First order low pass filter ECSE-2010ssawyer/CircuitsSpring2020_all/... · 2016. 4. 25. · 5 1 stORDER HIGH PASS FILTER 2 2 c H(j ) Gain ω ω ω ω = = + 0.01 .1 1 10 100 ωωc

1

ECSE-2010

Lecture 24 continued

� First order low pass filter

� First order high pass filter

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inv

R

CCv

+

−

C C inAs 0 (DC): Z ; v vω → → ∞ =

C CAs : Z 0; v 0ω → ∞ → =

AC

inv

R

CCv

+

−

Let's look at this in the s-Domain

C

in No Initial Stored Energy

V (s)Find H(s)

V (s)=

AC

2

C c

c

V (s) 1 RC 1H(s)

V(s) s 1 RC s 1 s

ωττ ω

= = = =+ + +

FIRST ORDER FILTERS

c

c

H(j ) H(j ) /j

ωω ω θω ω

= =+

c

2 2c

H(j ) Gainωω

ω ω= =

+

1

c

tan Phase Shiftωθω

−= − =

c

2 2c

H(j ) Gainωω

ω ω= =

+

c 10

1At 10 , Gain ; 20log (Gain) 20 dB

10ω ω= ≈ ≈ −

c 10

1At 100 , Gain , 20log (Gain) 40 dB

100ω ω= ≈ ≈ −

c 10

1At 1000 , Gain , 20log (Gain) 60 dB

1000ω ω= ≈ ≈ −

Gain decreases at a Slope 20 dB decade= −

1st ORDER LOW PASS FILTER


c

2 2c

H(j ) Gainωω

ω ω= =

+

0

cω ω.1 1 10 100.01

10log scale

1020log H

Gain in dB

•3−

20−

•

•

•40−

Asymptotes

20 dB/decade−

Bode Plot of Gain

••

c

2 2c

H(j ) Gainωω

ω ω= =

+

0

cω ω.1 1 10 100.01

10log scale

1020log H

Gain in dB

•

20−

•

•

•40−

Asymptotes

20 dB/decade−

Bode Plot of Gain

••

Passband Stopband


3

1

c


−= − =


cFor "low frequencies" ( ): 0ω ω θ<< →0

cFor "high frequencies" ( ): 90ω ω θ>> → −0

cFor : 45ω ω θ= = −

C

Most of the change in occurs for .1 10ωθω

≤ ≤

C

Corner Frequencies at .1, 10ωω

=

00

cω ω.1 1 10 100.01

10log scale

θ

•

045−•

•

•

090−

Asymptotes

••


1

c


−= − =

•

Bode Plot of θ


R

Coutv

+

−inv

+

−

out c

in c

V (s) 1 RCH(s)

V (s) s 1 RC s

ωω

= = =+ +

Low Pass Filter=>

Low Frequencies "Pass"; High Frequencies "Stopped"

BETTER 1st ORDER LOW PASS

invR

C

1R FR

outv

F

1

cF

1 c

R 1 RCH(s) (1 )

R s 1 RC

R (1 )

R s

ωω

= ++

= ++

CCV+

CCV−

4

1st ORDER HIGH PASS FILTER

out

in c

V (s) s sH(s)

V (s) s 1 RC s ω= = =

+ +High Pass Filter=>

inv

+

−R

C

outv

+

−

High Frequencies "Pass"; Low Frequencies "Stopped"

10As 0, Gain 0, 20log (Gain)ω → → → −∞

2 2c

H(j ) Gainωω

ω ω= =

+

cc 2 2

c

.01At .01 , Gain .01

(.01)

ωω ωω

= = ≈+

1020log (Gain) 40 dB= −


2 2c

H(j ) Gainωω

ω ω= =

+

c 10At .01 , Gain .01; 20log (Gain) 40 dBω ω= ≈ ≈ −

c 10At .1 , Gain .1, 20log (Gain) 20 dBω ω= ≈ ≈ −

Gain increases at a Slope 20 dB decade= +


c 10

1At , Gain , 20log (Gain) 3 dB

2ω ω= = = −


2 2c

H(j ) Gainωω

ω ω= =

+

0

cω ω.1 1 10 100.01

10log scale

1020log H

Gain in dB

3−

20−

•

•

•40−

Asymptotes20 dB/decade+

Bode Plot of Gain

••

5


2 2c

H(j ) Gainωω

ω ω= =

+

0

cω ω.1 1 10 100.01

10log scale

1020log H

Gain in dB

20−

•

•

•40−

Asymptotes20 dB/decade+

Bode Plot of Gain

••

Stopband Passband

0 1

c

90 tan Phase Shiftωθω

−= − =


0cFor "low frequencies" ( ): 90ω ω θ<< →

0cFor "high frequencies" ( ): 0ω ω θ>> →0

cFor : 45ω ω θ= = +

C

Most of the change in occurs for .1 10ωθω

≤ ≤

C

Corner Frequencies at .1, 10ωω

=

00

cω ω.1 1 10 100.01

10log scale

θ

•

045+•

•

•

090+

Asymptotes

••


0 1

c

90 tan Phase Shiftωθω

−= − =

•

Bode Plot of Phase Shift

BETTER 1st ORDER HIGH PASS

invC

R

1RFR

outvCCV+

CCV−

F

1

F

1 c

R sH(s) (1 )

