Successful application of Active Filters

By Thomas KuehlSenior Applications Engineer

andJohn Caldwell

Applications Engineer

Precision Analog – Linear ProductsTexas Instruments – Tucson, Arizona

A filter’s purpose in life

is to…

• Obtain desired amplitude versus frequency characteristics

or

• Introduce a purposeful phase-shift versus frequency response

or

• Introduce a specific time-delay (delay equalizer)

Common filter applications

Band limiting filter in anoise reduction application

Common filter applications

Common filter applicationsDelay equalization applied to a band-pass filter application

Filter Types

• Low-pass

• High-pass

• Band-pass

• Band-stop, or band-reject

• All-pass

Common filters employed in analog electronics

Filter Types

Low-pass High-pass

Band-pass Band-stop

A low-pass filter has a single pass-band up to a cutoff frequency, fc and the bandwidth is equal to fc

A high-pass filter has a single stop-band 0<f<fc, and pass-band f >fc

A band-pass filter has one pass-band, between two cutoff frequencies fl and fh>fl, and two stop-bands 0<f<fl and f >fh. The bandwidth = fh-fl

A band-stop (band-reject) filter is one with a stop-band fl<f<fh

and two pass-bands 0<f<fl and f >fh

fcfc

fl fh fl fh

Filter TypesAn all-pass filter is one that passes all frequencies equally well

The phase φ(f) generally is a function of frequency

T

Time (s)

0.0 500.0u 1.0m 1.5m 2.0m

Vol

tage

(V

)

-1.00

0.00

1.00

1st-order All-passf =1kHz, Td = 125usphase delay = -45deg

a b

T

Frequency (Hz)

1 10 100 1k 10k

Ga

in (

dB

)

-1m

0

1m

Frequency (Hz)

1 10 100 1k 10k

Ph

ase

[d

eg

]

-180

-90

0

-45deg @ 1kHz

1st-order All-passphase shift -45deg @ 1kHz

a

Phase-shift filter (-45º at 1kHz)

Time-delay filter (159us)T

Frequency (Hz)

1 10 100 1k 10k

Ga

in (

dB

)

0.0

5.0

10.0

15.0

Frequency (Hz)

1 10 100 1k 10k

Ph

ase

[d

eg

]

-450

-300

-150

0

-57.2 deg@ 1kHz

4-th order, 1kHz All-pass159us delay, Av = +4V/V

b

T

Time (s)

10.0m 10.5m 11.0m 11.5m 12.0m

Vol

tage

(V

)

-4

-3

-2

-1

0

1

2

3

4

4-th order, 1kHz All-pass159us delay, Av = +4V/V

ab

Filter Ordergain vs. frequency behavior for different low-pass filter orders

T

Frequency (Hz)

10 100 1k 10k 100k 1M

Ga

in (

dB

)

-80

-60

-40

-20

0

20

-160dB/dec

-120dB/dec

-80dB/dec

-40dB/decFilter Order

2nd 4th 6th 8th

Pass-band Stop-band T

Frequency (Hz)

250.00 538.61 1.16k 2.50k

Gain

(dB

)

-12

-9

-6

-3

0

3

fC (-3dB) 1kHz

typically, one active filter stage is required for each 2nd-order function

Filter Order

T

Frequency (Hz)

1 10 100 1k 10k 100k 1M

Gai

n (d

B)

-100

-80

-60

-40

-20

0

20

-2 slope(-40dB/dec)

-1 slope(-20dB/dec)

+2 slope(40dB/dec)

+1 slope(20dB/dec) 2nd-order Band-pass

2nd-order High-pass2nd-order Low-pass

2nd-order low-pass, high-pass and band-pass gain vs. frequency

Filter Responses

Response Considerations

• Amplitude vs. frequency

• Phase vs. frequency

• Time delay vs. frequency (group delay)

• Step and impulse response characteristics

Filter ReponsesCommon active low-pass filters - amplitude vs. frequency

T

Frequency (Hz)

100 1k 10k 100k

Ga

in (

dB

)

-80

-60

-40

-20

0

20

1kHz, 4th-order low-passresponses, Av = +5V/V

Bessel Butterworth Chebyshev (2dB) Gaussian Linear Phase (0.5deg)

Δ attenuationof nearly 30 dBat 1 decade

T

Time (s)

0.0 500.0u 1.0m 1.5m 2.0m 2.5m

Out

put

-500.0m

0.0

500.0m

1.0

1.5

100us pulse Bessel Butterworth Chebshev (2dB) Gaussian Linear phase (0.5deg)

T

Frequency (Hz)

100 1k 10k 100k

Pha

se [

deg]

-360

-270

-180

-90

0

Bessel Butterworth Chebyshev (2dB) Gaussian Linear phase (0.5deg)

Filter Reponses – phase and time responses1 kHz, 4th-order low-pass filter example

Phase vs. frequency Impulse response

Group Delay

Active filter topology and

response development

Why Active Filters?

