Side reader for R.H.Small “Direct-Radiator Loudspeaker ... · What are Thiele/Small parameters?...
Transcript of Side reader for R.H.Small “Direct-Radiator Loudspeaker ... · What are Thiele/Small parameters?...
©2013 Katsuyuki Tsubohara, All rights reserved.
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Last update: 13 May 2013
Side reader for R.H.Small “Direct-Radiator Loudspeaker System Analysis”
© 2013 Katsuyuki Tsubohara, All rights reserved.
Do not re-distribute anything in this document.
Disclaimer
This article may include miscalculations and/or wrong theories. The author accept no responsibility whatsoever for any direct or indirect damages, loss, prejudice or emotional distress caused by use of this document.
Contents
� Introduction � What are Thiele/Small parameters? � Equivalent circuit of loudspeaker driver � Step-by-step derivations of the equations shown in the paper � Further readings
©2013 Katsuyuki Tsubohara, All rights reserved.
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INTRODUCTION
This article is a side reader for the classic paper by R.H.Small “Direct-Radiator Loudspeaker System Analysis”. You
should purchase the original paper at first. It’s available on AES website. Don’t illegal download –I hope you support AES.
The methods and the technical terms described are de-facto standard now and help to design woofers, to read technical
documents and to access updated knowledge.
Supposed readers are around 2 year experienced driver engineers. If you are at a level as trainee, stop reading this and
assemble some drivers with parts available in your lab. It grows your practical skill far more. Because: Any engineer
without practical experience is useless even if he understands basic theories. It’s easy to obtain required LF performance by
following 4 equations. They are enough to calculate linear domain response of woofers and don’t need to understand the
background in usual cases.
MS
MS
MSMS
S M
K
CMf
ππ 2
1
2
1 == MS
MS
MS
TS C
M
DCR
lBR
Q ⋅+
= 22
1
[ ] [ ].25222 1007.4407 3 ltrDMSmDMSDoAS SCSCSCcV
MS××=×≅≡ ρ (MKSA units)
[ ]%
2
2224
2
222
1045.52 MSE
D
MSE
Doo
MR
SlB
MR
SlB
c××≅⋅= −
πρη
In addition, the analysis has an important assumption called “piston range”:
Wave length is enough longer than the diameter of the diaphragm
Thus the basic theories do not contribute to obtain good performance in mid and high frequency range. It’s clearly stated in
the paper, at “ASSUMPTION AND APPROXIMATIONS” paragraph.
In contrast to that the low-frequency response can be simply simulated by a few parameters, outside piston range, the
radiation phenomena is complex and the effects by the shape of diaphragm and the baffle are not ignorable. Also
diaphragms have many resonant modes in high frequencies also contribute to the performance in the range. It’s almost
impossible to be solved by simple calculation. Through the history of loudspeakers, many attempts using parametric method
to analyze behaviors of diaphragms have been done but have never been successful like methods by Thiele and Small.
I.E., a posteriori method, experimental guess is required to design Mid-Woofers, Full-ranges and Tweeters. In this era,
FEA simulation is available but importance of practical experiences has not been reduced at all. FEA calculates the response
but does not show how to improve the response. FEA is useful to save time by reducing number of times engineers create
actual samples, but to use it effectively, the guess by human is still required. That’s what makes loudspeaker design so
interesting.
In conclusion young engineers should experience cut-and-try approach. Basic theories are to be studied later. The author
learned it in the first year as a driver designer, but forgot in the next year and studied again 3 years later from scratch.
©2013 Katsuyuki Tsubohara, All rights reserved.
3 / 26
Listed below are motivations to learn from the Small’s:
� Enforce your skill with logical thinking
� Give clear technical information, especially in OEM scenes
� Understand de-facto standard technical terms and access updated knowledge
� Understand distortion phenomena by relating it to nonlinearities of the parameters (it’s only way to
understand how to reduce distortions of woofer drivers)
� Design enclosures (in this case, you should read all papers by Thiele and Small)
� Learn how to summarize technical report –the paper has only 13 pages but covers almost everything about
woofer system design.
� Learn how to sophisticate design approach (some kinds of definitions for LF parameters have been
established through history. The set defined by Thiele and Small has gotten the de-facto standard position by
its efficiency in design process)
You want to know the background? Congratulations, you have done 1st step to be a true pro.
The author wishes this article help the world filled with good music and nice sound.
What are Thiele/Small parameters?
The linear-domain response of a woofer driver is determined by 7 parameters, related to the specifications of the parts
(driver parameters):
MSM , MSR , MSC , Bl , eR , DS , ����
The translated parameters in useful definitions for enclosure designers (system parameters):
SF , MSQ , ESQ , ASV , oη , ( )ωjVCZ , ��
Above 2 sets have one-on-one relationship. So, when an enclosure designer shows a driver designer his requirements by the
system parameters, the driver designer can design a driver matches the requirements systematically. To avoid system
engineers order theoretically impossible drivers to driver designers, an equation for checking system parameters’ validity is
shown:
ASES
So V
Q
f
c⋅⋅=
3
3
24πη
©2013 Katsuyuki Tsubohara, All rights reserved.
4 / 26
Equivalent circuit of loudspeaker driver
The paper starts with an acoustic equivalent circuit of loudspeaker system. If you haven’t majored acoustics or electronics,
you may be fed up. In fact, the problem to be solved is just a simple forced oscillation and the motion equation is familiar to
people who learned dynamics. “Equivalent Circuit” sounds difficult but it’s just a rewritten motion equation.
