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active loudspeaker systems
Wolfgang Klippel
To cite this version:
Optimal Use of Software and Hardware Resources in Active
Loudspeaker Systems
Wolfgang Klippel
KLIPPEL GmbH, Dresden, Mendelssohnallee 30, Germany [email protected]
ABSTRACT
Traditional loudspeaker design requires a compromise between maximum SPL, sound quality, size, weight, cost and energy consumed by the audio device. Nonlinear adaptive control of the passive system using DSP and current monitoring in smart amplifiers equalizes the amplitude response, cancels undesired nonlinear distortion, actively protects the transducer against mechanical and thermal overload, stabilizes the voice coil position and copes with time-varying properties of the suspension. This paper discusses the consequences for the hardware design used with DSP in the active systems.
1. INTRODUCTION
This paper presents a new concept for designing hardware components for active loudspeaker systems that exploits the new degrees of freedom provided by DSP. The optimal use of all resources such as material, energy, and manufacturing effort becomes more important than a passive system that generates a linear transfer behavior and low distortion in the output signal. This new concept, also called Green Speaker Design [1], [2] uses the efficiency of the electro-acoustical conversion and voltage sensitivity as the key criteria for the speaker optimization. This causes a paradigm shift: Some nonlinearities inherent in the electro-dynamical loudspeaker are considered beneficial because they make the loudspeaker more efficient.
2. NONLINEAR ADAPTIVE CONTROL
This section summarizes the new opportunities provided by digital signal processing for loudspeakers:
audio stimulus Parameter Identification w(t) ul(t) i(t)
DSP
P current sensor transducer in enclosure Linearization Stabilization Equalization Protection Transducer Diagnostics p(t,r)Figure 1: Active loudspeaker system with adaptive, nonlinear control of the transducer
Figure 1 shows an adaptive, nonlinear control system that generates a desired linear transfer behavior between digital audio input w(t) and sound pressure
output p(t, r) over the lifetime of the loudspeaker system and reliably protects the transducer against mechanical and thermal overload. The first system is a linear filter for equalizing the frequency response. The following system is a time-variant, linear system for speaker protection. The nonlinear system just before the power amplifier linearizes and stabilizes the loudspeaker and cancels the nonlinear distortion in the output signal. The power amplifier has a constant gain and is preferably DC-coupled to avoid attenuating low-frequency compensation signals.
An electrical sensor (e.g. resistor) measures the transducer input current i(t) which is used for identifying the time variant properties of the transducer and adjusting the audio processing to the varying conditions. Thus, the nonlinear adaptive control system uses a feedback of updated transducer parameters P. The parameters P have a physical meaning, a high diagnostic value and can be stored in a memory.
H(f,r2) N p(r2) H(f,r3) H(f,r1) p(r1) p(r3) uD N u -uD ul mirror filter loudspeaker
-Figure 2: Active linearization and stabilization of the passive loudspeaker system by using the mirror filter derived from nonlinear loudspeaker modeling
The mirror filter at the end of the digital audio path
as shown in Figure 2synthesizes a compensation signal
-uD(t) that cancels the equivalent input distortion uD(t)
generated in the loudspeaker [3]. The subsystem N, connected in a feed-back loop at the loudspeaker input, models the nonlinear dynamics of the force factor Bl(x),
suspension stiffness KMS(x), mechanical resistance
RMS(v), inductance L(x,i) and other speaker
nonlinearities, which are lumped parameters depending on voice coil displacement x, velocity v, input current i and other state variables. The mirror filter [3] uses a copy of the subsystem N connected to the filter input
u(t) in a feed-forward arrangement to generate a linear
The active linearization of the loudspeaker should be
applied to a defined excursion range |x(t)| < xmech limited
by a mechanical limit xmech which is required to avoid
impulsive and other caused by bottoming, voice coil rubbing, buzzing and other mechanical limitations. This irregular distortion is not deterministic and cannot be cancelled by nonlinear control.
Modern amplifiers can provide more power than the transducer can usually handle. Active protection systems [5], [6], [7] have been developed that detect a critical state and attenuate the input signal before damage occurs. The protection system in Figure 1 uses the displacement and temperature of the coil reliably predicted by the mirror filter [8].
