Note: Descriptions are shown in the official language in which they were submitted.
background ox the Invention
_ . . .
Field of the Invention
The present invention relates to an improvement in means
for converting electrical signals into sound as to control with
high fidelity the acoustic response of a loudspeaker system. More
specifically, the invention relates to a loudspeaker system cross
over using passive or active circuit topology.
The background of the invention and the invention it-
self will be described with reference to the accompanying drawings,
in which:
Figure 1 illustrates a typical prior art loudspeaker
system;
Figure 2 illustrates plots of the complementary low-
pass and high-pass amplitude responses of the loudspeaker system
of Figure l;
Figure 3 is a circuit diagram illustrating a passive
embodiment of the invention;
Figure pa is a high-pass network illustrating the
tweeter topology employed in the circuit of Figure 3;
Figure 4b is a network of Figure pa with a zero of
transmission added;
Figure 4c is a pole-zero pattern or plot showing the
pi yielded by the network of Figure 4b;
Figure pa is a half-section band-pass network employed
in the mid-range topology of the circuit of Figure 3;
$
Figure 5b is the half-section network of Figure pa with
a first zero of transmission placed below the low frequency
crossover point and a second zero of transmission placed above
the high-frequency point;
Figure 5c is a pole-zero pattern showing the p-z yielded
by the network of Figure 5b;
Figure pa is a basic T-section network illustrating the
woofer or low-pass topology employed in the circuit of Figure 3;
Figure 6b is the network of Figure pa with a zero of
transmission added;
Figure 6c is a pole-zero pattern showing the p-z yielded
by the network of Figure by
Figure 7 illustrates the frequency response of the in-
dividual drivers of the circuit of Figure 3;
Figure 3 illustrates the composite frequency response of
the whole system of Figure 3, that is, the total acoustic sum of
all drivers on axis;
Figure 9 is a plot of the phase response of the system
of Figure 3;
Figure 10 is a circuit diagram illustrating a passive
embodiment of the invention with improved performance over that
of Figure 3;
Figure 11 is a pole-zero pattern illustrating the p-z
required for a perfect realization of the invention;
Figures aye, 12b and ].2c are pole-zero patterns thus-
treating the dominant p-z for the low-pass, band-pass, and high
I
pass filter circuits, respectively, of a general embodiment of
the invention;
Figure 12d is a pole-zero pattern showing a summation
of the dominant p-z of the pulsar plots of Figures aye, 12b
and 12c;
Figures aye and 13c are circuit diagrams and Figures
13b and 13d are pole-zero plots used in a mathematical derivation
of the p-z of tweeter and midrange networks similar to the embody
mint of Figure 3;
Figures lea, 14b and 14c are pole-zero patterns justify-
in the network topologies according to the invention;
Figure aye is a circuit diagram illustrating an active
circuit embodiment of the invention;
Figure 15b is a circuit diagram illustrating an active
circuit embodiment of the invention in detail;
Figure aye illustrates the amplitude response of the
individual drivers of the embodiment shown schematically in
Figure 10;
Figure 16~b) illustrates the overall on-axis frequency
response of the entire system of Figure 10;
Figure 16(c) illustrates the delay response of Figure 10;
Figure aye illustrates the amplitude response of the
individual drivers of the active embodiment shown schematically
in Figure lob;
Figure 17(b) illustrates the overall on-axis response
of the entire system of Figure 15b;
-pa-
Figure 18 illustrates a means whereby the acoustic wave
interference between drivers operating on adjacent frequency
bands may be quantized;
Figure lo illustrates the amplitude vs. frequency
response for the Figure 10 embodiment of the invention for response
Do . or So
measurements taken using different positions the microphone;
Figure lo illustrates the amplitude vs. frequency
response for a prior art loudspeaker system for response measure-
mints taken using different positions of the test microphone; and
Figures aye 20(b~ and 20(c) illustrate, respectively,
the amplitude vs. frequency response for (a) a loudspeaker driver
that is not connected to a crossover filter network, (b) the same
loudspeaker connected to a prior art crossover filter network
and (c) the same loudspeaker connected to a crossover filter net-
work of the present invention.
Description of the Prior Art and Definition of Terms
In the present state of the art individual loudspeakers
or drivers are not capable of accurately reproducing all audio
frequencies that are detectable by the human ear. Flight fidelity
loudspeaker systems have been realized in the prior art, however,
by dividing the audio frequency spectrum into two or more frequency
bands, and applying each of these portions of the audio spectrum
to a separate driver or group of drivers.
I 3L3~
For this purpose special electrical jilters,
called crossover networks, have been provided that allow the
different arrivers or groups of drives, each adapted for best
response to a particular range or band of frequencies, to be
5 combined in a single system capably of wide frequency coverage.
The crossover circuit directs the electrical signals of widely
varying frequency to the appropriate driver or group of
drivers in a multi driver loudspeaker system.
Crossover network topologies in general, belong
to three classifications according to the frequencies passed
and rejected, as follows:
(1) Low-pass for woofers/
I Band-pass for midranges, and
I ~igb-pass for tweeters,
where woofers are low frequency drivers and respond to the low
frequencies, midrange drivers respond to the midrange
frequencies, and tweeters are high frequency drivers and
respond to the high frequencies. Where more than one filter
is used, the frequency common to adjacent ranges or pass bands
is called the crossover frequency.
For "perfect fidelity it can be demonstrated
mathematically that a loudspeaker system crossover using
passive or active circuit topology must realize perfectly the
ideal all pass transfer function of Equation (1):
(1) f(s) - Cousteau Where s is the complex
frequency variable
s - + jaw
K and T are real f
positive constants; and
e = 20718
s' 3L~2
or
lo G(s) = Rest with s, R, T, and e
defined as above,
5 where whichever form the transfer function implied by Equation
(l) is relevant to a particular case will become clear when
the separate meanings of fuss) and Go are defined hereinafter
It is not possible, however, using current technology, to
perfectly realize the ideal all pass transfer function in
lo three-dimensional acoustic space with any known loudspeaker
system Accordingly, all real loudspeaker system
configurations or designs are based upon an approximation to
one or both forms of the ideal transfer function of Equation
(l) in three-dimensional space.
The simplest and commonest prior art
approximation to the ideal loudspeaker system is based upon a
"two-way" design using an assumed ideal woofer an tweeter
combined with a simple SdB/octave minimu~-pha~e r low-pass-
I ego
high-pass, cross-over network, as ill~tcr~ in Fig. l.
Mathematically, this approach takes the ideal transfer
function of Equation if) and attempts to reduce it to a
function that is independent of frequency, ideally a constant.
This is accomplished by expanding the ideal transfer function
of Equation (l) in a power series, as follows: when R = l,
then
to) fops) = east =
IT l+sT+(sT)~/21+(sT)~/31~...
Taking only the first term of this series of Equation I gives:
I Al =
lust
Those skilled in the art will recognize Equation (3j as the
simple one-pole low-pass transfer function
G 122~31~
f the term s in Equation S 3) is replace by
So
the complementary high-pass transfer function with cross-over
frequency 1 is obtainer:
I- 5 T I rlJ
(4) f2(s~ - 1 1 1 _ sty
lust s I l+ 1 lust
sty
Plots of the complementary amplitude response flus and us
of Equations (3) and (4) are given in Fig. 2.
. Equation I below shows that the sum of the simple
low pass function of Equation (3) and its complementary
high-pass function of Equation I is unity, that is, a
lo constant that is independent of frequency:
(5) f3(s3 = flus) = 1 sty = lust = 1
lust lust lust
If an ideal woofer were connected to a cross-over
network having the transfer response of Equation I and an
ideal tweeter were connected to a crossover network having the
transfer response of Equation (4), and the woofer and tweeter
were combined in a single system, the result would be a
"perfect loudspeaker system. Its amplitude response would be
perfectly flat for all frequencies and there would be no phase
shift at any frequency.
Serious problems arise, however when it is
attempted to construct a practical loudspeaker system
following the foregoing design procedure These problems
arise from three distinct causes:
flywheel) The woofer and tweeter do not have ideal
amplitude and/or phase characteristics.
~1.2) The loudspeaker drivers function in
three-dimensional acoustic space in which
the simple energy zillion of Equation to
is not valid at all points
(1.3) The gradual crossover slope ~6dB/octave)
allows too much bass energy to enter the
tweeter, and too much treble energy to enter
the woofer causing distortion.
Even with the reservations just mentioned the simple two-way
loudspeaker system of Figure 2 approaches the ideal to a
degree sufficient to achieve moderately satisfactory
performance
It should be noted that the transfer function us
as discussed here so far is that of the electrical cross-over
circuits alone, i.e., f(sj is defined as:
lay us = eon = Pus
e Us
wish in words, represents the ratio of electrical energy at
the output of a general crossover band pass filter circuit, or
combination of filter circuits including a complete crossover
system, to the electrical energy at the input, expressed as a
function of complex frequency.
