V

13.33

13.33 200 9 541 k

541 200 317 k541

4.5 0.25 4.75

DQ S

DQ n GS TN GS

G GS S G

G DD in DD

D DSQ S

I V

I K V V V

V V V V

RV V R VR R R

RR

R RR

V V V

R

= ⇒ = =

= − ⇒ = + =

= + = + ⇒ =

⎛ ⎞= ⇒ = ⋅ ⋅⎜ ⎟+⎝ ⎠

= ⇒ = Ω

= ⇒ = Ω+= + = + =

( )

( )( )( )( )

( )( )

3

3

midband

9 4.75 8.5 k0.51 1 1 7.96 ms

2 2 2 20

7.96 10 0.0398 F200 10

2 0.2 3.08 1.5 0.632 mA/V

0.632 8.54.08

1 1 0.632 0.5

D D

L LL L

LL in C C C

in

m

m Dv v

m S

R

f rr f

rr R C C CR

g

g RA A

g R

π π π

μ−

−= ⇒ = Ω

= ⇒ = = =

×= ⋅ ⇒ = = ⇒ =×

= − =

= = ⇒ =+ +

�A��

4.08

fL � 20 Hz

Phase

�90

�135

�180

fL

f

7.15

( )

( )

2 1 1 2.414 0.5

2.414 V2.414 5

2.59 k1

3 2.414 0.586 V5 0.59 4.41 k

1

DQDQ n GS TN GS TN

n

S

S S

D DSQ S

D D

II K V V V V V

KV

R R

V V V

R R

= − ⇒ = + = + =

= −− − −

= ⇒ = Ω

= + = − =−= ⇒ = Ω

b.

gmVgsVi

�

�

RG

RS

CC

RD RL

Vgs��

I0

( )

( )( ) ( )

( ) ( )( )

( )( )

0

0

0

1

1

1 1

11 1

Dm gs

D LC

igs

m S

m CD

i m S D L C

i

D L Cm D

m S D L D L C

RI g VR R

sCVVg R

I s g sCRV s g R s R R C

I sT s

V s

s R R Cg Rg R R R s R R C

⎛ ⎞⎜ ⎟⎜ ⎟= −⎜ ⎟+ +⎜ ⎟⎝ ⎠

=+

⎡ ⎤−= ⋅ ⎢ ⎥+ + +⎢ ⎥⎣ ⎦

=

+−= ⋅ ⋅

+ + + +

c.

( )

( ) ( )3

3

1 1 1 15.92 ms2 2 2 10

15.9 10 1.89 F4.41 4 10

L LL L

LL D L C C C

D L

f rr f

rr R R C C CR R

π π π

μ−

= → = = =

×= + ⇒ = = ⇒ =+ + ×

7.16 a.

( )

( )( )( )

( ) ( )( )( )

( ) ( )( )

2

2

2

2

9

9 0.5 12 4 4

6 23 15 0

23 23 4 6 153 V

2 62 2 0.5 3 2 1 mA/V1 1 12 0.923 k

SGD P SG TP

S

SG SG SG

SG SG

SG SG

m P SG TP m

o S om

VI K V V

R

V V V

V V

V V

g K V V g

R R Rg

−= = +

− = − +

− + =

± −= ⇒ =

= + = − ⇒ =

= = ⇒ = Ω

b. ( )0 L Cr R R C= +

c. ( )

( )3

3

1 1 1 7.96 ms2 2 2 20

7.96 10 0.729 F0.923 10 10

LL

C Co L

f rf

rC CR R

πτ π π

μ−

= ⇒ = = =

×= = ⇒ =+ + ×

7.17 a.

( )( )( )( )( )( )

( ) ( )

( )( ) ( )( )( )

( )( )

1 2

1

1

1

22

2

11 mA, 0.00833 mA120

|| 0.1 10.1 121 4 48.4 k

on 1

1 0.00833 48.4 0.7 121 0.00833 4

1 48.4 12 5.135

113 k

113 48.4 84.7113

CQ BQ

E

TH BQ TH BE BQ E

TH CC

I I

R R R

V I R V I R

R VR

RR

R R kR

β

β

= = =

= += = Ω

= + + +

⋅ ⋅ = + +

=

= Ω

= ⇒ = Ω+

b.

( )( )

0 0

0

0 0

1120 0.026

3.12 k1

80 80 k13.12 4 80 0.02579 4 80 25.6121

ErR R r

r

r

R R

π

π

β=

+

= = Ω

= = Ω

= = ⇒ = Ω

c.

( )( )

( )

0 23 6 3

3

0.0256 4 10 2 10 8.05 10 s1 1 19.8 Hz

2 2 8.05 10

L Cr R R Cr

f frπ π

− −

−

= += + × × × = ×

= = ⇒ =×

7.18 (a)

( )( ) ( )( )

( )( )

5 0.7 1.075 mA 1.064 mA4

10 1.064 2 1.075 43.57 V

1.064 40.92 mA/V0.026

100 0.0262.44 K

1.064

EQ CQ

CEQ

CEQ

CQm

T

T

CQ

I I

VV

Ig

V

VrIπβ

−= = =

= − −=

= = =

= = =

(b)

1 1

1 1 1 1

2 2

2 2 2

2440For , 200 40001 101

224.0 , 1.053 msFor , 2 47 49 K

49 ms

C eq S E

eq eq C

C eq C L

eq c

rC R R R

R r R CC R R R

R C

πβ

τ

τ

= + = ++

= = == + = + == ⋅ =

(c) ( )1 131

1 1 151 Hz2 2 1.053 10

f fπτ π −

= = ⇒ =×

7.19 (a)

( ) ( )

( )

3 12

8

8

2 47 10 10 10

1.918 10 s1 1 8.30 MHz

2 2 1.918 10

H C L L

H HH

R R C

f f

τ

πτ π

−

−

−

= = × × ×

= ×

= = ⇒ =×

(b)

( )

( )

( )

2

22

8

1 0.11 .2

1 100 1 .20.1

99 992 2 1.918 10

82.6 MHz

H

H

H

f

f

f

f

πτ

πτ

πτ π −

=+

⎛ ⎞ = = +⎜ ⎟⎝ ⎠

= =×

=

7.20 (a)

( )

( )( )( ) ( )( )

( )( )

( )( )