The protection system provides a flat response if no mechanical or thermal overload is detected. However, the filter generates a high-pass characteristic and attenuates low frequency audio components to keep the
peak displacement below a permissible peak value Xmax
[6]. Thermal protection requires an attenuation of signal components that significantly contribute to the real
input power PE.
Reliable overload protection of the loudspeaker is required for safely applying additional linear filtering
[9] with the transfer function Hequ(f) at the input of the
control system to generate an equalized overall system function defined as
( , ) ( ) ( , )
o equ
H f r H f H f r , (1)
which neither considers the protection system
responseHprot(f) nor the gain ga of the power amplifier
as illustrated in Fehler! Verweisquelle konnte nicht
gefunden werden.. The equalizer cannot change the
directivity of the following transducer but can align the
frequency response Ho(f, r) to a desired shape of the
high-pass characteristic at lower frequencies using generalized system functions (e.g. Butterworth).
Traditional alignment in the passive system design [10] accomplished this goal by searching for a particular transducer and cabinet with optimal properties. The active equalization provides more freedom for loudspeaker design because poles an zeros in the rational transfer function H(f,r) of the linearized passive system can be canceled and new ones can be generated
in the Ho(f, r). However, the equalization may increase
the power and voltage requirement of the amplifier and transducer.
The active control system discussed so far has a feed-forward structure and needs no additional sensor for monitoring the transducer states. Although the control system can be operated with fixed parameters, any mismatch between the control parameters and the actual loudspeaker parameters will deteriorate the
performance. Thus, the adaptive parameter
identification scheme [12] shown in Figure 1 is required to cope with unavoidable production variances, aging of the suspension [13], changing climate conditions and other external influences [14].
All lumped parameters of the nonlinear loudspeaker model can be identified by an inexpensive electrical sensor used for monitoring the input current i(t) [15]. The adaptive control system also simplifies the tuning process in active systems and provides accurate transducer parameters over the lifetime of the product while reproducing music or other audio signals. If the power amplifier is DC coupled, the control system can actively stabilize the transducer [16], compensate for external influences (e.g. air load, gravity) and keep the coil at the optimal rest position, giving maximum AC displacement and sound output.
The feed-forward structure of the mirror filter has low complexity, is always stable and generates no latency in the audio signal.
3. EFFICIENCY
As shown in the previous section, digital signal processing can significantly improve the quality of the reproduced sound, but the heat generated in the transducer limits the maximum output. Most of the electrical input power will be converted into heat, and only a small fraction (far below 1% in micro-speakers) will be radiated as sound power. The efficiency of the electro-acoustical conversion is a useful characteristic for assessing the properties of the loudspeaker and investigating the influence of the audio stimulus and the digital signal processing applied to the audio signal [1].
In general, the electro-acoustical efficiency is defined as the ratio
100% a e P P K (2)
between acoustic output power Pa and electric input
power Pe.
The real input power Pe can be calculated in the time
domain as the product of input current i(t) and open circuit amplifier output voltage (here called generator
voltage ug(t)) averaged over a measurement interval:
1
2 0 ( ) ( ) ( ) ( ) e g E u P u t i t Z f G f df f
³
(3)This is equivalent to the squared voltage spectrum
G2
u(f) of an arbitrary stimulus multiplied by the real part
of the electric input admittance ZE(f)-1.
The acoustic output power Pa can be determined by
integrating the far-field sound pressure over a closed surface S around the loudspeaker and expressing the result in the time and frequency domain as
2 2 2 0 0 0 ( ) 1 ( , ) 1 ( , ) ( , ) u a ref ref S G f P p t dS H f Sdf c c Q f U U f
³
r³
r r (4) 4. VOLTAGE SENSITIVITYtransducer in enclosure) is a useful characteristic for matching the transducer to the amplifier. The voltage sensitivity can be defined in a general way as the total sound pressure level in dB
, 0 2 2 2 0 0 ( ) 20 lg 1 10 lg ( ) ( , ) ref ref rms ref u r u ref p r L p G f H f r df p f § · ¨ ¸ © ¹ § · ¨ ¸ ©
³
¹ (5) at a given reference distance on-axis (typicallyrref = 1 m) with the reference sound pressure
p0 = 2∙10-5 Pa generated by an arbitrary stimulus with
known voltage amplitude spectrum Gu(f) corresponding
to a reference rms-value of the generator voltage
(typically uref = 1 V): 2 2 0 ( ) ref u u G f df f
³
(6)Efficiency and voltage sensitivity are related but are not identical [2]. Both characteristics contain the
on-axis transfer response H(f,rref) that generates a similar
curve shape at low and medium frequencies. However,
the electrical impedance ZE(f) increases the efficiency at
the resonance frequency fs. The directivity factor
Q(f,rref) usually reduces the efficiency at higher
frequencies.