Why do we have to use the equivalent circuit? It’s that there are far more electronics engineers than acoustics’ and we
should use their accomplishments like Spice.
Necessary basics are shown below.
If you don’t familiar about these, go to a bookstore. There are so many plain textbooks for electronics.
Precalculus of exponent function:
tjtj ejedt
d ωω ω= , .1
constej
dte tjtj +=∫ωω
ω
Forget about the integral constant by assuming initial condition zero. It just complexes.
Ohm’s law expanded to AC circuits:
( ) ( ) ( )tjt IZE ω=
Series impedance calculation:
K+++= 321 ZZZZ
Parallel impedance calculation:
K+++=321
1111
ZZZZ
Impedances of inductor (L), capacitor (C) and resistor (R):
( ) ( )
( ) ( )
( ) ( )
=
=
=
tRtR
tCtC
tLtL
RIE
ICj
E
LIjE
ω
ω1
i.e.
( )
( )
( )
=
=
=
RZ
CjZ
LjZ
jR
jC
jL
ω
ω
ω
ω
ω1
Let’s start. The definitions for the symbols are shown in Small’s. It’s skipped here (this is a side reader). Model of a
loudspeaker driver mounted on an infinite-baffle is shown in Fig.1 and Fig.2
Fig.1 Mechanical model Fig.2 Model of motor and amplifier
©2013 Katsuyuki Tsubohara, All rights reserved.
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Note) Radiation resistance ARℜ is small in direct-radiator systems and ignorable to analyze the movement of the
diaphragm.
The motion equation for Fig.1 is
( ) ( ) ( ) ( )ttDMStDMS
tDMS Blixdt
dRx
Cx
dt
dM +−−= 1
2
2
(I)
The solved Maxwell’s equation for Fig.2 is
( ) ( ) ( ) ( )tgEtDtg iRRxdt
dBle +=− (II)
Second member in left part is counter EMF which disturbs amplifier give energy into driver.
From Eq.(II)
( )( )
( )tDgEgE
tgt x
dt
d
RR
Bl
RR
ei
+−
+= (III)
By substituting Eq.(III) to (I)
( ) ( ) ( )( )
( )
+−
++−−= tD
gEgE
tgtDMStD
MStDMS x
dt
d
RR
Bl
RR
eBlx
dt
dRx
Cx
dt
dM
12
2
(IV)
Simplified Eq.(IV) is
( ) ( ) ( )( )
gE
tgtD
MStD
gEMStDMS RR
Blex
Cx
dt
d
RR
lBRx
dt
dM
+=+
+++ 122
2
2
(V)
By assuming the amplifier gives effective voltage ge and angular velocity ω
( ) ( )tj
jgtg eee ωω2= (VI)
It’s a forced oscillation problem and the solution for Dx is sinusoidal with angular velocity ω
If it’s strange, go to the bookstore again and buy a textbook for dynamics.
For physicians, familiar subject is displacement Dx but to deliver the equivalent circuit, velocity Du is to be used. Why?
–you will find later. Forget about the question. Dx is sinusoidal so Du is sinusoidal too,
( ) ( )tj
jDtD euu ωω2= (VII)
Note) The symbols in the analysis indicate effective values (RMS)
From (VII),
( ) ( ) ( )tDtDtD uj
dtux ∫ ==ω1
(VIII)
( ) ( ) ( )tDtDtD ujudt
dx
dt
d ω==2
2
(IX)
By substituting Eqs. (VII), (VIII) and (IX) into Eq.(V),
©2013 Katsuyuki Tsubohara, All rights reserved.
6 / 26
( ) ( ) ( ) ( )tggE
tDMS
tDgE
MStDMS eRR
Blu
Cju
RR
lBRuMj
+=+
+++
ωω 122
(X)
In the paper, “acoustic equivalent circuit” is used and “mechanical equivalent circuit” doesn’t appear. Nowadays, the
mechanical one is widely in use because for driver engineers, it’s easier to understand. So here express the mechanical
equivalent circuit first and then deform it to the acoustic one.
A series LCR circuit is shown in Fig.3. Don’t think why. You’ll find later.
Fig.3 Series LCR circuit
By assuming AC source voltage of angular velocity ω. Impedance of series LCR is
(XI)
By applying Ohm’s law
( ) ( ) ( ) ( ) ( )tttttotalt ICj
RILIjIZEω
ω 1++== (XII)
By comparison between Eq.(X) and Eq.(XII), you’ll find relationship of
( ) ( )tDt uI ⇔ , ( ) ( )tggE
t eRR
BlE
+⇔ , MSML ⇔ ,
++⇔
gEMS RR
lBRR
22
, MSCC ⇔
So they are same shape equations. Now mechanical equivalent circuit is delivered
Fig.4 Mechanical equivalent circuit of driver
Through analyzing the relationship between � and �, velocity of diaphragm by input voltage is calculated.
Next subject is the acoustic equivalent circuit.
In acoustics, the volume velocity of air driven by the diaphragm is more important than the velocity of diaphragm �.
Equation should be deformed using relationship:
( )( )
D
tDtD S
Uu = (XIII)
CjRLjZZZZ CRLtotal ω
ω 1++=++=
©2013 Katsuyuki Tsubohara, All rights reserved.