2

12 2

2

5

5 1 1.2 1.5 1.2 3 2.25

1.2 2.6 2.3 0 2.84

1.8

10 1.8 1.2 1.2 5.68

2 2 1 1.8 2.683 /

SGP SG TP

SG SG SG SG

SG SG SG

DQ

SDQ SDQ

m P DQ

o

VK V V

R

V V V V

V V V V

I mA

V V V

g K I mA V

r

−= +

− = − = − +

− − = ⇒ =

=

= − + ⇒ =

= = =

= ∞

(b) 1 1 0.3727

2.681.2 0.373 0.284

ism

i

R kg

R k

= = = Ω

= = Ω

For ( )( )61 1, 284 200 4.7 10 2.27C sC msτ −= + × =

For ( )( )3 3 62 2, 1.2 10 50 10 10 51.2C sC x msτ −= + × =

(c) CC2 dominates,

( )3 32

1 1 3.12 2 51.2 10dB

s

f Hzπτ π− −

= = =×

7.21 Assume 21 , 80 / , 0TN nV V k A Vμ λ′= = = Neglecting 200SiR = Ω , Midband gain is:

v m DA g R= Let 0.2 , 5DQ DSQI mA V V= =

Then 9 5 200.2D DR R k−= ⇒ = Ω

We need 210 0.5 /20

vm

D

Ag mA V

R= = = and 2 2

2n

m n DQ DQk Wg K I I

L′⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

or ( )0.0800.5 2 0.2 7.812

W WL L

⎛ ⎞⎛ ⎞= ⇒ =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Let

( ) ( )( )

( )( ) ( ) ( )

( ) ( ) ( )

1 2

2 2 2

1 2

2 1

1 2

44

1 31

9 9 2250.2 0.2 0.2

0.0800.2 7.81 1 1.80 9 92 225

45 , 180

180 45 36

1 1 7.96 107.958 10 or 2 2 200 200 36 10

DQ

DQ GS GS

TH

Si TH C C

C

R R kI

R RI V VR R

R k R k

R R R k

s R R C Cf

C

τπ π

−−

+ = = = Ω

⎛ ⎞ ⎛ ⎞⎛ ⎞= = − ⇒ = = = ⇒⎜ ⎟⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠⎝ ⎠= Ω = Ω

= = = Ω

×= = = × = + = ⇒+ ×

=

( )5

52 33

2

0.022

1 1 5.31 105.305 10 or 2.65 2 20 102 3 10 D L L L

F

s R C C C nFf x

μ

τπ π

−− ×= = = × = = ⇒ =

×

7.22 a.

( ) ( )( )

( ) ( )( )

2

2 1

2

1 2

2 1 2

1/1/

11

,A B

R sCT s

R sC RsR CT s

s R R C

r R C r R R C

+=

+ ++

=+ +

= = +

b.

1

0.0909

fAfB f

�T �

c.

( )( )( ) [ ]

( )

( )

3 12 72

3 121 2

6

7

6

10 100 10 10 s

10 1 10 100 101.1 10 s

1 1 1.59 MHz2 2 10

1 1 0.145 MHz2 2 1.1 10

A A

B

B

A AA

B BB

r R C r

r R R Cr

f fr

f fr

π π

π π

− −

−

−

−

−

= = × = =

= + = + × × ×= × =

≈ = ⇒ =

≈ = ⇒ =×

7.23

( )( )( )

( )( )

( )( )

( )( ) 3 3

10 0.7 0.00997 430 201 2.5200 1.995

200 0.0262.61

1.992.61 201 2.5 505

1 1 0.0106 2 2 15

0.5 505 430 10 232.7 10

BQ

CQ BQ

ib

sL

eq C C C

I mA

I I mA

r k

R k

sf

R C C C

π

τπ π

−= =+

= =

= = Ω

= + = Ω

= = =

= = + × = ×

Or 84.55 10 45.5 CC F nF−= × ⇒ 7.24

( ) ( )( )( )

( ) ( )( )

( )3

10 (300) 0.7 201 1 0.0186 mA 3.7126 mA200 0.026

1.40 K3.71

1 1.40 201 1 202.4 K0.1 300 202.4 121 K

1 1 1 7.958 10 s2 2 2 20

7.958 10

BQ BQ BQ CQ

T

CQ

i E

eq S B i

L eq C

L LL

LC

eq

I I I I

VrI

R r RR R R R

R C

ff

CR

π

π

β

β

τ

τπτ π πτ

−

= + + ⇒ = ⇒ =

= = =

= + + = + == + = + =

= ⋅

= ⇒ = = = ×

×= =3

3 0.0658 F121 10 CC μ

−

⇒ =×

7.25

( ) ( )

( )( )

( )( )

( ) ( )1 2

1 2

2

1 2

3 1 3 2

1.2 1.2 0.6

1.2 5 2.5 1.2 1.2

2.5 0.7 0.319 0.6 101 0.0531.9 100 0.026

0.0815 31.9

1 and so that 2C C

TH

TH CC

BQ

CQ

C C dB C dB C

R R R k

RV V VR R

I mA

I mA

r k

f f C f C

π

τ τπτ − −

= = = Ω

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠−= =

+=

= = Ω

>> = <<

Then, for ( )3 1 2dB C Cf C C− ⇒ acts as an open and for ( )3 2 1dB C Cf C C− ⇒ acts as a short circuit.

( )3 22

1252dB Cf C Hzπτ− = = , so that

( )2 21 0.006366

2 25 eq Cs R Cτπ

= = =

where

( )( )( )

( ) ( )( )

1 2

2 2

2

1 2 1 1

1

1

81.5 600 30010 50 10 50 2.787

101

0.0063712.64 504 12.6

1 Assume an open81.5 101 50 5132 100 100 0.006366 0.6366

Seq L E

eq C C

ib E C

ib

eq C

eq S

r R R RR R R

R C C F

R r R CR

s R CR R R

π

π

β

μ

β

τ τ

⎛ + ⎞= + ⎜ ⎟+⎝ ⎠

⎛ + ⎞= + = + ⇒⎜ ⎟

⎝ ⎠

= Ω ⇒ = ⇒ =

= + += + = Ω

= = = == +

1 1

300 600 5132 837.2

0.6366So 760 837.2

TH ib

C C

R

C C Fμ

= + = Ω

= ⇒ =

7.26 From Problem 7.25 0.6 K, 31.9 mA, 81.5 TH CQR I rπ= = = Ω

2 11 and

2C C fτ τπτ

= so ( ) ( )3 2 3 1dB C dB Cf C f C− −

Then ( )3 2 1dB C Cf C C− ⇒ acts as an open circuit and for ( )3 1 2dB C Cf C C− ⇒ acts as a short circuit.

( )

( )( ) ( )( )3 1 1

1

1 1

1 1

2 1

2

120 Hz 0.007958 s2

1 81.5 101 50 10 923.2

300 600 923.2 663.7

0.007958 12 F663.7

100 0.7958 s

81.5 60010 501 101

dB C CC

ib E L

C eq S TH ib

C C

C C

THeq L E

f C

R r R R

R R R R

C C

r RR R R

π

π

τπτ

β

τ

μ

τ τ

β

− = = ⇒ =

= + + = + = Ω

⇒ = + = + = Ω

= ⇒ =

= =

⎛ ⎞+ +⎛= + = +⎜ ⎟+ ⎝⎝ ⎠

2

2 2

10 50 6.748 15.95 0.7958 0.050 F15.95

eq

C C

R

C C

⎞⎜ ⎟

⎠= + = Ω

= ⇒ =

7.27 a.

