Chapter 10 Exercise Solutions EX10.1

( )1

0

0

on 10 0.715

0.62 mA

0.622 21 1

750.604 mA

BEREF

REF

REF

V VI

RI

II

Iβ

+ − −= =

=

= =+ +

=

EX10.2

( ) ( )

( )( )

( )( )

1

0

20 0 2 0

0 0

00

on 5 0.7 512

0.775 mA0.775 0.7549 mA

2 21 175

10.02 0.7549 0.0151 mA and

4 265 kΩ 265 0.7549 200 V0.0151

BEREF

REF

REF

CECE

AA A

V V VI

RI

II

VI I V rr I

Vr V VI

β

+ −− − − − −= =

=

= = =+ +

ΔΔ = = Δ = Δ ⇒ =

Δ

= = = ⇒ = ⇒ ≅

EX10.3

( ) ( )

( ) ( )

1

0

0 1

01 2 1 2

3 1 2 3

33 3

2 on 9 2 0.712

0.6333 mA

0.6333 0.6331 mA2 21 1

1 75 760.6331 mA

8.44 A

16.88 A

0.222 A1

BEREF

REF

REF

C

B B B B

E B B E

EB B

V VI

RI

II

I I

II I I I

I I I I

II I

β β

μβ

μ

μβ

+ − −= =

=

= = =+ +

+= =

= = ⇒ = =

= + ⇒ =

= ⇒ =+

EX10.4

( )( )

00

0 0

1 1

1 2 0 1 2

ln

0.026 0.75ln ln 3.54 kΩ0.025 0.025

5 0.7 5.73 kΩ0.75

0.025 3.54 88.5 mV

REFE T

T REFE E

BE BE E BE BE

II R VI

V IR RI I

R R

V V I R V V

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞ ⎛ ⎞= = ⇒ =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠−= ⇒ =

− = = ⇒ − =

EX10.5

( )

( ) ( )

00

0 00

5 0.7 50.775 mA

12

ln

0.7756 0.026 ln 16.6 A

REF REF

REFE T

I I

II R VI

I II

μ

− − −= ⇒ =

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞

= ⇒ ≅⎜ ⎟⎝ ⎠

EX10.6

( )( )

( )

00

02 2

20

020

2

0 02 2

ln

0.026 0.70ln 3.465 kΩ0.025 0.025

0.025 0.9615 mA/V0.026

150 0.026156 kΩ

0.025100 4000 kΩ

0.025|| 3.47 ||156 3.39 kΩ1 400

REFE T

E E

m mT

T

A

E E

m E

II R VI

R R

Ig gV

VrI

VrI

R R rR r g R

π

π

β

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞= ⇒ =⎜ ⎟⎝ ⎠

= = ⇒ =

= = =

= = =

′ = = =′= + = ( )( )

0

0 2 00

0 1 0.962 3.39

17.04 MΩ1 3 0.176 A

17,040C

R

dI dV dIR

μ

+⎡ ⎤⎣ ⎦=

= ⋅ = ⇒ =

EX10.7

( )

1

0101 02 0 1 2

0101 01

...

... and ...

11 1

REF R BR B BN

R N BR B B BN

REF

I I I I II

I I I I I I I I

I NI I N I

β

β β

= + + + +

= = = = = = = = =

⎛ ⎞ ⎛ ⎞+= + + = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( )

( )

01 02 0

01

So ...11

10.9011

501 11

50 0.911 1 50

0.91 1 50 1

0.94.55 4

REFN

REF

II I IN

INI

N

N

N

N N

β

= = = =++

= =++

++ =

⎛ ⎞+ = −⎜ ⎟⎝ ⎠

⎛ ⎞= − −⎜ ⎟⎝ ⎠

= ⇒ =

EX10.8

( )

( ) ( )

( ) ( )

2 2 2

2 22 2 2

2

2

2 2

1 2 3

2

0.20 0.020 3 2 10

DS GS TN GS GS

n oxO n GS TN GS TN

V sat V V V V V VC WI K V V V V

LW WL L

μ= = − = − ⇒ =

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

( )

( ) ( )

21

1

1 2

2

1 1

2

0.5 0.020 3 2 25

n oxREF GS TN

GS GS

C WI V VL

V VW WL L

μ⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠=

⎛ ⎞ ⎛ ⎞= − ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( )

( ) ( )

3 1

23

3

2

3 3

10 3 7

2

0.5 0.020 7 2 1

GS GS

n oxREF GS TN

V V V VC WI V V

LW WL L

μ

+= − = − =

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

EX10.9 a. ( )

( )

2

20.020 0.080 11.5 V all transistors

REF n GS TN

GS

GS

I K V V

VV

= −

= −=

b.