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By substituting this into Eq. (X)
( ) ( ) ( ) ( ) ( )tggE
tDDMS
tDDgED
MStD
D
MS eRR
BlU
SCjU
SRR
lB
S
RU
S
Mj
+=+
+++
ωω 122
(XIV)
To understand acoustic phenomena, “pressure” is better than “force”.
To relate right part to pressure, dividing both parts by DS then
( ) ( ) ( ) ( ) ( ) ( )tgDgE
tD
DMS
tD
DgED
MStD
D
MS eSRR
BlU
SCjU
SRR
lB
S
RU
S
Mj
+=+
+++ 22
22
22
1
ωω (XV)
Defining acoustic parameters as following
2DMSAS SRR ≡ ,
2DMSAS SMM ≡ ,
2DMSAS SCC ≡
Then
( ) ( ) ( ) ( ) ( ) ( )tgDgE
tDAS
tD
DgE
AStDAS eSRR
BlU
CjU
SRR
lBRUMj
+=+
+++
ωω 1
2
22
(XVI)
Replacing the electronic symbols to the acoustic symbols as following then the acoustic equivalent circuit is delivered.
( ) ( )tDt UI ⇔ , ( ) ( ) ( )tgDgE
t eSRR
BlE
+⇔ , ASML ⇔ , ( )
++⇔ 2
22
DgE
ASSRR
lBRR , ASCC ⇔
Fig.5 Acoustic equivalent circuit of driver
Note) Descripting L, C and R by derivation and integral as shown below, it’s able to obtain the equivalent circuit
without using complex analytics. But without understanding complex analytics, it’s difficult to understand the
paper.
( ) ( )tLtL Idt
dLE = , ( ) ( )∫= dtI
CE tCtC
1, ( ) ( )tRtR RIE =
Now you’re at the start-line to read the classic.
R U ready?
©2013 Katsuyuki Tsubohara, All rights reserved.
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Step-by-step derivations of the equations shown in the paper
Because the paper is an academic article not a textbook, detailed deformations for the equations are skipped. For most of
us, it’s usual to take hours to obtain only one equation by ourselves. To “battle” against equations is meaningful to be a
physician or a mathematician. Great Scott we audio engineers in 21st century are always pressed for time and it’s too heavy.
But it’s certain that the best way to understand background of a technical paper is to obtain the equations by ourselves. So
here given are step-by-step deformations.
This article is just a typing of author’s hand calculations. The deformations are not optimized completely and you’ll find
his bad habitual manners. The author wishes it to be corrected by somebody smarter.
Notes)
� Numbers of equations are obverse to Small’s.
� Refer to the paper when figure with number appears in texts.
� Refer to the paper for the definitions of symbols.
� By the author’s cup of tea, � = �� is always assumed.
� By the author’s glass of beer, ≡ is used in definition equations.
� Practical equations are indicated by (‡)
� If you are a driver engineer and the mechanical parameters are familiar than the acoustic ones, keeping following 3
equations in mind is suggested. ��� would be the only parameter with letter AS used in daily work.
2
2
2
DASMS
DASMS
DASMS
SRR
SMM
SCC
=
=
=
i.e. 2
2
2
DMSAS
DMSAS
DMSAS
SRR
SMM
SCC
≡
≡
≡
(†)
� Some equations are shown in both parameters of acoustic and mechanical using Eqs. (†).
� For the sake of clarifying infinite-baffle cases and in-enclosure cases,
���� and ���� are replaced by �������, �������,����������� and �����������.
First subject is to simulate frequency response of bass-reflex or passive-radiator systems. See Fig.1.
Symbols U are obverse to volume velocity of air. Obviously,
LPDo UUUU ++= (1)
Note) This works as Kirchhoff’s 1st theorem. In case systems have equivalent characteristics of Kirchhoff’s theorems
and Ohm’s Law, it can be written by equivalent circuit. It’s able to descript these kinds of systems simpler using
matrix calculations, but it’s not the subject of this side reader.
L+++= kjin iiii , L+++= rqpn eeee , nnn ize =
Fig.2 is an equivalent circuit for enclosed systems. It seems complex but the analysis starts from simpler case
(Infinite-Baffle) and you’ll understand later. Don’t worry.
By basic theories of acoustics, the radiated sound power is
©2013 Katsuyuki Tsubohara, All rights reserved.
9 / 26
ARoA UP ℜ= 2 (2)
Note) This equation has the same shape of electric power’s RIP2=
In piston range ka ≪ 1
( )coAR πωρ 22≅ℜ (3)
Note) By Eq.(2) and Eq.(3), flat F-resp. equals to constant acceleration in piston range. That’s why all direct radiator
systems have flat energy response in the range.
Note) It’s so time-consuming to obtain Eq.(3). Shown below is the rigorous calculation taken from L.L.Beranek
“Acoustics” (McGraw-Hill, New York, 1954).
Radiation Impedance for One Side of a Plane Piston in Infinite Baffle
a : Radius of Piston
k: wave number ( λπ2≡k )
( )( )
( )
( )
kaW
where
WWWK
WWWWJ
Kk
cj
ka
JcajXZ
W
W
kaoka
oMMM
2
753533
2
8642642422
21
22
7
2
53
1
222
7
22
5
2
3
1
212
212
≡
⋅⋅+
⋅−=
⋅⋅⋅−
⋅⋅+
⋅−=
+
−=+ℜ=
K
L
π
πρρπ
To be a true tweeter pro, at least the knowledge how above behaves is required. Without the knowledge,
you’ll never find why surface-driven plane drivers have HF cutoff. Google it, you’ll find some graphical
plots of radiation impedance.
