special notes: filter design methods spectral power...

Special Notes: Filter Design Methods

Spectral Power Responses

For jwTjwTjwT exp

What is jwT

Magnitude (must be the same!):

N

n n

n

n

M

m m

R

Q

i i

w

w

w

w

p

ww

z

w

KjwT

1

222

1

2

1

2

211

1

Phase:

N

n

n

n

n

M

m m

Q

i i

w

w

w

w

ap

waR

z

wajwT

12

111

2

tantan2

tan

N

n

n

n

n

M

m m

Q

i i

w

w

w

w

ap

waR

z

wajwT

12

111

2

tantan2

tan

Therefore jwTjwT and 0 jwTjwT

Then, what is jwTjwTjwTjwTjwTjwT expexp

or 2jwTjwTjwT

This is a power term (notice the square), so we use 10*log to create decibels. It is the same result!! Note, this works for (s,-s) too!!

2sTsTsT

This is the generic form for defining the magnitude response of a filter! Why … The poles and zeros are symmetric about the jw axis of the s-plane! Therefore, the LHP and RHP elements can be separated into T(s) and T(s) and guarantee marginal stability!!

The Butterworth Lowpass Filter

nnn

ww

jwTjwT2

01

1

nn

nn

nnn

ws

wsjwj

ssTsT

2

0

2

0

22

011

1

1

1

1

1

Characteristic Eq. 0112

0

ssw

sn

n

Frequency normalized 011 2 nn sss

Reference: M.E. Van Valkenburg, Analog Filter Design, Oxford Univ. Press, 1982 ISBN: 0-19-510734-9

Solving for the Butterworth Filter poles:

Filter in jw nnn

ww

jwTjwT2

01

1

Laplace

nn

nn

nnn

ws

wsjwj

ssTsT

2

0

2

0

22

011

1

1

1

1

1

Characteristic Eq. 0112

0

ssw

sn

n

Normalize 011 2 nn sss

For n odd:

0111 2 nnn sssss

Roots at mjs n 2exp12

n

jms

exp

Let s be the LHP poles and s be the RHP poles

For n even:

0111 2 nnn jsjssss

Roots at jmjs n 2exp12

n

jmjs

2

2exp

Let s be the LHP poles and s be the RHP poles

Matlab Code (BW_Filter_Example.m) % BW Filter generation demonstration % close all clear all Rin=1; Rload=1; Rmatch=1; PBfreq=1; PiW=logspace(log10(PBfreq)-2,log10(PBfreq)+2,1024); colorseq=['b' 'g' 'r' 'y' 'm' 'c']; ii=0; PolesRange=6:-1:1 for BWn=PolesRange ii=mod(ii,6)+1; denP=roots([((-1/(PBfreq^2))^(BWn)) zeros(1,2*BWn-1) 1]) [Y,I] = sort(real(denP)); denPsort=denP(I) den=poly(denPsort(1:BWn)); figure(1) plot(real(denP),imag(denP),sprintf('%cx',colorseq(ii)) ); title('Power Magnitude Poles') grid on; hold on; num = [PBfreq^(BWn)]; zpi=abs(roots(num)); ppi=abs(roots(den)); BWsys=tf(num,den) [PiMAG, PiPHASE]=bode(BWsys,PiW); figure(2) semilogx(PiW, dBv(squeeze(PiMAG)),colorseq(ii) ); grid on; hold on; title('Power vs. Frequency') xlabel('Freq (rad/sec)'); ylabel('Magnitude dB'); plotv=axis; axis([plotv(1) plotv(2) -120 10]) figure(3) semilogx(PiW, (squeeze(PiPHASE)),colorseq(ii) ); grid on; hold on; title('Phase vs. Frequency') xlabel('Freq (rad/sec)'); ylabel('Phase'); axis([plotv(1) plotv(2) -max(PolesRange)*90 15]) pause end

Results

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Power Magnitude Poles

10-2

10-1

100

101

102

-120

-100

-80

-60

-40

-20

0

Power vs. Frequency

Freq (rad/sec)

Mag

nitu

de d

B

10-2

10-1

100

101

102

-500

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

Phase vs. Frequency

Freq (rad/sec)

Pha

se

What if we want to change the frequency …

nn

nnn

ws

wjs

sTsT2

0

2

011

1

1

1

Just change the natural frequency, 00 2 fw ; the center frequency is simply scaled!