( ) ( ) ( )( ) ( )

4 3 1

4 4 4

4 4 4 4 4

4 4

1.5 1.5 5 2 V2 1.5 3.5 V

min sat and 1.5 1 0.5 So min 3.5 0.5 min 3.0 V

G GS GS

S G GS

D S DS DS GS TN

D D

V V V VV V V

V V V V sat V V VV V

−= + + = + − = −= − = − − = −= + = − = − == − + ⇒ = −

c. ( )

( )( )( ) ( )( )

( )( )( )

0 04 02 04

02 040

0 0

11 1 2500 kΩ

0.02 0.0202 2 0.080 1.5 1 0.080 /2500 2500 1 0.080 2500 505 MΩ

m

m n GS TN m

R r r g r

r rI

g K V V g mA VR R

λ

= + +

= = = =

= − = − ⇒ == + + ⇒ =

EX10.10

( ) ( ) ( ) ( )2For : min 2 V min min 5 2 5 min 3 VDS P S DS SQ v V V v V= = ⇒ = − = − ⇒ = −

( ) ( ) ( )( )

( ) ( )

0 2 2 0

2

10 1

1

21

11

1

1 0.5 1 0.15 2 0.65 mA

1

0.65 0.80 12

0.0986 0.197 V2

0.197 3 min 3.2 V

DSS DS

GSDSS

P

GS

GSGS

GS I S I I

I I v I

vI IV

v

v v

v V V V V

λ= + = + ⇒ =

⎛ ⎞= −⎜ ⎟

⎝ ⎠

⎛ ⎞= −⎜ ⎟−⎝ ⎠

= ⇒ = −−

= − − = − − ⇒ = −

2 10, = = −gs gs XV V V

11

02 01

X XX m X

V V VI g Vr r

−= + + (1)

1 11

01

Xm X

D

V V V g VR r

−+ = (2)

101

1

01

1

1 1

X m

D

V gr

V

R r

⎛ ⎞+⎜ ⎟

⎝ ⎠=+

101 01

10 02 01

01

011

02 01

01

102 01

01

1 11 1 1

1 1

11 1 1

1 1

11 1

1 1

mX

mX

D

m

D

Dm

D

gr rI g

V R r rR r

rgr r

R r

Rgr r

R r

⎛ ⎞+⎜ ⎟

⎝ ⎠= = + + −+

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

01 10 02 01

1 1 1For D mR r gR r r

⎛ ⎞⇒ ≅ + +⎜ ⎟

⎝ ⎠

For Q1: ( )1 1

1

1

2 0.82 0.1971 12 2

0.721 mA/V

DSS GSm

P P

m

I Vg

V Vg

⎛ ⎞ −⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠=

( )( )00

00

1 1 10.3 kΩ0.15 0.65

1 1 1 0.721 0.915 1.09 kΩ10.3 10.3

rI

RR

λ= = =

= + + = ⇒ =

EX10.11

a.

( )

2

3

2 212

exp

0.5 10ln 0.026 ln 0.521 V10

EBREF S

T

REFEB T EB

S

VI IV

IV V VI

−

−

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞ ⎛ ⎞×= = ⇒ =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

b. 1 15 0.521 8.96 kΩ

0.5R R−= ⇒ =

c. Combining Equations (10.79), (10.80), and (10.81), we find

( )

2

02

12 3

12 4

1exp 1

1

2.512.5 10010 exp 1 0.5 10

0.521100 1100

1.025 10 exp 5.098 10 exp 4

EC

APCEoIS REF

T AN EB

AP

I

T

I I

T T

VVVVI I

V V VV

VV

V VV V

− −

− −

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

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

⎛ ⎞ ⎛ ⎞× = × =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠8.974 10 0.521 VIV× ⇒ =

d.

