Application Report
SLAA857 – August 2018

SmartPA Speaker Protection Algorithm
Susan Xue, Supriyo Palit, Peter Wei, Andrew Du .............................................. China HUAWEI Sales Team
ABSTRACT
SmartPA has seen increased usage in personal electronic in recent years. SmartPA is used often in smart
phones to make full use of the relatively smaller speaker and get better sound quality with maximum
volume.
SmartPA system consists of• Power amplifier with current and voltage sensing:• SmartPA speaker protection algorithm which protects the speaker from excursion and temperaturedamage (This is based on the mechanical and electrical characteristics of the speaker)
This article discusses the speaker modeling and protection based on the TI SmartPA algorithm.

1
2
3
4
5

Contents
Speaker Basics .............................................................................................................. 2
Speaker Impedance and Excursion Model ............................................................................... 4
Speaker Model Measurement .............................................................................................. 6
SmartPA Temperature and Excursion Protection Algorithm ........................................................... 9
References .................................................................................................................. 13
List of Figures

1

Energy Transfer in Speaker ................................................................................................ 2

2

Typical Speaker Structure .................................................................................................. 2

3

Electro-Motive Model of Speaker .......................................................................................... 3

4

Speaker Characterization by PPC3 and Learning Board 2 ............................................................ 6

5

Impedance and Excursion Simulation Result with Matlab Based on Speaker Model .............................. 7

6

Impedance and Excursion Characterization Result with PPC3 and LB2............................................. 8

7

Smart amp Algorithm ........................................................................................................ 98

Excursion Protection System in Smart Amp Algorithm ................................................................. 99

Excursion Approximation in Low Frequency Range ................................................................... 10

10

Speaker Model Update Algorithm ........................................................................................ 11

11

Thermal Protection in Smart Amp Algorithm ........................................................................... 11

12

Typical Thermal Model of Speaker ...................................................................................... 12

13

Thermal Characterization Method in PPc3 and Learning Board 2................................................... 12
List of Tables

1

Main Parameters in Speaker Electro-Motive Model ..................................................................... 3

2

Thiele/Small (TS) Parameter ............................................................................................... 5

3Main Physical Parameters of Iphone 7 Speaker ........................................................................ 7

4Main Thermal Parameters of Iphone 7 Speaker ....................................................................... 12

SLAA857 – August 2018
Submit Documentation Feedback

SmartPA Speaker Protection Algorithm
Copyright © 2018, Texas Instruments Incorporated

1Speaker Basics

www.ti.com

Trademarks
All trademarks are the property of their respective owners.

1

Speaker Basics

1.1

Speaker Structure
In a speaker, similar to motor driver, current flows through windings and magnetic field is created.
Windings (voice coil) move in magnetic field because of the magnetic force. The cone membrane (all themoving parts including diaphragm, frame, suspension, and so forth) is attached firmly to the windings andwill move the same way, leading to sound. This is the basic principle of a speaker. It is a type ofelectroacoustic transducer.

Electrical energy

Voice
Coil

Mechanical energy

Cone
MembraneSound energy

Figure 1. Energy Transfer in Speaker

F BlI
VBEMF Blv

(1)
(2)

Figure 2. Typical Speaker Structure2

SmartPA Speaker Protection Algorithm

SLAA857 – August 2018
Submit Documentation Feedback
Copyright © 2018, Texas Instruments Incorporated

Speaker Basics

www.ti.com

1.2

Classical Electro-Motive Modeliu+Heat :
P = i 2 x Re±

Cms

Electro Motor

Re(T)Bl x v++

F = Bl x i

±

±

x/Cms

Rms

Mms

~SPL
(accel.)V

Figure 3. Electro-Motive Model of Speaker
Table 1. Main Parameters in Speaker Electro-Motive Model
Electro-Motive Parameter

Description

Unit

Rms

Mechanical damping factor

N × s/mMms

Mechanical massg

Cms

Mechanical compliancemm / NBl

Force factor

T×m

RE

DC resistance of the speakerΩuAmplifier voltage

Vi

Voice coil current

Av

Velocity of membranem/sFMechanical force in magnetic field

N

X

Membrane excursionm

Table 1 describes how the speaker converts electrical energy to mechanical energy. Speakers act as a
type of transformer. The difference from regular electro-magnetic transformer is that the secondary side ismechanical dimension instead of electrical dimension. The primary side of the system can be taken as anelectrical circuit, where excitation signal u is audio input. In this closed circuit, part of voltage drop is
across the DC resistance, while the other part of voltage drop results from back electromotive force(induced from windings cutting the magnetic field, for example, BackEMF).