( )

( )

2

0.5 0.8 1.8 0.5

5 5 1.8 6.4 k0.5 0.5

4 1.8 2.2 V5 2.2 5.6 k

0.5

D n GS TN

DGS TN

n

GSS S

D DSQ S

D D

I K V V

IV V VK

VR R

V V V

R R

= −

= + = + =

− − − −= = ⇒ = Ω

= + = − =−= ⇒ = Ω

(b)

( )( )( )( )

( )

( )( ) ( )

( )

3 6

2

2

36

3

3

2 2 0.5 0.5 1 /

6.4 10 5 10

3.2 10 s1 1 4.97 Hz

2 2 3.2 10

6.4 10 5 101 1 1 6.4

4.32 10 s1 1 36.8 Hz

2 2 4.32 10

m n D

A S S

A AA

SB S

m S

B BB

g K I mA V

r R C

f fr

Rr Cg R

f fr

π π

π π

−

−

−

−

−

−

= = =

= = × ×

= ×

= = ⇒ =×

⎡ ⎤⎛ ⎞ ×= = ×⎢ ⎥⎜ ⎟+ +⎢ ⎥⎝ ⎠ ⎣ ⎦= ×

= = ⇒ =×

c. ( )

( )

1

1 11

m D S Sv

Sm S S

m S

g R sR CA

Rg R s Cg R

+=

⎡ ⎤⎛ ⎞+ +⎢ ⎥⎜ ⎟+⎝ ⎠⎣ ⎦

As RS becomes large ( )

( ) 1

m D S Sv

Sm S S

m S

g R sR CA

Rg R s Cg R

→⎡ ⎤⎛ ⎞

+⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

( ) 1

11

m D Sm

v

Sm

g R s Cg

As C

g

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦=

⎛ ⎞+ ⎜ ⎟

⎝ ⎠

The corner frequency ( )

12 1/B

m S

fg Cπ

= and the corresponding 0Af →

( )( )

( )63

2 2 0.5 0.5 1 /

1 31.8 Hz12 5 10

10

m n D

B B

g K I mA V

f fπ −

−

= = =

= ⇒ =⎛ ⎞ ×⎜ ⎟⎝ ⎠

7.28

a. ( )

( )1 and

2 1E S E

B BB S E

R R r Cf r

r R r Rπ

ππ β+

= =+ + +

( )( )

For 01

0.7 101.86 mA

575 1.84 mA125 1.85 mA

E ES B

E

EQ

CQ

CQ

R r CR rr R

I

II

π

π β

ββ

= =+ +

− − −= =

= ⇒ == ⇒ =

( )1For 200 Hz 0.796 ms

2 200B Bf rπ

≤ ⇒ ≥ =

rπαβ so smallest Br will occur for smallest .β ( )( )

( )( )( )( )

33

75 0.02675; 1.06 k

1.845 10 1.06

0.796 10 57.2 F1.06 76 5

EE

r

CC

πβ

μ−

= = = Ω

×× = ⇒ =

+

b. ( )( )

( )( )( )( )( )

( )( )( )

( )

3 6

3

3 6

125 0.026For 125; 1.76 k

1.855 10 1.76 57.2 10

0.797 ms1.76 126 5

1 1 199.7 Hz Essentially independent of .2 2 0.797 10

5 10 57.2 10 0.286 sec

1 1 0.556 Hz In2 2 0.286

B

B BB

A E E

A AA

r

r

f fr

r R C

f fr

πβ

βπ π

π π

−

−

−

= = = Ω

× ×= =

+

= = ⇒ =×

= = × × =

= = ⇒ = dependent of .β

7.29 a. Expression for the voltage gain is the same as Equation (7.66) with 0.SR = b.

( )1

A E E

E EB

E

r R CR r Cr

r Rπ

π β

=

=+ +

7.30

( )( )

5 0.7 1 mA 0.99 mA4.3

100 0.0262.626 K

0.99

E C

T

CQ

I I

VrIπβ

− = = =

= = =

( )( )

( )( )( )( )

( )( )

( )

( )

3 6 2

3 3 64

3 3

2

34

4.3 10 5 10 2.15 10 s

4.3 10 2.626 10 5 101.292 10 s

1 2.626 10 101 4.3 10

1 1 7.40 Hz2 2 2.15 10

1 1 1.23 10 1.23 kHz2 2 1.292 10

A E E

E EB

E

AA

BB

r R C

R r Crr R

fr

fr

π

π

π π

β

π π

− −

−−

−

−

= = × × = ×

× × ×= = = ×

+ + × + ×

= = =×

= = = × =×

( )( )( )

( )( )( )( )

0

100 20.458

1 2.626 101 4.3

100 276.2

2.626

Cv w

E

Cv w

RA

r R

RAr

π

π

ββ

β

→

→∞

= = =+ + +

= = =

�A��

76.2

0.458

7.40 Hz 1.23 kHz f 7.31

( ) ( )

( )

( ) ( )

3 12

8

8

10 5 10 15 10

5 10 s1 1 3.18 MHz

2 2 5 10

10 0.7 0.93 mA, 0.921 mA10

0.921 35.4 mA/V0.026

35.4 5 10 118

H L C L

H

H HH

EQ CQ

m

v m C L v

r R R C

r

f fr

I I

g

A g R R A

π π

−

−

−

= = × × ×

= ×

= = ⇒ =×

−= = =

= =

= = ⇒ =

7.32

( )2

1 2

166 10166 234

4.15 V

G DDRV V

R R⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠

=

( )

( )( )( )

2

2

2

4.15 0.5 0.5 4 4

0.25 3.15 0 3.55 V

G GSD n GS TN

S

GS GS GS

GS GS

V VI K V V

R

V V V

V V

−= = −

− = − +

− = ⇒ =

( ) ( )( )2 2 0.5 3.55 21.55 /

m n GS TN

m

g K V Vg mA V

= − = −=

0

0

1 10.5 0.5 0.6451.55

0.282 k

Sm

R Rg

R

= = =

= Ω

( )

( )

( )

0

86

8

30

1 and 2

15 MHz 3.18 10 s2 5 10

3.18 10 121 pF0.282 4 10

L L H

H

L LL

r R R C fr

f r

rC CR R

π

βωπ

−

−

= =

≈ = ⇒ = = ××

×= = ⇒ =×

7.33

(a) Low-frequency

��

Vs

RSCC

RB

�

�

V� r�

gmV�

RC RL

Vo

Mid-Band

��

Vs

RS

RB V� r�

gmV�

RC RL

Vo�

�

High-frequency

��

Vs

RS

RB V� r�

gmV�

RC RL CL

Vo�

�

(b)

fL fH

�Am�

f (c)

