Chapter 11(Analog)

8/10/2019 Chapter 11(Analog)

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Chapter 11 Filters

Section 11.1 The Basic Principles of FiltersA filter allows only some signals to go through. A low pass filter only allows signals

with low frequencies to go through and a high pass filter only allows high frequency

signals to go through. A band pass filter is a circuit which allows signals whose

frequencies are within a certain range to go through.

Fig. 11.1 1 shows a low pass filter.

Fig. 11.1 1 An RC low pass filter

!n this circuit" suppose that the frequency of the input signal is high. The capacitor

will be almost short circuited. Thus #out v . This means that no high frequency signals

can go through. $n the other hand" if the frequency of the input is low" the capacitor is

almost open circuited. Thus" inout vv and the circuit is a low pass filter.

Fig. 11.1 % shows a high pass filter.

Fig. 11.1 % An RC high pass filter

!n this circuit" when the frequency of the input signal is low" the capacitor is almost

11 1


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open circuited and thus a &ery small current will flow. Thus" #= iRvout . $n the other

hand" if the frequency is high" the capacitor will be almost short circuited and inout vv .

Thus" this is a high pass filter.

Fig. 11.1 ' shows a band pass filter. !t is a combination of a high pass filer and a

low pass filter. Since neither low nor high frequency signals can pass through" this is a

band pass filter.

vin vout

R C

C R

Fig. 11.1 ' An RC band pass filter

Filters can also be designed by using inductors" capacitors and resistors. Aninductor is short circuited when the frequency is low and open circuited when the

frequency is high. The reader can thus easily see that the circuit in Fig. 11.1 ( is a low

pass filter" that in Fig. 11.1 ( is a high pass filter and that in Fig. 11.1 ) is a band pass

filter.

11 %


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vin

C

L

R

vout

Fig. 11.1 ( An LCR low pass filter

vin

C

L

R

vout

Fig. 11.1 ) An LCR high pass filter

11 '


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vin

C

L

R vout

Fig. 11.1 * An LCR band pass filter

Section 11.% The Transfer Functions of Some

Filters!n the abo&e section" we briefly introduced the basic concept of filters. +e now discuss

the transfer functions of filters which describe the relationships between out v and inv in

terms of frequency.

,et us redraw the low pass filter in Fig. 11.1 1 again here as in Fig. 11.% 1.

Fig. 11.% 1 The redrawing of an RC low pass filter

+e shall denote f j %= by S . The transfer function of the filer is as follows-

11 (


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SRC

SC R

SC vv

S Ain

out

+=

+==

11

1

1

/

/11.1 1

%%%1

1

11

/

C R

RC j j A

+=

+= /11.% %

From 0quation /11.% % " we can see why this is a low pass filter. As#./" = j A " which means that high frequency signals cannot go through. !t is

customary to denote RC

1# = and RC f %

1# = . c f is called the critical frequency in

this circuit. ote that

1#/ma2 === A A

and ma2# %1

/ A A ==

That is" when # = " the gain of the filter is reduced to %1

of its ma2imum &alue. +e

also say that it is reduced to its '3B &alue.

Fig. 11.1 % shows the transfer function of a low pass filter.

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Fig. 11.% %

,et us redraw the high pass filter in Fig. 11.1 % as in Fig. 11.% '.

Fig. 11.% ' The redrawing of an RC high pass filter

!t is easy to deri&e the transfer function of this high pass filter to be as follows-

SRC SRC

S A += 1/ /11.% '

and%%%

%%% 11

1

1/C R

C R

RC

j A

+=+= /11.% (

From 0quation /11.% ( " we can see that as #/"# == j A " and

ma2/1/ j A A === . Thus" this circuit is a high pass filter. +e may again let

11 *


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RC 1

# = and RC f

%1

# = . +hen # = " the gain of the filter is reduced to %1

of its

ma2imum &alue.

The transfer function of this high pass filter is now illustrated as in Fig. 11.% (.

