CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
OPTIMAL CROSSOVER DESIGN
FOR LOUDSPEAKERS
A Graduate Project submitted for partial
satisfaction of the requirements for the
degree of Master of Science in
Electrical Engineering
by
Steven Edward Anderson
May, 1982
The graduate project of Steven Edward Anderson
is approved:
ii
CONTENTS
Introduction
Loudspeaker Pressure Response
Constant Voltage Filters
Cascaded Butterworth and Butterworth Filters
Loudspeaker System Design
Loudspeaker System One
Loudspeaker System Two
Loudspeaker System Three
System Test Results
Appendix D
Appendix E
-
Loudspeaker Transfer Functions
Theoretical System Transfer Functions
Appendix F - Theoretical Transfer Functions
iii
LIST OF FIGURES
Figure 1
Effect of Non-Coincident Sound
Waves on Pressure Response
Figure 2
3rd Order Butterworth Asymmetrical
Constant Voltage
Figure 3
3rd Order Butterworth Symmetrical
Constant Voltage
Figure 4
3rd Order Synchronously Tuned
Asymmetrical Constant Voltage
Figure 5
3rd Order Small Symmetrical
Constant Voltage
Figure 6
3rd Order Small Asymmetrical
Constant Voltage
Figure 7
Cascade of Two LP Butterworth
2nd Order Filters
Figure 8
Cascade of Two HP Butterworth
2nd Order Filters
Figure 9
Summed Response of Two Cascaded
2nd Order Butterworth Filters
Figure 10
3rd Order Butterworth LP Filter
Figure 11
3rd Order Butterworth High Pass
Filter
Figure 12
Summed Response of HP and LP
3rd Order Butterworth Filters
Figure 13
Nearfield Frequency Response,
Peerless TP 165F 1n .85 ft3 Enclosure
Figure 14
Peerless TP 165F in .85 ft3 Box
With 3 dB Peaking Filter
Figure 15
Peerless PMT-30V, 3KHz -4th
Order Filter in Cascade
iV
Figure 16
3
AUDAX MHD 12P 1n 35 1n Enclosure
Figure 17
AUDAX MHD 12P in 35 in3 Enclosure
With 300 Hz Filter
Figure 18
System One Crossover
Figure 19
System One
Figure 20
AUDAX MHD 12P 1n 260 in 3 Enclosure
Figure 21
AUDAX MHD 12P in 260 in3 Enclosure
With 228 Hz Filter
Figure 22
System Two Crossover
Figure 23
System Two
Figure 24
Peerless PMT-51 Near field
Relative Frequency Response
Figure 25
Peerless PMT-51 Nearfield
Relative Frequency Response
With 380 Hz Filter
Figure 26
System Three Crossover
Figure 27
System Three
Figure 28
Ground Plane Response System 1
Figure 29
Ground Plane Response System 2
Figure 30
Ground Plane Response System 3
v
ABSTRACT
OPTIMAL CROSSOVER DESIGN
FOR LOUDSPEAKERS
by
Steven Edward Anderson
Master of Science in Electrical Engineering
A multi-driver loudspeaker system is discussed.
Methods of dividing the input frequency spectrum between
loudspeaker drivers are presented.
It is proposed that an
optimal crossover is one in which the acoustic recombination
of the signal in the listening area is a perfect replica of
the input signal.
Loudspeaker pressure response
~s
discussed when two
loudspeaker drivers are separated by a finite distance on a
common acoustic baffle.
The pressure response of the loud-
speaker system is shown to depend not only on the crossover
transfer functions but also the loudspeaker transfer functions.
It is s,hown that by using two non-coincident drivers
vi
to reproduce the audio spectrum, an angle-off-axis dependent
time delay is introduced.
Constant voltage filters and their relation to real
driver frequency responses are discussed.
worth and Butterworth filters
Cascaded Butter-
are related to real driver
parameters with the cascaded Butterworth shown to cause the
two drivers to interact over the smallest bandwidth.
Three loudspeakers were designed demonstrating that
the loudspeaker transfer function could be designed into the
overall crossover transfer function by careful driver selection and enclosure tuning.
were demonstrated of this
Low distortion driver/enclosures
type.
