NORTH READING, MASSACHUSETTS 01864 USA

MODELING THE KLIPSCHORN FOLDED BASS HORN

Part I

Presuming an electrical equivalent circuit model of the folded bass horn
behavior is available, it would be a useful tool to analyze such a modification.
The model could also be used to predict, for example, the behavior of the horn
for a given driver.

In PART I, the analysis considers the electrical behavior of the loudspeaker drivers used and how that behavior changes when the drivers are installed and operated in the horn. I attempt to demonstrate that the complex impedance of the folded bass horn can be accurately simulated using a Beranek model modified to account for the frequency dependence of the horn throat and voice coil resistances. In PART II, the model is used to examine how the horn response is effected by various driver parameters. The effect of various crossover network topologies is also considered.

In PART I, the analysis considers the electrical behavior of the loudspeaker drivers used and how that behavior changes when the drivers are installed and operated in the horn. I attempt to demonstrate that the complex impedance of the folded bass horn can be accurately simulated using a Beranek model modified to account for the frequency dependence of the horn throat and voice coil resistances. In PART II, the model is used to examine how the horn response is effected by various driver parameters. The effect of various crossover network topologies is also considered.

FACTORY DRIVER PARAMETERS

As shown in the plot the SQ magnet driver, which has seen considerable use, has
a lower Fs and higher impedance magnitude maximum at resonance. When compared to the RD magnet
unit, the suspension compliance is much higher (qualitatively confirmed by
lightly pushing down on the dust-cap glue line).
The drivers are shown below. The frames are formed sheet stock and, except for paint,
identical. The SQ magnet is attached to the frame by a
couple of machine screws (there might be some glue holding it down too). The RD magnet is
glued to the frame (no visible fasteners). The SQ magnet unit also has a nice PWK
badge adhered to the bottom-plate. A serial number for the driver is
stamped into the badge.

The drivers have the same cone assembly part number (451520-2) as is shown
below. The moving mass (Mms) for
this assembly is specified by Eminence to be 0.0786kg. Thus, presuming Mms is the same
for each, **Qes** is determined
using the method outlined by Small [1]. The parameters needed for the
horn analysis, derived from the impedance magnitude plots, are shown in the table.

The difference
in Fs shown in the magnitude plot can be accounted for by the suspension compliance
difference. Given 30+years of production separates the two driver birth
dates, compliance differences are to be expected.

The Beranek Model

The approach taken here is something of a *
perturbed analysis*. The
complex impedance of the two factory drive units are first measured in
free-air. The drivers are then installed in the bass horn and the
measurement repeated. A SPICE generated
equivalent electrical circuit is then fit to each response and, through differences,
the horn parameters derived. Simulation accuracy is assessed by
superimposing the complex impedance plot of the equivalent electrical circuit prediction
against the measured, complex impedance response. Plots in the complex
frequency domain, *s*, are made using
the Nyquist plot. By super-imposing the predicted impedance over the
measured impedance, a assessment of simulation accuracy is possible.

Before going into the measurements, shown below is the electrical equivalent circuit used here to provide guidance for the modeling approach. It's essentially the same model that Beranek shows in Fig. 9.3 on p. 262 in Acoustics [2] except the circuit is transformed into the electrical domain. The component labels are the same Keele uses in his paper [3]. The relationship for each component related to horn loading are found in that paper also (note that Keele wrote the paper when employed at Klipsch). The driver free air component labels are the ones used by Small [1].

Before going into the measurements, shown below is the electrical equivalent circuit used here to provide guidance for the modeling approach. It's essentially the same model that Beranek shows in Fig. 9.3 on p. 262 in Acoustics [2] except the circuit is transformed into the electrical domain. The component labels are the same Keele uses in his paper [3]. The relationship for each component related to horn loading are found in that paper also (note that Keele wrote the paper when employed at Klipsch). The driver free air component labels are the ones used by Small [1].

There are factors that contribute to complicating the analysis thus making
direct correlation to the elements defined in the theoretical model a
challenge. Departures from the ideal horn model include the obvious
such as horn folding, bifurcated horn path, a back chamber consisting of
three internally connected cavities, the driver mounting scheme and the
multiple flare, "rubber-throat" approach used by PWK [4]. Less obvious
departures include leakage losses at the driver gasket, leakage across front
and back sides of the diaphragm through the suspension and leakage through
at the driver access panel.