The assumption has been implicitly made so far in
this discussion that the transfer function of the loudspeaker
drivers defined immediately hereinafter) is either unity, or
an "ideal delay and thus may be ignored. This is, at best,
only approximately true. The transfer function of a
loudspeaker driver is an electroacoustical quantity and may be
defined as the ratio of the sound pressure at a point in the
listening environment to the electrical energy input to the
driver terminals; this expression being a ratio of terms in
complex frequency:
lo I = P - clue
Thus one can consider the overall transfer function -
,
AL
,~_
of a complete speaker system taken as a whole - crossover plus
loudspeaker drivers - which would be the product of the two
above mentioned transfer functions and would be defined as:
(to) Gas) = oh Pi = Pi
eye Q2(s~
The transfer function G(s) represents the ratio of acoustic
sound pressure at a particular point in the listening
environment to the electrical energy applied to the input
terminals of the speaker system, both as a function of complex
frequency. The terms Pus and Q2(s~ will contain the poles
and zeros pi of the loudspeaker drivers as well as the p-
of the crossover elements.
Well-designed loudspeaker drivers possess a band of
frequencies in which the amplitude response is flat to desired
accuracy, and phase response is linear to desired accuracy -
such loudspeaker drivers may be referred to as n ideal" and
will possess an ~lectroacoustical transfer function so which
can be considered to be unity or a constant. Loudspeaker
drivers will be considered to possess ideal delay - i.e., to
approximate Equation I to a high degree of accuracy within
their respective frequency bands of best performance - unless
specifically stated otherwise. Hereinafter transfer
functions designated as us will always relate to the
crossover only, while transfer functions designated G(s) will
always relate to the crossover filter circuits plus the
loudspeaker drivers, with definitions as set forth
hereinbefoxe.
Those skilled in the art will understand that the
units of the right-hand terms contained in Equation (lo) are
voltages and currents which relate to the electrical circuit
ox the crossover network under consideration. The units of
the right-hand terms of Equations lo and (to) are voltages,
currents, forces, velocities, pressures, and volume
Go ~22~312
-or-
velocities, which relate to the electrical circuit
crossover), the mechanical circuit (the loudspeaker driver,
and the acoustical circuit (the air surrounding the
loudspeaker driver). For a numerical solution to specific
forms of Equations (lb) and Sly), or any other equation
contained herein with mixed electrical, mechanical, and/or
acoustical circuits t such as Equation to) expressed
hereinafter, the method of dynamic analogies" must be
employed. This method is amply treated in the text
"Acoustics", Chapters 3 and 5, by Leo Beranek, McGraw ill
- 1947. hereinafter, this method of dynamic analogies will be
implicitly assumed to have been applied whenever an nacousticR
sum is discussed with respect to an equation containing
"mixed, i.e. electroacoustical units.
Returning to the earlier discussion, it is observed
that closer approximations to the ideal transfer function of
Equation (1) have traditionally been realized by taking more
terms of the infinite series of Equation (2), again attempting
to reduce the-result to a function independent of frequency,
and then using such function as a basis for design
considerations. Some of these methods have been treated in
the prior art an in particular, in several issues of the
Audio Engineering Society Journal, specifically in the
following articles: "Constant Voltage Crossover Network
Design, Richard Small, January, 1971; "Active and Passive
Filters in Loudspeaker Crossover Networks", Ashley and
Kaminsky, June 1971; and PA Novel Approach to Linear Phase
Loudspeakers Using Passive Crossover Networks," I. Backgaard,
May 1977.
When such prior art loudspeaker topologies are
considered with reference to the poles and zeros of the
resultant system input-output transfer function, it is found
that all poles and zeros that is all p-z, tend to disappear
I
In genera] most prior art loudspeaker designs or
configurations have utilized acoustic summations which caused
the disappearance of as many p-z as possible in the summation
while tending toward some good approximation of the transfer
function of Equation (1). The present invention takes an
opposite approach, that is, retaining all, or as many as
possible, of the p z of the individual elements in the final
summation. also crucial to the method of this invention is
the inclusion of transmission Eros in the design of the
crossover filter circuits These transmission zeros, taken
together with retention in the total loudspeaker system of the
dominant poles inherent in the separate crossover filter
circuits allows the approximation of the transfer function of
Equation I to a high degree of accuracy, while also
overcoming the problems, mentioned above (I 1.2 and 1.3),
of the prior art loudspeaker designs.
Summary of the Invention
A descriptive name will be given to the present
invention; operative embodiments of the same will sometimes be
referred to hereinafter as "infinite-slope" speaker systems.
A general object of the invention is to provide an
improvement in a loudspeaker system crossover using passive or
active circuit topology which accurately approximates the
ideal transfer function of Equation (1).
Another object of the invention is to provide an
improved method for converting varying electrical signals into
sound involving the utilization of crossover circuits that
maintain all or as many as possible of the dominant poles of
the individual crossover filter circuit transfer functions
defined by equation (lo) herein before.
A further object of the invention is to provide an
improved method of converting varying electrical signals into
sound in which the poles of the transfer function of any
filter circuit of the loudspeaker system may not be repeated
in the transfer function of any other filter circuit of the
loudspeaker system, thereby satisfying a necessary criterion
assuring that the individual crossover filter circuits will
possess mutually exclusive frequency passbandsO
I
A further object of the invention is to provide such
an improved method for converting varying electrical signals
into sound wherein the individual crossover filter-loudspeaker
driver combinations comprising the invention are caused to
function independently of each other in forming the total
acoustic sum, or total acoustic output, by the use of two or
more distinct "brick-wall" amplitude functions having separate
mutually exclusive frequency pass bands which pass bands in
addition possess very high band-edge amplitude vs. frequency
to response slopes. Brick-wall amplitude functions are defined
in the text "Circuit Theory and Design" Chapter 5, by John L.
Stewart, John Wiley and Sons, Inc., New York, 1956.
A more specific object of the invention is to
provide an improved crossover network for a loudspeaker system
wherein mutually-coupled coils are used in order to enhance
the steepness of the pass band band-edge amplitude vs.
frequency response slopes.
Still another object of the invention is to provide
an improved loudspeaker crossover network embodying all-pass
delay equalization ox one or more filter circuits in order to
achieve a more accurate approximation to the ideal transfer
function of Equation ~13.
further object of the invention is to provide an
improved loudspeaker system in which the crossover circuit
parameters an driver placement are so adjusted that the band-
width of audible frequency bands of mutual acoustic
interference are reduced to less than 1/3 octave
Another object of the invention is to provide an
improved loudspeaker system in which the electrical parameters
30 of the crossover network include transmission zeros placed at
frequencies just outside the pass bands of the individual
crossover filter circuits Lopez, band-pass, and whops
in order to achieve very high pass band band-edge amplitude V5.
frequency response slopes.
I
In accomplishing the foregoing and other objectives
of the present invention active-or passive topologies may be
employed. Since passive topologies are most commonly used in
loudspeaker system crossovers, the discussion that follows is
concerned mostly with passive circuitry Active circuit
embodiments of the invention however, will also be discussed.
The present invention achieves a new approximation
to the ideal transfer function in a novel manner. Particular
attention is paid to the problems mentioned above (1~1, 1.2,
an 1.3~. The invention will be shown to:
tl.4) Greatly minimize the undesirable effects of
non-ideal driver amplitude and phase
response on total system performance.
(1.5) Minimize the acoustic wave interference
between arrivers at the crossover
frequencies.
(1.6) Reduce total system harmonic and
inter modulation distortion.
The method of the invention takes into consideration
20 two ideas:
(1.7) The total system performance must relate to
an "acoustic sum of arriver energies in
three-dimensional space.
~1.8) The ideal transfer function of Equation (1)
is approached using embodiments of the
present invention which operate by
acoustically summing two or more
approximations to ideal brick wall
amplitude functions, all having separate
mutually-exclusive frequency pass bands. If
these amplitude functions are carefully
chosen, the acoufitic sum will not only
approach flat amplitude V5. frequency
I
characteristic but will also approach a
linear-phase us frequency characteristic
Jo with, at most an ambiguity of phase of + on Jo
where n = 0, 1, 2, 3, ...) at the Rhodes
crossover frequencies. Observe that if no
the drivers are absolutely as
well as relatively in phase, and no phase
ambiguity exists.
In accordance with the invention, a ~uasi-infinite
crossover bondage pow is achieved my employing crossover
circuit topologies which have zeros of transmission placed at
frequencies outside their respective band edges.
Mutually-coupled coils may be used in order to improve
out-of-band attenuation, in passive embodiments. Both
minimum phase and non-m;nimum-phase topologies are Lydia
In-band amplitude and phase characteristics of the loudspeaker
drivers may be considered as part of the overall system
transfer function. Crossover filter pass band band-edge
amplitude vs. frequency response slopes in embodiments of the
present invention approach 100 dB/octave with minimum
out-of~band attenuation being greater than 40 dub. This is in
sharp contrast to the monotonic 6, 12, or 18 dB/octave
crossover slopes commonly used in the prior art loudspeaker
systems
I There is no fixed way to embody the invention. Any
combination of active or passive crossover topologies
satisfying the requirements of the idea statement of 1~8 above
and fitting into the described three classifications of
general crossover circuit topologies, while approximating the
transfer function of Equation I will work. Crossover
topologies in an operative invention embodiment will possess
filter circuits having I separate and mutually exclusive
frequency pass bands and (2) at least one transmission zero in
AL
each filter circuit. Those skilled in the art will recognize -the
aforementioned to be characteristic to one class of brick-wall
amplitude functions.