1 138.460.026 1923

1 1 1 1 0.01 0.01100 100

Tv v

AN AP

VA A

V V

⎛ ⎞−⎜ ⎟ − −⎝ ⎠= = = ⇒ = −

++ +

EX10.12

0 02

0.8 30.77 mA/V0.026

80 100 kΩ0.8

CQm

T

A

CQ

Ig

VVr rI

= = =

= = = =

a. ( )( ) ( )[ ]

0 1 0 02 1

0 02

|| ,|| 30.77 100 ||100 1538

m i

v m v

V g V r r V VA g r r A

π π= − == − = − ⇒ = −

b. ( )

( )( ) ( )0 02|| ||

1540 770 770 30.77 50 || 50 || 25 50 kΩ2

v m L

v L L L

A g r r R

A R R R

= −

= − = − − = − ⇒ = ⇒ =

EX10.13 (a) Neglecting effect of λ and RL

( )( )

2

20.40 0.25 1

Then 2.265 V

O REF n IQ TN

IQ

IQ

I I K V V

V

V

= = −

= −

=

b. ( )( )

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

0 020

0 02

1 1 125 kΩ0.02 0.4

2 2 0.25 2.26 1 0.632 mA/V

|| 0.632 125 ||125 39.5m n IQ TN

v m v

r rI

g K V V

A g r r A

λ= = = =

= − = − =

= − = − ⇒ = −

c. ( ) ( )( )0 0239.4|| || 0.632 62.5 || 62.5 || 31.25 62.5 kΩ

2v m L L L LA g r r R R R R= − − = − ⇒ = ⇒ =

TYU10.1

( )

( )

( )

0

0

1

1

1

For 0.75 mA

2 21 0.75 1100

0.765 mA

on

5 0.7 50.765

12.2 k

REF

REF

BEREF

I

I I

I

V V VI

R

R

R

β

+ −

=

⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

=

− −=

− − −=

= Ω

TYU10.2

( )( )

0

0 00

0 2 00

10 0.7 20.717 mA

120.717100 139.5 kΩ

0.7171 4 0.0287 mA

139.5

REF

REF

A

CE

I

I IVr rI

I V Ir

−= =

≈ =

= = ⇒ =

Δ = Δ = ⇒ Δ =

TYU10.3

( ) ( )

0 0

03 3

3 3 3

1 0.50 0.4996 mA2 21 1

1 50 51

9.99 A

1 0.5096 mA

REF

B B

E C E

I I I

II I

I I I

β β

μβ

ββ

= ⋅ = ⇒ =⎛ ⎞ ⎛ ⎞

+ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠

= ⇒ =

⎛ ⎞+= = =⎜ ⎟⎝ ⎠

32 2 1

21 2 1 2

0.5096 0.490 mA22 11

50

9.80 A

EC C C

CB B B B

II I I

II I I I

β

μβ

= = ⇒ = =⎛ ⎞ ⎛ ⎞++ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

= = ⇒ = =

TYU10.4

( ) ( )

( ) ( )1

2

2 21 31 1 1 3

1 3

1 1 1

52 2

40 17.3 0.98 3.98 1.814 V38 2.70

n nREF GS TN GS TN

GS GS GS

k kW WI V V V VL L

V V V

′ ′⎛ ⎞ ⎛ ⎞= − = − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞⎛ ⎞ − = − ⇒ =⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

( )( )20.040 17.3 1.814 0.982

0.241 mA

REF

REF

I

I

⎛ ⎞= −⎜ ⎟⎝ ⎠

=

( )

( )( )

221 2

2

2

20.042 6.92 1.814 1.0

20.0963 mA

nO GS TN

O

k WI V VL

I

′ ⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

=

TYU10.5 a. From Equation (10.52),

( )

( ) ( )

( )( )[ ] ( )( )

1

1

1 12

3 3112 1210 1.8

3 31 112 12

0.5 1 0.510 1.81 0.5 1 0.5

3.93 V also 3.93 V

12 0.020 3.93 1.8 1 0.01 3.93 1.13 mA

GS

GS

GS DS

REF REF

V

V

V V

I I

⎛ ⎞−⎜ ⎟

⎜ ⎟= × + ×⎜ ⎟

+ +⎜ ⎟⎝ ⎠

−⎛ ⎞ ⎛ ⎞= + ×⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠= =

= − + ⇒ =⎡ ⎤⎣ ⎦

b. ( )( )

( )( )