Fig. 11.% ( The transfer of the RC low pass filter /0quation 11.% (

,et us now consider the band pass filter in Fig. 11.1 '. +e redraw Fig. 11.1 ' as in

Fig. 11.% ).

vin vout

R C

C R Z 2

Z 1 4

411 5


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Fig. 11.% ) 1 Z and % Z in the RC band pass filter

SC

SRC

SC R Z

SRC

R

SC R

SC R

Z

Z Z

Z

v

v

in

out

+=+=

+=

+=

+=

11

11

1

%

1

%1

1

Thus"

%%%

%%%

1'

1'

11

1/

C R RC

S S

RC

S SRC C RS

SRC SRC R

SC SRC

SRC

R

S A

++=

++=

+++

+=

/11.% )

,et RC 1

# = . Then

%##

%#

'/

++=

S S

S S A /11.% *

and %#

%%%%#

#

6//

+= j A /11.% 5

!t can be easily shown that in this case" the gain is ma2imi7ed when # = " as shown in

Fig. 11.% *. Perhaps it should be noted that the term # has different meanings for

different filters. For both low pass and high pass filters" when # = " the gain is

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reduced to%

1 of its ma2imum &alue while in the band pass case" when # = " the

gain is ma2imi7ed.

A

#

Fig. 11.% * The transfer of the RC band pass filter /S9$:,3 B0 A/;+

!n summary" we ha&e the following three transfer functions-

1. Low Pass -

RC S

RC S A1

1

/+

= /11.% 8

%. High Pass - RC

S

S S A

1/

+=

/11.% 6

'. Band Pass -%%

% 1'/

C R RC

S S

RC

S

S A++

= /11.% 1#

By letting RC

1# = " we ha&e-

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1. Low Pass: #

#/

+=

S S A /11.% 11

%. High Pass: #/

+= S S

S A /11.% 1%

'. Band Pass: %##

%#

'/

++=

S S

S S A /11.% 1'

Finally" we ha&e-

1. Low Pass: %

#

%1

1/

+= j A

/11.% 1(

%. High Pass -%

%#1

1/

+= j A

/11.% 1)

'. Band Pass - %#

%%%%#

#

6//

+= j A /11.% 1*

Section 11.' An RLC Filter !n the abo&e section" we introduced low pass" high and band pass filters. !n the

following" we shall introduce a circuit containing resistor" capacitor and inductance" as

shown in Fig. 11.' 1.

11 1#


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vin

C

L

R

Fig. 11.' 1 An RLC circuit

By choosing the output terminals in different ways" this circuit can be used as low

pass" high pass and band pass filter as shown in Fig. 11.' %.

Fig. 11.' % Three filters out of the RLC circuit

,et us see why the circuit will beha&e differently when different output terminals are

11 11


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chosen.

Case 1- The capacitor is chosen as the output. Then the output &oltage will be almost #

when the frequency is large and will be equal to the input &oltage as the frequency is low.

Thus this is a low pass filter.

Case %- The inductor is chosen as the output. Since the beha&ior of an inductor is


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LC L R

S S

LC SRC LC S R

SC SL

SC vv

in

out

1

1

11

1

1

%%

++=++=++

=

Case 2: The High Pass Filter

vin

C

L

R

vout

Case 3: The Band Pass Filter

11 1'

vin

C

L

Rvout


14/56

LC L R

S S

L R

S

SRC LC S SRC

RSC

SL

Rvv

in

out

111 %% ++=++=++

=

!n Summary" we ha&e the following three transfer functions-

1. Low Pass:

LC L

RS S

LC v

v

in

out

1

1

% ++= /11.' 1

%. High Pass: LC L

RS S

S

v

v

in

out

1%

%

++= /11.' %

'. Band Pass -

LC L

RS S

L

RS

v

v

in

out

1% ++= /11.' '

,et LC

1# = and

C L

RQ

1= . Then Q L R # = . Thus" we ha&e the following three

transfer functions-

1. Low Pass - %#

#%

%#/

++==

QS S v

vS A

in

out

/11.' (

%. High Pass - %#

#%

%

/

++

==

QS S

S

v

vS A

in

out

/11.' )

'. Band Pass -%

##%

#

/

++==

QS S

QS

v

vS A

in

out /11.' *

Finally" we ha&e the following-

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1. Low Pass -%

%#

%%%%

#

%#

/

/

Q

j A

+=

/11.' 5

%. High Pass -%

%#

%%%%

#

%

/

/

Q

j A

+=

/11.' 8

'. Band Pass -

%

%#

%%%%

#

#

/

/

Q

Q j A

+= /11.' 6

The physical meaning of Q is now gi&en- Q is defined as follows-

B A

Q ==cycle perlostenergy

circuit,C= aninstoredenergyma2imum% /11.' 1#

%

%1

m LI A = /11.' 11

%%1

%1 %%

mm RI T RI B == /11.' 1%

Thus" we ha&e

R L

RI

LI Q

m

m

==

%%1

%1

%%

%

/11.' 1'