( )( )

12 0.7 11.3 1

1.13 100 0.026

2.3 1.13

1.13 43.46 /0.026

BQ

CQ

m

I AM

I mA

r k

g mA V

π

μ−= =Ω

=

= = Ω

= =

( ) ( )

( )( )

( )( )

0

1000 2.343.46 5.1 500

1000 2.3 1

2.2943.46 5.05 1532.29 1

43.7

Bm m C L

s B S

m

m dB

R RVA midband g R RV R r R

A

A dB

π

π

⎛ ⎞= = − ⎜ ⎟⎜ ⎟+⎝ ⎠

⎛ ⎞= − ⎜ ⎟⎜ ⎟+⎝ ⎠

⎛ ⎞= ⇒ =⎜ ⎟+⎝ ⎠=

( ) ( ) ( )

( ) ( ) ( )

3 6

2

3 12

8

1 , or 1 1000 2.3 10 10 102

3.29 10 4.83

1 , 5.1 500 10 10 102

5.05 10 3.15

L L S B C LL

L L

H H C L L LH

H

f R R r C

s f Hz

f R R C

s f MHz

πτ τπτ

τ

τ τπτ

−

−

−

−

= = + = + × ×

⇒ = × ⇒ =

= = ⇒ = × ×

= × ⇒ =

7.34 a.

��

ViRD RLVsg gmVsg

CL

V0

�

�

( )

( ) ( )( )

( ) ( ) ( )

0 0

0

1

1

1

1

11

m sg LL

sg i

v m D Li L

D LL

m

D LL

v m D LD L L

V g V R RsC

V V

V sA s g R R

V s sC

R RsC

gR R

sC

A s g R Rs R R C

⎛ ⎞= ⎜ ⎟

⎝ ⎠= −

⎛ ⎞= = − ⎜ ⎟

⎝ ⎠⎡ ⎤

⋅⎢ ⎥⎢ ⎥= − ⎢ ⎥

+⎢ ⎥⎢ ⎥⎣ ⎦

= − ⋅+

b. ( )D L Lr R R C=

c. ( )

( )

3 12 8

8

10 20 10 10 10 6.67 10 s

1 1 2.39 MHz2 2 6.67 10H H

r r

f frπ π

− −

−

= × × × ⇒ = ×

= = ⇒ =×

From Example 7.6, gm = 0.705 mA/V ( ) ( )( )0.705 10 20 4.7v m D L vA g R R A= = ⇒ =

7.35 Computer Analysis 7.36 Computer Analysis 7.37 Computer Analysis 7.38

( )

( )3

12

2

1 38.46 mA/V0.026

38.46 102 10 2 10510 MHz

510 4.25 MHz120

mT

CQm

T

T

T

T

gf

C C

Ig

V

f

f

ff f

π π μ

β β

π

β

−

−

=+

= = =

×=+ ×

=

= = ⇒ =

7.39

( )

( )

( )( )

39

12

3

12 9

5000 MHz 33.3 MHz150

2

0.5 19.23 mA/V0.026

19.2 105 102 0.15 10

19.2 100.15 0.612 pF2 10 5 10

0.462 pF

T

mT

m

ff f

gf

C C

g

C

C

C

β β

π μ

π

π

π

β

π

π

π

−

−

−

−

= = ⇒ =

=+

= =

×× =+ ×

×+ = =×

=

7.40

a. 2000 MHz 13.3 MHz150

Tff fβ ββ= = = =

b.

( )

( )

( )

2

2 2

1501 /

150 101 /

1501 22510

224 13.33 224 199.6 MHz

fe

fe

hj f f

hf f

ff

f f f

β

β

β

β

=+

= =+

⎛ ⎞ ⎛ ⎞+ = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

= ⋅ = ⇒ =

7.41 (a)

( )

0

1 1

11

1 1

where

11

11

11

m L

i i

bb

i ib b b b

V g V R

rrsC sr CV V V

r rr r sr CsC

r rV Vr r sr r C r r s r r C

π

ππ

ππ

ππ

π

π π

π π π π

= −

+= ⋅ = ⋅

++ +

⎛ ⎞⎛ ⎞= ⋅ = ⋅⎜ ⎟⎜ ⎟⎜ ⎟+ + + +⎝ ⎠⎝ ⎠

So ( ) ( )( ) ( )

0

1

11v m L

i b b

V s rA s g RV s r r s r r C

π

π π

⎛ ⎞⎛ ⎞= = − ⎜ ⎟⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠

(b) Midband gain: ( )( )100 0.026 12.6 , 38.46 /

1 0.026mr k g mA Vπ = = Ω = =

(i)

( )( )1 1

For 100 2.638.46 4 148.1

2.6 0.1

b

v v

r

A A

= Ω

⎛ ⎞= − ⇒ = −⎜ ⎟+⎝ ⎠

(ii)

( )( )2 2

For 500 2.638.46 4 129.0

2.6 0.5

b

v v

r

A A

= Ω

⎛ ⎞= − ⇒ = −⎜ ⎟+⎝ ⎠

(c) ( )3 11 ,

2dB bf r r Cπτπτ− = =

(i)

( ) ( )3 12 101 3

For 100

0.1 2.6 10 2.2 10 2.12 10 751 b

db

r

s f MHzτ − −−

= Ω

= × × = × ⇒ =

(ii)

( ) ( )3 12 102 3

For 500

0.5 2.6 10 2.2 10 9.23 10 173 b

dB

r

s f MHzτ − −−

= Ω

= × × = × =

7.42 (b)

( )( )( )( ) ( )

4

4 6

2 24 6

10 kHz 10

2500 1 10 1.333 10200

1 10 1.333 10

200 2500 33.3 2700 33.3

i

f

jZ

j j

−

−

= =

− ×= +

+ ×

= + − = −

(c)

( )( )( )( ) ( )

5

5 6

2 25 6

100 kHz 10

2500 1 10 1.333 10200

1 10 1.333 10

200 2456 327 2656 327

i

i

f

jZ

Z j j

−

−

= =

− ×= +

+ ×

= + − = −

(d)

( )( )( )( ) ( )

6

6 6

2 26 6

1 MHz 10

2500 1 10 1.333 10200

1 10 1.333 10

200 900 1200 1100 1200

i

i

f

jZ

Z j j

−

−

= =

− ×= +

+ ×

= + − = −

7.43 a. ( )1M m LC C g Rμ= +

b.