Three separate active crossovers were constructed to
enable the tests of the three loudspeaker systems.
A cross-
over was designed using standard Butterworth two pole sections that could be switched to a number of different combinations for all three proposed systems.
Overall system response tests were performed that
verified theory with respect to a symmetrical radiation
the vertical plane.
~n
Off axis response abberations at high
frequencies were shown to exist due to the angle-off-axis
time delay.
v~~
INTRODUCTION
Loudspeakers are generally not designed as systems but
as add-ons
make
Speaker designers must
to existing equipment.
their systems stand alone to
amplifier unit.
interface to any power
Also a loudspeaker system designer must
choose between many design parameters and optimize as many
parameters as possible.
A loudspeaker system consists of multiple
loudspeakers,
each of which is optimized to operate over a particular
audio bandwidth.
An electrical network is used to divide
the electrical signal into separate bandwidths since most
loudspeaker systems employ multiple loudspeaker drivers.
These
filter networks are generally high pass and low pass
filters
and are referred to as crossover networks.
filter networks
These
in practical loudspeakers must handle large
amounts of electrical power.
Large gauge
w~re
must be used
on inductors to maintain driver damping and large capacitors
must be used to prevent breakdown due
to overvoltage.
Within the last several years active crossovers have
been introduced as
These
filters
an alternative to high level
filtering.
preceed the power amplifier enabling filtering
to be done on relatively low level signals.
vantage is that the
A further ad-
loudspeaker characteristic impedance is
then buffered by the power amp from the crossover filter.
1
2
While active crossovers eliminate some problems associated
with high level crossovers they are still generally add-ons
to existing systems.
Since a crossover is dividing the electrical signal
into several bandwidths, an optimal crossover design would
allow for the acoustic recombination of the signal in the
listening area to be a perfect replica of the input signal.
Several papers
[1 ],
(z],
[3],
[4] have discussed var1.ous
crossovers with respect to acoustic accuracy and loudspeaker
interaction.
It is the purpose of this paper to analyze
these crossovers with respect to loudspeaker parameters and
design and measure three loudspeaker systems.
LOUDSPEAKER PRESSURE RESPONSE
Figure 1 shows two loudspeaker drivers separated by a
and offset on an acoustic plane by distance d
distance d
A measuring device or listener is at position p 1 .
acoustic pressure at point p
1
The total
(assuming a sinusoidal input)
is given by
Eq.
1
where
PL(jw)= complex pressure of the low
frequency driver
and
PH(jw)= complex pressure of the high
frequency driver.
Since loudspeakers are vibrating mechanical systems and
can be represented by an analogous electrical network,
the
acoustic pressure output can be defined in terms of a transfer function and input drive signal.
Then
Eq.
where
2
FL(jw)= low frequency driver transfer function
FH(jw)= high frequency driver transfer function
eL= low frequency input drive voltage
eH= high frequency input drive voltage
3
4
An electrical network is employed that divides the 1nput signal between the low frequency and high frequency
loudspeakers.
The input drive voltages to the loudspeakers
can be represented by
e.=G (jw)ein
Eq.
3
e =G (jw)ein
Eq.
4
1
H
L
H
where
GL(jw)= low frequency crossover transfer function
GH(jw)= high frequency crossover transfer function
and
ein= system input drive voltage
The total acoustic pressure at point p
1
can be represented
as
Eq.
5
It can be seen from Equation 5 that the pressure response of
a
two way loudspeaker system depends not only on the cross-
over transfer functions but also the loudspeaker transfer
functions.
If the measuring device or listener 1s moved away from
point p
1
to,
say,
driver to point p
p
2
2
,
a difference 1n path length from each
exists.
It was determined
[z]
that the difference 1n path
length as a function of the angle from which the path
lengths were equal could be represented as
Eq.
6
5
In this simplified case d =0.
2
The path length differ-
ence corresponds to a delay of one driver with respect to
the other and is defined as
t= d sin
1
c
7
Eq.
where
c= speed of sound 1n a1r
By using two non-coincident drivers
full audio spectrum,
to reproduce the
an angle-off-axis dependent time delay
has been introduced.
