North Reading Engineering

INTRODUCTION
What prompted me to think about modeling the Klipschorn folded bass horn was this unpublished manuscript written by PWK related to the history of corner horns (here). In the article, PWK refers to modifications made to the original bass horn configuration that enlarge the back chamber volume of the horn. The modification adds the volume of air trapped within the top and bottom "sinuses" of the horn (a top section sinus is shown in the photo below) to the volume of the back chamber.

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 effect of various factory crossover network designs alter the horn response.

FACTORY DRIVER PARAMETERS
Consider first the free-air impedance magnitude and phase angle of two factory loudspeaker drivers, the K33E square magnet driver (SQ) and round magnet driver (RD) shown below.

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 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].

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 (m³) 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.

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, V_f, will have to do however and L_cec represents the electrical equivalent inductance of the compliance associated with this air volume. R* represents the electrical equivalent resistive losses (Ohm) associated with radiation of the back side of the driver diaphragm into the back chamber and interconnecting cavities. Electrical equivalent resistive losses due to the driver suspension elements are represented by R_es and, in parallel with R*, account for all dissipation losses reflected back to the input terminals.

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) + j0) and ranges from zero at very low frequencies to a maximum resistance of (St (BL)²/(r_oc Sd²)) where r_o is the density of air (1.21kg/m³), c is the speed of sound (343 m/s), S_t is the throat area (m²) of the constricted throat (0.025m²), Sd is the effective diaphragm area of the K33E driver (0.089m²) 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 BL². 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, 2pf. 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,

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.

The parameters R_eand 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_eand 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 R*_vc element alters the predicted response.

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*_orepresents the resistance increase associated with frequency dependent losses at a frequency, f_owhich, for the case of the K33E, is about 3.3 Ohms at 1000Hz.

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

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.

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.

With values for the driver R_e, L_vc, C_mes, L_ces,R_es and function form for R*_vc assigned, the remaining component values of the electrical equivalent circuit for the bass horn are determined. The SQ magnet K33E driver is considered first and its parameters given below,

The components L_ceb, C_met and R* determine how the driver free-air resonance peak is changed by horn loading. To determine the component values used in the simulation, a curve fitting routine comparing the real and reactive parts of the measured impedance to the real and reactive parts of the model impedance was used. For this step, the elements comprising L_cecand R_et are short circuited. The plot on the left is the impedance magnitude of the model with the component values for L_ceb, C_met and R* that provide the best fit to the large resonance peak measured in the horn response. The plot also shows how the functional form of R*_vccontributes to the resistive part of the curve. For this situation, the simulated reactance is super-imposed over the measured reactance and shown at the plot to the right.

An approximation for the back chamber volume, V_b, can be determined directly from the simulation derived value of L_ceb using the following relationship:

which yields a value for of 0.867m³ or about 3.0ft³. Since V_bis a constant, the ratio L_ceb / BL² must also be constant. Thus, the magnitude of L_ceb is directly proportional to the BL-product of the motor assembly. Note that discrepancies between the calculated volume of the rear chamber and the actual volume are likely. Causes include measurement errors associated with obtaining driver electro-mechanical parameters and the accuracy of the fitting routine used to fit the model simulation to the measured response.

By appropriate substitutions, the relationship is also used to correlate the front chamber volume, V_f, to the magnitude of L_cec.

Next, the n and f* parameters that best fit the functional form of the radiation admittance, 1/R_et(abs(s)), to the resistive part of the throat impedance are determined (the frequency range identified earlier by the dashed ellipse, see above). For this step, the element comprising L_cec shorted. The lower left plot shows how the contribution of the functional form of the model radiation resistance alters the model impedance shown above. In the plot right, the measured reactance is compared to the simulated reactance.

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 (the peaks seen in the red measured response plot).

The last element to consider is the magnitude of L_cec. To get some sense of the effect L_cec has on the both the simulated impedance and equivalent electrical power dissipation across the R_et, the model was run over a range of L_cec values shown in the plots. The result of the simulation series is shown below.

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 L_cec (i.e. increasing the volume of the front chamber) enhances horn output to about ~350Hz (whether this revision to the horn is realized experimentally I will leave to others to verify experimentally).

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 0.0008m³ or about 0.03ft³ (50in³). The simulation shows that power dissipation across the radiation resistance at 450Hz is about -12dB relative to the radiated power at 100Hz for L_cec =0.1mH. Note that the expression shown earlier correlating L_ceb to V_b is the same used to solve for the the volume equivalent of L_cec.

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 L_cec)and the real part of the experimentally determined complex impedance.

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.

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 R_et. The dark red plot is the output of the K33E operating as a large woofer in a sealed enclosure. Note that the red plot is not a model result; I simply traced out the response of what I believe is the response of the system below the horn cutoff frequency based on the measured response.

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.

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".

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.