Thus, in accordance with a broad aspect of -the invention,
there is provided a multi driver loudspeaker apparatus comprising
at least two loudspeaker drivers, a crossover filter network
having at least two crossover filter circuit means, said
loudspeaker drivers being each connected to a respective crossover
filter circuit means of the crossover filter network, at least
I one of the crossover filter circuit means of the crossover filter
network including an additional circuit structure means for
generating at least one zero or near zero transmission of response
in the transfer function of said crossover filter circuit means
for frequencies just outside the pass band of said crossover
filter circuit means.
Description of the Preferred Embodiments
For a better understanding of the preferred embodiments
of the invention, reference is first made to the loudspeaker
circuit diagram of Figure 3. In Figure 3 electrical signals
representing sound to be reproduced appear at the output terminals
of an exterior component (not shown) such as a suitable amplifier.
One output terminal, Al, of -the amplifier is connected to ground
potential and the other output terminal, A, is connected to a
conductor 12 of a
- I -
--15--
loudspeaker crossover circuit 10. Conductor 12 is common to
the high-pass, band-pass, arid low-pass crossover filter or
networks 14, 16, and 18 of circuit 10.
As shown in Fig. 3, a tweeter 20 is connected to the
output of the high pass filter circuit 14 and responds to
electrical signals in the range of 7 K~z to 20 KHz. The
filter circuit I and tweeter 20 form a first crossover filter-
loudspeaker driver combination. A delay network 22 is
provided between the tweeter 20 and the output of the
high-pass filter circuit 14. The purpose of the delay network
22 is explained hereinafter.
The band-pass filter circuit 16 includes a lower
midrange section aye and an upper midrange section 16b. An
upper midrange driver 24 is connected to the output of top
upper midrange section 16b~ forming therewith a second
crossover filter-loudspeaker driver combination Similarly, a
lower midrange driver 26 is connected to the output of the
lower midrange section aye, forming therewith a third filter
loudspeaker combination. Lower midrange driver 26 responds to
electrical signals in the range of 175 Ho to 800 Ho. Upper
midrange driver 24 responds to signals in the range of 800 I
to 7 KHz.
A woofer 28 is connected to the output of the
low-pass filter circuit 18 and responds to electrical signals
in the range of frequencies from 20 to 175 Ho. The filter
circuit 18 and woofer 28 form a fourth crossover filter-
loudspeaker driver combination.
In accordance with the invention, the four
separate filter circuit-loudpseaker driver combinations
operate independently of each other in forming the total
acoustic output of the system.
The tweeter topology selected for the Fig. 3
embodiment is based upon the high-pass network illustrated in
~2~3~
-16- .
Fig. pa and, as shown, includes a series connection from a
first input terminal, indicated as conductor 12, of capacitor
Of and C2 and a terminating resistor R to a conductor 32 that
leads to a second input terminal G with an inductor Lo
connected between the junction of capacitors Of and C2 and the
conductor 32~ It is noted that most passive crossover
networks are synthesized on the assumption that the network is
terminated by a pure and constant resistance
A zero of transmission is added to the high-pass
network of Fig. aye as is illustrated in Fig. 4b, by
connecting an inductor Lo and a capacitor C3 in series between
the junction of capacitor C2 and resistor R to the conductor
32. Mutual coupling between the inductors Lo and Lo is not
used here as it would cause the out-of-band input impedance to
become excessively low. The network of Fig 4b yields the
poles and zeros, that is, pi shown in the pole-zero pattern
of Fig. 4c~ .
Midrange topology selected for the midrange drivers
24 and I is based upon the simple half-section band-pass
network illustrated in Fig. pa. As shown, Fig. Spa includes an
inductor Lo and a capacitor C4 connected in series with a
resistor Al between a first input terminal 12 and a conductor
34 leading to a second input terminal G. There is also
provided a parallel connection of an inductor Lo and capacitor
Us between the junction of capacitor C4 and resistor Al and
the conductor 34.
Zeros of transmission are added to the network of
Fig. pa as is shown in Fig 5b. A single zero of transmission
is provided for the lower midrange circuit aye by the addition
in parallel to the inductor Lo and capacitor as of an inductor
Lo and a capacitor C6 which are connected in series. Mutual
inductance is provided, as shown in Fig. 3, between inductor
Lo and I This zero of transmission is placed above the
lower midrange crossover frequency.
~2~3~
-17-
For the upper midrange circuit 16b, there is further
provided in parallel to the inductor Lo and capacitor Us an
inductor Lo and a capacitor C7 which are connected in series.
Mutual coupling is provided between inductors Lo and Lo, as
shown in Fig r 3 7 in order to improve the attenuation of the
higher frequencies above the upper midrange.
More than one zero of transmission may be used for
the midrange or band-pass frequencies. In the Fig. 3 upper
midrange crossover, one is placed below the low frequency
point point and the other above the high frequency point.
These zeros of transmission are generally placed one octave
below the lower midrange crossover frequency, and one-half
octave above the higher midrange crossover frequency. General
p-z for the network of Fig. 5b are given in the pole-zero
pattern of Fig. So.
Tune low-pass topology selected for the Fig. 3
embodiment is the basic T-section network as shown in Fig. pa
and includes series-connected inductors Lo and Lo and a
terminating resistor R2 between first input terminal 12 and a
conductor 36 leading to a second input terminal G. The
network also includes a capacitor Cog, one terminal of which is
connected to the junction of inductors Lo and Lo an the other
terminal to the conductor 36.
A zero of transmission is added to the network of
I jig. pa as is illustrated in Fig. 6b by the connection of an
inductor Lug and a capacitor Cog in series between the junction -
of inductor Lo and resistor R2 and the conductor 36. The zero
ox transmission thus added to the network as shown in Fig. 6b
yields the p-z shown in the pole-zero pattern of Fig. 6c. The
zero of transmission will usually use mutual coupling of
inductances Lo and Lug, as is illustrated in Fig. 3, at 18.
This zero of transmission is placed one to two octaves above
the woofer crossover frequency.
~;~2~L2
-18-
By way of illustration and not limitation it is
noted that in the embodiment of the invention of Fig 3 the
electrical parameters of the individual components of the high
pass, upper midrange, lower midrange and low pass sections 14,
16b, aye, and 18, respectively, are as follows:
High pass section 14 Low ass section 18
P
Lo - 125 micro henries Lo - 6 millionaires
Lo 320 micro henries Lo - 5.1 millionaires
C1 - 2 micro farads Lug - 3.3 millionaires
- 10 C2 - 2 micro farads C8 - 220 micro farads .
R - 27 ohms Cog - 47 micro farads
Upper midrange section 16b Lower midrange section 16
Lo - 275 micro henries Lo - 1~75 millionaires
Lo - 1.07 millionaires . Lo - 3~82 millionaires
Lo - 1~92 millionaires Lo - 1.07 millionaires
Lo - 190 micro henries C4 - 100 micro farads
C4 - 12 micro farads Us - 22 micro farads
c5 - 6.8 micro farads C6 - 22 micro farads
C6 - 56 micro farads Al - 15 ohms
20 C7 - 2 micro farads
Al - 15 ohms
R2 - 3 ohms
The frequency response of the individual drivers of
the Fig. 3 system is illustrated in the chart of Fig. JO This
chart represents close microphone tests on individual drivers,
with all other drivers open or shorted. As those skilled in
the art understand the term "close microphone refers to a
technique for measuring the sound output of a loudspeaker
diaphragm by placing the microphone very close to the
diaphragm, typically 1/4" to 1/2~.
It is noted that the small loop under the tweeter
response curve in Fig. 7 is a spurious response from the upper
I 2
--19--
midrange arriver 24. This spurious response is a fault of the
driver 24 and can be eliminated by placing a I Jo inductor
in series with capacitor Us of Fog 3. The spurious response,
however is of little or no consequence, being about 15 dub down
and inaudible.
The chart of Fig 7 and that Allah of Fig. 8, now to
be describer were run 4 inches per minute.
The composite frequency response of the whole
system, that is, including the total acoustic sum of all
drivers on axis, is illustrated in Fig. 8. It is evident that
the amplitude response given in Fig 8 is a good approximation
to the amplitude criterion of the transfer function of
Equation I Consideration must still be given, however as
to how well the embodiment of Fig. 3 approaches the ideal
phase characteristic implies in Equation (1).
Fig. 9 is a plot of the phase response of the
embodiment of Fig. 3. Ideally in order to satisfy the ideal
phase requirements implicit in Equation (1), the phase
response shall be a straight line throughout the entire
frequency range. A compromise may be made however, by
allowing the phase response to exhibit changes in slope and/or
discontinuities at the crossover frequencies. In Fig. 9, the
phase response exhibits a constant slope above 1000 Ho while
breaking into a steeper slope below 1000 Ho and then becoming
I irregular at low bass frequencies.
The phase curve of Fig 9 implies the existence of a
relative time difference of about 800 microseconds between
modern frequencies (175-1000 Ho and high frequencies Lowe
- on zoo I ) . Acoustical researchers are in general
disagreement as to whether small acoustical time delay errors
of the order of 1 my.) are audible Whether such errors are
audible or not is moot, as the present invention permits such
errors to be essentially eliminated.
.
-20~
The invention embodiment of Fig 3 uses an all pass
; delay network for correction ox phase response. Thus, the all
pass delay network 22 in series with the tweeter matches
the delay of the tweeter 20 to what of the upper midrange 24.