��

�

�

V� r�C� CM gmV�

RL

V0

RB��RS rb

• Vi RB

RB � RS

0

1

Let

1

1

m L M i

i Bi

B SB S b

V g V R C C C

rsC RV V

R Rr R R r

sC

π π

π

π

π

= − + =

⎛ ⎞= ⋅⎜ ⎟+⎝ ⎠+ +

( ) ( )( )

( )( )

1

1

1

1

1

ov

i

i

iBm L

B S

iB S b

i

Bm L

B S i B S b

V sA s

V s

rsC

rsCRg R

R R rsC R R r

rsC

rRg RR R r sr C R R r

π

π

π

π

π

π π

=

⎡ ⎤⋅⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎛ ⎞

= − ⎢ ⎥⎜ ⎟+⎝ ⎠ ⎢ ⎥⋅⎢ ⎥

+ +⎢ ⎥⎢ ⎥+⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞

= − × ⎢ ⎥⎜ ⎟+ + + +⎢ ⎥⎝ ⎠ ⎣ ⎦

( )

( )( ) ( )

( ) ( )

Let

1

1

11

eq B S b

Bv L

B S eq eq i

L Bv

eq B S eq i

R R R r

RA s RR R r R s r R C

R RA sr R R R s r R C

π π

π π

β

β

= +

⎡ ⎤⎛ ⎞ ⎢ ⎥= − ×⎜ ⎟ ⎢ ⎥+ ⎡ ⎤+ +⎝ ⎠ ⎢ ⎥⎣ ⎦⎣ ⎦⎛ ⎞−= ⋅ ⋅⎜ ⎟+ + +⎝ ⎠

c. ( )1

2H

eq i

fr R Cππ

=

7.44

1 2High Freq. , , short circuitsC C EC C C⇒ →

�

�

V� r� C�

C�

gmV� RC RL

V0

R1��R2IS

I0

( ) ( )

( )( )( ) ( )

( )( )

36

1 2

5 192.3 mA/V0.026

192 10250 102 2

122.4 pF 5 pF, 117.4 pF

1

5 1 192.3 1 1 485.8 pF

117 485 603 pF200 0.026

1.04 k5

5 1.04 0.8

CQm

T

mT

M m C L

M

i M

eq

Ig

Vg

fC C C C

C C C C

C C g R R

C

C C C

r

R R R r

π μ π μ

π μ μ π

μ

π

π

π

π π

−

= = =

×= ⇒ × =+ +

+ = ⇒ = =

= +

⎡ ⎤= + ⇒ =⎣ ⎦= + = + =

= = Ω

= = =

( )( )

( )

3 12

7

7

61 k

0.861 10 603 10

5.19 10 s1 1 307 kHz

2 2 5.19 10

eq ir R C

f frπ π

−

−

−

Ω

= ⋅ = × ×

= ×

= = ⇒ =×

7.45

( )

( )( )

( )( )

1 2

2

1 2

60 5.5 5.04 k

5.5 15 1.26 V5.5 60

1.26 0.7 0.0222 mA5.04 101 0.22.22 mA100 0.026

1.17 k2.22

2.22 85.4 mA/V0.026

TH

TH CC

BQ

CQ

m

R R R

RV VR R

I

I

r

g

π

= = = Ω

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠−= =

+=

= = Ω

= =

Lower 3 – dB frequency:

( )( )

( )

1

1 2

3 6 4

4

2 60 5.5 1.17 2.95

2.95 10 0.1 10 2.95 10 s

1 1 540 Hz2 2 2.95 10

L eq C

eq S

L

L LL

r R C

R R R R rk

r

f fr

π

π π

− −

−

= ⋅

= += + = Ω

= × × = ×

= = ⇒ =×

Upper 3 – dB frequency:

( ) ( )

( ) ( )( )( )

( )( )

36

1 2

3 12

7

85.4 10400 102 2

34 pF; 2 pF 32 pF1 2 1 85.4 4 685 pF

32 685 717 pF2 60 5.5 1.170.644 k

0.644 10 717 10

4.62 10 s1

2

mT

M m C M

i M

eq S

eq i

H H

gfC C C C

C C C CC C g R C

C C CR R R R r

r R C

f fr

π μ π μ

π μ μ π

μ

π

π

π π

π

−

−

−

×= ⇒ × =+ +

+ = = ⇒ == + = + ⇒ =

= + = + == =

= Ω

= ⋅ = × ×

= ×

= ⇒ = 344 kHz

7.46

( ) ( )

( )( )

( )( )

1 2

2

1 2

600 55 50.38 K

5515 15 1.2595 V600 55

1.26 0.7 0.00222 mA50.4 101 20.2217 mA100 0.026

11.73 K0.222

0.2217 8.527 mA/V0.026

TH

TH

BQ

CQ

m

R R R

RVR R

I

I

r

g

π

= = =

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠−= =

+=

= =

= =

Lower – 3dB Fig.

( )( )

1 11

3 6 3

R ;

0.50 50.38 11.73 10.0 K

10 10 0.1 10 10 s

L e c eq S THq

L

C R R R rπτ

τ − −

= = +

= + =

= × × =

( )( )

( )

1 11

3 6 3

32

R ;

0.50 50.38 11.73 10.0 K

10 10 0.1 10 10 s

1 1 159 Hz2 2 10

L e c eq S THq

L

L L

C R R R r

f f

πτ

τ

πτ π

− −

−

= = +

= + =

= × × =

= = ⇒ =

Upper – 3dB Fig.

( ) ( )

( ) ( )( )

( )

36

12

2

32

8.527 10 400 102 2 102

3.393 pF 1.393 pF

1 2 1 8.527 40 684 pF

1.393 684 685.4 pF

0.5 50.38 11.73

50.38 0.480 0.4750 K

R . 0.4750 10 685.

mT

M m C

T M

eq S TH

H eq T

gf

CC C

C C C

C C g R

C C C

R R R r

C

ππ μ

π μ π

μ

π

π

ππ

τ

−

−

×= = = ×+ ×+

+ = ⇒ =

= + = + =⎡ ⎤⎣ ⎦= + = + =

= =

= =

= = × ( )

( )

12

7

7

4 10

3.256 10 s1 1 489 KHz

2 2 3.256 10H HH

f fπτ π

−

−

−

×

= ×

= = ⇒ =×

7.47

( )( )

( )( )

( )

2

6

12

2

402 , 15 60 /10

2 60 100 154.9 /

154.9 10 44.8 2 0.5 0.05 10

mT

gs gd

m n D n

m

T T

gfC C

g K I K A V

g A V

f f MHz

π

μ

μ

π

−

−

=+

⎛ ⎞= = =⎜ ⎟⎝ ⎠

= =

×= ⇒ =+ ×

7.48

( )( )

( )

( ) ( )

2

2

2

or then 2

2So

2

mT

gs gd

m n Gs T

DD n Gs TN Gs TN m n D

n

n D n DT

gs gd gs gd

gfC C

g K V V

II K V V V V g K IK

K I K If

C C C C

π

π π

=+

= −

= − − = =

= =+ +

(a) ( )( )