VBEMF

Blv

u

1u R e T(3)

Similar to a transformer, the power delivered to secondary is:P1

VBEMF u i Blv u i(4)Corresponding impedance is:

R1

Blv
i

(5)

The secondary side is a mechanical system. The power is proportional to ampere force and velocity ofmembrane.

P2

Fuv

Bli u v

P1

(6)

Take v as voltage and F as current, the equivalent mechanical resistance is:

R2

v
F

v
Bl u i(7)

The equivalent turns ratio is:TR

Bl u v
v

Bl(8)

According to principle of transformer, ratio of R1 and R2 is:

SLAA857 – August 2018
Submit Documentation Feedback

SmartPA Speaker Protection Algorithm
Copyright © 2018, Texas Instruments Incorporated3

Speaker Impedance and Excursion Model

R1
R2

TR 2

Bl

www.ti.com

2(9)

For a transformer, turns ratio is the most important parameter. It describes how the voltage and power istransferred from one side to the other side. Get back to the speaker “transformer system”, the “turns ratio”
Bl is the most important parameter to determine the sensitivity from electrical system to mechanicalsystem. In other words, Bl decides how fast Mms will be moving. So obtaining Bl is a critical step for thecharacterization of a speaker.
Now let’s try to focus on the mechanical system alone. Take the moving part (voice coil and membrane)
as an object, and the mass of this moving part is Mms.
Three kinds of forces are imposed on Mms:
1. Elastic force FC (measured by compliance Cms, the higher Cms, the smaller elastic force)
2. Viscous resistance force FR measured by Rms (a higher Rms indicates larger resistance and damping)
3. Outside force from ampere force F (Bl × i according to electro-magnetic principle)

x
C ms

(10)

FR

R ms v(11)F

Bl u iFC(12)

According to Newton’s Second Law,

M ms a

2

F

FC

FR

Bl u ix
C ms

R ms v(13)

Speaker Impedance and Excursion Model
For certain input level and frequency, impedance and excursion (excursion doesn’t mean absolute
excursion, it describes transfer function from input voltage to excursion, with a unit of mm/V) of speaker
are dependent on first five parameters in Table 1. See Equation 14 and Equation 15 for details.

Bl
Bl
sM ms
Z exc sR ms

Bl
sR E §
¨ sM
ms
¨
©

s

2

Z BMEF s

2

M ms
1
sC ms

s2

s

R ms
M ms

1
§
¨R
¨ ms
©

Bl ·
¸
RE ¸
¹
2

·
1 ¸
sC ¸
¹

1
M msC msBl
M msR E §
¨
¨ 2
¨s
¨
¨¨
©

(14)

1
Bl
Rms
RE

s
Mms

2

·
¸
¸
1
¸
M msC ms ¸
¸¸
¹
(15)

Above the electrical and mechanical parameters are basic physical parameters in a speaker. However,they are not easy to measure once speaker is manufactured. Also, they still can’t describe clearly enoughto help us to understand how the system works. T/S parameters are introduced to better describe theresonance system as a whole.
Take speaker as an oscillation system and equations above can be rewritten as below:
4

SmartPA Speaker Protection Algorithm

SLAA857 – August 2018
Submit Documentation Feedback
Copyright © 2018, Texas Instruments Incorporated

Speaker Impedance and Excursion Model

www.ti.com

Bl

Z BMEF s2s
Zs

M ms § 2·s
Z s2 ¸
¨s
¨
¸
Q ms
©
¹
Bl
s
M msR E § 2·
Zss
Z s2 ¸
¨s
¨
¸
Q ms
©
¹

Z exc s

(16)

(17)

This system oscillates at certain frequency, resonance frequency

fs

Zs

1

2S

2S M msC ms(18)

Impedance transfer function is also limited by mechanical quality factor:

Q ms

1
R ms

M ms
C ms

(19)

Equation 19 shows how each element in mechanical system has effect on the mechanical quality factor.
Higher Qms causes the bandwidth to narrow, which causes the system to oscillates with less damping.
The lower Qms leads to an opposite effect.
While impedance is influenced mainly by mechanical part, excursion value is measured by both electricaland mechanical system. Total quality factor Qts is introduced here.