A detailed discussion of the theory and design of delay
networks will not be undertaken hero For those skillet in
the art, the relevant theory is disclosed in the text "Network
Analysis and Synthesis," by OF Duo, John Wiley, 1962, pages
315 to 321~ -
It is noted that at low frequencies (175 I and
lower) the ear becomes less sensitive to phase and
directionality Thus irregularities in phase inherent in low
bass loudspeaker response are relatively insignificant The
present invention finds particular application above the low
bass frequency region that is, frequencies above 175 Ho.
An embodiment of the invention with improved performance over that of Fig 3 is shown schematically in Fig.
10. Amplitude response of individual arrivers within the
system is shown in Fig. aye Overall amplitude response of
the entire system is shown in Fig. 16(b). It is clear that
the crossover pass band slopes are steeper than whose of Fig. 7
while the frequency overlap between adjacent pass bands is
reduced in Fig aye over that of Fig. 7. The embodiment
shown schematically in Fig. 10 functions better than that of
Fig. 3 in fulfilling the intent of this invention according
to statements 1~4 through 1.6 inclusive as set forth
hereinbefoxe.
An active embodiment of the invention with
performance superior to that of the passive embodiment of Fig.
30 10 is shown schematically id Figs. aye and lob. Amplitude
response of individual drivers within the system is shown in
Fig. aye Overall amplitude response of the entire system
is shown in Fig 17(b~ It is clear thaw the crossover
-21-
pass band slopes are steeper than those of Fig. 10, while the
frequency overlap between adjacent pass bands is reduced in Fig.
aye over that of Fig. 10, Although the active embodiment of
Figs. aye and 15b of the invention do outperform the two above-
mentioned passive embodiments of Figs. 3 and 10, it should be noted
that the active embodiment is one of greater complexity. The
circuitry used in active embodiments of the invention will be
briefly described.
In the circuit diagram on Fig. aye there is illustrated
an active circuit embodiment 35 of the present invention including
a tweeter 37, a midrange driver 38, and a woofer 40. Electrical
signals representing sound to be reproduced appear at the output
terminals A and A ox a component (not shown) such as an amplifier.
The terminal A is connected to the conductor 42 of a loudspeaker
crossover circuit 44. Conductor 42 is common to the active high-
pass, band-pass and low-pass crossover filters indicated at 50, 48
and 46. Filters 50, 48 and 46 may comprise suitable electronic
devices such as operational amplifiers incorporating the crossover
p-z, as described herein before, of the present invention for the
tweeter 37, midrange driver 38, and woofer 40. As shown a
separate power amplifier 52, 54 and 56 is provided for amplifying
the output of a respectively associated filter 46, 48 and 50. The
outputs of amplifiers 52, 54 and 56 are applied, respectively, to
the tweeter 37, midrange driver 38 and woofer 40.
Figure 15b illustrates the circuitry of Fig aye in
detail. The design and method of operation of active filter air-
cults to realize any p-z are sufficiently well explained, for those
-aye-
skilled in the art, in the reference "Handbook ox Operational
Ampler Circuit Design," by David F. Stout and edited by Milton
Kaufman, McGraw-Hill, 1976.
The phase response of the invention embodiments of
Figs. 10 and aye and 15b can be shown by phase and time
measurements familiar to those skilled in the art to fulfill
the intent of the invention as contained in the idea statement
1.8. Acoustical measurements taken on the aforementioned
embodiments show all drivers operating in the same relative
phase at the crossover frequencies with, at most phase
rotations of on radians. Translated to delay error, this
phase shift never exceeds a maximum delay error of one
millisecond, for frequencies above 175 I Delay response for
one preferred invention embodiment, that shown in Fig. 10, is
given in Fig. 16(c3~
It should also be noted that as a result of steeper
crossover filter pass band band-edge slopes, the improved
invention embodiments, especially those of Figs. 10, lea, and
15b will possess small wave interference between arrivers
operating on adjacent frequency bands. Acoustical researchers
have not determined a precise value of such aforementioned
wave interference at which its audible effects become
2Q objectionable. however, it will be evident to twos swilled in
the art that the present invention allows the level of wave
interference to be reduced arbitrarily to a level such that
drivers on adjacent frequency bands can be considered to
radiate sound energy independently of each other, that is, the
wave interference is rendered to be, at most, just barely
audible.
Acoustical researchers are not in agreement as to
the definition of audibly objectionable wave interference with
respect to the sound radiated from a multi-driver loudspeaker
system. This inventor has experimentally determine that if
the overlap", defined immediately hereinafter, in amplitude
responses at the -lode points of loudspeaker drivers sharing a
common crossover frequency is greater than 1/3 octave, the
interference becomes audible, and thus, objectionable.
-23-
Fig. 18 illustrates the definition of acoustic wave
interference. The amplitude responses 60 and 62 of two
drivers sharing a common crossover frequency We are shown
The overlap region shown at 64 becomes just barely audible at
a point 10 decibels below the flat portions of the amplitude
responses shown at 10 and 20~ If this width of the same
overlap region shown at 64 in Fig 18 is greater than lJ3
octave, centered on the crossover frequency We, the acoustic
wave interference is arbitrarily, but necessarily, considered
by the inventor to become objectionable.
Acoustic wave interference effects in multi driver
loudspeaker systems are related to the steepness of the
crossover filter pass band band-edge amplitude vs. frequency
response slopes If the aforementioned slopes, shown at 65
and I in Fig. 18, become steeper, the width ox the overlap
shown at 64, which is a measure of the energy causing acoustic
wave interference, will decrease. Slopes in excess of 40
- dB/octave will result in an overlap shown at 64, of less than
1/3 octave in typical operative invention embodiments, with
acoustic wave interference being just barely audible. This
would yield acceptable porphyrins for an operative invention
embodiment. Slopes greater 70 dB/octave, and approaching
100 dB/octave as in the better embodiments result in
inaudible acoustic wave interference, with superior
performance, as the width of the overlap region, shown at 64
in Fig 18, becomes much less than 1/3 octave, too narrow for
the human ear to perceive.
The separate crossover filter-loudspeaker driver
combinations comprising an invention embodiment which share a
common crossover frequency, for which typical amplitude vs.
frequency responses in the crossover region are illustrated in
Fig. 18, will be defined as functioning "effectively"
independently of each other if the acoustic wave interference
I 2
becomes just barely audible. This will correspond to
crossover filter pass band band-edge amplitude us frequency
response slopes of about 40 dB/octave, and up to about 70
dB/octave.
The aforementioned crossover filter-loudspeaker
driver combinations will be defined as functioning
absolutely" independently of each other, if the acoustic wavy
interference becomes inaudible. This will correspond to
crossover filter pass band band-edge amplitude vs. frequency0 response slopes equal to or greater than about 70 ds/~ctave~
In either of the aforementioned situations for
"effective or "absolute" independence, the separate crossover
filter-loudspeaker driver combinations will comprise an
operative invention embodiment and will be considered to
function independently of each other, without qualification
since the lesser condition of effective independence, defined
herein before, describes an acceptable or useful invention
embodiment. The aforementioned distinction between effective
and absolute independence is made merely in order to
distinguish between "acceptable and "superior" invention
embodiments, respectively. In summary, an operative or useful
embodiment of the present invention wit exist, if at least at
one system crossover frequency, and preferably at all system
crossover frequencies it more than one such crossover
frequency exists, the width of the overlap region shown at 64
in Fig. 18 is, at most, 1/3 octave, which will correspond
generally to a crossover filter pass band band-edge amplitude
vs. frequency response slope of about 40 dB/oct~ve.
Acoustic wave interference effects in multi driver
loudspeaker systems cause the tumbrel balance, or Sweeney
quality" of a single, or monophonic speaker system to vary as
the listener moves about in the room in which the speaker
system is sounding There will exist a different amplitude
-25
vs. frequency response for each and every listening position
in the room. For a stereo pair of speakers, the
aforementioned effect of acoustic wave interference still
exists. In addition, acoustic wave interference within the
individual speaker systems comprising a stereo pair will cause
a blurring or "lack of focus" to the stereo image, because the
accurate amplitude and phase information necessary to
psycho acoustically reconstruct the stereo image becomes
distorted by the same acoustic wave interference
Sonic problems in loudspeaker systems caused by
acoustic wave interference are common knowledge to those
familiar with the art Designers of prior art loudspeakers
have attempted to minimize the effects of acoustic wave
interference in one of two ways:
(1) Elimination of the crossover network
entirely, and devising loudspeaker systems
which function by operating a single large
loudspeaker driver or plurality of small
loudspeaker drivers over the entire audio
frequency range;
for example, in a speaker system presently marketed as the
"901~ by the Bose Corporation of Framing ham, Massachusetts,
VISA or
(2) Use of a conventional crossover system
in the prior art as described herein before
taken together with physical location of the .
loudspeaker drivers comprising the total
loudspeaker system having their acoustic
centers as close together as practicable;
for example as in a speaker system presently marketed as the
"AURELIUS by the Acoustic Research Corporation of oared,
Massachusetts, U.S.A.