5. LOUDSPEAKER MODELING
A lumped parameter model [1] can be used to describe
the on-axis transfer response H(f,rref), the electric input
impedance ZE(f) and the frequency dependent efficiency
of the passive loudspeaker system comprising an electro-dynamical transducer mounted in a baffle, mounted in a sealed or vented enclosure using mechanical resonators. This modeling is also valid at higher amplitudes, where the force factor product, suspension stiffness and voice coil inductance are not
constant values Bl, KMS and LE, but become nonlinear
functions Bl(x), KMS(x) and LE(x), respectively,
depending on instantaneous voice coil displacement x. For a particular stimulus generating a voice coil displacement x with the probability density function
pdf(x), an effective force factor
( ) ( )
Bl Bl x pdf x dx
f
f
³
(7)
an effective stiffness of the suspension
( ) ( ) MS MS K K x pdf x dx f f
³
(8)and an effective voice coil inductance
( ) ( ) E E L L x pdf x dx f f
³
(9) can be introduced. --55 -4-4 -3-3 -2-2 -1-1 -0-0 11 22 33 4 5<< Coil in X [mm] coil out >>
0.00 0.50 1.00 1.5 0 1 P D F [1 /mm]
Probability Density Function
MUSIC
maximal al displacement
Figure 3: Probability Density Function pdf(x) of voice coil displacement x for a typical audio signal (e.g. music)
Figure 3 shows an example of the pdf(x) of typical audio material representing most music genres and speech. This bell-shaped characteristic shows that the coil is close to the rest position most of the time, and high peak values are relatively rare. Thus, from a
statistical perspective, the effective parameters Bl|Bl,
MS MS
K |K LE |LEare close to the nonlinear parameter
values found at the rest position x=0, and the small signal efficiency and voltage sensitivity are useful approximations for higher amplitudes.
6. PRACTICAL SYSTEM DESIGN
This section describes the procedure of the Green
Speaker Design [1] from the perspective of system
integration as illustrated in Figure 4. The target is to specify all components and to ensure the optimal use of the resources. A set of physical characteristics will be defined for transducer design as discussed in a second related paper.
Target
Performance StimulusTypical
Passive System Design Restrictions (geometry, size, cost,…) Parameter Identification Prototyping Active System Evaluation Optimize System Components SPLmax Ho(f, r) Gw(f), CFu VB, ST
umax, imax, Pe,max
lumped parameters control parameter sample Transducer Manufacturer
6.1 Target Performance Specification
The design process starts with the specification of the following characteristics:
x maximum sound pressure level SPLmax(r) at a
distance r on-axis in accordance with IEC 60268-21 [11], while
x reproducing a broadband audio stimulus (music, speech), that is typical for the particular application, characterized by a long-term
amplitude spectrum Gw(f) and a crest factor
CFw, and
x providing a desired overall frequency response
Ho(f, r) and requiring neither thermal nor
mechanical protection (Hequ(f)=1).
Thus, the maximum SPL output can be expressed as
max 0 0 20lg a ( ) ( , ) w o g SPL G f H f r df p f § · ¨ ¸ ©
³
¹ (10)using the amplification gain ga at the control input as
shown in Fehler! Verweisquelle konnte nicht
gefunden werden..
The desired frequency response shape of the overall
transfer function Ho(f, r) can be specified by simple
characteristics such as cut-off frequency fc, quality
factor, alignment type (e.g. Butterworth) or by poles and
zeros in a rational representation of Ho(f, r).
The design process has to consider additional restrictions and particular requirements in the target application such as
x available box size (volume VB),
x total radiation area ST including diaphragm
area SD, passive radiator area SR, and panel size,
x maximum peak voltage umax,
x maximum peak current imax,
x maximum electrical power Pe,max and
x other constraints (price, weight, …).