M ms

1

Q ts

Bl

R ms

2

C ms

RE

(20)

Qts reveals quality factor of the combination of electrical and mechanical system. Since they are in series,and electrical quality factor is introduced as Qes,

Q ts

Q esQ ms
Q es

Q ms

(21)

We have Qes as below:

Q es

RE
Bl

M ms
2

C ms

(22)

Table 2. Thiele/Small (TS) Parameter
Driver model parameter (Thiele/Small)

Description

Unit

fs

Resonance frequency of electro-motive
systemHz

Qts

Total quality factor/

Qms

Quality factor of mechanical system/

Qes

Quality factor of electrical system/

SLAA857 – August 2018
Submit Documentation Feedback

SmartPA Speaker Protection Algorithm
Copyright © 2018, Texas Instruments Incorporated5Speaker Model Measurement3

www.ti.com

Speaker Model Measurement
Based on speaker model specified above, we use the characterization tool in PPC3 (PurePath Console 3)
to measure each parameter. (Details about PPC3 is available in Smart Amp Quick Start Guide provided in
the Reference section.)

Figure 4. Speaker Characterization by PPC3 and Learning Board 2
First, Sweep speaker with a certain range of frequencies (usually 3KHz). Get the impedance curve. Fitting
the impedance curve into model as shown in Equation 16:

Z s

RE

Z BEMF s

a

k
s2

s
bus

c

(23)

As the DC resistance, RE can be obtained directly:

RE

a

(24)

According to the relation between T/S and physical parameters,

Bl

2

k
M ms
Zs

(25)

1

c

M msC ms

Zs

R ms

Q ms

M ms

(26)

b
(27)

There are three formulas, but 4 parameters Mms, Cms, Rms and Bl are unknown. Bl should be input
before all the other 3 parameters are derived. It can be either from speaker vendor or from laser
measurement. Once Bl is known, Mms, Cms and Rms can be derived from Equation 25, Equation 26, andEquation 27.
6

SmartPA Speaker Protection Algorithm

SLAA857 – August 2018
Submit Documentation Feedback
Copyright © 2018, Texas Instruments Incorporated

Speaker Model Measurement

www.ti.com

Bl
M ms

C ms

2

k

(28)

1
c u M mskc u BlBl

R ms

b u R ms

bu

2

(29)

2

k

(30)

Then according to Equation 17, excursion curve can be fit based on the five physical parameters of
speaker.
For Iphone 7 speaker, below in Table 3 is the characterization result of different physical parameters:
Table 3. Main Physical Parameters of Iphone 7 Speaker
Driver model parameter(physical)

Description

Unit

Value

Rms

Mechanical damping factor

N × s/m

0.305

Mms

Mechanical mass

g

0.116

Cms

Mechanical compliancemm / N

0.21

Bl

Force factor

T×m

0.841

RE

DC resistance of the speaker

Ω

7.35

And according to Equation 14 and Equation 15, impedance and excursion simulated with matlab. Figures
as below Figure 5:

Figure 5. Impedance and Excursion Simulation Result with Matlab Based on Speaker Model
TS parameters can be derived based on the five physical parameters:

fs

1
2S M ms Cms

1
2S 0.116 u 0.21 u 101020 Hz6

SLAA857 – August 2018
Submit Documentation Feedback(31)

SmartPA Speaker Protection Algorithm
Copyright © 2018, Texas Instruments Incorporated

7

Speaker Model Measurement

Q ms
Q ts

Q ts
Q es

1
R ms

www.ti.com

M ms
C ms

1
0.116
0.305 0.21

(32)

1
0.84120.305
7.35
Q msQ es

Q ms Q es
Q msQ ts
Q ms

Q ts

2.43

0.116
0.21

1.85
(33)

(34)

2.43 u 1.85
2.43 1.85

7.75
(35)

The calculation of TS parameters and simulation curves of impedance and excursion match withcharacterization results well.

Figure 6. Impedance and Excursion Characterization Result with PPC3 and LB2

8SmartPA Speaker Protection Algorithm

SLAA857 – August 2018
Submit Documentation Feedback
Copyright © 2018, Texas Instruments Incorporated

SmartPA Temperature and Excursion Protection Algorithm

www.ti.com

4

SmartPA Temperature and Excursion Protection Algorithm
A block diagram of the SmartPA algorithm is shown below:

Input
OutputPower/Excursion
Protection+

SmartPAAmplifier

ZEXC(S)
P(T)
Pilot
Tone

Xmax

ThermalController

I/V

ThermalEstimator

Tmax
SpeakerModel
SmartPA Algorithm

Figure 7. Smart amp Algorithm
The speaker temperature and the speaker model is derived from I/V sense. This is used to adaptively
modify the input signal in the Power/Excursion protection block so that the output provided to the speaker
is within the excursion and thermal limit of the speaker. The excursion and thermal protection algorithm is
discussed in detail in the following sections.