Section 11.( The Significance of the Second

$rder Transfer Functions-

!n Section 11.%" we ha&e three transfer functions" e2pressed in 0quations /11.% 11 "

/11.% 1% and /11.% 1' . Both /11.% 11 and /11.% 1% are first order transfer functions

11 1)


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while /11.% 1' is a second order transfer functions. The transfer functions we obtained

in Section 11.' are all second order transfer functions" as shown in 0quations /11.' ( to

/11.( * . !n this section" we shall e2plain why we need second order transfer functions.

,et us consider the low pass filter as an e2ample. 0quation /11.% 11 is a first order

transfer function for low pass filters. +e rewrite its magnitude e2pressed in 0quation

/11.% 1( as follows-

%#

%

1

1/

+= j A

/11.( 1

Fig. 11.( 1 is a plot of the abo&e function.

|A/ j? |

1

#.5#5

#

Fig. 11.( 1 The transfer function of an RC low pass filter /0quation 11.( 1

0quation /11.' ( e2presses a second order transfer function of a low pass filter. !ts

magnitude function" e2pressed in 0quation /11.' 5 " is now rewritten as follows-

11 1*


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( )

%

%#

%%%%

#

%#

/Q

j A

+=

/11.( %

From /11.( % " we can pro&e the following-

Case 1 -%

1#


18/56

The proof of the abo&e equations can be easily obtained and will not be gi&en in this

boo>. !n Fig. 11.( %" we plot the second order transfer of the low pass filter e2pressed in

0quation /11.( % for different ?@s as follows-

Fig. 11.( % 0quation 11.( % for different Q@s

!f we compare the second order transfer function for low pass filter" as e2pressed in

0quation /11.( % with the corresponding first order transfer function" we can easily see

the difference between these two transfers. The second order transfer function pro&ides

an additional parameter to control the ma2imum magnitude and the sharpness of the

transfer function of the low pass filter. For the first order low pass filter" only # can be

used to control the bandwidth. For the second order transfer function" Q plays a critical

role. As discussed abo&e" a &ery small ? gi&es a rather narrow bandwidth and we usually

assume that%

1>Q .

11 18


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+e ha&e seen the significance of Q . !t is now appropriate to e2plain the physical

meaning of Q . For this =,C circuit"

C

L

RQ

1= /11.( 8

ote that the circuit is a series connection of resistor" inductance and capacitor. !f the

circuit only contains inductance and capacitor" it will cause resonance. The e2istence of

the resistor will dampen the oscillation caused by the resonance. The smaller the

resistance is" the more the circuit will tend to oscillate. !f the resistance is &ery large" the

circuit has a small tendency to oscillate. But" as seen in 0quation /11.( * " a smallresistance corresponds to a large Q. Thus a high Q means that # is closer to ma2 .

This will be made clearer when we e2amine the band pass filter.

As for the high pass filter" the situation is the same as that of the low pass filter. +e

shall now discuss the band pass filter. ,et us rewrite the transfer function of the =,C

band pass filter" e2pressed in 0quation /11.' * " as below-

%#

#%

#

/

++==

QS S

QS

v

vS A

in

out /11.( 6

!ts magnitude function" as e2pressed in 0quation /11.' 6 " is as follows-

%

%#

%%%%

#

#

/

/

Q

Q j A

+= /11.( 1#

!n this case" we can easily pro&e that

11 16


20/56

#ma2 = /11.( 11

and 1ma2 = A /11.( 1%

+e would li>e to >now the frequencies where A drops to %1 of its ma2imum

&alue. To do this" we ha&e to sol&e the following equation-

( ) %%

#%

%

%#

%%%%

# % QQ

=+

Thus" we ha&e to sol&e two equations-

#%##% =+

Q /11.( 1'

#%##% = Q /11.( 1(

There are four solutions for the abo&e two equations. The following two satisfy the

condition that they ha&e to be positi&e-

%##

%(

11

% QQ++=

/11.( 1)

%##

1 (1

1% QQ

++= /11.( 1*

Q#

1% = /11.( 15

The abo&e discussion is illustrated in Fig. 11.( '.

