Copyright © 2012 by Wayne Stegall
Updated November 3, 2012. See Document History at end for
details.
Warp Filter
Highpass
notch
filter
eliminates
dangerous
LP
warp
signals
Introduction
With 60dB or more of DC gain, a phono preamplifier can amplify record
warps into subsonic energy dangerous to woofers. Because of this
some proposed long ago to amend the RIAA equalization curve to add a
20Hz
highpass pole to filter out subsonics. However this firstorder
pole will only suppress warp energy by 31.1261dB. The filter
circuit proposed here has a secondorder pole and a notch at the
offending frequency to nearly eliminate the problem altogether.
First try
Figure
1:
Desired
warp
filter
topology


Refer to the frequency plot of
figure
3 below for the following description. The circuit of
figure 1 above operates in the
following manner. In the passband above 20Hz C
_{1} passes
all signals unaltered. As the frequency drops to near the
cutoff frequency, L
_{1} interacts with C
_{1} and R
_{1}
to create the usual secondorder highpass pole at 15Hz. At this
point C
_{2}, because of its larger value, only couples L
_{1}
with the rest of the circuit. As the response falls due the pole
the impedance of c rises as the zero approaches to begin the greater
suppression of the notch. When the impedance of C
_{2}
becomes equal to that of L
_{1} their opposite polarity results
in zero impedance producing the zero magnitude of the notch.
Passing the notch the impedance of C
_{2} exceeds that of L
_{1}
allowing the magnitude to return to a lower stopband ripple level after
which the response drops again in the normal way.
Direct sdomain analysis of circuit produces a thirdorder transfer
curve that does not readily lend itself to a solution.
An empirical solution
It is more desirable then to break the problem down into two
parts. If you can assume that C
_{2} >> C
_{1},
then
C
_{2} can be removed
from the pole analysis allowing separate calculation of pole and
zero. Then the problem reduces to two parts:
 Design circuit with C_{2} removed for desired pole.
 Calculate C_{2} to give desired zero.
 Simulate and make adjustments if necessary.
Design for pole parameters
Figure
2:
Circuit
for
pole
design 

First setup and reduce sdomain transfer equation for the simplified
circuit:
(2)


sLR


sLR


H(s) =


=






sL
+
R
+
s^{2}LC_{1}R
sC_{1} 

(3)

H(s) = 
s^{2}LC_{1}R
s^{2}LC_{1}R
+
sL
+
R

(4)


s^{2} 

H(s) =




s^{2} + s 

1
C_{1}R 

+

1
LC_{1} 

Now the fully reduced transfer equation of
equation 4 is in the form of the
standard secondorder highpass transfer curve.
(5)


s^{2} 

H(s) =




s^{2} + s 

ω
Q


+ ω^{2} 

where the characteristic parameters are derived as follows.
(7)

Q
ω 
= C_{1}R,
therefore Q = ωC_{1}R 
Combining
equations 6 and
7 gives:
Which reduces to:
Desiring a Bessel Q, I determined that f
_{pole} for a cutoff of
20Hz is slightly above 15Hz.
Choose 15Hz for pole frequency, then calculate the radian frequency.
(10)

ω_{pole} = 15Hz × 2π =
94.2478 radians/s

The secondorder Bessel Q is known as:
(11)


1 


Second order Bessel Q =


≈ 0.57735





Choose R = 100kΩ
Rearrange
equation 7 to solve
for C
_{1}.
(12)

C_{1} = 
Q
ω_{pole}R 
=

0.57735
94.2478 × 100kΩ 
= 61.2588nF

Round up to nearest 5% value
C
_{1} =
62nF
Rearrange
equation 6 to solve
for L:
(13)

L =

1
ω_{pole}^{2}C_{1} 
=

1
(94.2478)^{2} × 62nF 
= 1.81579kH

This huge inductance value will require an active inductor design at a
later stage.
Design for zero parameters
Calculate the zero required to cancel a 33⅓ RPM warp:
(14)

f_{zero} = 
33.3333 cycles/min
60 s/min

= 0.555555Hz, therefore ω_{zero}
= 2π × 0.555555Hz = 3.49066 radians/s 
An equation like
equation 6
defines the notch frequency relative to L
and C
_{2}. Rearrange it and calculate C
_{2}.
(15)

C_{2} =

1
ω_{zero}^{2}L 
=

1
(3.49066)^{2} ×
1.81579kH 
= 45.198µF

Round up to nearest 5% value
C
_{2} =
47µF
At this point our initial assumption that C
_{2} >> C
_{1}
is correct, validating this shortcut design method.
The necessity to round C
_{2 }to the nearest standard value
requires
recalculation of exact L.
(16)