The volume of air pushed out from enclosure equals to the volume vacuumed out from enclosure, thus
Bo UU −= (4)
Obviously, when an enclosure has separated chambers
,321 LBBBo UUUU ++= (5)
Defining input power by “Consumed power at the resistor when the driver in the system is replaced by a resistor having the
same resistance equals to the DCR of voice coil”,
( ) EEg
gE
gE
gg
gE
E
EE R
RR
eR
RR
ee
RR
R
RP
2
2
22
1
+=⋅
+=
⋅
+≡ (6)
Note) Today, input power is usually defined by nominal impedance. It gives approx. 10% to 20% less value.
By defining efficiency by ratio of radiated acoustic power and above input power,
©2013 Katsuyuki Tsubohara, All rights reserved.
10 / 26
( )Eg
EgARo
EEg
g
ARo
E
A
Re
RRU
RRR
e
U
P
P2
22
2
2 +⋅ℜ=
+
ℜ=≡η (7)
Note) Modern amplifiers have very small output impedance. If cable is short enough, $ is ignorable.
To start from simple case, infinite-baffle condition is now assumed in Fig.2 and
No air leakage,
0≅LU i.e. ∞≅ALR
No port or passive radiator
0=PU i.e. ∞=APR
Infinite volume of chamber
∞≅ABC
No acoustic damping
0≅ABR
To simplify, counter EMF and mechanical loss are summed
( ) 2
22
DEg
ASATSRR
lBRR
++≡ (8)
Then Fig.3 is delivered. Calculating %&,
Note) The form of equation would be strange for engineers who have not studied about active filters. To separate
efficiency and shape of frequency response curve, following is required (normalization).
12
)(lim =∞→
ωω
jIBG i.e. ( ) 12
≅⇒>> ωωω jIBs G
( )
( )
( ) ( )
( )( )
( )
( )( )
( )
( ) )(
2
2
2
2
2
1
1
1
11
ω
ω
ω
ωωω
ω
ωωω
ω
ωωω
ωω
jIBASDEg
g
ASASASAT
ASAS
ASDEg
g
ASASASAT
AS
DEg
g
ASASASAT
AS
DEg
g
ASAS
ATDEg
g
jo
GMSRRj
Ble
CMjCRj
MCj
MSRRj
Ble
CMjCRj
Cj
SRRj
Ble
CMjCRj
Cj
SRR
Ble
MjCj
RSRR
Ble
U
⋅+
≡
++⋅
+=
++⋅
+=
++⋅
+=
++⋅
+=
(9)(10)
From Eq.(9)
©2013 Katsuyuki Tsubohara, All rights reserved.
11 / 26
( ) ( )( ) ( ) ( )
( ) ( )2
2222
222
2
ω
ωω
ω
ωω
jIB
ASDEg
g
jIBjIBASDEg
g
ASDEg
g
oo
o
GMSRR
lBe
GGMSRRj
Ble
MSRRj
Ble
UU
U
⋅+
=
⋅+−
⋅+
=
=
∗
∗
From Eqs. (2), (3) and (7)
( )
( ) ( )( )
( )2
22
22
2
222
2222
222
2
2
ω
ω
ω
πρ
πωρ
ω
η
jIB
ASDE
o
Eg
EgojIB
ASDEg
g
j
GMSR
lB
c
Re
RR
cG
MSRR
lBe
⋅⋅=
+⋅⋅⋅
+=
(11)
Note) For driver engineers, calculations by the mechanical parameters are shown below. It is enough to calculate SPL
curve of a driver on an infinite-baffle in piston range.
'& =(&2*+
∙-./.
$01�.2��
. =(&2*+
∙-./.1�
.
$023�.
( ) ( ) ( )
( )( )
( )( ) ( )
( )( ) ( ) ( )( )
( ) ( )( )
( ) 22222
224
22
224
22
224
2
2
2
2
2
1
11
11
11
ATASASAS
ASAS
ATASASASATASASAS
ASAS
ATASASASATASASAS
ASAS
ATASASAS
ASAS
ATASASAS
ASAS
jIBjIBjIB
RCMC
MC
RCjMCRCjMC
MC
RCjMCjRCjMCj
MC
RCjMCj
MCj
RCjMCj
MCj
GGG
ωωω
ωωωωω
ωωωωω
ωωω
ωωω
ωωω
+−=
−+−+−=
+−+−++=
+−+−−⋅
++=
= ∗
Thus
©2013 Katsuyuki Tsubohara, All rights reserved.
12 / 26
( )
( )
( )
( )
21
222
2222
224
21
2
2
22
2422
2
222
4
2424
21
22222
224
1
1
1
+++−
=
+++
−
⋅=
+−=
EgMSMSMSMS
MSMS
DEgD
MSDMS
D
MSDMS
D
MSDMS
ATASASAS
ASAS
jIB
RR
lBRCMC
MC
SRR
lB
S
RSC
S
MSC
S
MSC
RCMC
MC
G
ωω
ω
ωω
ω
ωωω
ω
To calculate impedance curve, deforming Fig.3 by basic theories for electric circuit are shown below.
By separating $�4 to $�� and counter EMF,
By applying dual transformation
By applying Thieving’s theorem to the current source and the left resistor
�-/5$ 6 $071�
-./.