11 %#


21/56

Fig. 11.( ' 0quation /11.( 1# for different Q@s

From the abo&e discussion" we can see that the larger Q" the smaller the bandwidth.

+e may conclude that the second order transfer function gi&es us more fle2ibility to

design a filter than the first order transfer function.

Section 11.) 02periments with the ,C= Filter ALL FIGUR ! " # T$ B LAB L #

%&eri'ent 11.()1 The Low Pass Filter

The filter circuit is as shown in Fig. 11.) 1.

11 %1


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L =40mH

C =25nF

R=1k

vin

1

2

3

vout

Fig. 11.) 1 The LCR low pass filter circuit for 02periment 11.) 1

The critical frequency )1#%)1#(#%

1%

1% 6'

##

===

LC f 97. The

program is shown in Table 11.) 1 and the gain &s frequency cur&e is shown in Fig. 11.)

%.

Table 11.) 1 Program for 02periment 11.) 1

,C=

11 %%


23/56

.P=$T0CT

.$PT!$ P$ST

.,!B c- mm#'))&.l TT

.: P=$T0CT

.op

, 1 % (#m9

C ' # %)nF

= % ' 1>

Din 1 # AC 1

.AC 30C 1## 1 1####>

.P,$T AC D3B/'

.0 3

!n the abo&e" there is an instruction as follows-

AC 30C 1## 1 1####>

The meaning of the abo&e instruction as follows-

30C means in decimal" 1## means sampling for e&ery 1##97 and 1 1####>

means that we sample from frequencies 1 to 1####>.

11 %'


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25/56

Table 11.) % Program for 02periment 11.) %

,C=

.P=$T0CT

.$PT!$ P$ST

.,!B c- mm#'))&.l TT

.: P=$T0CT

.op

, 1 % (#m9

C ' # %)nF

= % ' 1##

Din 1 # AC 1

.AC 30C 1## 1 1####>

.P,$T AC D3B/'

.0 3

11 %)


26/56

Fig. 11.) ' The LCR low pass filter transfer function with Q enlarged

As e2plained in the abo&e section" the increasing of the Q &alue will ma>e the cur&e ha&e

a sharp ma2imum.

%&eri'ent 11.()3 The #e*reasing o+ the Q),al-e +or the Low Pass Filter

!n this e2periment" we decrease the &alue of ? &alue by increasing the &alue of

resistor from 1## ohms to ' ohms. !n this case" it can be shown that Q is around #.('

which is smaller than%

1. The program is in Table 11.) ' and the gain &s frequency

cur&e is in Fig. 11.) (. As can be seen" this ? &alue creates a flat cur&e. !n fact" this is

usually called a ma2imally flat cur&e.

Table 11.) ' Program for 02periment 11.) '

,C=

11 %*


27/56

.P=$T0CT

.$PT!$ P$ST

.,!B c- mm#'))&.l TT

.: P=$T0CT

.op

, 1 % (#m9

C ' # %)nF

= % ' '>

Din 1 # AC 1

.AC 30C 1## 1 1####>

.P,$T AC D3B/'

.0 3

Fig. 11.) ( The LCR low pass filter transfer function with a small Q

11 %5


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%&eri'ent 11.() The LCR Band Pass Filter

!n this e2periment" we tested the performance of the ,C= band pass filter. The

circuit is shown in Fig. 11.) ). The program is displayed in Table 11.) ( and the gain &s

frequency cur&e is shown in Fig. 11.) *.

L =40mH

C =25nF

R =3k

vin

1

2

3

vout

Fig. 11.) ) The LCR band pass filter for 02periment 11.) (

Table 11.) ( Program for 02periment 11.) (

,C=

11 %8


29/56

.P=$T0CT

.$PT!$ P$ST

.,!B c- mm#'))&.l TT

.: P=$T0CT

.op

, 1 % (#m9

C % ' %)nF

= ' # '>

Din 1 # AC 1

.AC 30C 1## 1 1####>

.P,$T AC D3B/'

.0 3

Fig. 11.) * The LCR band pass filter transfer function

11 %6


30/56

%&eri'ent 11.()( The In*reasing o+ the Q),al-e +or the Band Pass Filter

!n this e2periment" we increased the Q &alue of the circuit by reducing the &alue of

= from ' ohms to 1## ohms. The program is in Table 11.) ) and the gain &s frequency

cur&e is shown in Fig. 11.) 5. As can be seen" the bandwidth is decreased.

