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
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