• Inductor size, weight and cost for low frequency LC filters are often prohibitive

• Magnetic coupling by inductors can be a problem

• Active filters offer small size, low cost and are comprised of op-amps, resistors and capacitors

• Active filter R and C values can be scaled to meet electrical or physical size needs

RS1 1k L1 225m

C1 220n-

+ VpasRL1 1k

R1 2.72k R2 19.8k

-

+

IOP1

C1 10n

C2 47n

-

+ Vact

+

VG1

-

+

-

+VCV1

RL2 1k

1kHz Passive LP

1kHz Active LP

Source

Impedance Load

Impedance

T

Vact Vpas

Passive and Activereponses are identical

Frequency (Hz)

10 100 1k 10k 100k 1M

Gai

n (d

B)

-100

-80

-60

-40

-20

0

20

Gain vs. FrequencyPassive and Activereponses are identical

Vact Vpas

T

Frequency (Hz)

10 100 1k 10k 100k 1M

Pha

se [

deg]

-180

-135

-90

-45

0

Phase vs. Frequency

Vact Vpas

A comparison of a 1kHz passive and active 2nd-order, low-pass filter

• At fc C1 & C2 impedances are equal to R1 and R2 impedances. Positive feedback is present and Q enhancement occurs

• Higher Q is attainable with the controlled positive feedback localized to the cutoff frequency

• Q’s greater than 0.5 are supported allowing for specific filter responses; Butterworth, Chebyshev, Bessel, Gaussian, etc

Comparing 2nd-order passive RC and an active filters

Resource: Analysis of Sallen-Key Architecture, SLOA024B,July 1999, revised Sept 2002, by James Karki

• Cascaded 1st-order low-pass RC stages

• Overall circuit Q is less than 0.5

• Q approaches 0.5 when the impedance of the second is much larger than the first; 100x

• Common filter responses often require stage Qs higher than 0.5

-

+

IOP1R1 4.64k R3 14k

C1

22n

C2 2.2n

R2 9.31k

Vo_MFB

+

VG1

-

+

IOP1

R4 10kR3 10k

C3

10n

R2 13.7kR1 2.1k

C2

22n

Vo_SK

+

VG1

Two popular single op-amp active filter topologies2nd-order implementations

Multiple Feedback (MFB) low-pass

• supports common low-pass, high-pass and band-pass filter responses

• inverting configuration

• 5 passive components + 1 op-amp per stage

• low dependency on op-amp ac gain-bandwidth to assure filter response

• Q and fn have low sensitivity to R and C values

• maximum Q of 10 for moderate gains

Sallen-Key (SK) low-pass

• supports common low-pass, high-pass and band-pass filter responses

• non-inverting configuration

• 4-6 passive components + 1 op-amp per stage

• high dependency on op-amp ac gain-bandwidth to assure filter response

• Q is sensitive to R and C values

• maximum Q approaches 25 for moderate gains

Popular Active Filter Topologies2nd-order implementations

Pass Z1 Z2 Z3 Z4 Z5

Low R1 C2 R3 R4 C5

High C1 R2 C3 C4 R5

Band R1 R2 C3 C4 R5

Pass Z1 Z2 Z3 Z4 Z5

Low R1 R2 C3 C4 na

High C1 C2 R3 R4 na

Band R1 C2 R3 R4 C5

Component type for each filter topology

Poles and Zero locations in the s-plane establish the filter gain and phase response

third-order low-pass transfer function k

H(s) = s3/ω1ω2

2+ s2(ω1ω2+1/ω22)+s(1/ω1+1/ω2)+1

ω = 2πf, k = gain

Response pole σ jω

Butterworth Real -1.0 0

Complex -0.5 0.866

Complex conj. -0.5 -0.866

1 dB Chebyshev Real -0.455 0

Complex -0.227 0.888

Complex conj. -0.277 -0.888

Bessel Real -1.346 0

Complex -1.066 1.017

Complex conj. -1.066 -1.017

σ

jω

Transfer function roots plotted in s-plane

All pole filter responses s = -σ ± jω

Third-order low-pass transfer function from Burr-Brown, Simplified Design of Active Filters, Function Circuits – Design and Application, 1976, Pg. 228Response table data from, High Frequency Circuit Design, James K. Hardy, Reston Publishing Company, 1979, table 4A-4, Pg. 152

Complex frequency and active filtersthe s-plane provides the amplitude response of a filter

Damping factor (ζ) determines amplitude peaking around the damping frequency fd

ζ = cos θ

Q the peaking factor is related to ζ by

Q = 1/(2ζ)