I
-26-
The present invention presents a novel and superior
crossover method which rectifies the aforementioned sonic
difficulties caused by acoustic wave interference,
specifically those problems relating to tumbrel balance and
stereo image. Loudspeaker systems embodying the present
invention possess separate crossover filter-loudspeaker driver
combinations which function independently of each other in
their contribution to the total system acoustic output,
thereby exhibiting negligible acoustic wave interference among
the same crossover filter-loudspeaker driver combinations
comprising the total loudspeaker system, thereby producing
within the entire listening environment a very uniform
amplitude us. frequency sonic energy response yielding both
an accurate tumbrel balance and a clear and well-focused
stereo image anywhere within the aforementioned listening
environment. Also, as an added benefit from the present
invention, mentioned herein before in idea statements (1.4) and
(1.6~ and repeated here, sonic difficulties resulting from the
transmittal of energy to loudspeaker drivers outside their
respective frequency bands of best performance are eliminated.
Figs. lo and lo illustrate the superior
performance of the present invention over prior art, with
regard to the projection of a uniform amplitude vs. frequency
sonic energy into the space in front of the loudspeaker system
box, shown at 68. Due to the elimination of acoustic wave
interference effects among the separate crossover
filter-loudspeaker driver combinations comprising loudspeaker
systems embodying the present invention, more specifically the
invention embodiment schematically depicted in Fig 10, this
same invention embodiment exhibits nearly identical 1/3 octave
pink noise amplitude vs. frequency responses, shown at 67 it
Fig lea, for three different positions of the test
microphone, said positions being shown at 69/ 70 and 71, for a
-27-
box to microphone position of one meter. The same test
results for a prior art loudspeaker system, of about the same
size, shaper and cost of the aforementioned invention
embodîmentJ namely the model "50" made by the Speakérlab
Company of Seattle, Washington, USE is shown at Fig. lob.
Variations in amplitude vs. frequency response of this same
prior art system, shown at 72, for the aforementioned same
three microphone positions are clearly apparent.
Figs. aye 20(b~ and 20~c3 illustrate the superior
performance of the present invention over prior art, with
regard to the reduction of loudspeaker driver performance
deficiencies stated in idea statements (1.4) and (1.6~. Fig.
aye shows the amplitude vs. frequency response for a
high-quality 6 1/2" bass-midxange river when the same is
operated over the entire audible frequency range without the
use of a crossover filter. The frequency region of best, or
ideal" driver performance is shown at 73. Above the
frequency shown as lo cone breakup occurs, which is the main
cause of the irregular frequency response shown at 74.
Audibly unpleasant "sonic colorations and high distortion
occur in the same frequency region shown at yo-yo phenomena well
known to those skilled in the art. Figure 20b shows the
amplitude vs. frequency response of this same 6 1/2" driver
when operated over the entire audible frequency range using a
common prior art crossover filter topology, specifically a
low-pass filter having a 6 dB/octave pass band band-edge slope.
The frequency region of best driver performance, shown at 73,
is unaffected. The frequency region of poor driver
performance, shown at 75, is clearly evident, and will be
30 audible as sonic coloration and distortion. Figure 20c shows
the amplitude vs. frequency response of the some 6 1/2R driver
when operated over the entire audible frequency range but
using n infinite slope low-pass filter topology of the present
I
-28-
invention. The frequency region of best driver performance,
shown at 73, is unaffected but response in the frequency
region of poor driver performance, shown at 76~ is eliminated
and rendered inaudible.
For those skilled in the art, it should become
evident upon study of the amplitude responses of invention
embodiments riven here in Figs. 7, 16 and 17 that the
amplitude us frequency response shapes characteristic to the
separate crossover filter-loudspeaker driver combinations of
the present invention are much closer to the ideal brick-wall
response shape, than exists in prior art. Thus the present
invention will, by virtue of its unique "infinite-slope"
crossover filter topology, improve over prior art, by
permitting the mitigation of certain performance deficiencies
in loudspeaker systems, namely the deficiencies caused by
acoustic w Ye interference, and also deficiencies caused by
operation of loudspeaker drivers at frequencies outside the
same driver's respective frequency bands of best performance.
The reader is especially referred to the prior art designs
given in this disclosure in order to make the comparisons just
implied.
By way of illustration and not limitation it is
noted that the electrical parameters of the individual
components of the invention embodiments depicted in Figs. 10
and 15b are as follows:
1. Embodiment of Fig 10:
Lit - 5~1 my Coo - 470 micro farads
L12 - 5.1 my Oil - 220 micro farads
L13 - 3.2 my clue - 100 micro farad
L14 - 8.4 my C13 - lo micro farads
~15 - 1.55 my C14 16 micro farads
I
I
L16 1.7 my C15 36 m;crofara~s
L17 - 757 micro henries C16 16 mlcrofarads
Lo - 757 micro henries C17 - 14 micro farads
~19 - 2.1 my C18 10 micro farad
L20 320 micro henries Cog - 3.3 microEaraas
L21 540 micro henries C20 - 3.0 micro farads
L22 215 micro henries C21 - 1.0 microfaraas
L23 170 micron C22 1.68 micro farads
L24 - 540 micro henries C23 - 24 micro farads
Roll 30 ohms
Al - 30 ohms
R13 15 ohms
2. Embodiment of Fig. 15b:
R21 - lo ohms C31 0.13 micro farads
R22 - lo ohms C32 0.033 micro farads
R23 lo ohms C33 0.16 microfaraa~
R24 - 22k ohms C34 0.0016 micro farads
R25 - ok ohms C35 1027 micro farads
R26 - 22k ohms C36 0.127 micro farads
R27 lo ohms C37 0.127 micro farads
R28 - lo ohms C38 0.127 micro farads
Rug 7.2k ohms C3~ - OKAY micro farads
R30 ~50k ohms C40 0.0013 microfaradq
R31 -2.8k ohms C41 0.~2 micro farads
R32 lo ohms C~2 - 437 picofarads
~33 -lo ohms C43 0.343 micro farads
R3~ k ohms C44 - 0.02 micro farads
R35 -22k ohms C45 0.02 m;crofarads
R36 ~22k ohms C46 0~02 microfarad8
R37 - 22k ohms ~47 1137 picofarads
-30~
R3 10K ohms C48 - 0.89 micro farads
R - 10K ohms
R40 3.9K ohms All operational amplifiers
R41 - 155K ohms Signetics NOAH or equiva-
R42 620 ohms lent.
R43 - 22K ohms
R44 - 22K ohms
~45 22K ohms
R46 10K ohms
R4 10K ohms
Theoretical Basis for the Invention
It is believed that a better understanding of the
function of the crossover circuits of Figs. 3 and 10 (or other
topologies based upon the concepts of the present invention) will
be had upon an examination of the development of the character-
fistic p-z of the invention. A general reference that describes
the p-z concepts utilized according to the present invention is
circuit Theory and Design", (pp. 168-173) John L. Stewart,
John Wiley & Sons, Inc., New York, 1956. Other references that
are pertinent with respect to the network analysis are "Network
Analysis and Synthesis", First Edition, (pp. 320 and 321) Franklin
F. Queue John Wiley & Sons, Inc., New York, 1962; and "Linear
Network Analysis", (pp. 284 and 285) S. Swish and N. Balabanian,
John Wiley & Sons, New York, 1963.
A pole-zero pattern or p-z as is illustrated in Fig. 11
would be those for a perfect realization of the invention.
-aye-
These are the p-z of Equation (6) below;
(See the above mentioned reference by Stewart, pages
96-97 and pages 168-172; see also Swish and Balabanian, page 284
Fig. 27).
it. -`
-31- ~%~
I f4(s) = essay = l-sU+~sU)2~2l - (Sue!+...
east l+sT~sT~2/2l + (sty.
Taken to infinity in both U and T, real positive constants,
the above function will yield an infinite vertical row of
poles in the left hand s-plane and an infinite vertical row
of zeros in the right-half s-plane, as illustrated in Fig. 11.
If the series in Equation (6) is terminated in a finite number
of terms, with U = T, the following Equation 57) is obtained:
' -
(7) f5(s~ = east = east = l-$t~(sT~2/2! - (sty!
east STY ( sty 2 Jo + ( sty
which is a standard approximation to the Alps transfer
function with the series of Equation I terminated after
three terms. If the zeros in Equations I and (7) are
omitted the function remains all-pass for an infinite
expansion - Equation (63 becoming Equation (2) - and becomes
low-pass for a finite truncation equation I without its
Eros], both merely having half the delay at any frequency
(see page 285 of the reference "Linear Network Analysis"
referred to above). When the zeros of Equation lo or I art
removed, the earlier approximation based on Equation (2) is
obtained:
(2) fops) = east =
1 eat l~sT~tsT~/2l Steele+...
It is noted that Equations I and I) are identical if U = T
in Equation (~) except for the content multiplier in the
exponential Both represent an all-pass transfer function,
the addition of the retooled plane zeros in Equation (6)
merely doubles the delay. Since Equations (2) and I above
%
-32-
each represent an approximation to the ideal all-pass transfer
function, then any linear combination of them could also
approximate an Alps transfer function. If a loudspeaker
system is constructed such that its acoustical output may be
represented mathematically by multiplying the electrical input
by some linear combination of Equations (63 and I there
will be obtained a good approximation to the ideal loudspeaker
- system. It is noted that Equation (5) represents the simplest
possible such approximation. This invention considers a novel
and more accurate approximation.