The circle radius of the poles defines the cut-off frequency.

nnn

n

nnsw

wsTsT

220

20

1

Design approach:

1. Determine the order of the filter you want. What attenuation do you need at the 010 w

point? (There are plenty of curves, like those above, if the value you need comes before t 10x the cutoff frequency.)

2. Generate the Butterworth Coefficients on the unit circle for w=1

3. Scale the poles by the desired frequency (remember that w=1 is in radians/sec, therefore multiply by 00 2 fw .

Active Audio Frequency Filters

An active lowpass filter implementation of a 1st order Butterworth filter

+

-R1

R2

VinOP-Amp

Vout

C2

The transfer function for this circuit is

221

2

1 CsRR

R

sVin

sVout

1

2

R

RGMaxGain

220

1

RCw

To tune the circuit ….

An alternate approach:

+

-

R1

Ra

+Vdc

-Vdc

V1OP-Amp

Vout

C1

Rb

111

1

CsRR

RR

sVin

sVout

b

ba

b

ba

R

RRGMaxGain

110

1

CRw

To tune the circuit ….

A Second Order Butterworth Lowpass Filter

Let’s do the math for the second order system, for n=2 and 10 w .

22

0

222

0

22

11

1

1

1

ws

wjs

sTsT

1212

1

1

122422

ssssssTsT

12

1

12

12222

sssTand

sssT

For sT2 12

1

12

1222

ssss

sT

A second order underdamped system with 22 or 707.02

1

21

211, 2

21 jss

After frequency scaling

221, 002

0021wjwwwss

How to make a second order LPF for audio ….

Sallen-Key Circuit Lowpass Filter

An active lowpass filter implementation of a unity gain Friend Circuit, also referred to as a Sallen-Key circuit as described in: Walter G. Jung, IC OP-Amp Cookbook, Howard W. Sams Co. Inc, Indianapoli, IN, 1974.

+

-

R1 R2+Vdc

-Vdc

V1OP-Amp

Vout

C2

R3

R4

C1

The transfer function for this circuit is (a generic second order filter equation is also shown)

22

2

22

2121

1

213

4

22

1

12

1

2121

1

3

43

1 wsws

wK

RRCCRCR

R

RCRCss

RRCCR

RR

sV

sVout

2121123

411211

1

3

43

1 2 RRCCsRCRRRCRCsR

RR

sV

sVout

Letting CCC 21 and RRR 21 and 3

43

R

RRK

3

43

R

RRKMaxGain

RCw

1

2

3 K

Function Derivation

The circuit derivation assumes a perfect op-amp, with infinite gain, infinite input impedance, and zero output impedance, non-limiting power supplies and voltage drops, and no frequency response considerations.

The circuit derivation follows:

221

12

2

1

1

12 CsVo

R

Vp

R

VCs

RRV

2

21

2

1

R

VCs

RVp

VoRR

RVn

43

3

Letting VnVp

2

21

2

1

43

3

R

VCs

RRR

RVo

Vo

RR

RCsRV

43

21132

22112

2

1

12

2

1

1

12 CsVo

RCsR

V

R

VCs

RRV

21

12

211

1

1

12 CsVo

R

VCs

RCs

Cs

RV

1

122

211

1

1

1

43

2113

R

VCsVoCs

RCs

Cs

RRR

RCsRVo

1

12

1

21211211211

43

3 2

R

VCsVo

R

RRCCsRCsRCsRCs

RR

RVo

1

1

1

21213

41211211

43

32

R

V

R

RRCCsR

RRCsRCsRCs

RR

RVo

1

21213

41211211

1

3

43

2

V

RRCCsR

RRCsRCsRCs

R

RRVo

2121

1

213

4

22

1

12

1

2121

1

3

43

12

RRCCRCR

R

RCRCss

RRCCR

RR

V

Vo

Letting CCC 21 and RRR 21 and 3

43

R

RRG

22

2

13

1

1

RCRC

Gss

RCG

V

Vo

Resulting in

3

43

R

RRGMaxGain

RC

w

1

And 2

3 G

Note that for a stable system 31 G

Implying that 3240 RR

Multiple Feedback (MFB) Circuit Lowpass Filter

An active lowpass filter implementation of a multiple feedback circuit (MFB), that is may also be referred to as a derivative of the Sallen-Key Filter.