= 1/(2cos θ)

The damping frequency fd is related to the un-damped natural frequency fn by

fd = fn (1- ζ2 )½

= fn [1- 1/(4Q2)]½ (rect form)

fd = fn sin θ (polar form)

The pole locations

p1, p2 = -ζ fn ± j fn (1-ζ2)½Adapted from High Frequency Circuit Design, James K. Hardy, Reston Publishing Company, 1979, Appendix 4A-1, s-PLANE

s-planeComplex frequency plane

The damping frequency fd approaches the undamped natural frequency fn as the Q increases

Adapted from High Frequency Circuit Design, James K. Hardy, Reston Publishing Company, 1979, Appendix 4A-1, s-PLANE

-3 dB

fd

fn

Q ζ fd Hz

10 0.05 998.7

5 0.10 995.0

2.5 0.20 978.3

1.67 0.30 953.9

1.25 0.4 916.5

1.0 0.5 866

0.83 0.60 800

0.707 0.707 704.0

The stage Q (1/2ζ) affect the time and phaseresponses of the filter

Increasing Q higher peaking High Q = longer settling time

Decreasing Q - more linear phase change

Adapted from Google Images: gnuplot demo script: multiplt.dem

Filter sections are cascaded to produce the intended response the cutoff frequency fc , gain, roll-off, etc is the product of all stages

• The overall response at Vo is the product of all filter stage responses

• Each stage has unique Av, fn, Q

• The resulting filter has an fc of 10 kHz, with a pass-band gain of 10 V/V

• The 10 kHz pass-band bandwidth is defined by the 1 dB ripple

• stage 3 gain was manipulated to be low because of its high Q

• Doing so relaxes the GBW requirement – more on this later

1 dB Chebyshev, 6th-order LP filter, fc = 10 kHz, Av = 10 v/v

T

Frequency (Hz)

1k 10k 100k

Ga

in (

dB

)

-40

-20

0

20

40

stg 1 stg 2 stg 3 all

Active filter synthesis programsto the rescue!

• Modern filter synthesis programs make filter development fast and easy to use; no calculations, tables, or nomograms required

• They may provide low-pass, high-pass, band-pass, band-reject and all-pass responses

• Active filter synthesis programs such as FilterPro V3.1 and Webench Active Filter Designer (beta) are available for free, from Texas Instruments

• All you need to provide are the filter pass-band and stop-band requirements, and gain requirements

• The programs automatically determine the filter order required to meet the stop-band requirements

• FilterPro provides Sallen-Key (SK), Multiple Feedback (MFB) and differential MFB topologies; the Webench program features the SK and MFB

• Commercially available programs such as Filter Wiz Pro provide additional, multi-amplifier topologies suitable for low sensitivity, and/or high-gain, high-Q filters

Operational amplifierGain-bandwidth (GBW) product

26

Operational amplifier gain-bandwidth productan important ac parameter for attaining accurate active filter response

BW= 220kHz

Gain= 100 = 40dB

In this example, for anygain from 0dB to Avol.GBW Gain BW=

where GBW -- Gain Bandwidth in HzGain -- closed loop voltage gainBW -- Bandwidth in Hz

For example

Gain 100=

Closed Loop Bandwidth is calculated:

BWGBW

Gain=

22MHz

100= 220kHz=

Op-amp gain-bandwidth requirements

The active filter’s op-amps should:

• Fully support the worst-case, highest frequency, filter section GBW requirements

• Have sufficiently high open-loop gain at fn for the worst-case section

The operational-amplifier gain-bandwidth requirements

TI’ s FilterPro calculates each filter section’s Gain-Bandwidth Product (GBW) from:

GBWsection = G ∙ fn ∙ Q ∙ 100

where: G is the section closed-loop gain (V/V) fn is the section natural frequency Q is stage quality factor (Q = 1/2ζ)

100 (40 dB) is a loop gain factor

The operational-amplifier gain-bandwidth requirements

Op-amp closed loop gain error

• The filter section’s closed-loop gain (ACL) error is a function of the open-loop gain (AOL) at any specified frequency

* equivalent noise gain ACL

• For example, select AOL to be ≥100∙ACL at fn for ≤1% gain error

AOL / ACL*

Gain error ∆

Gain error %

104 10-4 0.01

103 10-3 0.10

102 10-2 1.00

101 10-1 10.0

The operational-amplifier gain-bandwidth requirement an example of the FilterPro estimation

FilterPro’s GBW estimation for the worst-case stage yields:

GBW = G ∙ fn ∙ Q ∙ 100

GBW = (2V/V)(10kHz)(8.82)(100) = 17.64MHz

vs. 16.94 MHz from the precise determination – see Appendix for details

Let FilterPro estimate the minimum GBW for a 5th-order, 10 kHz (fc) low-pass filter having a Chebyshev response, 2 V/V gain and a 3 dB pass-band ripple

Operational amplifier gain-bandwidth effectsthe Sallen-Key topology

• The operational amplifier gain-bandwidth (GBW) affects the close-in response

• It also affects the ultimate attenuation at high frequency

-

+ +

U2 OPA340

-

+ Vo

R5 5.11k R6 931

C3 22n

C4 2.4n

R7 2.49k R8 22.6k

+

VG1

+

VS1 5

+

VS2 2.5

Sallen-Key - Butterworth 10 kHz, 2nd-order low-pass, Av = +10 V/V

T

Frequency (Hz)

1k 10k 100k 1M 10M

Ga

in (

dB

)

-60

-40

-20

0

20

40

OPA170 GBW 1.2MHz OPA314 GBW 2.7 MHz OPA340 GBW 5.5 MHz OPA140 GBW 11 MHz

Op-amp fH Hz dBOPA170 90 k -21.8OPA314 110 k -23.5OPA340 260k -38.1OPA140 428 k -44.3

FilterPro GBW 7.1 MHz

Operational amplifier gain-bandwidth effectsthe Multiple Feedback (MFB) topology

• The MFB shows much less GBW dependency than the SK

• Close-in response shows little effect

• Insufficient GBW affects the roll-off at high frequencies

• The lowest GBW device (1.2 MHz) produces a gain deviation about 50-60 dB down on the response

• A GBW ≥ 7 MHz for this example provides near ideal roll-off

+

VS1 2.5

+

VS2 2.5

R1 1.13k R2 1.02k

C1 22n+

VG1

C2 1n

R3 11.3k

-

+Vo

-

+ +

U1 OPA340

Multiple Feedback - Butterworth 10 kHz, 2nd-order low-pass, Av = +10 V/V

T

Frequency (Hz)1k 10k 100k 1M 10M

Ga

in (

dB

)

-80

-60

-40

-20

0

20

40

OPA170 GBW 1.2 MHz OPA314 GBW 2.7 MHz OPA340 GBW 5.5 MHz OPA140 GBW 11 MHz

An active filter response issueWhat a customer expected from their micro-power, 50 Hz active low-pass filter (SK)

• A customer designed a 50 Hz low-pass filter using the FilterPro software:

– Gain = 1 V/V

– Butterworth response (Q = 0.71)

– The Sallen-Key topology was selected

• The FilterPro and TINA-TI simulations using ideal operational amplifiers models produced ideal results

• Note FilterPro recommended an operational-amplifier with a gain-bandwidth product (GBW) of 3.55 kHz

-

+ VLP_IR4 100k

R1 2.94k R2 1.58k

C1 2.2u C2 1u

-

+

IOP1

+

VG1

50 Hz Butterworth Low-pass with ideal operational-amplifier

T

Frequency (Hz)

1 10 100 1k 10k

Pha

se [

deg]

-180

-135

-90

-45

0

2nd-order Butterworth -90 deg at 50 Hz

T

Frequency (Hz)

1 10 100 1k 10k

Gai

n (d

B)

-100

-80

-60

-40

-20

0

20

2nd-order Butterworth -3 dB at 50 Hz

An active filter response issueWhat the customer observed with a micro-power, 50 Hz active low-pass filter (SK)

• Normal low-pass response below and around the 50 Hz cutoff frequency

• The -40 dB/dec roll-off fails about a decade beyond the 50 Hz cutoff frequency

• The gain bottoms out at about 775 Hz and then trends back up

• Note that the OPA369 does meet the minimum GBW specified by FilterPro, 3.55 kHz . Its GBW is about 8 to 10 kHz

T

Frequency (Hz)

1 10 100 1k 10k 100k

Gain

(dB

)

-60

-40

-20

0

10 kHz-9.5 dB10 kHz-9.5 dB

775 Hz-41.8 dB

Sallen-Key LP2nd-orderfc (-3dB) = 50 Hz

V1 3-

+ Vsk_2ndR3 100k

-

++

3

1

5

4

2

U1 OPA369

R1 2.94k R2 1.58k

C1 2.2u C2 1u

+

VG1

+

VS1 1.5

OPA369 50 Hz Butterworth Low-pass Filter

GBW~8 kHz

T

Frequency (Hz)

100m 1 10 100 1k 10k 100k

Zo

(ohm

s)

10

100

1k

10k

100k

1M

0nA 200n

400n 600n 800n

1uA

The real operational amplifier can have a complex open-loop output impedance Zo

For the OPA369 FET Drain output stage Zo:

• Changes with output current

• is low, <10 Ω and resistive below 1 Hz

• increases from tens-of-ohms to tens, or hundreds of kilohms, from 10 Hz to 10 kHz

• Is complex, resitive plus inductive (R+jX), from 1 Hz to 10 kHz

• Becomes resistive again above 10 kHz, the unity gain frequency

• The hi-Z behavior is reduced by closing the loop but Zo still alters the expected filter response

R+j0

R+j0

R+jX

Unity gain

-

+

OPA369

C1 2.2u

R1 2.94k R2 1.58k

C2 1u

+

VG1

-

+ VLPRL 100k

Complex Zo

Net affect on response due to operational-amplifier complex Zo a result similar to low GBW fold-back, but with added peaking

OPA369 Zo-related altered response

• Adding a load resistor may reduce the peaking but doesn’t resolve roll-off fold-back

• The output offset-related current flow through RL significantly reduces Zo

• If the operational amplifier has low offset the Zo can remain high and the problem remains

• The added load resistor may draw more current than the op-amp defeating the purpose of using a ultra-low power op-amp

T

Frequency (Hz)

10 100 1k 10k 100k 1M

Gai

n (d

B)

-50

-40

-30

-20

-10

0

10

Rload 100k 1M 10M 100M 1G

V1 3-

+ Vsk

-

++

3

1

5

4

2

U1 OPA369

R1 2.94k R2 1.58k

C1 2.2u C2 1u

+

VG1

+

VS1 1.5

V2 0

RL 1G

OPA369 50 Hz Sallen-Key LP

Active filter sensitivity to source impedanceand components

The effect of source impedance on filter response

• Most active filter designs assume zero source impedance

• Source impedance appears in series with the filter input

• The impedance will affect the filter response characteristics

• The multiple-feedback topology can develop gross gain, bandwidth and phase errors

• The Sallen-Key maintains its pass-band gain better, but the cutoff frequency and Q can change

• Actual results will vary with the RC values and pass-band characteristics

• Active filters maintain their response when preceded by a low impedance source such as an op-amp amplifier

5 kHz Butterworth Low-pass, G = 10 V/V

V+/2 V+

V+

V+/2

V+/2

V+

V+/2

V+/2

R1 511 R2 665

R3 5.11k

C1 3.6n

C2 82n

-

+ +

U1 OPA340

-

+ +

U2 OPA340

R7 2.49k R8 22.6k

R9 1.02k R10 187

C5 24n C6 220n

+

VS1 2.5

+

VS2 2.5

Rs 50

+

VG1

-

+ Vmfb

SW

1 S

W2

-

+ Vsk

Multiple-Feedback

Sallen-Key

signal source with Rs

The affect of source impedance on filter response5 kHz Butterworth Low-pass, G = 10 V/V

T

Frequency (Hz)

100 1k 10k 100k

Ga

in (

dB

)

0

6

12

18

24

Vsk 0 Ohm Vsk 250 Ohm Vsk 500 Ohm Vsk 1k Ohm Vmfb 0 Ohm Vmfb 250 Ohm Vmfb 500 Ohm Vmfb 1k Ohm

MFB Rs = 1000, Av = 3.4 V/V

MFB Rs = 500, Av = 5.1 V/V

MFB Rs = 250, Av = 6.7 V/V

MFB Rs = 0, Av = 10 V/V

Sallen-Key

SK Av = 10 V/V

MFB

Component sensitivity in active filtersa vast subject of its own

• Passive component variances and temperature sensitivity, and amplifier gain variance will alter a filter’s responses: fc ,Q, phase, etc.

• Each topology and filter BOM will exhibit different levels of sensitivity

• Mathematical sensitivity analysis provides a method for predicting how sensitive the filter poles (and zeros) are to these variances

• The sensitivity analysis for a filter topology is based on the classical sensitivity function

• This equation provides the per unit change in y for a per unit change in x. Its accuracy decreases as the size of the change increases

• An example an analysis - if the Q sensitivity relative to a particular resistor is 2, then a 1% change in R results in a 2% change in Q

• The 1970’s Burr-Brown, “Operational Amplifiers” and “Function Circuits” books provide the sensitivity analysis for many MFB and SK filter types

• A modern approach is to use a circuit simulator’s worst-case analysis capability and assigning component tolerances relative to projected changes

• Low tolerance/ low drift resistors (1% and 0.1%, ±20 ppm/°C) and low tolerance/ low drift C0G and film capacitors (1% to 5%, ±20 ppm/°C) will reduce sensitivity compared to other component types

• Often, filters having two or three op-amp per section have low sensitivity

xx

yy

Sx

yx Δ

Δ

Δlim 0

Component sensitivity in active filtersa MFB band-pass filter component tolerance simulation