Given the two acoustic pressures as before except now
measured at point p
p( jw)
2
P ( j w) = [c L ( j w)
II F
+ [cH(jw)l
ft
L ( j w) J
~) + ¢ (a , w ~
j w) +
JFH(jw)J/cH(jw)+~H(jw)J
Eq.
8
where
c
As d
2
is increased to some value other than zero
(either positive or negative)
ative delay between drivers
on axis at p .
1
the angle a at which the rel-
is zero is some other value than
The effect of delaying one driver with res-
pect to the other is the shifting of the vertical radiation
pattern up or down.
Pressures will add in phase at point p
whenever
¢(a,w)= ±n2'IT
E q.
9
2
6
and subtract when
¢(a,w)=±(2n+1)'TT
Eq.
10
where n= 0, 1, 2 · ·
the strongest interaction between the drivers will occur at
the crossover frequency where both contribute equal amp1itude signals.
For frequencies which are much higher or
lower than the crossover point the radiation pattern
~s
determined only by the radiation pattern of the driver that
is active in that range
[2].
CONSTANT VOLTAGE FILTERS
It was shown
~n
the previous section that the total
acoustic pressure at point p
1
was
the vector sum of the
cascade of the low frequency loudspeaker and crossover
transfer function and the cascade of the high frequency
loudspeaker and crossover transfer function.
Let
E q.
11
Eq.
12
Eq.
13
Eq.
14
Then from Equation 5
PT(jw)fL (jw)+HH(jw)] ein
If we let
then the acoustic pressure at point p
1
is an exact replica
of the input signal.
Equation 14 is defined as a constant voltage crossover
network.
A general transfer function H(s) can be expressed as
n b
n-1 . . . . b
1
N(s) s + n-ls
+
+ 1s+
H(s) D(s)
n
n-1 . . . .
s +an-ls
+
+a 1 s+ 1
Network functions
15
for the constant-voltage crossovers
are formed by dividing D(s)
nomials NL(s) and NH(s)
Eq.
in Equation 15 into two poly-
such that
7
8
Eq.
16
Eq.
17
Eq.
18
then
HL(s)=NL(s)
D(s)
HH(s)=NH(s)
D(s)
There are two classes of constant voltage crossover networks:
asymmetrical and symmetrical.
Figures 2 through 6
illustrate examples of proposed useful constant voltage netI t was shown
[3
J
that one o f the more des -
irable characteristics of all constant voltage networks was
that of maintaining an accurate system transient response.
A second desirable characteristic which applies to a
third order
asymmetrical crossover exclusively is an in-
sensitivity to the mid frequency driver transfer function
phase response.
The phase lead of a mid frequency loud-
speaker modeled as a second order high pass filter
(Appendix D) causes sometimes large anomalies
speaker system amplitude response
in the loud-
[1 J.
Figure 2 shows the amplitude and phase response of a
3rd order Butterworth asymmetrical constant voltage filter.
The high pass section is a standard 3rd order Butterworth
filter.
However, the low pass section displays two inter-
esting characteristics.
First, at the filter corner fre-
quency (f =300 Hz) a +4 dB peak occurs which reduces system
n
9
headroom and second,
the response is not down 10 dB until
two and three quarter octaves from f
a sufficient
Figure 2,
n
.
If,
for example,
in
phase lead or phase lag to cause 0
degrees or 180 degrees phase shift between the two coradiating sources were introduced due to off-axis characteristics of non-coincident drivers,
the following off axis
pressure response (PT(t)) would result.
pT(t)= 20 log [ log -1 (O)+log -1
= 2.15 dB
pT(t)= 20 log ~log -1 (0)-log -1
= -2.87 dB
~~
Eq.
19
~~~
Eq.
20
This indicates the off-axis response could vary greater
than plus or minus 2 dB over a three and one half octave
bandwidth.
This variation in frequency response as it re-
lates to a symmetrical radiation pattern was first shown by
Linkwitz
131.
Figures 3, 4,
5 and 6 demonstrate additional types of
constant voltage filters.