5$ 6 $071�. $�� 8�� 2��
5$ 6 $071��-/
5$ 6 $071�
.
-./.
2�� 8�� 1
$��
©2013 Katsuyuki Tsubohara, All rights reserved.
13 / 26
By replacing the voltage source using transformer
Now the source voltage is � , the output of amplifier. Focused on the electric impedance, it’s equivalent to
Defining electric parameters as following, Fig.4 is delivered
830� ≡�9
:
�:;:2��, <=0� ≡
�:;:
�9: 8��, $0� ≡
�:;:
�9: ∙
>
?@A
Note) Using the mechanical parameters,
830� 3BA
�:;:, <=0� -./.83�, $0�
�:;:
?BA
Because of the ignored radiation impedance, 1� does not contribute to the motion of diaphragm thus does not
appears in the electric impedance.
By defining C� as resonant frequency of Fig.4 then
� ∙1�
-/
1�.
-./.∙ $
1�.
-./.∙ $0
8�� 2��
1
$��
�
1:1�
-/
1�.
-./.∙ $
1�.
-./.∙ $0
2�� 8��
1
$��
�
$ $0
1�.
-./.2��
-./.
1�. 8��
-./.
1�. ∙
1
$��
©2013 Katsuyuki Tsubohara, All rights reserved.
14 / 26
( )
MSMS
D
MSDMS
ASAS
D
ASDAS
CESMES
sss
MC
S
MSC
MC
S
lBC
lB
SM
LC
FT
=
⋅=
=
⋅=
===
2
2
2
22
22
2
222 211 πω
(12)
The Q factors are
2
22
22
2
2
22
22
21
DAS
DAS
ASAS
ASAS
DAS
DAS
ASAS
ESMESS
MS
SR
lB
lB
SM
MC
MC
SR
lB
lB
SM
MC
RC
Q
⋅⋅=
⋅⋅=
≡ ω
ASASS
ASASASAS
RC
RCMC
ω1
1
=
⋅=
MSMSS
D
MSDMSS
RC
S
RSC
ω
ω
1
1
22
=
⋅=
(13)
EMS
S
EDAS
S
EMESS
ES
RlB
M
RlB
SM
RC
Q
⋅⋅=
⋅⋅=
≡
22
22
2
ω
ω
ω
(14)
For driver designers, forms without ω� works better because they are directly related to the specifications of the parts
MS
MSEES
MSMS
MSMS
ClB
MRQ
CR
MQ
22=
=
(‡)
©2013 Katsuyuki Tsubohara, All rights reserved.
15 / 26
Note) Q factor is ratio of energy stored and dissipated. To get physical image, considering via the mechanical equivalent
circuit would be helpful. Shown below is the mechanical acoustic circuit ignoring output impedance of amplifier;
$.
Q factor in series LCR circuit is defined by (inverse of parallel circuit case):
CRR
CLQ
oω11
=≡−
E0� and E3� are delivered directly through above.
Note that counter EMF acts like resistance but no energy is dissipated. It just disturbs energy input into driver.
Too big Magnet reduces the force around C�.
The definition of ��� is
��� ≡ (&+.8�� (15)
If “Volume of air having same acoustic compliance…” sounds too academic, “Volume of closed enclosure which
gives 41% higher resonance” would be nice.
From the electric equivalent circuit,
( )( ) ( )ESCES
CESESESMESCES
ESCESMESjRLC RLj
LjRRCLj
RLjCjZ
ESCESMES ωωω
ωωω
++=++=−2
1 11
( )
( )
( ) 12
2
+⋅+
⋅⋅+=
++⋅+=
ES
CESMESCES
ES
CES
ESE
CESESESMESCES
CESESE
jVC
RL
jCLj
R
Lj
RR
LjRRCLj
LjRR
Z
ωω
ω
ωωω
ω
�-/
$0
-./.
$0
$3� 83� 23�
©2013 Katsuyuki Tsubohara, All rights reserved.
16 / 26
( )
( )
( )
( )
( )
+++=
+⋅+
⋅⋅+=
+⋅+
⋅⋅+=
+⋅+
⋅⋅+=
+⋅+
⋅⋅+=
1
11
1
1
1
1
22
22
22
22
2
MSSs
MSSESE
MESESSSs
MESESSS
ESE
MESES
MESCESSs
MESES
MESCESS
ESE
MESES
CESSs
MESES
CESS
ESE
MESES
CESCESMESMESCES
MESES
CESCESMES
ESE
QTjTj
QTjRR
CRTjTj
CRTj
RR
CR
CLTjTj
CR
CLTj
RR
CR
LTjTj
CR
LTj
RR
CR
LLCjCLj
CR
LLCj
RR
ωωω
ωωω
ωω
ωω
ω
ωω
ω
ωω
ω
(16)
Today, measurement tools using laser are used in loudspeaker manufactures. They can measure displacement of diaphragm
directly and the method described in “Measurement of Driver Parameters” is not widely in use now. Modern amplifiers
have ignorable output impedance and “Measurement of Amplifier Source Impedance” has lost its importance.
So, let’s skip.
E0� definition is “the Q $0 acting alone i.e. with $ = 0”. In condition cable resistance and output impedance of
amplifier are not ignorable, it’s obvious $0 should be replaced by $0 6 $ by according to the electric equivalent circuit.