( )

3 6

12

0.2 10 20 1033.6 MHz

0.5 0.1 10T Tf fπ

− −

−

× ×= ⇒ =

+ ×

(b) ( )( )

( )

3 6

12

0.2 10 250 10118.6 MHz

0.5 0.1 10T Tf fπ

− −

−

× ×= ⇒ =

+ ×

(c) ( )

( )

3

912

0.2 1010 17.8 mA

0.5 0.1 10D

D

II

π

−

−

×= ⇒ =

+ ×

7.49

( )

( ) ( )

( )( )

( )2

2

2 22

Then 2

2

mT

gs gd

gs gd ox

n oxm n GS TN GS TN

n ox GS TN

Tox

n GS TNT

gf

C C

C C WLCCWg K V V V V

LW C V VLf

WLCV V

fL

π

μ

μ

πμ

π

=+

+ =

⎛ ⎞⎛ ⎞= − = −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞ −⎜ ⎟⎝ ⎠=

−=

(a) ( )

( )24

450 0.52.49 GHz

2 1.2 10T Tf f

π −= ⇒ =

×

(b) ( )

( )4

450 0.5111 GHz

2 0.18 10T Tf fπ −

= ⇒ =×

7.50 a.

( )( )( )( )1

5 1 3 15 10 95 pF

M gd m o D

M M

C C g r R

C C

′= +

⎡ ⎤= + ⇒ =⎣ ⎦

b. ( )( )( )( )

( )

3 12 6

6

10 10 50 95 10 1.45 10 s

1 1 110 kHz2 2 1.45 10

i gs Mr r C C

r

f frπ π

− −

−

= +

= × + × = ×

= = ⇒ =×

7.51

( ) ( )Eq. 7.1042

mT

gsT gdT

gfC Cπ

=+

Let ( )20 and 3gdT gsT oxC C WLC⎛ ⎞= = ⎜ ⎟

⎝ ⎠

2 22

n oxm n D D

C Wg K I IL

μ⎛ ⎞ ⎡ ⎤= = ⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠

( )

122

So 22 3

123

2

32 2

n ox D

T

ox

n ox D

ox

n DT

ox

WC IL

fW LC

WC IL

L W C

If

L W C L

μ

π

μ

π

μπ

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= ⋅

= ⋅

7.52 (a)

( )( )( ) ( )

8

4 2

12

400 7.25 1010

2 2

1.45 10 /

mm

m S

m n GS TN

n oxn

n

ggg r

g K V V

C WKL

K A V

μ −

−

′ =+

= −

⎛ ⎞×⎛ ⎞⎛ ⎞ ⎜ ⎟= =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠= ×

( )( )( )

( )( )

4 3

3

33

3

3

For 5 V

2 1.45 10 5 0.65 1.26 10 A/V

0.80 1.01 10 A/V

1.26 101.01 101 1.26 10

1 1.26 10 1.25 196

GS

m

m m

S

S S

V

g

g g

r

r r

− −

−

−−

−

−

=

= × − = ×

′ = = ×

×× =+ ×

+ × = ⇒ = Ω

b.

( )( )

( )( )

4 3

3

3

3

For 3 V

2 1.45 10 3 0.65 0.6815 10 A/V

0.6815 101 0.6815 10 196

0.60 10 A/V

Reduced by 12%

GS

m

m

m

V

g

g

g

− −

−

−

−

=

= × − = ×

×′ =+ ×

′ = ×

≈

7.53 a.

�

�

rS

gmVgs

RLRi

Vgs

IiI0

0 and m gs gs i i m gs SI g V V I R g V r= = − so 1

i igs

m S

I RVg r

=+

Then 0

1m i

ii m S

I g RAI g r

= =+

b. As an approximation, consider

g�mVgs

Ii Vgs Ri CgsT CM

I0

�

�

RL

In this case

( )0 1

1i m ii i gsT M

IA g RI sR C C

′= = ⋅+ +

where ( )1 and 1

mM gdT m L m

m s

gC C g R gg r

′ ′= + =+

c. As rS increases, CM decreases, so the bandwidth increases, but the current gain magnitude decreases. 7.54

( )

( ) ( )( )

2

1 2

225 10225 500

3.10 V2 2 1 3.10 22.207 /

GS DD

GS

m n GS TN

m

RV VR R

Vg K V Vg mA V

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠== − = −=

a.

�

�

gmVgsVgs CgsT

Ri

RD

CgdT

V0

R1��R2Vi��

b.

( ) ( ) ( )( )1 1 1 2.207 5

12 pFM gdT m D

M

C C g R

C

= + = +⎡ ⎤⎣ ⎦=

c. ( )( )

( )( )

( )

( )( )( )

1 2

1 2

3 12 8

8

1 2

1 2

1 500 225 1 155 0.9936 k

0.9936 10 5 12 10 1.69 10 s

1 1 9.42 MHz2 2 1.69 10

155 and 0.994 155 1

2.2 5 0.994 10.9

i gsT M

i

H H

m gs Dv gs i i i

i i

v v

r R R R C C

R R R

r

f fr

g V R R RA V V V V

V R R RA A

π π

− −

−

= +

= = = Ω

= × + × = ×

= = ⇒ =×

−= = ⋅ = ⋅ =

+ += − ⇒ = −

7.55

( ) ( )

( )( )

( )( )

1 2

2

1 2

0

33 22 13.2 k

225 5 2 V22 33

2 0.7 0.00261 mA13.2 121 40.3138120 0.026

9.94 k0.3138

0.3138 12.07 mA/V0.026100 318 k

0.3138

TH

TH

BQ

CQ

m

R R R

RVR R

I

I

r

g

r

π

= = = Ω

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠−= =

+=

= = Ω

= =

= = Ω

a.

( )

( )3

6

2

12.07 102 2 600 10

3.20 pF; 1 pF 2.20 pF

mT

m

T

gf

C C

gC C

f

C C C C

π μ

π μ

π μ μ π

π

π π

−

=+

×+ = =×

+ = = ⇒ =

( )( ) ( )( )

1

1 1 12.07 318 4 5

27.6 pF

M m o C L

M

C C g r R R

C

μ⎡ ⎤= +⎣ ⎦⎡ ⎤= +⎣ ⎦

=

b.