Notice in all cases the high
frequency and low frequency drivers interact over a bandwidth of greater than two and one half octaves.
CASCADED BUTTERWORTH AND BUTTERWORTH FILTERS
A crossover network has been proposed to allow for a
symmetrical radiation pattern when two non-coincident
drivers are radiating into the same acoustic space
(Figure 1) near the crossover region [3].
Three criteria
have to be met to provide a symmetrical radiation pattern
at crossover.
1)
The phase difference between drive signals have to be
zero.
2)
The output amplitude from the high-pass and low-pass
section has to be 6 dB down at the crossover frequency.
3)
The high-pass and low-pass filters must have identical
group delay.
Linkwitz has proposed that the cascade of two identical
Butterworth filters meet these requirements
3
Let
= (
2
1 )2
Eq.
21
Eq.
22
s +\'21+1
=
{.z+~s+J
And
4
s +1
2 \f.:
(s +v2s+l)
10
Eq.
2
23
11
Then
l+w
=
4
l+w
'~
3 -w)
(w 4 -4w 2 +1)-2y2(w
4
= ---4
Eq. 24
1+w
Figures 7, 8 and 9 show the amplitude and phase response for a low-pass, high-pass and summed, cascade of two
Butterworth 2nd order filters respectively.
From Figures 7
and 8 notice the driver interaction is only over a bandwidth
one half octave wide.
Figures 10, 11 and 12 show the amplitude and phase
response for a low-pass, high-pass and summed,
Butterworth filter.
third order
This type of filter is widely used in
both passive and active crossovers for loudspeaker systems.
The summed phase response (Figure 12)
is identical to the
summed phase response of the cascade configuration (Figure
9).
The radiation pattern for
using this
two non-coincident drivers
type of crossover, however,
is non-symmetrical.
From Figures 10 and 11 the driver interaction is over a
bandwidth of one octave.
p •
LOUDSPEAKER SYSTEM DESIGN
Three loudspeaker systems were designed to determine
the validity of the theory presented in the preceeding
sections.
All three loudspeaker systems used the same low
frequency and high frequency drivers.
The same low fre-
quency tuning and high frequency crossover points were used
for all three systems.
The low frequency loudspeaker chosen for the system
Figure
design was the Peerless TP 165F 6.5 inch long throw.
13 shows the relative nearfield frequency response and also
shows an extremely low second harmonic distortion.
drive~/enclosure
The
tuning is detailed in Appendix D and is
shown to be -3 dB down relative to mid-band at approximately
70 Hz.
400 Hz.
Mid-band frequencies
~n
this case are from 100 Hz to
Figure 14 shows the same driver/enclosure with the
+3 dB peaking filter as detailed in Appendix D.
increase in second harmonic distortion.
Notice an
This is the price
that must be paid for forcing a driver to work beyond its
linear region.
The mid-band sensitivity of this driver is
89 dB-SPL for one (1) watt of electrical input power,
so
reasonable listening levels may be achieved at very low
distortion in quiet areas such as living rooms.
This peak-
ing filter allows the system to be-demonstrated as a full
range loudspeaker.
This loudspeaker could have also been
12
13
tuned higher as a second order Butterworth filter.
A second
second order Butterworth filter could have then been placed
in cascade reducing the second order harmonic distortion to
levels less than illustrated in Figure 13.
This system
would have required a very low frequency (VLF)
loudspeaker
to be a full range system and is beyond the scope of this
paper.
The high frequency driver selected was the Peerless
PMT-30V,
one (1)
response
~s
this driver
inch nominal dome.
shown in Figure 15.
~s
550 Hz.
The nearfield frequency
The resonant frequency of
An upper crossover frequency of
3 KHz was chosen as an optimal frequency for all three
system configurations.
This frequency was chosen to be at
least two octaves above the resonant frequency of the high
frequency driver so as
to minimize the driver transfer
function phase interaction with the crossover filters.
LOUDSPEAKER SYSTEM ONE
Figure 16 shows the relative nearfield response of an
AUDAX MHD 12P mid frequency driver in a 35 cubic inch enclosure.