The compensated Q, E0 is defined by
( )
E
EGES
EgMESS
E
R
RRQ
RRC
Q
+⋅=
+≡ ω
(21)
In the same way when mechanical loss is added by absorbing materials,
ATASS
T RCQ
ω1≡
(27)
Note) The most familiar Q factor for driver engineers, E4� does not appear in the paper. It’s because of “System
Analysis”.
©2013 Katsuyuki Tsubohara, All rights reserved.
17 / 26
ESMS
ESMS
AS
AS
ASAS
ASE
AS
ASE
AS
AS
AS
AS
AS
ASEAS
AS
ASEAS
AS
DE
AS
AS
AS
DE
AS
TS
C
M
RC
M
lB
R
C
M
lB
R
C
M
R
C
M
lBRRC
M
lBRRC
M
SR
lBR
C
M
SR
lBR
Q
+=
+
⋅=
⋅+
=
+≡
1
1
1
1
22
22
22
22
2
22
2
22
(‡)
See Fig.6. It’s a simplified acoustic equivalent circuit for “system” described in Fig.2.
Note) In this era, Many Spice simulators are available for free and we can simulate any complex system. Fig.6 works as
basic in the approach.
Obviously,
Eg
gg RR
Blep
+=
(23)
( )AS
ASATjAS CjMjRZ
ωωω
1++= (24)
ASABAB Cj
RZω
1+= (25)
AAABAS
g
D
ZZZ
p
U
+=
Thus
©2013 Katsuyuki Tsubohara, All rights reserved.
18 / 26
( )
( ) gjsystemAS
g
AAABASABAS
g
AA
AA
AAABAAASABAS
AAAB
AAAB
AA
g
AAAB
AAABAS
AAAB
AA
AAABAS
g
AAAB
AA
DAAAB
AA
o
pGMj
pZZZZZ
pZ
Z
ZZZZZZ
ZZ
ZZ
Z
p
ZZ
ZZZZZ
Z
ZZZ
p
ZZ
Z
UZZ
Z
U
⋅⋅=
⋅++
=
⋅⋅++
+⋅+
=
⋅
++
⋅+
=
+⋅
+=
⋅+
=
−
−
ωω1
1
1
1
1
( )
( )AAABASABAS
AS
g
oAS
jsystem
ZZZZZ
Mj
p
UMj
G
++=
≡
ω
ω
ω
(26)
Note) To understand intuitively why ����������� has the shape of ω2��GH
IJ , discussing about Fig.2 would be helpful.
Ignoring the leakage (when enclosure has leakage, it’s defect) and assuming ω ≫ ��,
>>
>>
CjMj
RMj
ωω
ω
1
i.e.
≅
≅
01
0
Cj
R
ω
Thus the only impedance to consider in the case ω ≫ �� is jω2�� so
( )AS
gjo Mj
PU S
ωωω
ω → >>
The air in an enclosure is compressing and expanding, not moving. So M�� does not include any M. So
����������� is always normalized. Describing mathematically,
©2013 Katsuyuki Tsubohara, All rights reserved.
19 / 26
( )
( )
( )
1
0101
1
1
1
1
1
lim
lim
lim
lim
=
⋅++=
⋅++=
++=
=
−
−
∞→
−
∞→
−
∞→
∞→
ASAP
AS
AP
AS
AB
AS
AS
AS
AB
AS
AS
AS
APABASABAS
jsystem
jsystem
MM
Mj
ZMj
Z
Mj
Z
Mj
Z
Mj
Z
Mj
ZZZZZ
G
G
ω
ωωωω
ω
ω
ω
ωω
ωω
i.e. ( ) 1≅⇒>> ωωω jsystems G
Note) When leakage is not ignorable
( )ALAB
ALjsystem RR
RG
+=
∞→ ωωlim
This shows leakage causes low SPL in piston range. By understanding radiation phenomena outside piston range,
you’ll find SPL does not decrease by leakage in case wave length is smaller than diameter of diaphragm and
enclosure is large enough. This knowledge is useful to find out the cause of low SPL defect. The most important
thing to analyze defects is accumulated past trouble records of course, but sometimes backing to basic theories
solves the problem faster.
Next subject is to deliver frequency response of driver on infinite-baffle by method of filter design.
Eqs.(10), (12) and (27) are shown here again
The author thinks it’s waste of time academic textbooks always require flipping pages again and again.
( )( )
( ) ASASASAT
ASASjIB
CMjCRj
MCjG
2
2
1 ωωω
ω ++≡
(10)
ASAS
S
S MCT ==2
2 1
ω (12)
ATASST RC
Qω
1≡ (27)
In summary,
©2013 Katsuyuki Tsubohara, All rights reserved.
20 / 26
( )
( )( )
( )( ) 1
1
22
22
2
2
++=
++=
TSS
S
ASASSS
ASAT
ASAS
jIB
QTjTj
Tj
CMjCR
j
MCj
G
ωωω
ωωω
ω
ωω
(28)
Eq.(28) is 2nd-order HPF function widely known by electronics or DSP engineers as
( )( )
( ) ( ) 1122
22
++=
OO
OjIB
TjaTj
TjG
ωωω
ω
(29)
Much knowledge about designing 2nd-order HPF has been obtained by electronics engineers, that’s why we use “equivalent
circuit” to design loudspeakers.