T

Frequency (Hz)

15k 20k 25k 30k 35k

Gai

n (d

B)

-12.0

-6.0

0.0

6.0

12.0

18.0

14 dB

fc27.6 kHz22.6 kHz

+/- 5% resistors+/-5% capacitors

25 cases

T

Frequency (Hz)

15k 20k 25k 30k 35k

Gai

n (d

B)

-12.0

-6.0

0.0

6.0

12.0

18.0 fc

25.4 kHz24.4 kHz14 dB

+/-0.1% resistors

+/-2% capacitors 25 cases

VCC 5

-

+ +

U2 OPA320

+

VS1 2.5

+

VG1

R1 5.76k

R2 147

C1 2.2n

C2 2.2n

R3 57.6k

-

+ Vbp

25 kHz Bessel Bandpass Q = 10, AV = 5 V/V

Componenttolerance

Center Freqvariance

Gainvariance

5% resistors5% capacitors

Δ 5 kHz Δ 1.7 dB

0.1% resistors2% capacitors

Δ 1 kHz Δ 0.3 dB

Noise and distortion considerationsin active filters

Comparison of Filter Topologies: Noise Gain• “Noise gain” is the amplification applied

to the intrinsic noise sources of an amplifier

• Sallen-Key and Multiple Feedback Filters have different noise gains– Different RMS noise voltages for the

same filter bandwidth!

• TINA-TI is a useful tool for determining the noise gain of a complex circuit.

– Insert a voltage generator in series with the non-inverting input of the amplifier

– Ground the filter input– Perform an AC transfer characteristic

analysis

Measuring the noise gain of a Sallen-Key low pass filter

Measuring the noise gain of a MFB low pass filter

Noise Gain Comparison

• FilterPro was used to design 2, 1kHz Butterworth lowpass filters

– 1 Sallen-Key topology– 1 Multiple Feedback topology

• The signal gain of both circuits was 1

• Tina-TI was used to determine the noise gain of the circuits from 1Hz to 1MHz

• Within the passband, the MFB filter has 6dB higher noise gain

– This is because it is an inverting topology

• Noise gain above the corner frequency quickly decreases

• The noise gain for both circuits peaks at the corner frequency of the filter

1 10 100 1000 10000 100000 10000000

1

2

3

4

5

6

7

8

9

10

Noise Gain Comparison of 1kHz Butterworth Lowpass Filters

Multiple Feedback Sallen Key

Frequency (Hz)

No

ise

Gai

n (

dB

)

Sallen-Key Multiple Feedback

Noise Gain at the Filter Corner Frequency• The magnitude of the noise

gain peak is dependant upon the Q of the filter– Higher Q filters have

higher peaking in their noise gain.

• The peak in noise gain may significantly affect total integrated noise– This depends on how wide

a bandwidth noise is integrated over

• The table displays the total integrated noise of 1kHz Sallen-Key lowpass filters of different Q’s– OPA827 simulation model– 100 Ohm resistors used in

all circuits (only capacitors changed)

1 10 100 1000 10000 1000000

1

2

3

4

5

6

7

8

9

10

Noise Gain Comparison of 1kHz Sallen-Key Lowpass Filters

Bessel (Q: 0.58) Butterworth (Q: 0.707) Chebyshev 1dB (Q: 0.957)

Frequency (Hz)

No

ise

Gai

n (

dB

)

Topology QNoise Voltage

(2kHz Bandwidth)

Noise Voltage (20kHz

Bandwidth)

Noise Voltage (200kHz

Bandwidth)Bessel 0.58 249.8 nVrms 615 nVrms 1.733 uVrms

Butterworth 0.707 303.1 nVrms 628.4 nVrms 1.737 uVrmsChebyshev 1dB 0.957 393.1 nVrms 693.7 nVrms 1.762 uVrms

Noise considerations in an active filter an OPA376 inverting amplifier is compared in a 2nd-order low-pass

T

Frequency (Hz)

10 100 1k 10k 100k 1M 10M

Out

put

nois

e (V

/Hz½

)

5n

50n

500nOPA376 output noiseAv = -10 V/V

Vamp

Vlp

frequency Amp en (nV/√Hz)

Filter en (nV/√Hz)

ratio

10 kHz 91 119 0.76 : 1

100 kHz 89 15 6 : 1

1 MHz 42 7.4 5.6 : 1

-

++

4

3

5

1

2

U1 OPA2376

V1 2.5

V2 2.5

R1

10k

Vamp

+

VIN

R2 1k

R3 10k

-

++

4

3

5

1

2

U2 OPA2376

V3 2.5

V4 2.5

R4

10k

Vlp

R5 1.13k

R6 11.25k

R7 1.02k

C1

22n

C2 1n

Inverting amplifier vs. a 2nd-order Butterworth,10 kHz LP filter (Av = -10 V/V in both cases)