Referring to the schematic,**C**_{mes}
is the electrical capacitance (F) equivalent
associated with the driver moving mass**, L**_{ceb} is the
electrical inductance (H) equivalent of the compliance contribution associated with the
air volume
(m^{3}) contained
in the back chamber, **V**_{b}, in contact with the
rear of the driver cone.
A photo of a bass horn factory build is shown below. The pink border
highlights the boundary between the back chamber volume and the folded horn.
The small blue arrows identify the "*means of coupling the top and bottom
sinuses*" to the chamber housing the driver.
**L**_{ceb} plus the
electrical inductance equivalent of the driver
suspension compliance, **L**_{ces}, contribute to the total
electrical inductance reflected back thru the motor to the input terminals and
represents the total effective compliance of the driver
diaphragm. Compliance is a ratio of diaphragm
displacement for a given
amount of force (m/N). It's the reciprocal of stiffness. For example, a high
compliance suspension has low stiffness.

Referring to the schematic,

The small notches shown in the picture above add **V2** and **V3** to
**V1**. If you
own a Klipschorn and wish to convince yourself that these access ports actually
exist, a wire coat hanger, a Philips screwdriver and a flashlight are the tools you
need. Remove the woofer access panel and focus your attention on the
location shown in the photos below. Fashion a hook from the coat hanger
and probe around the area shown in the second photo from the left. If you
don't find it or it doesn't exist, you're either not looking in the right place,
your Klipschorn is a very early production unit,
or it's a fake(!).

The Klipschorn bass horn does not have a front chamber per se (although early versions of the horn did). The volume of air between the front of the driver cone and the baffle board,

*Throat Impedance*

The throat impedance (*Z-throat*) consists of the radiation resistance,
**R**_{et},
in series with the electrical equivalent of the capacitance associated with the throat air mass in contact
with the front of the driver cone, **C**_{met }
. The magnitude of
**R**_{et}
is a function of frequency (i.e. **R**(*f*) +
*j*0) and ranges
from zero at very
low frequencies to a maximum resistance of (**St
**(**BL**)^{2}/(*r*_{o
}*c*** Sd**^{2}))
where *r*_{o}
is the density of air (1.21kg/m^{3}),
*c*
is the speed of sound (343 m/s), **S**_{t}
is the throat area (m^{2}) of the constricted throat (0.025m^{2}),
**Sd**
is the effective diaphragm area of the K33E driver (0.089m^{2}) and
**BL** is the BL-product of the woofer motor (T-m). As is
evident in the relationship, the maximum radiation resistance is proportional to
**BL**^{2}. Using the SQ and RD magnet parameters shown above, a maximum value for
**R**_{et }
is between 1.1 to about 1.2 Ohms.

Although relatively small in magnitude, the output of the horn is determined*
entirely* by the
radiation
resistance!
Frequency dependent
components are simulated in SPICE using the g-component with a transfer
function for a given component specified as
an admittance, *Y*(*s*).
Thus, the Laplace transform for a frequency dependent resistor is =1/**R**(abs(*s*)),
for an inductor =1/(*s
***L**(abs(*s*)))
and for a capacitor =*s ***C**(abs(*s*))
where *s* is the complex frequency
domain variable with modulus equivalent to the angular frequency, 2p*f. *
The trick however is to either derive, approximate or experimentally determine
the functional form.
To execute a
*continuous* SPICE mode for all
values of *s*, the bass horn radiation resistance
must be expressed as an admittance,
1/**R**_{et}(abs(*s*))
that is also continuous for all values of *s*.
The functional form of the relation used must also be one that allows fitting to the measured
resistive part of the throat impedance. Here, a two parameter fitting model was
developed,

Although relatively small in magnitude, the output of the horn is determined

where
*n* is a
fitting exponent restricted to odd integer values (*n** *= 3, 5, 7, 9...) and
*f*** *
a fitting frequency (Hz).* *
The
two plots below show how changing one parameter, leaving the second constant,
changes the shape of the function.

In the plot at left, changing *
n *
while
keeping *
f** fixed changes the slope
of the frequency dependence (i.e. the "steepness"). In the
right, increasing (or decreasing) the magnitude of
*
f* *
at a fixed value of
*n*
"shifts" the frequency
dependence to higher (or lower) values along the frequency axis. The small arrows shown in
each plot identify the
frequencies where the simulated acoustic output of the horn is a maximum
(in the actual Klipschorn bass horn, a peak in the output sensitivity is
measured between 100-200Hz).
Unlike the theoretical horn (i.e. one of "infinite" length), the radiation resistance of a
finite sized horn is not zero at the
cutoff frequency and useful output at, and below, the cutoff observed [5].
With this in mind, the two
parameter model was developed to provide some room for
adjustment to correlate the simulated behavior to the measured data both near
the horn cutoff and at the frequency where output sensitivity of the actual bass
horn peaks.