( ) ( )( )( )( )

220

11

0 0

/ 1/ 1

1 0.01 261.13 0.555 mA12 1 0.01 3.93

DSREF

DS

W L VI I

W L V

I I

λλ

+= × ×

+

+⎡ ⎤⎛ ⎞ ⎣ ⎦= × × ⇒ =⎜ ⎟ +⎡ ⎤⎝ ⎠ ⎣ ⎦

c. 2For 6 V 0.576 mADS oV I= ⇒ = TYU10.6

( ) ( )2 21 1 3 3n GS TN n GS TNK V V K V V− = −

( )

( )( )

( )( ) ( )( )

1 3

1 3

3 1

1 1

1 1

0.102 20.25

2 0.6325 210

2 0.6325 10 0.6325 21.6325 7.06 4.325 V

GS GS

GS GS

GS GS

GS GS

GS GS

V V

V VV V

V VV V

⎛ ⎞− = −⎜ ⎟⎜ ⎟

⎝ ⎠− = −

= −− = − −

= ⇒ =

( ) ( ) ( )2 21 1 0.25 4.325 2 1.35 REF n GS TN REFI K V V I mA= − = − ⇒ =

( ) ( )( )2 22 13 3 0.25 4.325 2

3 4.05 O n GS TN

O REF O

I K V VI I I mA

= − = −= ⇒ =

TYU10.7

( ) ( )

( ) ( )

( )( )

2 21 1 1 1 2

2 22 2

23 3

23 3

0.20 0.15 1 2.15

0.15 2.15 1 0.10 2

0.10 0.15 1 1.82

REF n GS TN GS GS GS

O n GS TN O

O n GS TN

GS GS

I K V V V V V V

I K V V I mA

I K V V

V V V

= = − = − ⇒ = =

= − = − ⇒ =

= −

= − ⇒ =

TYU10.8

( )( )

0

2

2

All transistors are identical250 A

0.25 0.20 1 2.12 V

REF

REF n GS TN

GS GS

I I

I K V V

V V

μ⇒ = =

= −

= − ⇒ =

TYU10.9

( )1 1 1For : 1D DSS DSQ i I vλ= +

( )2

22 2 2For : 1 1GS

D DSS DSP

vQ i I vV

λ⎛ ⎞

= − +⎜ ⎟⎝ ⎠

2 1 2 1 and GS DS DS DS DSv v v V v= − = −

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

21

1 1 2 1

1 22

11 1

21 1 1 1

So

1 1 1

1 0.1 1 1 0.1 3 0.12

1 0.1 1 0.25 1.3 0.1

DSDSS DS DSS DS DS

P

DSS DSS

DSDS DS

DS DS DS DS

vI v I V v

VI I

vv v

v v v v

λ λ⎡ − ⎤

+ = − + −⎡ ⎤⎢ ⎥ ⎣ ⎦⎣ ⎦

=

⎡ ⎤+ = − + −⎡ ⎤ ⎡ ⎤⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

+ = − + −

3 21 1 1This becomes 0.025 0.425 1.5 0.3 0DS DS DSv v v− + − =

1 2 2We find 0.2127 V, 2.787 V, 0.2127 VDS DS GSv v v= = = −

( ) ( )( )

( )( )

( )( )( )( )

1 1

0 02 01 2 02

2

02 04

0 0

1 2 1 0.1 0.2127

2.04 mA

12 22 0.21271 1

2 21.787 mA/V

1 1 5 kΩ0.1 2

5 5 1 1.787 5 54.7 kΩ

D DSS DS

D

m

DSS GSm

P P

m

DSS

i I v

i

R r r g r

I vgV V

g

r rI

R R

λ

λ

= + = +⎡ ⎤⎣ ⎦=

= + +

⎛ ⎞ −⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠=

= = = =

= + + ⇒ =⎡ ⎤⎣ ⎦

TYU10.10

a. ( )3

2 214

0.1 100.026 ln 0.557 V5 10EB EBV V

−

−

⎛ ⎞×= ⇒ =⎜ ⎟×⎝ ⎠

b. 1 15 0.557 44.4 kΩ

0.1R R−= ⇒ =

c.