+

-R1 R2

+Vdc

-Vdc

V1OP-Amp

Vout

C2

C1

R3

Figure 1. MFB Lowpass Filter

The transfer function for this circuit is

32211

311

211

111

32211

1

3

1 2

RRCCRCRCRCssRRCC

R

R

V

Vo

3221223213221

1

1

3

1 2 RRCCsRCRCRRRCsR

R

V

Vo

Resulting in

1

3

R

RKMaxGain 3221

1RRCCw

Letting CC 2 , CnC 1 , RRR 31 , RmR 2 , and 1G

22

2

222 2211

1

1 wsws

wK

CRmnsmRCsV

Vo

1 KMaxGain mnCR

w

1

mn

m

2

21

Function Derivation

The circuit derivation assumes a perfect op-amp, with infinite gain, infinite input impedance, and zero output impedance, non-limiting power supplies and voltage drops, and no frequency response considerations.

The circuit derivation follows:

31

11

3

1

2

1

1

12

R

Vo

R

VCs

RRRV

02

22

R

VVoCs

Combining 31

11

3

1

2

1

1

122

R

Vo

R

VCs

RRRRCsVo

1

1

3321

321132312122

R

V

R

Vo

RRR

RRRCsRRRRRRRCsVo

1

1

31

3212132312121 2

R

V

RR

RRRCCsRRRRRRCsRVo

3212132312121

3

1 2 RRRCCsRRRRRRCsR

R

V

Vo

32211

311

211

111

32211

1

3

1 2

RRCCRCRCRCssRRCC

R

R

V

Vo

Resulting in

1

3

R

RGMaxGain 3221

1RRCCw

And 311

211

111

2

1RCRCRCw

Higher Order Butterworth Lowpass Filters

Take multiple stages and cascade them!

Remember to determine the pole locations that each stage of the filter requires.

As a rule-of-thumb, you should select the order for the stages of your filter. If you look at the output of each stage, it will be the product of the transfer functions to that location! So, possible use those with damping factors closest to one before the smaller ones ….

Jim Karki, Texas Instruments, Active Low-Pass Filter Design, Application Report, SLOA049B, September 2002.

Note:

1. Real elements may not exactly match the values you select.

2. Components have a tolerance, they are within +/- some %!

3. If possible use cheaper components and one (or two) that are adjustable (potentiometers).

What do RF designers do?

Why might it be different?

Low Pass Filter, 3rd Order

The classic 3rd order LC Ladder Low Pass Filter.

C2

Vin Vout

L1 L3

Figure 2. LC Ladder 3rd Order Low Pass Filter

The circuit derivation assumes a source and load resistance. The source resistance is placed prior to the input voltage and the load is placed on the output.

For RRLRs and LLL 21 :

2211

21

2 LCs

RCs

R

Ls

Vin

Vout

Theoretical Derivation


The circuit node equations follow:

3

12

3

11

LsV

LsRLVout

1

1

3

1

3

12

1

12

LsRsVin

LsVout

LsCs

LsRsV

Solving for V2 and substituting:

RL

LsRLVoutV

32

1

1

3

1

3

12

1

13

LsRsVin

LsVout

LsCs

LsRsRL

LsRLVout

1

1

3

1

31

132313 2

LsRsVin

LsVout

LsLsRs

LsRsLCsLsLsRs

RL

LsRLVout

Vin

RLLs

RLLsRsLsRsLCsLsLsRsLsRLVout

3

1132313 2

RLLsRsLsRsLCsLsLsRsLsRL

RLLs

Vin

Vout

1132313

32

1323131323

322 LsRsLCsLsLsRsLsLsRsLCsLsRL

RLLs

Vin

Vout

13231122 22 LsRsLCsLsLsRsRLLCsRsRLCsRL

RL

Vin

Vout

1323212231 32 LLCsRsLCRLLCsRsRLCLLsRsRL

RL

Vin

Vout

RsRL

LLCs

RsRL

RsLCRLLCs

RsRL

RsRLC

RsRL

LLs

RsRL

RL

Vin

Vout

13232122

311

1

32

For RRLRs :