L =

1
ω_{zero}^{2}C_{2} 
=

1
(3.49066)^{2} × 47µF 
= 1.74617kH

SPICE results
SPICE results verify successful design outcome.
SPICE
model.
Figure
3:
Passive
filter
magnitude
response

Figure
4: Passive filter phase
response 


Figure 5:
Passive
filter magnitude notch detail 0.50.75Hz




Magnitude response at 0.555555Hz (33⅓ RPM warp) = 165.432dB
Magnitude response at 0.75Hz (45 RPM warp) = 59.288dB
Magnitude response at 20Hz = 2.59905dB.
I did not simulate distortion because it depends on final
component choice.
Add active inductor to design
The circuit that follows in
figure 5
adds an active inductor consisting
of U
_{1}, U
_{2}, R
_{1L}, R
_{2L}, R
_{3L},
C
_{4L}, and R
_{5L}. It is a form of a circuit
known as a
Generalized
Impedance
Converter. Concerns by those who prefer discrete circuits to
operational amplifiers are mitigated by the fact that the active
inductor is only in the signal path in the passband near the resonant
frequency.
Figure
6: Warp filter with
active inductor 

The equivalent active inductance is calculated by the formula:
(17)

L =

C_{4L} × R_{1L} ×
R_{3L} × R_{5L}
R_{2L} 
I choose C
_{4L} to be 1µF because it is the largest among
plastic capacitors to be of reasonable size and price. Then I
presume that the resistors would give the best noise result if they
were the same values. If all the resistors were set the same in
equation 17 above then
equation 17 above simplifies to:
(18)

L =

C_{4L} × R_{1L} ×
R_{1L} × R_{1L}
R_{1L} 
= C_{4L} × R_{1L}^{2} 
Rearrange
equation 18 and
solve for common resistor value.
(19)

R_{1L} =


=


= 41.7872kΩ 
Round to the nearest 1% value.
R
_{1L}, R
_{2L}
and R
_{3L} =
42.2kΩ
Now restore R
_{5L} to
equation
18 to calculate its similar adjustable value.
(20)

L =

C_{4L} × R_{1L} ×
R_{1L} × R_{5L}
R_{1L} 
= C_{4L} × R_{1L}
× R_{5L} 
Rearrange
equation 20 and
solve for R
_{5L}.
(21)

R_{5L} =

L
C_{4L} × R_{1L} 
=

1.74617kH
1µF × 42.2kΩ 
= 41.3784kΩ 
SPICE results
Again SPICE results confirm a successful design outcome.
SPICE
model with
active inductor.
Figure
7:
Passive/active
filter
magnitude
response 
Figure
8: Passive/active filter
phase
response 


Figure 9:
Passive/active
filter magnitude notch detail 0.50.75Hz 



Magnitude response at 0.555555Hz (33⅓ RPM warp) = 134.583dB
Magnitude response at 0.75Hz (45 RPM warp) = 59.2881dB
Magnitude response at 20Hz = 2.59943dB.
Because the GIC is supposedly a proven circuit, I used ideal op amps in
the simulation.
Application and Adjustments
In actual application, the proper operation of this passive circuit is
sensitive to the circuit context into which it is placed.
Therefore a buffer should drive it with near zero output impedance and
the output should drive near infinite input impedance in the buffer
that follows.
Adjust R
_{5L} to calibrate the correct notch frequency in the
manner you see best. I would suspect you would play a warped
record at 33⅓ RPM and then
 Measure C_{2} with an LCR meter and with same meter set l
to the exact value necessary. Depending on the exact C_{2}
value, you may have to recalculate L again as in the equations above.
 Connect an oscilloscope and adjust for the best null or
 Play it carefully through loudspeakers always monitoring for safe
volume until the woofer no longer responds to the warp signal.
Generalized
Impedance Converter (GIC)
For those curious about the active inductor itself, I include a little
information about the basis circuit itself.
^{1}
Figure
10:
GIC
can
emulate
many
other
impedances
other
than
the
active
inductor


Node B is always shown in the source document connected to
ground. There is no logical reason to think it has to be this way.
The virtual impedance created by the circuit of
figure 10 above is:
(22)

Z_{GIC} =

Z_{1}Z_{3}Z_{5}
Z_{2}Z_{4} 
^{1}Arthur B. Williams and Fred J.
Taylor, Electronic Filter Design Handbook (New York, 1988), p. 339 
341 "Generalized Impedance Converters."
Document History
September 22, 2012 Created.
September 23, 2012 Made a minor correction just prior to equation
15.
September 25, 2012 Corrected misspellings. I moved quickly
to
publish but I thought I did a spell check.
November 3, 2012 Corrected some grammer.