Table 11.) ) Program for 02periment 11.) )

,C=

.P=$T0CT

.$PT!$ P$ST

.,!B c- mm#'))&.l TT

.: P=$T0CT

.op

, 1 % (#m9

C % ' %)nF

= ' # 1##

Din 1 # AC 1

.AC 30C 1## 1 1####>

.P,$T AC D3B/'

.0 3

11 '#


31/56

Fig. 11.) 5 The LCR band pass filter transfer function with a larger Q

NEEDS TO BE LABELED

%&eri'ent 11.()/ The #e*reasing o+ the Q)0al-e +or the Band Pass Filter

!n this e2periment" we decreased the Q &alue by increasing the &alue of R to 1#

ohms. The program is in Table 11.) * and the gain frequency cur&e is shown in Fig.

11.) 8. As shown" the bandwidth is larger now.

Table 11.) * Program for 02periment 11.) *

,C=

.P=$T0CT

.$PT!$ P$ST

.,!B c- mm#'))&.l TT

.: P=$T0CT

.op

11 '1


32/56


33/56

from %)nF to '##nF. The program is shown in Table 11.) 5 and the gain frequency cur&e

is shown in Fig. 11.) 6. As shown" the critical frequency is decreased.

Table 11.) 5 Program for 02periment 11.) 5

,C=

.P=$T0CT

.$PT!$ P$ST

.,!B c- mm#'))&.l TT

.: P=$T0CT

.op

, 1 % (#m9

C % ' '##nF

= ' # 1##

Din 1 # AC 1

.AC 30C 1## 1 1####>

.P,$T AC D3B/'

.0 3

11 ''


34/56

Fig. 11.) 6 The decreasing of the critical frequency of the LCR band pass filter

Section 11.* Some Acti&e Filters

!n the abo&e sections" we only used passi&e components to design filters. But filters with

reacti&e components only will ha&e attenuation. To a&oid attenuation" we will employ

acti&e filters. Fig. 11.* 1 shows a typical low pass filter which employs a non in&erting

operational amplifier. !t is easy to see why this is a low pass filter because the =C circuit

itself is a low pass filter.

11 '(


35/56

-

+

vin

vout

R

C R a

R b

A

Fig. 11.* 1 A low pass filter built upon an operational amplifier

Fig. 11.* % shows another low pass filter with negati&e feedbac>.

11 ')


36/56

-

+vin v out

R 1

R 2

C

A

Fig. 11.* % A low pass filter with a capacitor connected the Eterminal and the output

terminal

+hy is this circuit a low pass filter ote that the capacitor is open circuited when

the frequency is low and the circuit becomes that shown in Fig. 11.* '. Thus the low

frequency signals may get through.

11 '*


37/56


38/56

-

+v in v out

R 1C

A

Fig. 11.* ( The filter in Fig. 11.* % in high frequency

Two high pass filters are shown in Fig. 11.* ) and Fig. 11.* *.

-

+

vin

vout

R

C

R a

R b

A

11 '8


39/56

Fig. 11.* ) A high pass filter based upon an operational amplifier

-

+vinvout

R 1 C

R 2

A

Fig. 11.* * Another high pass filter based upon an operational amplifier

Section 11.5 A General Case for Second $rder

Acti&e Filters

Fig. 11.5 1 shows a general case for second order acti&e filters. +e may obtain low pass"

high pass and band pass filters by gi&ing different components to Bi Z s.

11 '6


40/56

-

+

Z 1

Z 2

Z 3

Z 4 Z 5

Vout Vin V

A

.

Fig. 11.5 1 A general case of filters based upon an operational amplifier

,I" and ,$UT not *orre*t

,et us now find the transfer function for this general case circuit. ote that the

&oltage at the in&erting terminal is almost # for small signals. Thus" for ode A" we

ha&e-

#//

'%(1

=+++ Z

v Z

v Z

vv Z

vv out in /11.5 1

At the in&erting terminal"

#)'

=+ Z

v

Z

v out /11.5 %

Based upon 0quations /11.5 1 and /11.5 % " we ha&e-

.111111

1

('('%1)

'1

Z Z Z Z Z Z Z

Z Z vv

in

out

+

+++

= /11.5 '

A Low Pass Filter #eri0ed +ro' the General Case Filter

11 (#


41/56

Fig. 11.5 % shows a low pass filter deri&ed from the general case filter. !f the

frequency is high" C ) becomes short circuited. The operational amplifier becomes a

&oltage follower. !f the frequency is low" signals can go through. Thus the circuit is a

low pass filter.