The maximum peak voltage umax and peak current
imax are usually limited by the rail voltage and the FETs
in the H-bridge of the class-D power amplifier. The
maximum electrical power Pe,max can be supplied from
the amplifier to the transducer continuously (> 100h) without causing a thermal overload either in the amplifier output stage, power supply or in the voice coil [18].
Table 1: Overview of the pros (+) and cons (-) in the passive loudspeaker system used with nonlinear control
Passive Loudspeaker System + Control Smal l Size Smal l Radi at or Equalizat ion f<f 0 Sensitivity Effici ency Lo w C ost Closed-Box + + + - - + Vented-Box - + - + + + Passive Radiator + - - + + - Band-Pass - + + + + + Panel - - + + + +
6.2 Passive System Design
The second step in the design procedure determines the optimal design of the mechanical and acoustical components (panel, passive radiator, box, horn, transmission line, etc.) and the specification of the desired transducer properties. Table 1 gives an overview of the most popular passive system types and shows their pros (+) and cons (-) with respect to the target performance and their suitability for electrical control:
The closed-box system is the best candidate for realizing small loudspeakers at minimum cost but provides the lowest efficiency and sensitivity below the
system resonance f0. Active equalization can be applied
to low frequencies to shift the cut-off frequency fc down
by 1-2 octaves. This kind of bass extension cannot be realized in vented-box and passive radiator systems due to the steeper roll-off below resonance. However, the additional resonator is preferred in portable and professional applications where efficiency and voltage sensitivity are critical. The panel system represents any other distributed mode system using available components (e.g. display) as a radiator.
After selecting the optimal system type, the size and shape of the acoustical and mechanical elements (port, front volume, rear volume, panel) are designed for the
given box volume VB and total radiation area ST.
Contrary to the traditional passive design methods that optimizes the cabinet and other mechanical and acoustical elements for a given transducer to generate a desired frequency-response shape (alignment) and passband sensitivity, the Green Speaker Design optimizes the efficiency of the passive system while generating the desired output signal under the conditions defined in section 6.1. The active equalization at the input of the control system always
provides the desired overall transfer function Ho(f) as
illustrated in Figure 5. However, the power saved by the higher efficiency in the pass band is used for boosting lower frequencies. This corresponds with a virtual change of the transducer and enclosure parameters
0 10 20 30 40 50 60 70 80
Magnitude of Far Field Sound Pressure Response
[d B ]-[P a/ V ] H(f,r) passive speaker Ho(f,r) equalized speaker
Figure 5: Amplitude response of the loudspeaker with and without equalization
Numerical simulation can be used to find the optimal
values for the effective radiation area SD and other
lumped parameters in the passive system that give the maximum efficiency η for the typical stimulus with
voltage spectrum Gu(f).
With active alignment
K -30 -20 -10 0 10 Stimulus(f) Spectrum
relative input spectrum
[dB]
Stimulus Stimulus with alignment
Figure 6: Spectrum Gu(f) of terminal voltage of
loudspeaker with active alignment (thick line) at maximum SPL compared with the spectrum Gw(f) of the
stimulus (thin line) representing typical program material.
The voltage spectrum Gu(f) at the loudspeaker
terminals shown in the Figure 6 depends on the stimulus and the applied equalization and determines the amplifier and transducer requirements. If the electrical
input power Pe, or both peak upk and ipk values of
terminal voltage and input current exceeds the
permissible limit values Pe,max, umax and imax,
respectively, the efficiency of the passive system has to be improved by considering a nonlinear motor with small voice coil overhang, a soft suspension, additional resonators and other design choices ]. If only the voltage capability of the amplifier is not sufficient, an improvement of the voltage sensitivity expressed in dB
can be achieved by reducing the DC resistance RE.
Similarly, a higher RE value can reduce the peak current
ipk.
The peak excursion xpk shall not exceed the maximal
displacement xmech according to IEC standard 62458
[20], which describes the positive and negative limits of the useable working range without mechanical limiting, voice coil rubbing or other irregular behavior [18].
The performance of the passive loudspeaker system can be predicted in greater detail by large signal modeling using the linear, nonlinear and thermal parameters of an existing sample or using a virtual design choice developed by finite element analysis.
