COMM 704:COMM 704: Communication Systems · COMM 704:COMM 704: Communication Systems Lecture 4:...

COMM 704:COMM 704:Communication Systemsy

Lecture 4: Analog Filters (continued)Lecture 4: Analog Filters (continued)

Dr Mohamed Abd El GhanyDr. Mohamed Abd El Ghany, Department of Electronics and Electrical Engineering

[email protected]

Active Filter: Low-Pass FilterActive Filter: Low Pass Filter

Sallen Key LPFy

a

b

RRK +=1

CCRRK

V

2121221112

2

2121

1)111()(

CCRRS

CRK

CRCRS

CCRRVVsH

i

o

+−+++==

2Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering

COMM 704:Communication Systems Winter 2011


Sallen Key LPFy

G

Using the standard notation:

22 )()(

oo

LPF

wSQwS

GsH++

=

11

CCRR

2121

1CCRR

wo =

221112

2121

111CRk

CRCR

CCRRQ

−++=




Sallen Key LPFy

If no restriction is imposed on the gain K, we have five element values to design LPF circuit. Hence, we are free to make some arbitrary choices as follows:

Design 1: equal element values In this design, we letC1=C2=1F and R1=R2=R

Design 2: equal capacitance and equal feedback resistancesIn this design we chooseC1 C2 1F and R1 R2 R In this design, we chooseC1=C2=1F and Ra=Rb=R

Design 3: moderate-sensitivity Design 4: minimum sensitivityes g 3 ode ate se s t tydesignIn this design, we chooseC1=√3 Q and C2=1F and K=4/3

Design 4: minimum sensitivityIn this design, we chooseK=1 and R1=R2=1Ω




Definition of sensitivityy

The sensitivity of some performance measure y with respect to a network element value x is defined by:

dxdy

yxS y

x =

If y is a function of several variable [y=f(x1,x2,….,xn)], then the sensitivity of y with respect to xi is

d

i

iyx dx

dyyxS

i= Notes:

2121 yx

yx

yyx SSS +=

2121 / yx

yx

yyx SSS −=



xxx


Example:

Design a fourth-order butterworth low-pass filter using the cascade of two Sallen-key biquads using Design 1 and 2 respectively. Then let wo =2Пx1000 rad/sec and use 0.1 µF capacitors.Th li d t k f ti f th t l filt

1765367.0)( 2

11 ++

=SS

GsH

The normalized network function of the two low-pass filters are:

1765367.0 ++ SS

1847759.1)( 2

22 ++

=SS

GsH



Active Filter: High-Pass FilterActive Filter: High Pass Filter

First order HPF (using Inverting op-amp configuration) ( g g p p g )

RsH 1)( 2−=

SCR 1

1+

S

cwSSKsH+

−=)(

1

2

RRK =

CRwc

1

1=




First order HPF (using Non-Inverting amplifier) ( g g p )

2

3 )1()(

SRR

sH+

+=

11

1)(

CRS

sH+

+=

S

cwSSKsH+

=)(

2

31RRK +=

11

1CR

wc =




Sallen Key HPFy

a

b

RRK +=1

2

)( KSVsH o ==

2121112212

2 1)111()(

CCRRS

CRK

CRCRSV

sHi +

−+++

==




Sallen Key HPFy

2GS


22 )()(

oo

HPF

wSQwS

GSsH++

=

11

CCRR

2121

1CCRR

wo =

111222

2121

111CRk

CRCR

CCRRQ

−++=



Active Filter: Band-Pass FilterActive Filter: Band Pass Filter

Sallen Key BPFy

a

b

RRK +=1

SK

21321

21

12132311

2

11

)1111()(

CCRRRRRS

CRK

CRCRCRS

SCR

VVsH

i

o

++−++++==



Active Filter: Band-Pass FilterActive Filter: Band Pass Filter

Sallen Key BPFy

)( o SwG


22 )(

)()(

oo

o

BPF

wSQwS

SQ

GsH

++=

RR + 21 RR +

21321

21

CCRRRRRwo

+=

12132311

21321

1111CRk

CRCRCR

CCRRRQ

−+++

=



Active Filter: Band-rejection FilterActive Filter: Band rejection Filter

Sallen Key Band-Rejection Filtery j

a

b

RRK +=1

2 )1( SK

222

222

1)24(

)()(

CRS

RCKS

CRSK

VVsH

i

o

+−

+

+==



Active Filter: Band-rejection FilterActive Filter: Band rejection Filter

Sallen Key Band-Rejection Filtery j

22 )(


22

22

)(

)()(o

o

oBRF

wSQwS

wSKsH++

+=

1 1

RCwo

1= K

Q24

1−

=



Active Filter: Multiple-Feedback Biquads (MFB)

• The MFB topology is commonly used in filters that have high Q p gy y gand require a high gain