The driver/enclosure tuning is detailed in
Appendix D with Figure 16 clearly showing a 1.5 dB peak in
the response at approximately 320 Hz.
This first
loud-
speaker system is designed to use a standard 3rd order
Butterworth crossover.
A filter with a single pole at ap-
proximately 300 Hz is placed in cascade with this driver/
enclosure to produce the desired 3rd order Butterworth
response.
Figure 17 shows
the overall response of this
driver/enclosure with a 300 Hz single pole high pass and a
4th order 3 KHz
low pass filter
[z]
in cascade.
Figure 18 shows the measured electrical response of
the crossover used for system one.
Figure 19 shows the system one configuration.
Notice
the mid frequency driver is delayed with respect to the
low frequency driver.
The alignment of the driver acoustic
centers was accomplished using Time Delay Spectometry (TDS)Energy Time Curve (ETC) method.
14
LOUDSPEAKER SYSTEM TWO
Figure 20 shows
the relative nearfield response of the
AUDAX MHD 12P mid-frequency driver in a 260 cubic inch enThe driver/enclosure tuning is detailed in
closure.
Appendix D.
The driver/enclosure form a 2nd order Butter-
worth high pass filter.
A second identical Butterworth
filter is then placed in cascade to form a 4th order
Linkwitz filter
[2
J.
In this manner,
as
~n
system one,
the
driver transfer function is part of the overall crossover
transfer function.
The overall mid-frequency response with
a 2nd order Butterworth high pass at approximately 220 Hz
and a 4th order Linkwitz filter at 3 KHz is shown in
Figure 21.
The system two electrical crossover measured
response is shown in Figure 22.
is shown
~n
Figure 23.
15
The system configuration
LOUDSPEAKER SYSTEM THREE
Figures 24 and 25 show the relative nearfield response
of a Peerless PMT-51 2-inch dome midrange without and with
a 2nd order Butterworth filter in cascade respectively.
Notice in Figure 25 the remarkably low 2nd harmonic distortion with the 2nd order high pass filter in cascade.
This
figure indicates a 2nd harmonic distortion of less than
one (1) percent for a 2.83 VRMS (1 watt into nominal 8 ohms)
input signal.
The midband sensitivity for
this driver 1s
90 dB-SPL for 2.83 VRMS which is averaged from 500 Hz to
3 KHz.
Figure 26 shows the measured electrical crossover response for system three.
Figure 27 shows the configuration for loudspeaker
system three.
The same method (TDS-ETC) used in systems
one and two was used to align the acoustic centers of the
three drivers used in system three.
16
SYSTEM TEST RESULTS
Figures 28,
Systems one,
29 and 30 are the overall test results for
two and three respectively.
All tests were
made using the ground plane measuring technique
[s].
A
one-eighth-inch Bruel and Kjaer microphone was placed
four (4)
feet
from the loudspeaker system.
The measure-
ments were made at such a close range to optimize the signal
to noise since all measurements were made outside.
It can
be seen on Figures 28 and 29 that the ambient noise
~s
only
15 dB down from the mid-band signal.
Each loudspeaker system was placed on its side and
rotated about an imaginary axis between the low frequency
driver and mid frequency driver (See Figures 19, 24 and 27).
All systems were measured on-axis and plus/minus thirty (30)
degrees off
ax~s.
All systems were calibrated using pink
noise to set one watt of input power into the mid band
speaker.
A true RMS reading meter was used to measure a
nominal voltage of 2.83 vrms
watt into eight (8) ohms.
which corresponds to one (1)
The low frequency and high fre-
quency drivers were then adjusted using a one third octave
real time analyzer to measure a "flat" frequency response.
Since the loudspeaker systems were rotated about an
imaginary axis between the low frequency driver and the
mid frequency driver,
a corresponding increase or decrease
17
18
in the Sound Pressure Level (SPL) could be expected as the
driver changes distance from the measuring microphone.
measuring distances of four (4)
as +/-1 dB could occur.
cated in Figures 23,
and "low behind mid".
feet,
At
deviations as great
The measurements off axis are indi-
29 and 30 simply as "mid behind low"
The distances between drivers LS
shown Ln Figures 19, 24 and 27.