Voice Coil Impedance

The parameters **R**_{e }and **L**_{vc}
approximate the voice coil impedance and represent the voice coil DC resistance
(Ohm) and inductance (H), respectively. These parameters are typically
provided based on measurements obtained from free-air test conditions.
In practice however, the voice coil does not operate in "free-air" but within
the confines of the magnetic gap of the motor. This forces the inductance
of the voice coil to deviate somewhat from the inductance of an ideal inductor. Resistive losses too, are frequency
dependent and increase with increasing frequency. The behavior is examined
in detail by Vanderkooy [6] and Dodd [7].

The frequency dependent effects associated with the voice coil are pronounced at
higher frequencies (say above about 200Hz). To account for
additional, frequency dependent losses, a g-component identified as **R***_{vc }
was placed in series with **R**_{e }and **L _{vc}**. The
functional form was relatively straightforward to derive from the resistive part of the complex impedance of the bass horn. Below
200Hz however, its effect on
the simulated response is small. In the
modeling results I'll show how the

The functional form of the voice coil resistive losses, expressed as an admittance, is shown below. The fitting parameters, like the functional form for the radiation resistance, allows the modeler to "fine tune" the real part of the simulated response to the corresponding real part of the measured response.

The fitting exponent, *m*, varies
between 0.5 and 0.8, **R***_{o
}represents the resistance increase associated with frequency
dependent losses at a frequency, *f _{o}*

The revised Beranek model, incorporating the g-component elements, becomes

The schematic is the proposed equivalent electrical circuit model assumed for the Klipschorn bass horn.

**MEASUREMENTS**

*K33E Square and Round Magnet Drivers*

First, let's consider the impedance of the K33E drivers and measure how the impedance is changed when the drivers are loaded into the Klipschorn bass horn. The series of plots shown below will be the data used to develop the component magnitudes shown in the equivalent electrical circuit model.

The first plot shown above is the impedance magnitude of the two versions of the K33E shown
earlier. Super-imposed is the impedance magnitude plot of each driver
operating in the bass horn. Note how the large frequency response peak changes in size
between the free-air and horn loaded responses. The dashed ellipse highlights the bandwidth
(~50 to 400Hz) where driver output
sensitivity is enhanced, i.e. the frequency range where the horn "loads" the driver by the
effect of a significant increase in radiation resistance, **R**_{et}. The two plots below are the corresponding Nyquist plots of
each driver operating in free air and then operating after installing into the
bass horn. * *

Below is a plot of the reactance (imaginary part of the complex impedance) and resistance (real part of the complex impedance) as a function of frequency of both K33E drivers operating in the bass horn.

An over-plot comparing the resistive part of the impedance, shown above for each driver, is plotted below.

The accuracy of the simulations will be assessed based on how closely the model can replicate the measured responses shown.

**K33E SQUARE AND ROUND MAGNET EQUIVALENT CIRCUITS**

The plots shown below compare the simulated
complex impedance of the K33E
round and square magnet drivers to the measured responses. Four plots are
provided in the comparison, the impedance modulus, the real part, the reactive
part and the Nyquist plot. The schematic of the electrical
equivalent circuit used to simulate the responses for each driver are also shown.
The electrical capacitance due to the driver mass, **C**_{mes},
and the electrical inductance due to the driver suspension, **L**_{ces},
were derived from the compliance and BL-product magnitudes
provided in the table above. The resistive losses at resonance, **R**_{es},
were determined by iteration. The functional form of the frequency
dependent part of the voice coil impedance is shown as the g-component, **
R***_{vc}.** **

* *

With both real and reactive components derived from the complex impedance, the Nyquist plot
comparing simulations to the measured responses can be made. As
evidenced in the plots, the model component magnitudes shown in the schematics do a reasonable job
of predicting the free-air response of each driver.* *

**KLIPSCHORN FOLDED BASS HORN ELECTRICAL EQUIVALENT CIRCUIT**

With values for the driver **R _{e}**,

.

How the values were determined is now discussed.

**SOLVING FOR L _{ceb}, C_{met} AND R***

The components **L _{ceb}, C_{met} **and

An approximation for the back chamber volume, **V _{b}**, can be determined directly from the simulation derived value of

which yields a value for of 0.867m^{3} or about
3.0ft^{3}.
Since **V _{b }**is a constant, the ratio

By appropriate substitutions, the
relationship is also used to correlate the front chamber volume, **V _{f}**,
to the magnitude of

**SOLVING FOR R _{et}**

Next, the
*n* and
*f** parameters that best fit the
functional form of the radiation
admittance, 1/**R _{et}**(abs(

A plot of **R _{et}** as a function of frequency used is shown below, left. On the right is a closer
look at the relationship between the model radiation resistance and the
measured response.