Fig. 11.5 % A low pass filter deri&ed from the general case

" #! 4$#IFI #

From 0quation /11.5 ' " we ha&e-

)%('('1%

%

)%'1

11111

1

C C R R R R RC S S

C C R R

v

v

in

out

+

+++

= /11.5 (

By letting

)%('#

1C C R R

= /11.5 )

= 1

= '

= (

C%

C)

Din Dout

11 (1


42/56


43/56

VDD!VDD!

V BIA S3 =0.75

V B IA S67 =0V

V B IA S45 =-0.737V

V + V -vout

VSS!

H'

H1 H%

H(H)

H* H5

H8 H6

H1# H11

Fig. 11.5 ' The low pass filter with an operational amplifier for 02periment 11.5 1

!n this e2periment" .'.'"##(8.#"##(8.# (')% K R R C C ==== The

program is in Table 11.5 1 and the gain &s frequency cur&e is shown in Fig. 11.5 (.

Table 11.5 1 Program for 02periment 11.5 1

02periment 11.5 1

.P=$T0CT

.$PT!$ P$ST

.,!B c- fle2lm model tsmc H!I03#') mm#'))&.l TT

.: P=$T0CT

.op

11 ('


44/56

D33 D33J # 1.)D

DSS DSSJ # 1.)D

.G,$BA, D33J DSSJ

H1 ) Di * * PC9 +K1#: ,K%: mK'

H% ( DiL * * PC9 +K1#: ,K%: mK'

H' * DB' D33J D33J PC9 +K1##: ,K%: mK5

H( ) DB() DSSJ DSSJ C9 +K1#: ,K%:

H) ( DB() DSSJ DSSJ C9 +K1#: ,K%:

H* ' DB*5 ) DSSJ C9 +K1#: ,K%:

H5 D$ DB*5 ( DSSJ C9 +K1#: ,K%:

H8 ' ' 1 1 PC9 +K1#: ,K%: mK'

H6 D$ ' % % PC9 +K1#: ,K%: mK'

H1# 1 1 D33J D33J PC9 +K1#: ,K%: mK'

H11 % 1 D33J D33J PC9 +K1#: ,K%: mK'

DiL DiL # #&

DB!AS' DB' # #.5)&

DB!AS() DB() # #.5'5&DB!AS*5 DB*5 # #&

Din1 11 # AC 1

.AC 30C 1## 1 )###>

=1 11 1# '.'>

C% 1# # #.##(8u

=' Di 1# '.'>

=( 1# Do '.'>

C) Di Do #.##(8u

.P,$T AC D3B/Do

11 ((


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.0 3

Fig. 11.5 ( The transfer function of the low pass filter in 02periment 11.5 1

%&eri'ent 11. )2 The #e*reasing o+ #

+e increased the &alues of capacitors and thus decreased # according to 0quation

/11.5 ) . .1)% C C == The program is shown in Table 11.5 % and the gain &s

frequency cur&e is shown in Fig. 11.5 ). As can be seen" #

was significantlydecreased.

Table 11.5 % Program for 02periment 11.5 %

02periment 11.5 %

11 ()


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.P=$T0CT

.$PT!$ P$ST


.: P=$T0CT

.op

D33 D33J # 1.)D

DSS DSSJ # 1.)D

.G,$BA, D33J DSSJ

H1 ) Di * * PC9 +K1#: ,K%: mK'

H% ( DiL * * PC9 +K1#: ,K%: mK'

H' * DB' D33J D33J PC9 +K1##: ,K%: mK5

H( ) DB() DSSJ DSSJ C9 +K1#: ,K%:

H) ( DB() DSSJ DSSJ C9 +K1#: ,K%:

H* ' DB*5 ) DSSJ C9 +K1#: ,K%:

H5 D$ DB*5 ( DSSJ C9 +K1#: ,K%:

H8 ' ' 1 1 PC9 +K1#: ,K%: mK'

H6 D$ ' % % PC9 +K1#: ,K%: mK'H1# 1 1 D33J D33J PC9 +K1#: ,K%: mK'

H11 % 1 D33J D33J PC9 +K1#: ,K%: mK'

DiL DiL # #&

DB!AS' DB' # #.5)&

DB!AS() DB() # #.5'5&

DB!AS*5 DB*5 # #&

Din1 11 # AC 1

.AC 30C 1## 1 )###>

=1 11 1# '.'>

11 (*


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C% 1# # 1u

=' Di 1# '.'>

=( 1# Do '.'>

C) Di Do 1u

.P,$T AC D3B/Do

.0 3

Fig. 11.5 ) The transfer function of the low pass filter in Fig. 11.5 ' with &alues of

capacitors increased

A High Pass Filter #eri0ed +ro' the General Case Filter

Fig. 11.5 * shows a high pass filter deri&ed from the general case filter.