1323122312 322 LLCsRLLCsRCLLsR

R

Vin

Vout

or

R

L

R

LRCs

R

L

R

LRCsRC

R

L

R

Ls

Vin

Vout

312

3122

312

1

32

For LLL 21 :

2322 LCsRsRLLCsRsRLCLsRsRL

RL

Vin

Vout

RsRL

LCsLCs

RsRL

RsRLC

RsRL

Ls

RsRLRL

Vin

Vout

2

3221

For RRLRs and LLL 21 :

2211

21

221

21

22

32 LCs

RCs

R

Ls

R

LCsLCs

RC

R

Ls

Vin

Vout

Theoretical Derivation Pi-Filter

Vin Vout

L2

C1 C2

Figure 3. LC Ladder 3rd Order Low Pass Filter


For RRLRs and CCC 31 :

2211

1

2

1

2 CLs

R

LsRCs

Vin

Vout

Theoretical Derivation Pi-Filter


The circuit node equations follow:

2

123

2

11

LsVCs

LsRLVout

RsVin

LsVout

LsCs

RsV

1

2

1

2

11

12

Solving for V2 and substituting:

RL

RLCLsLsRLVoutV

3222

2

RsVin

LsVout

LsCs

RsRL

RLCLsLsRLVout

1

2

1

2

11

1322 2

RsVin

LsVout

RsLs

RsLCsLsRs

RL

RLCLsLsRLVout

1

2

1

2

212322 22

Rs

VinRLRsLs

RsRLRsRLCCLsRsCLRLCLs

LRLRsCLRsRLCLsRLLRsLsRsRL

Vout1

2

3121232

212322224223

22

Rs

VinRLRs

RsRLCCLsRsCLRLCLs

LRLRsCRsRLCsRLRs

Vout13121232

21332

RsRLCCLsRsCLRLCLs

LRLRsCRsRLCsRLRs

RL

Vin

Vout

3121232

21332

RLRs

RLRsCCLs

RLRs

RsCLRLCLs

RLRs

Ls

RLRs

RLRsCCsRLRs

RL

Vin

Vout

312

12322131

1

3

2

For RRLRs :

2

3121322

2131

1

2

1

32 RCCLsCCLs

R

LCRCRs

Vin

Vout

For CCC 31 :

RLRs

RLRsCCLsCLs

RLRs

LRLRsCsRLRs

RL

Vin

Vout

2222

1

1

32

For RRLRs and CCC 31 :

221

1

2

1

232 RCLsCLs

R

LRCs

Vin

Vout

2211

1

2

1

2 CLs

R

LsRCs

Vin

Vout

1.5 MHz Low Pass Filter, 7th Order, Coilcraft P7LP155

The 7th order elliptical LC Ladder Low Pass Filter.

Figure 4. Coilcraft LC Ladder Low Pass Filter

Manufacturer’s frequency response

Coilcraft LC Ladder Low Pass Filter

Test Analysis

A test circuit was built and tested using the network analyzer.

R is assumed to be 50 ohms, L1 and L3 were variable inductors in the range of 0.578 to 0.95 uH and C2 was 4 parallel 100pF (101K) capacitors or 400 pF.

Network Analyzer Measurements

References

[1] Walter G. Jung, IC OP-Amp Cookbook, Howard W. Sams Co. Inc, Indianapoli, IN, 1974.

[2] M.E. Van Valkenburg, Analog Filter Design, Oxford, 1982. ISBN: 0-19-510734-9.

[3] http://www.circuitsage.com/filter.html

[4] http://focus.ti.com/analog/docs/techdocs.tsp?contentType=8&familyId=78&navSection=app_notes

TI Application Notes: Slod006b Sloa093

TI Application Notes on Filtering Active Filter Design Techniques SLOA088 Analysis of the Sallen-Key Architecture (Rev. B) SLOA024 FilterPro MFB and Sallen-Key Low-Pass Filter Design Program SBFA001A Active Low-Pass Filter Design (Rev. A) SLOA049 Using the Texas Instruments Filter Design Database SLOA062 Filter Design in Thirty Seconds SLOA093 Filter Design on a Budget SLOA065 More Filter Design on a Budget SLOA096

special notes: filter design methods spectral power...

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