Since Systems 1 and 2 use the same midrange driver, a
comparison can be made with respect to symmetrical radiation
pattern at the crossover frequency.
In Figure 28 at approx-
imately 300 Hz the response labeled "mid behind low" is approximately 2 dB greater than the response "low behind mid".
In Figure 29 the opposite is true,
indicating a greater ten-
dency towards a symmetrical radiation pattern.
Further
studies need to be conducted in an anechoic environment
measuring polar response plots at these low frequencies
to
determine the true effects of a cascade type filter on the
radiation pattern using real
loudspeakers.
The symmetrical response is much more evident at higher frequencies as can be seen at approximately 2 KHz to
3 KHz in Figures 28,
29 and 30.
The measuring angle off
axis and upper crossover frequency were chosen such that a
complete cancellation would occur at the crossover point.
In all systems this cancellation occurs.
Since the transfer
function of the high frequency driver was not included in
19
overall crossover transfer function,
a phase shift is
present at the crossover frequency and causes a shift 1n
the cancellation frequency.
in Figures 28 and 30.
This phenomena 1s most obvious
20
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-BEFFECT OF NON-COINCIDENT SOUND WAVES
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REFERENCES
W.M. Leach, "Loudspeaker Driver Phase Response:
The Neglected Factor in Crossover Network Design,"
J. Audio Eng. Soc., vol. 28, pp. 410-421 (1980 June)
[2]
S.H. Linkwitz, "Active Crossover Networks for
Noncoincident Drivers," J. Audio Eng. Soc., vol.
pp. 2-8 (1976 Jan./Feb.).
24,
J.R. Ashley and A.L. Kaminsky, "Active and Passive
Filters as Loudspeaker Crossover Networks," J. Audio
Eng. Soc., vol. 19, pp. 494-502 (1971 June).
R.H. Small, "Constant-Voltage Crossover Network
Design," J. Audio Eng. Soc., vol. 19, pp. 12-19
(1971 January)
Gander, "Ground Plane Acoustic Measurement
[s J M.R.
of Loudspeaker Systems," Audio Engineering Society
Preprint, A.E.S. Convention 1980.
Richard C. Heyser, "Determination of Loudspeaker
Signal Arrival Times," J. Audio Eng. Soc., vol. 19,
pp. 734-742 (October 1971)
D.B. Keele, Jr., "Low-Frequency Loudspeaker
Assessment by Nearfield Sound-Pressure Measurement,"
J. Audio Eng. Soc. (April 1974)
R.H.
[8 J Part
pp.
Small, "Closed-Box Loudspeaker SystemsI:
Analysis," J. Audio Eng. Soc., vol. 20,
798-808 (1972 Dec.)
50
APPENDIX D
LOUDSPEAKER TRANSFER FUNCTIONS
ax~s
The on
pressure transfer function of a direct
radiating moving coil driver in a sealed enclosure is repre8
sen ted by
'
(s/w r)2
c
G(s) =
2
(s/w ) +(1/QTC)(s/wc)+l
c
Eq.
D-1
Eq.
D-2
Eq.
D-3
Wherf'l
w
1
----
c
VMATC AT
QTC =
1
RAT
(MATr
CAT
MAT = MAD + MAL
Eq. D-4
MAD = Acoustical mass of driver
CAT = Acoustical compliance of driver
RAT = Acoustical resistance of driver
MAL
=
Mass of
a~r
load on the
front of
the driver
In addition,
the following relationships may be used
to "tune" an enclosure for a given loudspeaker.
f
f£. =
k
(a+l)2
E q.
s
51
D- 5
52
Eq. D-6
Where
= total Q of system at f
c
including all
system resistances
f
f
= resonance frequency of closed box system
c
a~r
= free
s
resonant frequency of driver
VB = net internal volume of enclosure
VAS =volume of air having same acoustic
compliance as driver suspension.
= total Q of driver at f
s
considering all
driver resistances.
The following are the design steps to tune the low
frequency and mid frequency speaker/enclosures to produce
the desired transfer function.