MFBMFB

Multiple Feedback Low-pass

Multiple Feedback High pass

Multiple Feedback Band pass

Multiple Feedback

band-rejectionLow-pass biquad

High-pass biquad

Band-pass biquad

band-rejection biquad




Multiple Feedback Low-pass Biquad

21323211

2

2131

1)111(1

1

)(

CCRRS

RRRCS

CCRRVVsH

i

o

++++−==

21323211 CCRRRRRC

22 )()(

oo

LPF

wSQwS

GsH++

=

2132

1CCRR

wo =

3

2

2

3

1

322

1 1

RR

RR

RRRC

CQ++

=




Multiple Feedback High-pass Biquad

3212

2

2

1

1)(SCCCS

SCC

VVsH

i

o

++++−==

3221322 CCRRS

CCRSi ++

3221

1CCRR

wo =3

322

321

CCRCCC

Qwo ++

=




Multiple Feedback bandpass Biquad

2

21

]11[11

1

)(

RR

SCR

VVsH

i

o

+−==

11

212

31

2212

2 )11(CCRRRS

CRCRS +++

212

31

11

CCRRRw o

+=

11w

2212

11CRCRQ

w o +=

12

/1/)(

CCRRjwH o +

=



12 /1 CC+

Active Filter: Multiple-Feedback Biquads (MFB)Multiple Feedback band-rejection Biquad

22

2

222

1)24(

)/1()(

CRS

RCKS

CRSKVVsH

i

o

+−

+

+−==

w 1=

CRRC

RCw o =

KQ

241−

=K24

a

b

RRK += 1



Active Filter: Multiple-Amplifier Biquads

The step from single-amplifier to multiple-amplifier biquadratic filter sections id l b fit Th t i t t b fit f llprovides several benefits. The most important benefits are as follows:

-Reduced passive element spread, i.e., the ratio between the largest and smallest values of resistors and/or capacitors can be reduced compared to the single-amplifier case-Multiple-amplifier biquads often provide lower sensitivities to both passive and active

tcomponent-Most of the filter parameters can be tuned independently.-Designing multiple-amplifier filters, often the values of capacitors can be chosen freely. The filter parameters are then determine by resistors, which is less costly than by

itcapacitors.Note:• On the other hand, these benefits must be paid for by increased space requirements and increased power dissipation. However, today there are low-cost and low-power i t t d i it il bl ith t f hi Th f iintegrated-circuit op-amps available with up to four op-amps on one chip. Therefore, size and power dissipation are often no longer the main problem.• There are two very famous multiple amplifier biquads;

1. Tow-Thomas Filter (TT), 2. Kerwin- Huelsman- Newcomb Filter (KHN)



Multiple-Amplifier Biquads: Tow-Thomas Filter

By the direct analysis three outputsBy the direct analysis, three outputs are available;• inverting lowpass output (-VLP)• noninverting lowpass output (VLP)• inverting bandpass output (V )

R2

• inverting bandpass output (VBP)-VLP

VLPVBP

1 S1

1w = w o 1= 21 / CCRQ

213211

2

14

1)1(CCRR

SCR

S

SCR

VV

in

BP

++−=

213211

2

2142

1)1(CCRR

SCR

S

CCRRVV

in

LP

++=

2132 CCRRw o =

RCQ=

32

211 RR

RQ =

4

3)0(RRH LP =

4

1)(RRjwH oBP =



4

Multiple-Amplifier Biquads: Tow-Thomas FilterThe Tow-Thomas Filter has the

VLPVBP

following advantages:Its Q can be adjusted by varying a single resistance –> R1 without di t bi -VLP

VLPdisturbing wo

The gain can be varied by varying a single resistance -> R4 without disturbing w and Qdisturbing wo and QThe lowpass output can be either inverting or noninvertingThe sensitivity of the Tow-Thomas ybiquad are generally comparable to those of the KHN biquad



Multiple-Amplifier Biquads: Kerwin- Huesman- Newcomb FilterBy the direct analysis three outputsBy the direct analysis, three outputs are available;• noninverting High-pass output (VHP)• inverting band-pass output (-VBP)• noninverting low pass output (V )• noninverting low-pass output (VLP)

VLPVBPVHP

66532

212143

65

5

4

1)()]1)()([(

)1)()((

RSRRRS

CCRRRRRR

RR

VV

in

LP

++

+

++

=

Vin

2121511435

)()])()([(CCRRR

SCRRRR

Sin ++

+

66532

1143

65

5

4

1)()]1)()([(

)1)()((

RSRRRS

SCRRR

RRRR

VV

in

BP

++

+

++

−=

VV 1=

Notes:

2121511435

)()])()([(CCRRRCRRRR +

66532

2

43

65

5

4

1)()]1)()([(

))((

CCRRRRS

CRRRRR

RRS

SRRRR

RR

VV

in

HP

+++

+

++

=

BPLP VCSR

V22

−=

HPBP VCSR

V11

1−=



2121511435 CCRRRCRRRR +

Notes:Notes:The general transfer function of the second order active filter in the s-plane has the following form:

22

322

1

)()(

o wSwS

KSKSKsH++

++=

)( owSQ

S ++

K1 = K2 = 0 Low-Pass

K1 = K3 = 0 Band-Pass

K2 = K3 = 0 High-Pass

K1 = 1 K2 = 0 and K3 = band-reject2wK1 = 1, K2 = 0, and K3 = band-reject

K1 = 1, K2 = , and K3 = All-Pass

ow

Qwo− 2

ow



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Transcript of COMM 704:COMM 704: Communication Systems · COMM 704:COMM 704: Communication Systems Lecture 4:...