( )

( )

2

00

2

14 3

14 4

1exp 1

1

2.512.5 1005 10 exp 1 0.1 100.557100 1100

5.125 10 exp 1.019 10

exp

EC

CEI APS REF

EBT AN

AP

I

T

I

T

I

T

VVV VI I

VV VV

VV

VV

VV

− −

− −

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

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

⎝ ⎠⎛ ⎞

× = ×⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

91.988 10 0.557 VIV= × ⇒ =

d.

10.026 1923

1 1100 100

v vA A−

= ⇒ = −+

TYU10.11 a. ( )

( )

21

20.25 0.20 1 2.12 VREF p SG TP

SG SG

I K V V

V V

= +

= − ⇒ =

b. From Equation (10.89) ( ) ( )

( )( )( ) ( )( )

( )( )

2

2

2

1

1 0.015 10 2.12 0.2 15

0.030 0.25 0.030

0.15 1.12 0.8 1 2.10 V

P SG n I TNDSO o

n P REF n P

I

I I

V V K V VV V

I

V

V V

λ

λ λ λ λ

+⎡ ⎤+ − −⎣ ⎦= = −+ +

+ − −= −

= − − ⇒ =

c. ( )( )

( )( )( )

2

2 0.2 2.10 1.058.7

0.25 0.030

n I TNv

REF n P

v v

K V VA

I

A A

λ λ− −

=+

−= − ⇒ = −

TYU10.12 (a) ( )

( )

21

280 50 1 2.26 REF p SG TP

SG SG

I K V V

V V V

= +

= − ⇒ =

(b) ( ) ( )

( )( )( ) ( )( )

( )( )( )

2

2

2

1

1 0.015 10 2.26 50 15

0.030 80 0.030

20.83 1 32.2 2.243 V

p SG n I TNDSo o

n p REF n p

I

I I

V V K V VV V

I

V

V V

λ

λ λ λ λ

+⎡ ⎤+ − −⎣ ⎦= = −+ +

+ −⎡ ⎤ −⎣ ⎦= −

− = ⇒ =

(c) ( )( )

( )( )( )( )

2 2 50 2.243 151.8

80 0.030n I TN

v vREF n p

K V VA A

I λ λ− − − −

= = ⇒ = −+

TYU10.13 a.

0

0 0

02 02

0.5 19.2 mA/V0.026120 240 kΩ0.5

80 160 kΩ0.5

Cm m

T

AN

CQ

AP

CQ

Ig g

VV

r rI

Vr rI

= = ⇒ =

= = ⇒ =

= = ⇒ =

b. ( ) ( )[ ]0 02|| || 19.2 240 ||160 || 50 631v m L vA g r r R A= − = − ⇒ = − TYU10.14

( )( )1 2

1 2

1

11mA, 38.46 mA/V0.026

100 0.0262.6 K

180 80 K1

120 120 K1

12.6 80 0.0257 K38.46

C m

O O

O

O

I g

r r

r r

r

R

π π

= = =

= = =

= = =

= =

= =

( ) ( )[ ]

( )( )

1

1 1

2

For 9.3 K

9.3 0.0257 1 0.924 K

1 2.6 0.924 0.779 K

80 1 38.46 0.779 2476.7 K

O E

E

O

R

R R R R

R

R

=

′ = + = + =

′′ = + =

= + =⎡ ⎤⎣ ⎦

( ) ( )( ) ( )( )2|| 38.46 120 || 2476.7 38.46 114.54404

v m O O

v

A g r RA

= − = − = −= −

For 100 K38.46 114.5 100 2053

L

v

RA

== − ⎡ ⎤ = −⎣ ⎦

[ ]For 10 K

38.46 114.5 ||10 354L

v

RA

== − = −

TYU10.15

1 2 and identical o REFM M I I⇒ = a.

( )( )

( ) ( )( )

( )( )

( )( )

2

2

0 00

0 00

0.25 0.2 12.12 V2 2 0.2 2.12 1 0.447 mA/V

1 1 400 kΩ0.01 0.25

1 1 200 kΩ0.02 0.25

O n I YN

I

I

m n I TN m

n nn

p pp

I K V V

VVg K V V g

r rI

r rI

λ

λ

= −

= −== − = − ⇒ =

= = ⇒ =

= = ⇒ =

b. ( )( )[ ]

0 02|| ||0.447 400 || 200 ||100 25.5

v m L

v v

A g r r RA A

= −= − ⇒ = −