11 (5


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Fig. 11.5 * A high pass filter deri&ed from the general case

A needs to 5e added.

The transfer function of the high pass filter is as follows-

(')%'(('

1

)

%

(

1%

1111/

C C R RC C C C C

RS S

C C

S S A

+

+++

= /11.5 6

By letting(')%

#1

C C R R= /11.5 1#

++= ('1

('

%

)

C C C C C

R RQ /11.5 11

and(

1# C

C vv

Ain

out ===

" /11.5 1%

R2

R!

C 1

C "

C #

vin v

out

11 (8


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we ha&e-

%

#

#%

%#/

++

=S

QS

S AS A

/11.5 1'

0quation /11.5 1' is almost e2actly the same as 0quation /11.' ) " which is the transfer

function of another high pass filter.

A Band Pass Filter #eri0ed +ro' the General Case Filter

Fig. 11.5 1( shows a band pass filter deri&ed from the general case filter.

Fig. 11.5 1( A band pass filter deri&ed from the general case

!t can be easily seen that this circuit is a band pass filter. ote that high frequency

signals cannot go through because of C ' and low frequency signals cannot go through

because of C ( . The transfer function of this band pass filter is-

R1

R2

R!

C "

C #

vin v

out

11 (6


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++

++

=

%1(')(')

%

(1

111111

1

/

R RC C RC C RS S

C RS

S A /11.5 1(

By letting

+=

%1(')#

111 R RC C R

/11.5 1)

+

+=

('

('

%1)

11C C

C C

R R RQ /11.5 1*

and('

'

1

)#

#C C

C R R

vv

Ain

out

+== = /11.5 15

we ha&e-

%#

#%

##

/

++=

S Q

S

S Q

AS A

/11.5 18

%&eri'ent 11. )3 The Band Pass Filter #eri0ed +ro' the General Case Filter

!n this e2periment" we set K R R R $ C C 1'")# )%1(' ===== . The program is in

Table 11.5 ' and the gain &s frequency cur&e is in Fig. 11.5 1).

Table 11.5 ' Program for 02periment 11.5 '

02periment 11.5 '

.P=$T0CT

.$PT!$ P$ST


11 )#


51/56


52/56

=) Di Do 1'>

.P,$T AC D3B/Do

.0 3

Fig. 11.5 1) The transfer function of the band pass filter in Fig. 11.5 '

%&eri'ent 11. ) The #e*reasing o+ #

and nlarging o+ the Bandwidth

+e decreased # by setting .)(' $ C C == This caused a decreasing of the ?

&alue and an enlargement of the bandwidth. The program is in Table 11.5 ( and the gain

&s frequency is in Fig. 11.5 1*. As can be seen" # is made smaller and the bandwidth

is now larger.

11 )%


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.AC 30C 1## 1 1#########>

=1 11 1# 1'>

=% 1# # 1'>

C' Di 1# #.##)u

C( 1# Do #.##)u

=) Di Do 1'>

.P,$T AC D3B/Do

.0 3

Fig. 11.5 1* The band pass filter with # decreased

Section 11.8 The Sallen and ey Filters

11 )(


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There are other acti&e filters. !n this section" we shall introduce the Sallen and ey

filters. The Sallen and ey filters employ positi&e feedbac>. Fig. 11.8 1 shows a Sallen

and ey low pass filter.

-

+

vinvout

R 1

C 2 R a

R b

C 1

R 2

Fig. 11.8 1 A Sallen and ey low pass filter

!t is ob&ious that this is a low pass filter because the high frequency will short circuit thecapacitor C %.

Fig. 11.8 % shows a Sallen and ey high pass filter. !t is ob&ious that the low

frequency signals cannot go through as they will be bloc>ed by the capacitors.

11 ))


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Documents

Chapter 11(Analog)