R s 1 RC

R s (1 )

R s ω

= ++

= ++

6

1. Find poles, zeros

2. In each region between sequential poles/zeros, can use H(jω) or H(s)

a) Draw line in first region based on three possibilities based on frequency range

b) Repeat for every region

3. Add corrections at poles/zeros

a) n*pole n(-3db) correction at that pole

b) n*zero n(+3db) correction at that zero

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We have three possibilities

Note: Can ALSO analyze using H(s) instead of

H jω( ) ∝ ωn+n 20⋅

db

decslope

H jω( ) ∝constant K, constant 20 log⋅ K⋅

H jω( ) ∝1

ωn n− 20⋅db

decslope

Hjω( )

1. Crossing a n*pole is a change of phase of –n*90 deg (absolute change)

a) Changing over approximately two decades

b) Specifically, 0.1 and 10 times ωc

2. Crossing a n*zero is a change of phase +n*90 deg (absolute change)



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1. Crossing a n*pole is a change of phase of –n*90 deg (absolute change)



2. Crossing a n*zero is a change of phase +n*90 deg (absolute change)




SLA have slopes of +/- n*45 deg per decade slope (+ is a zero, - is a pole)

7

ECSE-2010

Lecture 19.2

� Cascaded filters/Parallel filters

� First order Bandpass filter

� First order Bandstop (Notch filter)

[email protected] www.rpi.edu/~sawyes

cL

cH cL

sH(s) K

s s

ωω ω

= + +

cL cH

H H L L

Let's Design Such that

R C R C

ω ω⇒

�

�

cL

2 2 2 2cH cL

H(j ) Kωωω

ω ω ω ω

= + +

High Pass Low Pass

>

>

dBK

ωcHω cLω

10log scale

Gain in dB20 dB/decade+

BANDPASS FILTER

cH.1ω cL10ω

dBK 20−

20 dB/decade−

Passband

Stopband Stopband

cL cHω ω�

cL cHBandwidth ω ω= −

>

8

stFor Low Frequencies Looks Like a 1 Order Low Pass⇒

cLAL H

B cL cH

R sH(s) H (s) H (s) 1

R s s

ωω ω

= + = + + + +

stFor High Frequencies Looks Like a 1 Order High Pass⇒

cH cL

L L H H

Let's Design Such that

R C R C

ω ω⇒

�

�

>

>

dBK

ωcLω cHω

10log scale

Gain in dB

BANDGAP OR NOTCH FILTER

cH cLω ω�

dB20

dec−

Stopband

Passband Passband

dB20

dec+

cL cHω ω

cH cLBandwidth ω ω= −

>


a) ( ) ( )1000

1000

+=

ssH


a) ( ) ( )1000

1000

+=

ssH

9


-40

-30

-20

-10

0

Mag

nitu

de (

dB)

101

102

103

104

105

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

>> sys=tf([1000],[1 1000]);>> h=bodeplot(sys);grid>>


( ) ( )2

2

1000

1000

+=

ssHb)


( ) ( )2

2

1000

1000

+=

ssHb)


-80

-60

-40

-20

0

Mag

nitu

de (

dB)

101

102

103

104

105

-180

-135

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

>> sys=tf([1000000],[1 2000 1000000]);>> h=bodeplot(sys);grid

10


( ) ( )( )1000010010

2

++=

ss

ssHc)


( ) ( )( )1000010010

2

++=

ss

ssHc)


( ) ( )( )1000010010

2

++=

ss

ssHc)


( ) ( )( )1000010010

2

++=

ss

ssHc)

11


-60

-40

-20

0

20

40

Mag

nitu

de (

dB)

100

101

102

103

104

105

106

0

45

90

135

180

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

>> sys=tf([10 0 0],[1 10100 1000000]);>> h=bodeplot(sys);grid>> setoptions(h,'MagLowerLimMode','manual','MagLowerLim',-60)


2) Bode plot-multiple stages

R1

100

R21k

L10.1

1

2

L2

0.001

12

U1

OPAMP

+

-

OUT

R310k

R4

90k

0

00

VoutVin

H1(s) H2(s) H3(s)

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12


-30

-20

-10

0

10

20

Mag

nitu

de (

dB)

101

102

103

104

105

106

107

108

-90

-45

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

>> sys=tf([10000000 0],[1 1001000 1000000000]);>> h=bodeplot(sys);grid

Documents

First order low pass filter ECSE-2010ssawyer/CircuitsSpring2020_all/... · 2016. 4. 25. · 5 1 stORDER HIGH PASS FILTER 2 2 c H(j ) Gain ω ω ω ω = = + 0.01 .1 1 10 100 ωωc