Noise considerations of an active filter an OPA376 inverting amplifier is compared in a 2nd-order band-pass application

-

++

4

3

5

1

2

U1 OPA2376

V1 2.5

V2 2.5

R1

10k

+

VIN

R2 1k

R3 10k

-

++

4

3

5

1

2

U2 OPA2376

V3 2.5

V4 2.5

R4

10k

Vbp

R5 723.4

R6 14.47k

R7 38.1

C1 22n

C2 22n

Vamp

Inverting amplifier vs. a 2nd-order Butterworth,10 kHz BP filter, Q = 10 (Av = -10 V/V in both cases)

frequency Amp en (nV/√Hz)

Filter en (nV/√Hz)

ratio

1 kHz 93 23 4 : 1

10 kHz 91 1390 1 :15

100 kHz 90 16 5.6 : 1

1 MHz 42 8.2 5.1 : 1

T

Q = 10

Q = 5

Frequency (Hz)

10 100 1k 10k 100k 1M 10M

Out

put

nois

e (V

/Hz½

)

1.5n

15.0n

150.0n

1.5u

Q = 10

Q = 5

OPA376 output noiseAv = -10V/V

Vamp

Vbp Q = 10

For Q = 10

Total Harmonic Distortion and Noise Review

• Total Harmonic Distortion and Noise (THD+N) is a common figure of merit in many systems– Intended to “quantify” the amount of

unwanted content added to the input signal of a circuit

– Consists of the sum of the amplitudes of the harmonics (integer multiples of the fundamental) and the RMS noise voltage of the circuit

– Often presented as a (power or amplitude) ratio to the input signal

• Harmonics of the fundamental arise from non-linearity in the circuit’s transfer function.– Integrated circuits AND passive

components can cause this

• Intrinsic noise is created in integrated circuits and resistances

100(%)

22

22

f

i ni

V

VVNTHD

Vi: RMS voltage of the ith harmonic of the fundamental (i=1,2,3…)

Vn: RMS noise voltage of the circuitVf: RMS voltage of the fundamental

0 2000 4000 6000 8000 10000 12000 14000-160

-140

-120

-100

-80

-60

-40

-20

0

Spectrum of a 500Hz Sine Wave

Frequency (Hz)

Am

pli

tud

e (d

B)

Fundamental

Harmonics

Noise

Distortion from Passive Components

• A 1kHz Sallen-Key lowpass filter was built using an OPA1612, and replaceable passive components.– Component values were

chosen such that both C0G and X7R capacitors were available

– Thin Film resistors in 1206 packages were used

• An Audio Precision SYS-2722 was used to determine the effects of capacitor type on measured THD+N– THD+N was measured from

20Hz to 20kHz– Harmonic content of a 500

Hz sine wave was also compared

• THD+N is noise dominated– Increases as the filter

attenuates the signal

20 200 2000 20000-120

-100

-80

-60

-40

-20

0

20Frequency Response and THD+N

1206 C0G Filter ResponseFrequency (Hz).

TH

D+

N (

dB

V)

0 2000 4000 6000 8000 10000 12000 14000-160

-140

-120

-100

-80

-60

-40

-20

0

Spectrum of a 500Hz Sine Wave

Frequency (Hz)

Am

plit

ud

e (d

B)

Capacitor Dielectric Effects

• 1206 C0G capacitors were replaced with 1206 X7R capacitors and the THD+N was re-measured– Signal level was 1Vrms– All caps are 50V rated– Minimum of 15dB

degradation of THD+N inside the filter’s passband

– Maximum of almost 40dB degradation of THD+N

• The spectrum of a 500Hz sine wave was also compared– X7R shows a large

number of harmonics – Odd order harmonics

dominate the spectrum

20 200 2000 20000-120

-110

-100

-90

-80

-70

-60

-50

-40Distortion Comparison of Different Capacitor Types

1206 X7R 1206 C0GFrequency (Hz)

TH

D+

N (

dB

V)

0 5000 10000 15000 20000 25000-160

-140

-120

-100

-80

-60

-40

-20

0 Spectrum of a 500Hz Sine Wave

Frequency (Hz)

Am

pli

tud

e (d

B)

Package Size Effects

• 1206 X7R capacitors were replaced with 0603 X7R capacitors and distortion was measured again– Signal level was 1Vrms– All tested capacitors have

a 50V rating

• Distortion increases for smaller package sizes!

• The spectrum of a 500Hz sine wave was again examined– Both odd and even order

harmonics dominate– Odd order harmonics still

dominate– 0603 capacitors produce

harmonics above 20kHz!