LOW FREQUENCY LOUDSPEAKER/ENCLOSURE
The manufacturers design parameters are as
f
s
follows:
= 53.0 Hz
v as = 15.8
X
l0- 3 (m 3 )
c ms = 7.27
X
10
-4 (m/n)
Qts = 0.42
S
D
=
3
2
12.45 x l0- (m )
The loudspeaker
~s
a nominal 5.5 inch piston.
If the
system is to be tuned to a 4th order Butterworth alignment,
53
then the following transfer function must be implemented.
H( s)
=
s
4
2
2
2
2
Es+0.3827) +0.9239 ] Es+0.9239) +0.3827 )]
from [8] we can tune the loudspeaker/enclosure to any
reasonable response desired.
Choosing to let
2.0(.9239)
=
2(,;
from
=
1
Q
= .54
then QTC
this relation holds until a
from above
a
f
c
=
.66
=
68.29
~4.0
also
=
v as
vB
so
a
=
.85 ft
3
Notice the first second order term has to be implemented
via an electrical network.
This network has the transfer
function
H(s)
=
s
2
(s+0.3827)
The peaking of this filter
Peaking
where
=
~s
2
+0.9239
defined as
20 log M
p
Mp
=
1
21
2
'fl::i 2
54
In the above case
Peaking
=
and
= 81
f
p
3.01 dB
Hz
MID FREQUENCY LOUDSPEAKER/ENCLOSURE
The manufacturer's design parameters are as follows:
=
210 ± 25 Hz
Qts
=
.65
V
=
4 3
3
7.827 x 10- m (.027 ft )
=
2,200 Hz
f
s
as
f
0
Tunina_!! -
Butterworth 2nd Order
QTC = .707
So
a
=
.183
f
c
=
228 Hz
VB
=
v
VB
=
260
as
a
Tun in~ iF 2
-
~n
3
Butterworth 3rd Order
A third order Butterworth high pass function
form
H( s)
=
s
3
3
2
s +2s +2s+l
=
s
3
2
(s+l)(s +s+l)
~s
of the
55
For tuning_#2 the function
H(s) =
s
2
will be used to tune the
2
s +s+l
enclosure.
QTC
So
=
1.0
QTC
f
Qts
fs
c = (l+a) .k2
a = 1. 36
VB
=
34.94 1n
f
c
=
323 Hz
3
Here a single pole at 323 Hz must be cascaded with the
speaker/enclosure to produce the desired 3rd order transfer
function.
APPENDIX E
THEORETICAL SYSTEM TRANSFER FUNCTIONS
I
~stem
4
·--:---w....:1;.._~---------=--
H( j w) =
(w
4
1
-3.41w 2 +1)+j2.61(w-w 3 )
1
•
(1-2w
2
2
3
)+j(2w -w )
2
System II
w =w/2 (70); w =w/2 (228); w =w/2 (3000)
2
1
3
System III
w =w/2 (70); w =w/2 (380); w =w/2 (3000)
1
2
3
H(jw)=_
w1
3
(w 4 -3.41w 2 +1)+j2.61(w-w )
1
1
1
3
2(w-w )
The high frequency driver transfer function is not
included here.
56
APPENDIX F
THEORETICAL TRANSFER FUNCTIONS
The following are a list of transfer functions shown
in Figures 2 through 12:
Figure 2
Figure 3
Figure 10
2s 2 +2s+l
D(s)
1
=
H(s)H
=
H(s)
1
=
2s+l
D(s)
H(s)H
=
s 3 +2s 2
D(S)
H(s)
=
H(s)
1
3
s
D(s)
1
])(s)
s
3
D(s)
H(s)
sum
D(s)
Figure 4
H(s)
Figure 5
H(s)
1
1
=
=
2
3s +3s+l
--nrs->-
3.7s+l
D(s)
57
58
=
3
s +3.7s
D(s)
=
Figure 6
2
3.7s +3.7s+l
D(s)
H(s)H
=
Figure 7
3
s
i)(s)
1
DTS1"
Figure 8
Figure 9
H(s)
sum