By means of the present invention, a system
input-output transfer function is generated such that it may
be broken apart, using a method analogous to a
partial-fraction expansion, into separate low-pass, band-pass,
and high-pass representations, said representations each being
brick wall amplitude functions, the individual dominant p-z of
which can be represented as a linear combination of finite
forms of Equations (2) and (6). Stated in another manner,
there is developed by means of the present invention an
acoustic sum whose mathematical representation as a system
function has the form:
(8) Acoustic output = electrical input x P(s) = Go
Q(s)
where the loudspeaker drivers are assumed ideal and which is
equivalent to an approximation to an all-pass transfer
function in linear combinations of truncations of Equations
(2) and (6) whose quotient Squeeze can be broken apart,
similar to partial fractions, into two or more separate
quotients of polynomials in s, each of which will take the
form of the dominant p-z of brick wall filter transfer
function. These fractional expansions will take the
following forms:
-33-
(9) 2-way system Pluckily + P2(s)/Q2(
(10) 3-way system Pluckily P2(s)/Q2( ) 3 3
where extensions to 4-way, 5-way, etc. systems may also be
effected. In Equation (9) the terms Pluckily and P2(s)/Q2(s)
are arranged in such a manner that they represent the dominant
p-z of a low-pass and high-pass filter, respectively. Similarly
in Equation (lO),Pl(s)/Ql(s~ represents the dominant p-z of a low
pass filter; P2(s)/Q2(s) represents the dominant p-z of a band-pass
filter; and P3(s)/Q3(s) represents the dominant p-z of a high-pass
filter.
It has already been shown by Equation (5) that a very
simple loudspeaker system can be realized which satisfies Equation
(9). It is noted also that extensions ox this simple idea will
satisfy Equation (1) and higher-order systems. All of the prior
art designs, however, depend upon filter topologies that have
gradual crossover slopes (6, 12, or 18 dB/octave in various
combinations) which suffer from deficiencies (1.1, 1.2, 1.3)
mentioned herein before. Novel means are provided according to the
present invention to mitigate these aforementioned deficiencies.
In accordance with the invention, a loudspeaker system
is constructed with crossover filter circuits having very high
pass band band edge amplitude vs. frequency response slopes while
generating an overall system transfer function having p-z in the
forms implied by truncations of Equations (2) and (6). This is
effectively done by the embodiments of the invention illustrated
in Figs. 3, 10 and 15. Basic characteristics of any embodiment
of the invention, including those of Figs. 3, 10 and 15, are as
-aye
hollows:
(1.9) The crossover filter pass band band-edge
slope is large; it may be as high as 100
dB/octave, or even higher.
I
I
(2.0) At least two loudspeaker drivers are used,
and at least two separate mutually
exclusive frequency bands are covered
ire., one woofer and one tweeter comprise
the simplest possible embodiment.
(2.1) The crossover filter pass band slope and
arriver placement are adjusted so that wave
interference between adjacent frequency
bands is minimized by keeping the audibly
effective band-width of any such
interference to less than 1/3 octave.
(202~ The electrical parameters of the crossover
are adjusted such that a fractional
expansion of the system transfer function
will appear as in Equation (9) or (10) or
any higher-order extension of these. The
denominator polynomials QSs) will, in the
best embodiments, have different poles,
with no repeated common) poles, which it
characteristic to a class of separate an
distinct brick-wall amplitude functions
employing both mutually exclusive frequency
pass bands and transmission zeros.
(2.3) The electrical parameters of the crossover
network, in the best embodiments, are so
adjusted that all drivers in the
loudspeaker system operate in the same
relative phase, which allows phase
cola no
rotations of 2nll~at the crossover
frequencies, in accordance with statement
(1.8~ herein before.
When conditions (1~9) to (2.3) are met, an
embodiment according to the invention is realized. Stated
~213~2
-35-
differently, when a loudspeaker system is constructed that
satisfies condition (1.9) through (2.3), the p-z of its
overall system transfer function will satisfy Equation (2) or
Equation I or some linear combination of these.
A pictorial illustration of the p-z peculiar to a
general embodiment of the invention will clarify how the
concepts of conditions (2.1~ to (2.3) are realized Figs.
aye, 12b and 12c show the dominant p-z for the low-pass,
band-pass, and high-pass filter circuits of an embodiment.
The p-z at the origin and those having large negative real
parts (depicted herein before in Figs. pa, 6b and 6c; pa, 5b
and 5c; pa, 4b and 4CJ respectively) are ignored as they
contribute little to the system response inside the pass band.
When the p-z in Figs. lea, 12b and 12c are summed,
the result will have dominant p-z as shown in Fig 12d~ Since
the poles of Figs. aye, 12b and 12c are simple and distinct,
all of those poles appear in Fig. 12d. all dominant let hand
plane zeros inside the system total pass band response, in any
embodiment of the invention, will disappear in this summation.
Dominant right half plane zeros, if and as they
occur in any embodiment, will not disappear and instead will
always be accompanied by corresponding mirror image left hand
plane poles The simpler case, in which there are no dominant
right half plane zeros, is treated in this discussion.
This concept may be depicted mathematically, as
explained further hereinafter, but involves the use of certain
expressions which will first be derived with reference to
Figs. aye and 13c which respectively illustrate, in general, a
tweeter network and a midrange-n~twork embodying the
invention.
The following is a mathematical derivation for the
p-z of the tweeter and midrange networks. The tweeter network
is given as shown in Fig aye and has one zero of transmission
-36~ I
placed about one octave below its pass band. Solving for the
transfer function (output voltage vs. input voltage) of the
network of Fig aye, yields t assuming no loss in the ICKY
series circuit:
eon = LlClC?Rs3(L2C~s2+1)
AS5+BS4+Cs3+Ds2~Es~F
where the denominator coefficients do not appear explicitly as
functions of the circuit elements; which has p-z as shown in
Fig. 13b. It is noted-by reference Jo Fog. 13b that with no
resistance in the L2-C3 branch the zeros are positioned on the
jaw axis. On the other hand, with series resistance in the
L2-C3 branch, the zeros are displaced to the left hand plane.
The midrange crossover network is given as shown in
Fig. 13c and has one zero placed about 1/2 octave above its
upper cutoff frequency. Solving for its transfer function as
above yields:
.
eon = Cls21As2+B)
e so s5~s3~ so us
which has p-z as is illustrated in Fig. 13d. It is noted that
zeros appear in Fig 13d on the jaw axis with no series
resistance in the L-C branch, corresponding to infinite
band-edge slope The zeros move to the left for real circuit
elements having resistance, corresponding to finite band-edge
slopes.
The concept that all dominant left hand plane zeros
inside the system total pass band response will disappear in
the summation, as noted above, is depicted mathematically a
follows, where the following expressions are those obtained it
the foregoing derivation.
~22~3
-37-
Let eel = Cls2~As2+B) represent the p-z of a
e (six poles)
band-pass midrange filter circuit of an embodiment of the
invention.
Let e 2 = LlClC~Rs3(L?C~s~2~1) represent the p-z of
(five poles of eon)
a high-pass tweeter) filter circuit of an embodiment of the
invention. The poles of eon and eon will be assumed to be
mutually exclusive, i.e7, no common factors.
It is desired to sum these transfer functions in
such a manner as to realize the invention. The sum is formed
as in Equation I below:
(11) eo--eol~eo2 =
e e e
Cls2~As2~B~Poles of eo2)+L~ClC?Rs3~L~C~s2+1)SPoles of cot)
Eleven poles of cot and eon
where choice of the plus or minus sign depends on the relative
phase characteristics of the drivers.
A straightforward solution for the locations of the
p-2 in Equation ~11), or in any other summation of the form of
Equation lend peculiar to embodiments of this invention,
is very difficult. Fortunately such a direct solution is not
necessary. The essential disappearance of all dominant
lightened plane zero in a summation of the form of Equation
(11) r and the consequent realization of the invention, is
easily shown heuristically from simple energy considerations.
Refer to Figs. Audi and consider statement (2.4)
below:
(2.4) Since each filter circuit of any embodiment
of the invention has very steep pass band
amplitude vs. fxequenc~ response bounded
slopes, combined with mutually exclusive
~l22~L?3 1~2
I
ranges or bands of frequency coverage, the
energy at any given frequency in the total
system transfer function can be considered -
to be sensibly contributed by one, and only
one portion, it crossover
filter-loudspeaker driver combination
(low-pass, ban pass or high-pass) of the
total system.
The only frequencies on the jaw axis where the
aforementinea statement becomes an approximation is in the
small regions of frequency overlap at the crossover
frequencies, Observe that it the invention were perfect
having infinite bondage slopes - there would be no energy
overlap at the crossovers and statement (2.4) would be
absolutely true at all frequencies
It can be shown that if statement t2.43 is true, and
the total system frequency response is sensibly flat, as is
the case in Fig. 8, there can be no dominant transmission
zeros within the total system band pass. The heuristic proof
that summations of the form of Equation (113 imply dominant
p-z as in Figs. aye is as follows- Consider the summation
of Equation ~11) and all the p-z involved in this summation
(Figs. aye, 14b and 14c). Here all the p-z shown not just
the dominant ones. Fig. 14c contains all the poles of Figs.
aye and 14b, which is fairly straightforward/ as all these
poles appear in the denominator of Equation (11). These poles
must appear since each of them is a non-common factor of the
least common denominator formed in the summation.