By Eq.(11), the efficiency of the driver where ω ≫ �� is
22
22
2
2
2
222
22
22
2
2
2
MSDE
o
D
MSDE
Do
ASDE
o
o
MSR
lB
c
S
MSR
SlB
c
MSR
lB
c
⋅=
⋅=
⋅=
πρ
πρ
πρ
η
(30)(31)
From Eqs.(12) ,(14), (15), (30), the efficiency calculation by the system parameters is
22
22
2
22
22
2
1
2
1
ASDEAS
AS
ASDE
o
o
MSR
lB
cCc
V
MSR
lB
c
⋅⋅=
⋅⋅=
π
πρ
η
ASASES
ASS
ASDEAS
AS
ES
DASES
ASDEAS
AS
MCQc
Vf
MSRCc
V
Q
SMRf
MSRCc
VlB
3
223
2
223
22
2
2
2
=
⋅=
⋅=
ππ
π
©2013 Katsuyuki Tsubohara, All rights reserved.
21 / 26
ASES
S VQ
f
c⋅⋅=
3
3
24π
(32)
Substituting velocity of sound; 345m/s, where ��� is expressed in litters
ASES
So V
Q
f ⋅⋅×= −3
10106.9η (33)
Where ��� is expressed in NOP
ASES
So V
Q
f⋅⋅×= −
310107.2η
(34)
Note) Enclosure designers MUST check performance requests for driver designers via above.
Scientifically impossible requests do not work at all.
Discussion about linear domain is finished now. Next subject is large signal performances; nonlinearity and power
handling capacity. The description given by Thiele and Small is the first quantitative approach and updated knowledge is
based on it. One of the examples is “Loudspeaker Nonlinearities –Causes and Symptoms” by Wolfgang Klippel (available
on Klippel GmbH website).
First, assume driver has displacement limit: ����
Note) How to determine ���� is not shown in the paper and is still subject to debate (refer to the Klippel’s).
Small suggested to consider:
1) Damage to suspensions
2) FM distortion
3) Harmonic distortion or AM distortion
Using Google, many sites will be found explaining it as the length of Voice Coil out of Gap of motor, but it’s out
of date. Suspensions give more distortion in many cases. Klippel’s approach is performance-based (10%THD or
IMD) and works nicely for subwoofers, but because it uses near-field measurement, IMD in midrange or higher
caused by surround resonance or <���� is not taken account. Further discussions are still required.
Defining system parameter �� as moving air volume at ���� thus
maxxSV DD = (35)
The velocity of the diaphragm by assumption that the driver is linear and works precisely as the equivalent circuit shown in
Fig.6 is
©2013 Katsuyuki Tsubohara, All rights reserved.
22 / 26
gAAASABABAS
AAAB
gABAAABASAAAS
ABAA
g
ABAA
ABAAAS
ABAAAS
g
D
PZZZZZ
ZZ
PZZZZZZ
ZZ
P
ZZ
ZZZ
ZZZ
P
U
⋅++
+=
⋅++
+=
⋅
++
=
+=
1
1
%� is volume velocity of air i.e. velocity of the diaphragm is
D
DD S
Uu =
Displacement of diaphragm �� is calculated by the integral (note that it’s effective value),
( )
( )
( )
AAASABABAS
AAAB
ED
E
AAASABABAS
AAAB
EDEG
Eg
AAASABABAS
AAAB
DEG
g
D
D
jD
ZZZZZ
ZZ
RSj
BlP
ZZZZZ
ZZ
RSj
Bl
RR
Re
ZZZZZ
ZZ
SRR
Ble
Sj
uj
x
+++⋅⋅=
+++⋅⋅
+=
+++⋅
+⋅=
=
1
1
11
1
22
1
2
ω
ω
ω
ω
ω
( )
( ) ( )ω
ω
σ
ωω
ω
jsystemxPxE
jsystemx
E
MSE
AAASABABAS
AAAB
ASAS
ED
E
XkP
XkR
BlCP
ZZZZZ
ZZ
CjCj
RSj
BlP
2
1
2
1
22
1 11
≡
⋅⋅≡
+++⋅⋅⋅⋅=
(36)
Q0R:S��T� = UQ0 ∙ -/83� U$0V equals to the displacement of the diaphragm of a driver mounted on an infinite-baffle by DC
voltage equivalent to Q0.
Note) Peak value equals to effective value in DC case.
©2013 Katsuyuki Tsubohara, All rights reserved.
23 / 26
EMS
MSEG
g
PBlCx
xCRR
eBl
=
=−+
⋅ 01
To express S��T� by the system parameters, E0� and ��� are to be deformed to
ES
MSS
EMSESES Q
M
R
lBlBMRQ
ωω =⇔=22
22
222
DoASMSASoAS ScVCCcV ρρ =⇔=
Then
( )2
1
22
2
1
2
1
22
1222
=
=
⋅=
=
ESDoS
AS
ESS
MSMS
ES
MSS
E
MSPx
QSc
V
Q
CC
Q
M
R
lBC
ρωωωσ
(37)
( )ωjsystemX : normalized system displacement function
xk : system displacement constant of unity or less
( )AAASABABAS
AAAB
ASjsystemx ZZZZZ
ZZ
CjXk
+++⋅= 11
ωω (38)
Note) To understand why Eq.(36) has such shape, let’s check the behavior. Discussions are similar to �����������.
