unusual filter ckts
DESCRIPTION
Unusual Filter CircuitsTRANSCRIPT
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Some Interesting Filter Design Configurationsand Transformations Normally
Not Found in The General Literature
Speaker:Arthur Williams – Chief Scientist Telebyte Inc.
Tuesday September 27, 2005
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TOPICS
• Design of D-Element Active Low-Pass Filters and a Bi-directional Impedance Converter for Resistive Loads
• Active Adjustable Amplitude and Delay Equalizer Structures
• High-Q Notch filters
• Q-Multiplier Active Band-Pass filters
• A Family of Zero Phase-Shift Low-Pass Filters
• Some Useful Passive Filter Transformations to Improve Realizability
• Miscellaneous Circuits and “Tricks”
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Frequency and Impedance Scaling from Normalized Circuit
Frequency Scaling
frequency reference existingfrequency reference desiredFSF =
Normalized n = 3 Butterworth low-pass filter normalized to 1 rad/sec : (a) LC filter; (b) active filter; (c) frequency response
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Denormalized low-pass filter scaled to 1000Hz: (a) LC filter; (b) active filter; (c) frequency response.
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Impedance scaling can be mathematically expressed as
R′ = ZxR
L′ = ZxL
C’= CZ
Frequency and impedance scaling are normally combined into one step rather than performed sequentially. The denormalized values are then given by
L′ = ZxL/FSF
C’=Z x FSF
C
Impedance-scaled filters using Z=1K : (a) LC filter; (b) active filter.
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Design of D-Element Active Low-Pass Filters and a Bi-Directional Impedance Converter for Resistive Loads
Generalized Impedance Converters (GIC)
Z11= Z2Z4
Z1Z3Z5By substituting RC combinations for Z1 through Z5a variety of impedances can be realized.
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GIC Inductor Simulation
sCR1R3R5
R2
Z11=
If Z4 consists of a capacitor having an impedance 1/sC where s=јωand all other elements are resistors, the driving point impedance becomes:
The impedance is proportional to frequency and is therefore identical to an inductor having an inductance of:
CR1R3R5
R2
L=
Note: If R1 and R2 and part of a digital potentiometer the value of L can be digitally programmable.
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D Element
If both Z1 and Z3 are capacitors C and Z2,Z4 and Z5 are resistors,the resulting driving point impedance becomes:
R5
s2C2R2R4Z11=
An impedance proportional to 1/s2 is called a D Element.
C2R2R4R5
1s2DZ11= where: D =
If we let C=1F,R2=R5=1 Ω and R4=R we get D=R so:
1s2RZ11=
If we let s=јω the result is a Frequency Dependant Negative Resistor FDNR
1-ω2RZ11=
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D Element Circuit
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Rule
A transfer function of a network remains unchanged if all impedances are multiplied (or divided) by the same factor. This factor can be a fixed number or a variable, as long as every impedance element that appears in the transfer function is multiplied (or divided) by the same factor.
The 1/S transformation involves multiplying all impedances in a network by 1/S.
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The 1/S Transformation
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Design of Active Low-Pass filter with 3dB point at 400Hz using D Elements
Normalized Low-Pass filter 1/S Transformation
Realization of D Element Frequency and Impedance Scaled Final Circuit
Filter is Linear Phase ±0.5° Type
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Elliptic Function Low-Pass filter using GICs
Normalized Elliptic Function FilterC11 20 θ=75°N=11 Rdb=0.18dB Ωs=1.0353 60.8dB
Requirements: 0.5dB Maximum at 260Hz60dB Minimum at 270HzSteepness factor=1.0385
NormalizedElliptic Function Filter
1/S Transformation
Realization of D Elements
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Frequency and Impedance Scaled Final Circuit
Note: 1 meg termination resistor is needed to provide DC return path.
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Bi-Directional Impedance Converter for Matching D Element Filters RequiringCapacitive Loads to Resistive Terminations
Rs+ +
CGICRs
R RR R
Value of R ArbitraryRs is source and load resistive terminationsCGIC is D Element Circuit Capacitive Terminations
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Active Amplitude and Delay Equalizer Structures
Σ1
+ K
Fr Freq
Out
Freq
1
K=2
K=10
2
K=3
T(s)
In Out
Eout = T(s)K-1EInT(s)=1
K=2 corresponds to the all-pass case
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T(s) is a second-order bandpass network having a gain of +1 at center frequency Fr
ωr
Qs
+ ωr2s2 +
ωr
Qs
T(s)=ωr= √α2+β2
ωr= 2πFr
Q=ωr
2αAs a function of Q the circuit has a group delay as follows:
The peak delay is equal to
Tgd,max= ωr
4Q
Where K=2 and Q >2
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•This architecture can be used as an amplitude equalizer by varying K.
•If K=2 the circuit can be used as a delay equalizer by varying Q.
•By varying both K and Q this architecture can be used for both amplitude and delay equalization.
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5
2.414
2
1.707
1.25
2.414
1.707
2
There will be an interaction between amplitude and delay as K is varied to change the amount of
amplitude equalization.
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Adjustable Delay Equalizer
Tgd,max=4R1C
R2=R3=ωrC
1C,R,R′ and R″can be any convenient values
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Adjustable Delay and Amplitude Equalizer
By introducing a potentiometer into the circuit amplitude equalization can also be achieved The amplitude equalization at ωr is given by:
Adb= 20 Log (4K-1)
Where K from 0.25 to 1 covers an amplitude equalization range of - ∞ to +9.54 dB
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To extend the amplitude equalization range beyond +9.54 dB the following circuit can be used. The amplitude equalization at ωr given by:
Adb= 20 Log (2K-1)
where a K variation from 0.5 to ∞ results in an infinite range of equalization capability. In reality ± 12dB has been found to be more than adequate so K will vary from 0.626 to 2.49.