Proof of the existence of all of the zeros is a
little less obvious. We proceed using an argument based on
the aforementioned energy consideration (2.4) and an
examination ox the total system amplitude and phase response
Consider Fig. aye in which we seek to evaluate the
-39~
System amplitude and phase response at a point on the jaw axis
within the midrange pass band, say at wow The p-z at Fig. aye
imply that only the midrange driver is functioning in the
total loudspeaker system, i.e., the woofers and tweeters are
disconnected.
Now consider Fig. 14c~ Here, the tweeter has been
connected. There are now five more poles in the total system
response. Two of these are a dominant complex-coniugate
pair, which extends the system response into the
high-frequency (tweeter) region. The remaining three poles
are real and exist on the far left real axis, contributing
almost nothing to the total system amplitude response. They
must also not change the phase response, since we know from
(2.4) that connecting the tweeter has no effect on the system
midrange amplitude or phase response. Thus the total system
zeros of our example Equation (11~ must fall into positions
which will guarantee that the midrange phase response, as well
as the amplitude response is the same with the tweeter in the
system or out.
Examination of the numerator in Equation (11~ shows
that an so is common throughout, implying two zeros at the
origin. The highest order of s in the numerator is eleven, so
there are nine more zeros to account for. The positions of
these remaining nine zeros cannot be explicitly found without
25 expanding and factoring the numerator of (11). But we do know
enough to determine their approximate locations by induction.
What we do know about the zeros can be summarized as
hollows:
(2.5) None of the zeros are dominant, it they
cannot exist near the jaw axis within the
total system pass band.
(2.6~ The zeros exit in positions such that the
phase response defined within the
_40~ I
individual pass bands of Figs. aye and 14b
remain unchanged idyll in Fig. 14c.
A plausible set of locations for the remaining nine
zeros is shown in Fig. 14c. The four transmission zeros which
are dominant in Figs. aye and 14b move approximately
horizontally to the left as in Fig. 14c, becoming no longer
dominant. This supposed position of these zeros is supported
by observation; for example, when one carefully measures the
frequency response of an embodiment of the invention, very
slight irregularities in amplitude of a fraction of a dub can
be detected on the frequency response near the frequencies of
the zeros.
The remaining five zeros are acerbated near or on
the negative real axis. They must occupy positions such that
lo the phase response within the pass band of Fig 14c is the
same as that within the individual pass bands of Figs. aye and
14b. This may be confirmed by observation; in any embodiment
of the invention total system phase shift is unchanged within
any pass band when drivers associated with other pass bands
are connected or disconnected. The aforementioned statement
is exact it the crossover network frequency response band-edge
slopes are infinite, and becomes approximate in the frequency
band overlap of actual embodiments having non-infinite slopes.
Thus, there has been established, by intuitive
arguments supported by observation the crucial concepts which
underlie the operation of operative embodiments of the
invention. These concepts may be presented as corollary of
statement I
t2~7~ Jo The separate portions, or crossover
filter-loudspeaker driver combinations of
the invention (low-pass, band-passr Of
whops, function independently of each
other in their separate contributions to
~22~
-41-
the total system acoustic output, whereby
the degrees of "effective" or "absolute"
independence are define, as in Fig. 18 end
the textual explanation of this figure
given hereinbeEore.
(2.8) The approximation to absolute independence
among the same portions in (2~7) becomes
more accurate as the crossover jilter
pass band binge amplitude us frequency
response slopes approach infinity. If the
same band-edge slope becomes greater than
about 70 dB/octave, acoustic wave
interference becomes inaudible, and the
acoustic outputs ox the portions, so
crossover filter-loudspe~ker driver
combinations in 2~7 are considered to be
absolutely independent, as defined in Fig.
18 and the textual explanation of this same
figure given herein before
Statements I and (2.8), taken together with
statements (1.7) and (2.4), inclusive, mentioned herein before,
are characteristic to any embodiment of the invention, and
serve to define its intent and operation Summarizing the
foregoing and showing how the invention improves upon the
prior art systems, and specifically, how the present invention
solves the difficulties mentioned herein before, the following
is noted with respect to statements (1.4~ through (1.6),
inclusive:
Statement tl.4): Well-designed loudspeaker
drivers generally exhibit a band of freguen-
ales in which the amplitude response is
very flit, and in which the phase response
Jo
-42-
is relatively linear. According to the
invention, a driver is connected to a
crossover filter circuit (1.8) having a
"brick wall" passbana characteristic
corresponding to its frequency region of
best performance. Energy is transmitted
negligibly -to the arriver outside its
frequency band of best performance, so that
problems of irregular frequency an phase
response - i.e., non-ideal behavior -
originating from an individual driver
are minimized-
Statement (1.5): Acoustic wave interference
between drivers is caused by simultaneous
radiation at the same frequency from two or
more drivers, combined with spacing between
the acoustical centers of the arrivers of
about one-half wave length or more If the
width of these frequency bands of
simultaneous reedition, the overlap
region shown at 64 in Fig. I is
minimized, or reduced to zero r the same
wave interference is correspondingly
minimized or eliminated. The steep
I crossover filter pass band slopes
characteristic to this invention will
minimize the width of these frequency bands
of mutual interference to 1~3 octave or
less, with the width of these bands of
mutual interference tending towards zero as
the crossover slopes approach infinity.
statement (1.6) Loudspeaker drivers tend
to have low amplitude distortion only
-43-
within their frequency bands of flat
amplitude and linear phase response
Outside these bands, the distortion
generally rises rapidly. Since top
invention minimizes the energy transmitted
to the drivers outside their frequency
bands of best response, nonlinear
distortion in the total system will be
reduced
Finally, mention is made of two crucial discoveries
which clarify the nature of the invention. First, and most
important, is the discovery that a good approximation to the
ideal transfer response (1) can be realized by properly
summing two or more "brick wall" amplitude functions having
mutually exclusive frequency pass bands lying adjacent to one
another. By proper choice of system parameters such as
driver placement, type, the size also crossover topologies,
pass bands and slowest it is possible, by the methods of the
invention, to achieve the aforementioned good approximation to
the ideal system transfer function.
Also, observe that in forming the acoustic sum of
the sound outputs of the individual drivers comprising the
stymie as for example a sum in the mathematical form of
Equation to) or ~10), in general, no poles or zeros disappear
in the summation as was shown explicitly in the single case
described by Equation (11). This follows logically from
Statement I Contrast this aforementioned situation with
a speaker system based on concepts of which Equation (5) is
the simplest example. Here, all p-z disappear In general,
prior art speaker designs have utilized acoustic summations
which caused the disappearance of as many pi as possible in
the summation while tending towards some good approximation of
Gaussian (1). According to the prevent invention, the
~22~ Lo
-44~
opposite approach is taken; i.e., retaining all, or as many as
possible, of the p-z of the individual elements in the final
summation This latter approach also approximates Equation
I to a high degree of accuracy, while overcoming
shortcomings of the prior art.
There are two possible (but not necessarily all
inclusive) design approaches one may take in the realization
of loudspeaker systems using methods of the invention.
I Exact approach: The designer chooses the
dominant poles of a system transfer
function which approximates Equation (1) to
the desired degree of accuracy. This
involves picking as many terms in the
infinite series of Equation I as desire,
and is generally best realized as the p-z
peculiar to Bessel filter (Duo, "Network
Analysis and Synthesis", p. 343-348). Then
a loudspeaker crossover network with an
acoustic summation implicit as the
right-hand side of Equation (8) is designed
to have dominant poles as near as possible
to the aforementioned Bessel approximation.
The arithmetic will be cumbersome and will
require a computer program for an accurate
solution. The p z peculiar to the
loudspeaker arrivers themselves should by
considered as part of the analysis, these
p-z being either approximated for the
drivers assumed as ideal, or either
calculated or measured for thy driver
considered as non-ideal.
(2) Empirical approach: The designer chooses
p-z of "brick-wall" amplitude functions
-45-
which based ox previous experience, will
yield a good (although not necessarily
optimum) approximation to Equation (1) when
an acoustic summation is performed as in
Equation to). Driver p-z may or may not be
considered curing the initial steps in the
d~slgn. A prototype is then built and
tested, and adjustments made to the
crossover circuit components until the
performance of the speaker system - in
regard to the accuracy of its approximation
to Equation I - is satisfactory The
inventor used this empirical approach in
realizing the embodiments of the invention
discussed herein.
Empirical methods were used for design of the
invention embodiments schematically depicted in Figs 3, 10
and 15. Passive crossover filter circuits, i.e., as shown in
Figs. 3 and 10, were realized by utilizing an empirical
extension of the well-known "image-parameter n method,
described hereinafter. The active crossover filter circuit of
the embodiment shown in Fig. 15 was realized by first
determining the circuit element values using methods of the
text "Handbook of Operational Amplifier Circuits" mentioned
herein~efore, and then empirically adjusting said circuit
element values for optimum performance. To illustrate the
procedure for making an operative embodiment of the invention
more clearly, it will be treated in some detail in the
following
An empirical method was used to realize a preferred
embodiment of the invention schematically depicted in Fig. OWE
Numerical values for inductive and capacitative circuit
elements comprising the crossover filter circuits of the
-46-
aforementioned invention embodiment are calculated by use of
the well-known "image parameter" method, which is described
fully in the text "Electrical Engineering Circuits", by Ho
Swilling, John Wiley & Sons ~1957), Chapters 18 and 19.