( )

( )( )

( )( )

1 2

3 12

8

8

0

1 2

1 2

1 2

33 22 2

1.74 9.94 k 1.48 k

1.48 10 2.20 27.6 10

4.41 10 s1 1 3.61 MHz

2 2 4.41 10

33 22 9.94 5.67 k

5.675.67

eq M

eq s

eq

H H

m o C L

iS

r R C C

R R R R r r

R

r

r

f fr

V g V r R R

R R rV V

R R r R

R R r

v

π

π π

π

ππ

π

π

π

π π

−

−

−

= +

= =

= Ω ⇒ = Ω

= × + ×

= ×

= = ⇒ =×

= −

⎛ ⎞= ⎜ ⎟⎜ ⎟+⎝ ⎠

= = Ω

=

( )( )( )0

0.7392

318 4 5 2.18 k

12.07 0.739 2.1819.7

i i

C L

v

v

x

V V

r R R

AA

⎛ ⎞ =⎜ ⎟+⎝ ⎠

= = Ω

= −= −

7.56

( )

( )( )

( )( )

( )

1 2

2

1 2

0

3

40 5 4.44 k

5 10 1.111 V5 40

1.111 0.7 0.00633 mA4.44 121 0.50.760 mA120 0.026

4.11 k0.760

0.760 29.23 mA/V0.026

2

29.23 102 2 25

TH

TH CC

BQ

CQ

m

mT

m

T

R R R

RV VR R

I

I

r

g

rgf

C C

gC Cf

π

π μ

π μ

π

π π

−

= = = Ω

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠−= =

+=

= = Ω

= =

= ∞

=+

×+ = = ( )60 10

18.6 pF; 3 pF 15.6 pFC C C Cπ μ μ π

×

+ = = ⇒ =

a. ( )

( )( )1

3 1 29.2 5 2.5 149 pF

M m C L

M M

C C g R R

C C

μ ⎡ ⎤= +⎣ ⎦⎡ ⎤= + ⇒ =⎣ ⎦

For upper frequency:

( )

( ) ( )

1 2

3 12

8

4.11 40 5 0.5

0.405 k

0.405 10 15.6 149 10

6.67 10 s1 2.39 MHz

2

H eq M

eq S

eq

H

H HH

r R C C

R r R R R

R

r

f fr

π

π

π

−

−

= +

= =

= Ω

= × + ×

= ×

= ⇒ =

For lower frequency:

( )( )

1

1 2

3 6 2

0.5 40 5 4.11

2.64 k

2.64 10 4.7 10 1.24 10 s

1 12.8 Hz2

L eq C

eq S

eq

L

L LL

r R C

R R R R r

R

r

f fr

π

π

− −

=

= + = +

= Ω

= × × = ×

= ⇒ =

b.

39.5

fHfL

�A��

f ( )

( )( )( )

0

1 2

1 2

2.135 0.81022.135 0.529.23 0.8102 5 2.5

39.5

m C L

iS

i i

V

V

V g V R R

R R rV V

R R r R

V V V

A

A

π

ππ

π

π

= −

⎛ ⎞= ⎜ ⎟⎜ ⎟+⎝ ⎠⎛ ⎞= =⎜ ⎟+⎝ ⎠

=

=

7.57

( )

( )( )( )

( ) ( )( )( )

( ) ( ) ( )

( )( )

( )( )

2

2

2

2

2

0 0

9

2 1.2 4 4 9

2.4 8.6 0.6 0

8.6 8.6 4 2.4 0.62 2.4

3.512 V2 2 2 3.512 26.049 /

2 3.512 2 4.572 mA1 1 21.9 k

0.01 4.56

SGD P SG TP

S

SG SG SG

SG SG

SG

SG

m P SG TP

m

D

o

VI K V VR

V V V

V V

V

Vg K V Vg mA V

I

r rIλ

−= + =

− + = −

− + =

± −=

== + = −=

= − =

= = ⇒ = Ω

a. ( )( )( ) ( )( )

1

1 1 6.04 21.9 1 6.785 pF

M gdT m o D

M M

C C g r R

C C

= +

⎡ ⎤= + ⇒ =⎣ ⎦

b. ( )( )( ) ( )

( )

( ) ( )

3 12

8

0

2 100 10 10 6.78 10

3.29 10 s1 4.84 MHz

2

100102

1006.04 21.9 1102

5.67

H i G gsT M

H

H

H HH

m o D gs

Ggs i i

G i

v

v

r R R C C

r

r

f f

V g r R V

RV V VR R

A

A

πτ

−

−

= +

= × + ×

= ×

= → =

= − ⋅

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎝ ⎠⎛ ⎞= − ⎜ ⎟⎝ ⎠

= −

7.58

( ) ( )

( )

( )( )( )

( )( )( )

( ) ( )( )

2

1 2

2

2

2

2220 10 20 1022 8

4.67 V10 4.67

5.33 1 0.5 4 4

0.5 3.33 0

1 1 4 0.5 3.333.77 V

2 0.52 2 1 3.77 23.54 /

G

G

SGD P SG TP

S

SG SG SG

SG SG

SG SG

m p SG TP

m

RVR R

VV

I K V VR

V V V

V V

V V

g K V Vg mA V

⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠=

− −= = +

− = − +

− − =

± += ⇒ =

= + = −=

b. ( )( )

( ) ( )( )1

3 1 3.54 2 5 18.2 pF

M gdT m D L

M M

C C g R R

C C

= +

⎡ ⎤= + ⇒ =⎣ ⎦

a. ( )

( )( )1 2

3 12

8

0.5 8 22 0.461 k

0.461 10 15 18.2 10

1.53 10 s1 10.4 MHz

2

eq gsT M

eq i

H H

r R C C

R R R R

r

f frπ

−

−

= +

= = = Ω

= × + ×

= ×

= ⇒ =

c. ( )

( )

( )( )( )

0

1 2

1 2

5.87 0.92155.87 0.5

3.54 0.9215 2 5 4.66

m gs D L

gs i i gs ii

v v

V g V R R

R RV V V V V

R R R

A A

= −

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ = ⇒ =⎜ ⎟⎜ ⎟ +⎝ ⎠⎝ ⎠= − ⇒ = −

7.59

( )

( )( )

1000.5 mA 0.5 0.495 mA101

0.495 19.0 mA/V0.026100 0.026

5.25 k0.495

E CQ

m

I I

g

rπ

⎛ ⎞= ⇒ = =⎜ ⎟⎝ ⎠

= =

= = Ω

a. Input: From Eq. 7.107b

3 12

10

15.25 0.5 0.05 10 10 10101

2.43 10 1 656 MHz

2

P E S

H H

rr R R C

s

f frp

ππ π

π ππ

β

π

−

−

⎡ ⎤= ⎢ ⎥+⎣ ⎦⎡ ⎤= × × ×⎢ ⎥⎣ ⎦

= ×

= ⇒ =

Output: From Eq. 7.108b ( ) ( ) 3 12

10

100 1 10 10

9.90 10 s1 161 MHz

2

P B L

H HP

r R R C

f fr

μ μ

μ μμπ

−

−

= = × ×

= ×

= ⇒ =

b. gmV�

RS

V0

Vi RE RB RL��

V� r�

�

�

( )

( )

( ) ( )( )

( )( )( )

0

0

0

1 1 1

1 1 1195.25 0.5 0.05 0.05

41.19 200.4856

19 0.4856 100 1

9.14

m B L

im

E S

im

E S S

i

i

i

i

v

V g V R RV VV Vg V

r R R

VV gr R R R

VV

V VV VVVA

π

ππ ππ

π

ππ

π

π

π

= −− −

+ + + =

⎡ ⎤ −+ + + =⎢ ⎥

⎣ ⎦−⎡ ⎤+ + + =⎢ ⎥⎣ ⎦

= −= −

= − −

=

c. ( ) ( )( )12 3

8

15 10 1 100 10

1.485 10 s1 10.7 MHz

2Since 3 freq. dominated by .