Table 2: Characteristics of the original transducer (A) and the optimized transducer (B) with reduced voice coil height hc. Transducer Parameters A B Change Nominal impedance [Ω] 4 2 -50% Gap depth dg [mm] 5 5 0% Coil height hc [mm] 11 5.75 -47% Resistance RE[Ω] 3.31 1.74 -47%
Force factor Bl(0) [N/A] 4.6 3.5 -24%
Bl limited peak displacement XBl [mm]
4.5 3.5 -40%
Motor efficiency factor Bl(0)2/R
E[Ns/m] 6.39 7.04 +10%
Compliance Cms [mm/N] 0.66 0.67 +1%
Moving mass MMS [g] 10.4 8.6 -17%
Mass ratio Mratio 1.5 3 +100%
Resonance frequency fs [Hz] 61 67 +10% Passband efficiency ηPB [%] 0.267 0.433 +62% Voltage sensitivity in passband SPLPB,1Vrms, 1m [dB] 81.2 85.9 +4.7dB
For example, Table 2 shows the impact of a Green speaker optimization applied to an existing transducer [2]. Only the voice coil height was reduced by 47% while using the same wire and winding, magnet, diaphragm and suspension. This small change provided 62% more efficiency and 4.7 dB more sensitivity in the pass band. However, the smaller voice coil overhang increased the nonlinearity of the force factor Bl(x) that
generates at smaller displacement XBl= 3.5 mm about
10% intermodulation distortion. The optimized transducer needs adaptive, nonlinear control for canceling the undesired distortion.
6.3 Prototyping
6.4 Initial Parameter Identification
If the nonlinear control is activated at the passive loudspeaker system for the first time, a one-time identification process (similar to the Large Signal Identification [21]) is automatically activated. This measurement uses a broadband stimulus and provides all linear, nonlinear and thermal parameters of the transducer mounted in the passive system. It also determines the permissible working range based on generic protection values such as nonlinear parameter
variation (Blmin, Cmin), maximal increase of voice coil
temperature Tmax and maximal electrical input power
Pe,max.
6.5 Active System Evaluation
The identified transducer parameters are stored and used as initial control parameters after powering up the controlled loudspeaker system of any unit of the same type. The adaptive control with current monitoring
provides updated parameters and diagnostic
information about the internal state of the particular device while reproducing natural audio stimuli and artificial test signals.
The target performance is evaluated by monitoring
the input power Pe and the peak values of voltage and
current at the transducer terminals and measuring the generated SPL output while reproducing a multi-tone or
noise signal with a spectrum Gw(f) and crest factor CFw.
Active Speaker Characteristics +NC A +NC B Mean SPL [dB] 92.2 92.2 Displacement peak Xp [mm] 3.2 3.3 Displacement rms Xac [mm] 0.51 0.54 Crest factor CFX [dB] 15.9 15.9 Distortion Compensation EID [%] 11 30 (+19%) Voltage peak Up [V] 45.6 (-33%) 30.4 Voltage rms Uac [Vrms] 8.7 (-45%) 4.7 Crest factor CFu [dB] 14.3 16.2
Real input power Pe [W] 14.2 (-40%) 8.4
Temperature coil ΔTv
[K] +96 (-39%) +59
Table 3: Performance of the original loudspeaker (A) and the modified speaker (B) with nonlinear control (+NC) while reproducing a typical audio signal.
For example, Table 3 compares the performance of the original loudspeaker A and the modified speaker B of the example already discussed in Table 2. Both passive speakers are operated by adaptive nonlinear control and generate the linearized output (spectrum,
mean SPL, peak displacement Xp, crest factor CFx) for
a typical audio stimulus. However, the mirror filter has
to generates 20 % more compensation voltage to cancel the equivalent input distortion (EID) of the modified speaker B using the shorter voice coil. The shorter coil reduces by about 40% the rms voltage and power requirements of the amplifier and the voice coil temperature. Thus, the modified speaker can generate 3dB more SPL at higher frequencies than the original speaker using the same amplifier and generating the same increase in voice coil temperature.