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Simplified Adjustable Amplitude Equalizer
The following circuit combines a fixed Q bandpass section with a summing amplifier to provide a low complexity adjustable amplitude equalizer. The design equations are given by:
Q≥0.707
Where K will range from 0 to 1 for an infinite range of amplitude equalization.
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To compute the desired Q first define fb corresponding to one-half the boost(or null) desired in dB.
Then:
Q =fbb2 √ Kr
fr(b2 -1)
Where Kr = Log-1 ( Adb
20 ) = 10Adb/20
ffr
+X db
+X/2 db
fb
fb
frand b=
fr
fbor b=
ffrfb
-X dB
-X/2 dBWhichever b is >1
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High-Q Notch Filters
This circuit is in the form of a bridge where a signal is applied across terminals’ 1 and 2 the output is measured across terminals’ 3 and 4. At ω=1all branches have equal impedances of 0.707 ∠-45°so a null occurs across the output.
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The circuit is redrawn in figure B in the form of a lattice. Circuit C is the Identical circuit shown as two lattices in parallel.
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There is a theorem which states that any branch in series with both the ZA and ZB branches of a lattice can be extracted and placed outside the lattice. The branch is replaced by a short. This is shown in figure D above. The resulting circuit is known as a Twin-T. This circuit has a null at 1 radian for the normalized values shown.
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To calculate values for this circuit pick a convenient value for C. Then
R1=1
2πfoC
The Twin-T has a Q (fo/BW3dB ) of only ¼ which is far from selective.
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Circuit A above illustrates bootstrapping a network β with a factor K. If β is a twin-T the resulting Q becomes:
Q =
If we select a positive K <1, and sufficiently close to 1, the circuit Q can be dramatically increased. The resulting circuit is shown in figure B.
14(1-K)
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Bridged-T Null Network
CωrLQ/2ωrLQ/2
-ωrLQ/4C L
ωrLQ/4≡
at resonance
ωrLQ/4
Impedance of a center-tapped parallel resonant circuit at resonance is ωrLQtotal and ωrLQ/4 from end to center tap (due to N2 relationship). Hence a phantom negative resistor of -ωrLQ/4 appears in the equivalent circuit which can be cancelled by a positive resistor of ωrLQ/4 resulting in a very deep null at resonance (60dB or more).
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Adjustable Q and Frequency Null Network
Σ1
+1T(s)
InOut
T(s) can be any bandpass circuit having properties of unity gain at fr,adjustable Q and adjustable fr.
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Q Multiplier Active Bandpass Filters
If T(s) in circuit A corresponds to a bandpass transfer function of:
The overall circuit transfer function becomes:
OutIn
The middle term of the denominator has been modified so the circuit Q is given by Q/(1-β) where 0<β <1. The Q can then be increased by the factor 1/(1-β) . Note that the circuit gain is increasedby the same factor.
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A simple implementation of this circuit is shown in figure B.The design equations are:First calculate β from β= 1 - Qr
Qeff
where Qeff is the overall circuit Q and Qr is the design Q of the bandpass section.The component values can be computed from:
R1b=R1a
2Qr2-1
Where R and C can be conveniently chosen.
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A Family of Zero Phase Shift Low-Pass Filters
Attenuation of Normalized Chebyshev LPF
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
20.00
25.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
Freq Radians
dB
0.5dB
1dB
3dB
6dB
12dB
24dB
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note near-zero phase slope for high ripples
Phase of Normalized Chebyshev LPF
-180.00
-160.00
-140.00
-120.00
-100.00-80.00
-60.00
-40.00
-20.00
0.000.00 0.20 0.40 0.60 0.80 1.00
Freq Radians
Phase Degrees
Phase 0.5dB
Phase 1dB
Phase 3dB
Phase 6dB
Phase 12dB
Phase 24dB
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Transformation to Band-Pass Filter
amplitude
phase
Fo Freq
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Some Useful Passive Filter Transformations to Improve Realizability
NC2
C1
L
C2 N(N-1)
C2
LN2
Advantages :Reduces value of LAllows for parasitic capacity across inductor N=1+
C1
C2
N=1+CB
CA
CA
N
CA(1-1/N)CB
CA
LAN2LA
Advantages:Increases value of L
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Narrow Band Approximations
This transformation can be used to reduce the value of a terminating resistorand yet maintain the narrow-band response.
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The following example illustrates how this approximation can reduce the source impedance of a filter.
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Using the Tapped Inductor
An inductor can be used as an auto-transformer by adding a tap
Resonant circuit capacitor values can be reduced
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Intermediate branches can be scaled in impedance
Leakage inductance can wreak havoc
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Effect of leakage inductance can be minimized by splitting capacitors which adds additional poles
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Exponentially Tapered Network
Z2
Z1 Z3 Z5
Z4
ZN
Z2R2/N
Z1R1/N Z3R3/N Z5R5/N
Z4R4/N
ZNR
1Ω
1Ω
1Ω
RΩ
Method of exponentially tapering a network to a higher (or lower)load impedance value with minimal effect on response
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EOUTEOUT
SCR2
EOUT
SCR2R3
i =
EOUTEOUT
SCR2
EOUT
SCR2R3
i =L=CR2R3
State Variable (Biquad) Bandpass Filter
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Center tap
0.2432F
1.605F
1.605H
0.4864F
0.8025H
0.8025F1 ohm 1 ohm
1 ohm 1 ohm
1.6 sec-1%
-5%
1 1.22 Rad/sec
All-PASS DELAY LINE SECTION
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SINGLE TRANSFORMER HYBRID
N =1
N =0.707
N =0.707
R /2
R
R
R
R
+1V +
+
R
+0.5V
--+
-+0.5V
+0.5V
+2V