20 200 2000 20000-120

-110

-100

-90

-80

-70

-60

-50

-40Distortion Comparison of Different Capacitor Types

0603 X7R 1206 X7R 1206 C0G

Frequency (Hz)

TH

D+

N (

dB

V)

Capacitor Distortion and Signal Level

• As previously mentioned, capacitor distortion increases with electric field intensity– Worse at signal levels– Worse for smaller packages

• Changing the signal level is a simple way to determine the source of distortion

• If the circuit is noise dominated the plot will have a slope (m) of:

• Distortion from passive components will INCREASE with higher signal levels

0.001 0.01 0.1 1 10-120

-100

-80

-60

-40

-20

0

THD+N of a 500Hz Sine Wave vs. Signal Level

0603 X7R 1206 X7R 0805 C0G 1206 C0G

Signal Level (Vrms)

TH

D+

N (

dB

V)

Signal

noise

V

Vm

20 200 2000 20000-120

-110

-100

-90

-80

-70

-60

-50

-40Distortion Comparison of Different Capacitor Types

0603 X7R 1206 X7R 1206 C0G

Frequency (Hz)

TH

D+

N (

dB

V)

Capacitor Distortion Over Frequency• Tina-TI was used to measure

the voltage across each capacitor over frequency– The sum of the two voltages

is plotted in green– Diagram below indicates

measurement points

• The maximum voltage appears below the corner frequency– This also correlates well to

the measured peak in distortion

-

+

C122n

C2 10n

R1

7.87kR2

14.7k

+

VS

-+

VC1

-

+VC2

20 200 2000 200000

0.2

0.4

0.6

0.8

1

1.2Capacitor Voltages in a 1kHz Sallen-Key Lowpass Filter

VC2 VC1 Combined

Frequency (Hz)

Vo

ltag

e (V

rms)

Putting it all together

Achieving optimum active filter performance

Capacitors

• Use quality C0G or film dielectric for low distortion

• Type C0G has a low temperature coefficient (±20 ppm)

• Lower tolerance, 1-2%, assures more accurate response

• Higher order filters require ever lower tolerances for accurate response

Resistors• Use quality, low tolerance

resistors • 1 % and 0.1% reduce filter

sensitivity• Lower tolerance assures more

accurate response• Low temperature coefficient

reduces response change with temperature

• Higher order filters require ever lower tolerances for accurate response

Operational Amplifier• Use required GBW - especially for the Sallen-Key• Be sure to consider the amplifier noise• High Zo effects can distort response • Higher amplifier current often equates to lower Zo

and wider GBW• Consider dc specifications – especially bias

current

Signal source

• Zs→ 0 Ω

• An op-amp driver with low closed-loop gain often provides a low source impedance

Appendix

The source of the peakingamplifier complex Zo

• R+jX region of Zo exhibits Henries of inductance and kilohms of resistance

• Here a higher current, lower Zo op-amp has a complex Zo added to its output path

• Estimated R and L values have been taken from the OPA369 Zo curves

• Although the peaking frequency isn’t the same the mechanism is demonstrated

V1 3-

+ Vsk

R1 2.94k R2 1.58k

C1 2.2u C2 1u

+

VG1

+

VS1 1.5

V2 0

RL 1M

-

+ +

U1 OPA234E

Rout 10k Lout 2.2

OPA234 50 Hz Sallen-Key LP

Complex Zo R +jX

T

Frequency (Hz)

10 100 1k 10k 100k 1M

Gai

n (d

B)

-40

-30

-20

-10

0

10

OPA234 + R+jx IQ = 350 uA

The operational amplifier gain-bandwidth requirement - a more precise determination

For a 2nd-order stage in an nth-order filter:

GBW = 100•ACL(fc/ai)√[(Qi2-0.5)/(Qi

2-0.25)]

Where: ACL is the closed-loop gain (V/V)fc is the filter cutoff frequencyai is the filter section coefficientQi is the filter section Q

Example:

Determine the recommended GBW for a 5th-order, 10 kHz (fc) low-pass filter having a Chebyshev response, 2 V/V gain and a 3 dB pass-band ripple

Source: Op-amps for everyone, chapter 16, by Thomas Kugelstadt

The operational amplifier gain-bandwidth requirements - a more precise determination

Select constants from highest

‘Q’ stage

The operational amplifier gain-bandwidth required - a more precise determination

ACL = 2 V/V, fC = 10 kHz

Coefficients from table: ai = 0.1172, Qi = 8.82

GBW = 100 ACL(fC/ai)√[(Qi2-0.5)/(Qi

2-0.25)]

GBW = 100 2(10kHz/0.1172)√[(8.822-0.50)/(8.822-0.25)]

GBW = 16.94MHz