The cancelation of the nonlinear distortion can easily be checked by using standard measurement
techniques [18] for harmonic distortion,
intermodulation and impulsive distortion indicating voice coil rubbing, air leakage noise and other loudspeaker defects. 0 1 2 3 4 5 6 7 8 9 10 40 60 80 100 200 2nd Harmonic % Frequency [Hz] 0 5 10 15 20 25 30 35 40 3rd Harmonic %
B+LC
B+NC
B+LC
B+NC
40 60 80 100 200 Frequency [Hz]Figure 7: 2nd- and 3rd order harmonic distortion of the
optimized loudspeaker with linear control (B+LC) and nonlinear control (B+NC).
Figure 7 shows the active compensation of the harmonic distortion generated by a single excitation tone. Since the modified transducer B is carefully manufactured, the 2nd
-order distortion found under linear control is relatively small, indicating symmetrical stiffness KMS(x) and force
Fundamental (with linearization) 20 30 40 50 60 70
80 Sound pressure spectrum p(f) of reproduced multi-tone signal
dB B+LC B+NC Distortion Fundamental A+LC
Figure 8: Fundamental components and nonlinear distortion in a multi-tone test stimulus reproduced by the original loudspeaker (A) and the modified loudspeaker (B) with linear (+LC) and nonlinear control (+NC). The nonlinear control (+NC) also reduces the intermodulation distortion generated by a more complex stimulus (e.g. music) as shown for a sparse multi-tone stimulus in Figure 8. The distortion reduction achieved for the modified speaker B+NC are shown as thin solid and thick solid lines, respectively, while the dashed lines represent the nonlinear distortion found with linear control LC.
The modified speaker with linear control B+LC generates much higher distortion than the original speaker A+LC, but nonlinear control can reduce the distortion by more than 15 dB at low frequencies. Here, the nonlinear
motor with nonlinear control B+NC generates lower distortion than the original speaker A+LC that uses a much more linear motor. At 600 Hz and 1.3 kHz, there are nonlinear modal vibrations on the cone which cannot be canceled by the control structure, which is based on lumped parameter modeling.
Evaluating the distortion reduction in music or any other audio stimulus having a dense spectrum requires a measurement technique [23] that separates the nonlinear distortion (residuum) from the desired linear component. The residuum can be analyzed in the time and frequency domain and is the basis for perceptual modeling, distortion auralization and a systematic listening test [22] to determine audibility and annoyance of the remaining distortion components.
The adaptive loudspeaker system based on current monitoring allows a stand-alone self-test on a variety of program materials to check assumptions made in the design process while assessing aging of the suspension and the long-term performance of other critical hardware components.
6.6 System Optimization
The evaluation above provides valuable information for optimizing the target specification and selecting or designing better hardware components. If there is no time for significant changes, a modification of the protection parameter and the target alignment in the control software can be used to cope with unknown limitations in the hardware. This usually requires an additional iteration of the last steps in the development process. For example, voice coil rubbing occurring at higher amplitudes can be avoided by reducing the peak
displacement xmax in the mechanical protection system.
If the transducer has a lower power handling than expected due to blocked heat transfer in the cabinet, the thermal protection system can provide a fast interim solution until a more efficient transducer with improved heat transfer [8] is available.
7. CONCLUSION
Digital signal processing available in active audio devices changes the paradigm in passive transducer design. The electro-acoustical efficiency η is an important criterion for evaluating the transducer and system design with respect to the optimal use of all resources. Increasing efficiency is the key to generating more output in smaller loudspeaker systems and reducing power consumption in portable applications with limited battery capacity.
Efficiency η is not identical with the sensitivity
L1V,1m of the transducer. The sensitivity L1V,1m is
required to match the transducer to the peak voltage capabilities of the amplifier. The instantaneous values of efficiency and sensitivity depend on the particular stimulus supplied to the loudspeaker. Mean values of η
and L1V,1m can be calculated by using the frequency
dependent efficiency function η(f) and sensitivity
function L1V,1m(f) as well as the voltage spectrum Gu(f)
and probability density function pdf(u) representing a typical program material.
The lumped parameters of the transducer model measured or predicted by FEA are a suitable basis for evaluating design choices and investigating the influence on sensitivity and efficiency. The actual
resonance frequency fs and the quality factors QTS, QMS
and QES of the transducer play a minor role in Green
Speaker Design because those transducer characteristics are highly time-variant due to production variances, aging and external influences (climate) and can easily be compensated by adaptive control.
8. REFERENCES
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