Tables of equations, amplitude and phase response graphs,
schematic diagrams for filter circuit portions, and design
aids for the image-parameter method are given in many
reference books, one of which is reference Data for Radio
Engineers" r Fourth Edition, Stratford Press (1963), Chapter 6
As mentioned herein before, the present invention is
based upon the discovery by the inventor that a good and
useful approximation to the ideal transfer function I can ye
realized by properly summing two or more brick-wall amplitude
junctions having separate and mutually exclusive frequency
pass bands which pass bands when taken together encompass the
entire audible frequency range. Furthermore, the inventor has
discovered that ordinary image-parameter theory, augmented by
empirical methods may be used to realize suitable brick-wall
amplitude functions for crossover network filters used in
operative invention embodiments
Design of an infinite slope loudspeaker system
begins with the selection of proposed loudspeaker drivers. Two
or more of the same are needed in order to cover the audible
range of frequencies, i.e., I I to 20 kHz. Selection of the
I number of drivers required is based upon considerations of the
cost and size of the prospective loudspeaker system. For the
invention embodiment of Fig. 10, four loudspeaker drivers are
chosen, one woofer, two midranges and one tweeter. Amplitude
and phase measurements are made on all available and known
drivers; from these, drivers are chosen which have
performance, with respect to their amplitude and phase
response, which can be considered to be ideal over separate
frequency ranges which when taken together will encompass the
-47-
entire audible frequency range. These frequency ranges for
the four drivers chosen appear in Fig. 10 and also
hereinafter.
Crossover network filters having brick-wall
amplitude response characteristics are synthesized to possess
pass bands matching the frequency ranges of best performance of
the aforementioned drivers. This is accomplished using
enhanced image-parameter theory. If transmission zero
frequencies, mutual coupling of coils, and filter pass bands
are chosen and adjusted properly, the acoustic sum, i.e., the
total acoustic amplitude and phase response of the invention
embodiment will become an accurate approximation to the ideal
transfer function of equation I
The empirical design procedure begins with
computation, using image-parameter methods of initial
crossover network circuit element values. Calculations for
four separate pass bands are required, i.e., the crossover
network will have four separate filter circuits. Each filter
circuit will need at least one transmission zero, in this
case, six transmission zeros are incorporated into the
crossover network filter topology, one zero each for the
woofer low-pass filter and tweeter high-pass filter, and two
zeros each for the two separate midrange band-pass filters
Frequencies for filter pass bands and transmission zeros are
tabulated us:
DRIVER PASS BAND TRANSMISSION ZERO(S)
woofer low - pass 280 Ho
20 - 150
I ~2~3~2
lower band - pass 80 Ho
midrange 150 - 800 Hz1450 Ho
.
upper band - pass 550 Ho
midrange 800 Ho - 5 kHz 6000 I
tweeter High - pass 300U Ho
5 kHz - 20 kHz
In the above table frequency ranges for the
crossover filter pass bands correspond to the respective driver
frequency ranges of best performance. Frequencies for the
transmission zeros are determined by intuition based upon past
experience in working with the invention Using the data
tabulated herein before image-parameter methods are used to
calculate initial circuit element values. Two sets of
illustrative calculations are shown:
1. low-pass response for lower midrange driver:
lo = 150 Ho impedance = 8 ohms
Of= 1 = 1 = 132~f
war (2 ~)(150)18~
Lo = R = 8 = 8~48 my
wow Sly)
2. lower transmission zero for upper midrange driver:
lo = 800 Ho f = 550 Ho Impedance - 8 ohms
m = I - f 2/C2 _ Al _ ~550)2/~800)2 = 0,726
~49-
Lug = Lo = R = 8 = 2.19 my
m wcm (2~)(800)(.726)
C = m Ok m . ]. = _ .726
23 1_m2 l_m2 war (1 - .726 ) (2~)~800)(g)
= 38.1 of
It should be noted that some initial circuit element
values as just calculated will change later after empirical
adjustments. For example, C13 became 100 of, and L14 became 8.4 my
in the final circuit (Fig. 10), representing a slight change after
empirical adjustments. One component in the design required a
large change, C23 calculated above as 38.1 of became I of in
the final circuit (Fig. 10). All circuit element values given
in the table herein before for Fig. 10 represent the final empiric
gaily adjusted circuit element values for the same invention
embodiment.
Calculations for the remaining circuit element values
will not be shown here, but proceed in a manner similar to that
just illustrated, except for resistive elements Roll, R12, and
R13, which were determined purely empirically. These resistors
serve to damp the efficiency of the three higher frequency drivers,
matching the efficiency of the same to that of the woofer, such that
all four drives will sound with equal loudness, thereby producing
a flat total system amplitude response.
After computation of all reactive circuit element values
and selection of the damping resistors for the hither frequency
drivers, the crossover circuit is constructed, using components
and techniques common to the art. Mutual coupling of some coil
pairs is used where it has been empirically found to enhance the
, . Jo
I
-aye-
steepness of the pass band band-edge response. This same mutual
coupling of coil pairs is achieved by methods common to the art;
including winding the two coils on a common
'I
-50~ 73~
iron core or winding the two coils on separate cores and then
mounting them physically close together so their magnetic
fields interact to provide mutual coupling.
The completed crossover network along with the
loudspeaker arrivers is assembled in a cabinet or box, in a
manner common to the art, i.e., with the crossover network
mounted inside the box, the drivers arranged physically close
together in a vertical array on the front face of the box, and
suitable electrical connections being made between the
crossover network filter circuits and the respective
loudspeaker drivers, and a pair of input terminals provided
for connection of the loudspeaker system to a driving signal.
Amplitude, phase and delay measurements are made on
the completed speaker system, and the crossover circuit
reactive and resistive element values together with the mutual
coupling of coil pairs are empirically adjusted until the
system performance with respect to amplitude, phase, and delay
response is, as recognized by those skilled in the art, an
accurate and superior approximation to either form of the
ideal delay function of Equation (1).
More specifically, part of the acoustical test
procedure during empirical optimization and final evaluation
of an invention embodiment makes use of the Wilkinson n fast
Fourier transform" method. This same method employs a digital
computer to analyze the time domain impulse response of a
loudspeaker system. Software is used by this same method to
produce thy usual frequency domain amplitude,phaser and delay
spectra from the time domain impulse signal response of the
loudspeaker system under test. The aforementioned
computer-produced spectra of amplitude, phase, an delay
rosins are then confirmed by further measurements made
directly in the frequency domain, using sine-wave and
"pink-noise" test signals, together with wave analyzers, AC
-51~ I
voltmeters, oscilloscopes, and chart recorders, in a manner
common to the art.
Thus, there has been provided, according to the
invention a method of and a means for reproducing by a
loudspeaker system sounds from electrical signals representing
sounds to be reproduced. A plurality of crossover filter
circuits are provided to separate such signals into bands of
different frequencies there being established for each of the
crossover filter circuits very high pass band band-edge
frequency response slopes by means of transmission Eros
appropriately placed at frequencies just outside the pass
bands of the individual filter circuits while simultaneously
insuring that all or as many as possible of the dominant poles
of the individual crossover filter circuits remain in the
lo acoustic summation. this insures that each of the individual
crossover filter circuits will posses a brick-wall amplitude
vs. frequency characteristic so that as the same filter
circuits direct electrical energy to their respective
loudspeaker drivers over the frequency ranges of best
performance of the same loudspeaker drivers, each of the
individual crossover filter - loudspeaker driver combinations
so formed will function independently of each other in their
separate contributions to the total loudspeaker system
acoustic output. This same independence is characteristic to
operative embodiments of the present invention, in which
embodiments by virtue of this same aforementioned independence
the acoustic output of any one crossover filter - loudspeaker
driver combination will not audibly impinge upon the acoustic
output of any one another crossover filter-loudspeaker driver
combination, as implied in idea statements (2.4) t I and
I given herein before, thereby enhancing, by means of the
present invention, the fidelity ox response in the
reproduction of sounds encompassing the audible frequency
range from electrical signals representing said sounds.
~2~3~
-52-
Of particular significance in connection with top
transfer function employed according to the present invention
is the use of different poles in the best embodiments for the
low-pass, band-pass and high pass functions. For example, in
the empirical method based on image-parameter theory just
described, the individual low-pass) band-pass, and high-pass
functions will not in general share any common poles, because
each of the aforementioned functions are calculated, and later
empirically adjusted, independently of the others. It is
characteristic to independently chosen brick-wall amplitude
functions having mutual exclusive frequency pass bands, that
the denominator poles of their respective transfer functions
he generally different, as illustrated earlier in Fig. 14 and
the textual explanation of this same figure given hereinhefore.
The functions that have been used according to the proposals
of the prior art have mainly had the same poles in the
denominator parts or low-pass, band-pass and high-pass
functions. The prior art workers, in attempting to achieve a
closer approximation to the ideal transfer function, have
concentrated on the numerator parts of the functions, as is
evident from the three papers of the Journal of the Audio
Engineering Society mentioned hereinbe~orev
An example of a prior art crossover system having
crossover jilter transfer functions not possessing brick-wall
amplitude vs. frequency characteristics but in which different
poles appear in the denominator parts for low-pass and
high-pass functions is contained in U.S. Patent 2,612,558
issued to POW. Klipsch on August 13, 1946. However, crossover
designs in which different poles appear in the denominator
parts of the transfer functions of individual elements have
fallen into disfavor in recent times.