L L B

H L

r C R R

r

f fr

f f d B Cμ

π

−

−

= = × ×

= ×

= → =

< ⇒

7.60

( )

( )( )

20 0.7 1.93 mA10

100 1.93 1.91 mA1011.91 73.5 mA/V0.026100 0.026

1.36 k1.91

EQ

CQ

m

I

I

g

rπ

−= =

⎛ ⎞= =⎜ ⎟⎝ ⎠

= =

= = Ω

a. Input:

3 12

10

11.36 10 1 10 10 10101

1.327 10 s1 1.20 GHz

2

P E S

P

P PP

rr R R C

r

f fr

ππ π

π

π ππ

β

π

−

−

⎡ ⎤= ⋅⎢ ⎥+⎣ ⎦⎡ ⎤= × × ×⎢ ⎥⎣ ⎦

= ×

= ⇒ =

Output: ( ) ( ) 3 12

9

6.5 5 10 10

2.826 10 s1 56.3 MHz

2

P C L

P

P PP

r R R C

r

f fr

π μ

π

μ μππ

−

−

= = × ×

= ×

= → =

b. ( )

( )

( )( ) ( )

( )( )( )

0

0

0

1 1 1

1 1 173.51.36 10 1 1

75.34 0.0132773.5 0.01327 6.5 5

2.76

m C L

im

E S

im

E S S

i

i i

i

V g V R RV VV Vg V

r R R

VV gr R R R

VV

V V V VV V

Av

π

ππ ππ

π

ππ

π

π π

= −− −

+ + + =

⎛ ⎞+ + + = −⎜ ⎟

⎝ ⎠−⎡ ⎤+ + + =⎢ ⎥⎣ ⎦

= − ⇒ = −= − −

=

c. ( ) ( )( )12 3

8

.

15 10 6.5 5 10

4.24 10 s1 3.75 MHz

2Since , 3 frequency is dominated by

L L

L

r C R Rc

r

f fr

f fp dB Cπ

μ

−

−

= = × ×

= ×

= → =

<

7.61

( )

( )( )( )

( ) ( )( )( )

2

2

2

2

55

5 3 10 2 1

30 59 25 0

59 59 4 30 251.349V

2 30

GS D S

GSD n GS TN

S

GSGS GS

GS GS

GS GS

V I RVI K V V

R

V V V

V V

V V

+ =−

= = −

− = − +

− + =

± −= ⇒ =

( ) ( )( )2 2 3 1.35 12.093 /

m n GS TN

m

g K V Vg mA V

= − = −=

On the output: ( ) ( ) 3 12

9

5 4 10 4 10

8.89 10 s1 17.9 MHz

2

P D L gd

P

P PP

r R R C T

r

f fr

μ

μ

μ μμπ

−

−

= = × × ×

= ×

= → =

�

�

gmVgsRi

V0

Vi RS Vgs RD RL��

( )

( )

( )

( )( ) ( )

0

0

0

1 1

1 12.09310 2 2

0.1857

2.093 0.1857 5 4

0.864

m gs D L

i gsgsm gs

S S

igs m

S i i

igs

gs i

vi

v

V g V R R

V VVg V

R R

VV gR R R

VV

V VVAV

A

= −

− −+ + =

⎛ ⎞+ + = −⎜ ⎟

⎝ ⎠⎛ ⎞+ + = −⎜ ⎟⎝ ⎠=

= =

=

7.62

( )

( )( )( )( )

( ) ( )( )( )

( ) ( )( )

2

2

2

2

2

dc analysis

5 1 4 0.8

4 1.6 0.64

4 5.4 2.44 0

5.4 5.4 4 4 2.441.707

2 42 2 1 1.707 0.81.81 /

SGD P SG TP

S

SG SG

SG SG

SG SG

SG

m P SG TP

m

V VI K V VR

V V

V V

V V

V

g K V Vg mA V

+ −= = +

− = −

= − +

− − =

± += =

= + = −=

CgsT CgdT

�

�

gmVgsRi

RS Vgs RD RL

13 frequency due to :gsT eq S im

dB C R R Rg

⋅ =

( ) ( )12

12

1 4 0.5 0.246 k1.81

1 162 MHz2 246 4 10

Aeq gsT

eq

A

fR C

R

f

π

π −

=⋅

= = Ω

= =×

3 frequency due to gdTdB C−

( )

( ) 3 12

12

12 2 4 10 10

119 MHz

BD L gdT

fR R C

f

π

π −

=

=× ×

=

Midband gain

�

�

gmVgsRi

V0

Vi Vgs RS RD RL��

( )( )( )( )

0

1 1 41.81

1 1 4 0.51.81

0.492

0.492 1.81 4 2 1.19

Sm

gs i i

S im

i

m gs D L

v v

RgV V VR R

gV

V g V R R

A A

− −= ⋅ = ⋅

+ +

= −

= −

= ⇒ =

7.63

( )( )120 0.0263.059 k

1.0239.23 mA/Vm

r

g

π = = Ω

=

a.

( )

( ) ( )( )

( )

( )( )

( )

2 3

12 9

9

3 12

9

9

1Input: 2

2

0.1 20.5 28.3 3.06 0.096 k

96 12 2 2 10 1.537 10 s

1 103.6 MHz2 1.536 10

1Output: 2

15 10 10 2 10

6.67 101 23.9 MHz

2 6.67 10

H

s

eq

H

H

C L

H

fr

r R R R r C C

R

r

f

fr

r R R C

f

ππ

π π π μ

π

π

μμ

μ μ

μ

π

π

π

π

− −

−

−

−

−

=

⎡ ⎤= +⎣ ⎦

= = Ω

= + × = ×

= =×

=

=

= × × ×

= ×

= =×

b.

( )

( )( )

2 3

2 3

2 3 20.5 28.3 3.059 2.433 k

2.43339.23 5 10 125.62.433 0.1

m C LS

R R rA g R R

R R r R

R R r

A A

π

π

π

⎡ ⎤= ⎢ ⎥+⎣ ⎦

= = Ω

⎡ ⎤= ⇒ =⎢ ⎥+⎣ ⎦

c. 15 pFL LC C Cμ= > ⇒ dominates frequency response.