In the plot below, the real part of the model impedance magnitude shown upper right, is super-imposed over the real part of the measured, complex impedance of the bass horn response. As is evident in all of the plots, the model does not capture the reflections seen in the actual response (considered in PART II).

**SOLVING FOR L _{cec}**

The last element to consider is the magnitude of **L _{cec}**.
To get some sense of the effect

As seen in the left plot, the volume of the compression chamber between the front
of the cone and the throat baffle effects the magnitude of the impedance between
100~400Hz. By examination of the electrical equivalent circuit, **
L _{cec}** is a -6dB/octave low-pass filter with the throat
impedance as a load. As shown in the frequency response plot at right,
increasing the magnitude of

The best fit to the *experimentally measured impedance*
was determined to occur at **L _{cec}** ~0.1mH which
correlates to a front chamber volume of 0.0008m

**PUTTING IT TOGETHER**

*K33E Square Magnet Driver*

With each component in the electrical equivalent circuit
assigned either a value or functional form, the impedance response is plotted and compared to the experimental
results obtained for the SQ magnet K33E. Lower left, the impedance
magnitude of the bass horn is compared to the simulation at two values of
**
L _{cec}**. At right, a comparison between
the real part of the simulation (same values of

The horn simulated reactance with
**
L _{cec}** equal to a 0.1mH inductor is plotted over
the actual reactance of the horn, below left. Lower
right is a Nyquist plot of the complex impedance of the horn (red) with the simulated
(blue) complex impedance
super-imposed.

*Predicted Response*

Below shows a plot of the factory measured anechoic frequency response of the
Klipschorn bass horn. The plot is taken from [8]. Super-imposed over
the frequency response of the factory horn is the predicted frequency response of the horn based on
the model (blue).
The predicted response is the electrical power dissipation across
**R _{et}**. Note that a significant amount of the
output is related to reflections not considered in the functional form
representing

*K33E Round Magnet Driver*

The complete equivalent electrical circuit for the bass horn with the RD K33E magnet driver is shown in the schematic below.

A comparison between the round magnet frequency response simulation and square
magnet is shown for **L _{cec}**
=0.1mH. The model predicts that the low frequency response of the horn is
extended by using the SQ magnet driver and, given the difference measured in the
compliance between the two drivers, a result that should not come as a surprise.
As the suspension compliance of the RD magnet driver increases over time by use
(i.e. the so-called break in),
the low frequency response should improve. It may also suggest that
the factory response plot shown above was taken with a new, high compliance, RD
magnet K33E.

*Enlarging the Rear Chamber*

Getting back to Mr. Klipsch's question of enlarging the rear chamber,
**V _{b}**. From the same unpublished manuscript
discussed at the top of this page, Klipsch writes:

.

In the plot below, the electrical power dissipation (in dB equivalent) for three
rear chamber volumes is shown. The model predicts that reducing **V _{b}
**by 50% does increase output near 60Hz as Klipsch states. In the
acoustic domain, the response may be associated with a peak between 50 and 90Hz
and thus yielding the "spectacular bass rise".

**REFERENCES**

1. R. Small, Direct-Radiator Loudspeaker System Analysis, IEEE Transactions on Audio and Electroacoustics, Vol. AU-19, pp.269-281, 1971

2. L.L. Beranek, Acoustics, Published by the American Institute of Physics for the Acoustical Society of America, p.262, 1986 Ed.

3. D.B. Keele, Jr., Low-Frequency Horn Design Using Thiele / Small Driver Parameters*, AES preprint no. 1250, 1977 (*presented at the 57th Convention May 10-13, Los Angeles, CA 1977)

4. P.W. Klipsch, A Low Frequency Horn of Small Dimensions, Journal of Acoustical Society of America, p.138, Vol. 13, October, 1941

5.*
*D. J. Plach and P. B. Williams, Reactance Annulling
for Horn Loudspeakers, Radio-Electronic Engineering, p.15 February, 1955

6. J. Vanderkooy, A Model of Loudspeaker Driver Impedance Incorporating Eddy Currents in the Poles Structure, Journal of the Audio Engineering Society, Vol. 37, No. 3, March 1989

7. M. Dodd, W. Klippel, J. Oclee-Brown, Voice Coil Impedance as a Function of Frequency and Displacement, white paper no. 4, Klippel GmbH, Dresden, 02177, Germany (available at www.klipple.de)

8. R. Delgado, Jr,, P.W. Klipsch, A Revised Low-Frequency Horn of Small Dimensions, Journal of the Audio Engineering Society, pp.922-929, Vol. 48, No. 10, October, 2000.

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