Considering Fig.2 in condition of no leakage and ω ≫ ��,
0lim =∞→
AA
AB
Z
Zω
( )
0
0101
0001
111
11
lim
lim
lim
=⋅++
⋅+⋅=
⋅++
⋅+⋅=
+++⋅=
∞→
∞→
∞→
AS
AP
AB
AS
AS
AS
AB
AS
AS
AP
AB
ASAS
AS
APASABABAS
APAB
AS
jsystemx
Cj
ZZ
MjZ
MjZ
MjZ
Z
Z
MjMj
Cj
ZZZZZ
ZZ
Cj
Xk
ω
ωωω
ωωω
ω
ω
ω
ωω
So W������ is always LPF function. Next, considering ω ≪ �� case then
©2013 Katsuyuki Tsubohara, All rights reserved.
24 / 26
>>
>>
MjCj
RCj
ωω
ω
1
1
i.e.
≅≅
0
0
Mj
R
ω
So the impedances to consider are 1 �ω8��⁄ , 1 �ω8��⁄ and 1 �ω8�T⁄ thus
( )
ASAPAB
APAB
AB
AS
AP
AS
AB
AS
AP
AS
AB
AS
AP
AS
ASASABASAPASABASAPASASAS
ABASAPAS
ASABAPABAPAS
ABAP
AS
APASABABAS
APAB
AS
jsystemx
CCC
CC
CC
CC
CC
CC
CC
CC
ZCjZCjZCjZCjZCjZCj
ZCjZCj
ZZZZZZ
ZZ
Cj
ZZZZZ
ZZ
Cj
Xk
+++=
+⋅+
+=
⋅+⋅+⋅+=
+++⋅=
+++⋅=
→
→
→
→
ωωωωωωωω
ω
ω
ω
ω
ω
ωω
0
0
0
0
lim
1lim
11lim
lim
����������� is to be normalized, ( ) 1lim0
=→ ωω jX thus
ASAPAB
APABx CCC
CCk
+++=
In infinite-baffle cases, 8�� = ∞ and 8�T = ∞. In bass-reflex systems, 8�T = ∞. Both of these have unity W�.
When a bass-reflex system inputted LF signal below port resonant frequency, not only the sound is not
represented but also it causes harmonic distortion and IMD in higher frequency range. If it’s available, such signal
should be cut by an active HPF. Especially in 2-way or single driver systems, such IMD harms harmony and
pianists will never like the sound.
In infinite-baffle case, W� = 1, M�� = ∞ and M�� = 0 thus
( )ASAS
jIB ZCjX
11 ⋅=ωω
Applying the conditions to Eq.(26),
( )AS
ASjIB ZMjG
1⋅= ωω
From Eq.(12)
( ) ( ) ( ) ( ) ( )ωωω ωω jIB
S
jIB
ASAS
jIB GTj
GMCj
X222
11 ==
From Eq.(28)
©2013 Katsuyuki Tsubohara, All rights reserved.
25 / 26
( ) ( ) ( ) 1
122 ++
=TSS
jIBQTjTj
Xωωω
(39)
Note) This is 2nd order LPF function
Then the condition that peak displacement reaches ���� is delivered from Eq.(36)
( ) ( ) maxmax2
1
2 xXP jsystemPxE =⋅ ωσ
Q0? defined by the condition is
( ) ( )
2
max
max
≡
ωσ jsystemxPx
ERXk
xP
(40)
Using system parameters, by substituting Eq.(35) and (37) then
( ) ( )( )( )
( )
( )2
max
2
22
2
max
22
2
2
2
max
22max
12
2
1
1
2
1
ω
ω
ω
πρ
ρπ
σ
jsystemxAS
DESSo
jsystemxAS
ESDoS
D
D
jsystemx
Px
ER
XkV
VQfc
XkV
QScf
S
V
Xkx
P
⋅=
⋅⋅⋅=
⋅⋅= −
(41)
Q�? defined by Eq.(32) and (41) is
( )
( )2
max
2
243
2
max
2
22
3
32
4
4
ω
ω
ρπ
πρπη
jsystemx
DSo
jsystemxAS
DESSo
ES
ASS
ERo
AR
Xk
Vf
c
XkV
VQfc
Qc
Vf
P
P
⋅=
⋅⋅=
≡
In infinite-baffle cases, W� = 1 thus
( )( )
2
max
22
ω
πρjIBAS
DESSoIBER
XV
VQfcP ⋅=
(44)
( )( )
2
max
2434
ω
ρπ
jIB
DSoIBAR
X
Vf
cP ⋅=
(43)
As described above, modern measurement tools deliver parameters by displacement. So “APPENDIX” is also skipped here.
So that’s all.
©2013 Katsuyuki Tsubohara, All rights reserved.
26 / 26
Further readings
� Articles on J. Audio Eng. Soc.:
R.H.Small
“Closed-Box Loudspeaker Systems Part I: Analysis”
“Closed-Box Loudspeaker Systems Part II: Synthesis”
“Vented-Box Loudspeaker Systems Part I: Small-Signal Analysis”
“Vented-Box Loudspeaker Systems Part II: Large-Signal Analysis”
A.N.Thiele
“Loudspeakers in Vented Boxes: Part I”
“Loudspeakers in Vented Boxes: Part II”
� L.L.Beranek “Acoustics” (McGraw-Hill, New York, 1954)
� W. Klippel “Loudspeaker Nonlinearities –Causes and Symptoms” on Klippel GmbH website
Thank you.
K.Tsubohara