4.1

Excursion Protection Algorithm
For excursion protection, excursion is first calculated from the input signal based on the speaker model.
Then the excursion will be compared with the maximum excursion before deciding if protection kicks in. If
the excursion exceeds limit Xmax, input will be attenuated otherwise the input signal will pass throughunchanged. Since over excursion could possibly damage the system in a short time, a time delay isinserted in feedforward signal chain, to make sure excursion estimation and comparison are finished
before signal is fed to protection.
Feedforward (input attenuation)
Delay
OutputInput
ZExc (s)excursion

Excursion
limiter

Xmax

Figure 8. Excursion Protection System in Smart Amp Algorithm
Equation 15 is the excursion model when input voltage is 1 V. So total excursion Exc(s) should be,

Exc su s u Z Exc s(36)

If switched to time domain by doing inverse Laplace transform on ZExc(s), final excursion is convolution of
input signal and L–1(ZExc(s))

SLAA857 – August 2018
Submit Documentation Feedback

SmartPA Speaker Protection Algorithm
Copyright © 2018, Texas Instruments Incorporated

9

SmartPA Temperature and Excursion Protection Algorithm

www.ti.com

x t u L 1 Z Exc s

Exc t

(37)

Figure 9. Excursion Approximation in Low Frequency Range
This is how excursion is calculated and protected.
As the speaker keeps working, speaker model can change, which will have effect on the estimation
accuracy. So the speaker model should be updated dynamically to ensure proper protection. Figure 9shows how speaker model is updated. This update is finished in feedback module.
The speaker model update is mainly based on impedance model. If speaker output voltage and current isV(s) and I(s) for frequency domain, respectively, then:I s

VBEMF sZ BEMF s

VBEMF sV s

(38)

I s RE(39)

And VBEMF(s) is the backemf.
Equation 38 is converted into time domain by inverse Laplace transform as below,

VBEMF tI t u L 1 Z BEMF s(40)

As is shown in Figure 9, an adaptive filter is used to update the speaker impedance model. Output voltage
and current of the speaker are detected as V(t) and I(t), respectively using the sensing circuit of theSmartPA amplifier. Back-EMF is estimated as VBEMF_est(t) based on original impedance model according toEquation 40, and VBEMF_est(t) is compared to the actual backemf VBEMF_est(t) (=V(t) — I(t)RE) from detection
and the error e(t) is obtained:e t

VBEMF _est t

VBEMF t(41)

The speaker model parameters are modified dynamically till e(t) is small enough to be close to zero, whichmeans the model is accurate enough to forecast the impedance and excursion.

10

SmartPA Speaker Protection Algorithm

SLAA857 – August 2018
Submit Documentation Feedback
Copyright © 2018, Texas Instruments Incorporated

SmartPA Temperature and Excursion Protection Algorithm

www.ti.com

Speaker Model
Parameters
I(t)VBEMF_est(t)VBEMF(t)L-1(ZBEMF(s))= V (t) ± I (t) REe(t)Figure 10. Speaker Model Update Algorithm
TS parameters are calculated using updated physical parameters. For actual backemf we need tomeasure RE. At very low frequencies, the backemf is almost zero so Equation 39 becomes,0

V s

I s RE(42)

hence,

v t
RE

I t

(43)

A pilot tone at low frequency (for example 60Hz.) is used to estimate RE × RE is also used for thermal
protection which is discussed in the next section.

4.2

Thermal Protection Algorithm
Thermal protection algorithm is divided into three main parts. Thermal estimator, thermal controller andpower protection. Temperature T is first estimated from I/V feedback using pilot tone based on the relation
between DC resistance and temperature, as is shown in Equation 43 and Equation 44. Then T iscompared with temperature limit Tmax to decide if it has exceeded the limit.
Power Protection

I/V
T
Thermal Controller

Thermal
Model

Tmax

Thermal Estimator

D

Figure 11. Thermal Protection in Smart Amp Algorithm

R T

R e TAD T

TA R T(44)

R R e TA
T

TA

R
D

(45)

The second important part is thermal controller. It controls the power protection block based on
temperature. This is done using a thermal model of the speaker. For the speaker, thermal model is shownin Figure 11. Voice coil and magnet pole are both included. Thermal resistance and thermal capacitanceare introduced to model the transient temperature behavior and it can predict how temperature changes
as time goes by under different input power.
SLAA857 – August 2018
Submit Documentation Feedback

SmartPA Speaker Protection Algorithm
Copyright © 2018, Texas Instruments Incorporated

11

SmartPA Temperature and Excursion Protection Algorithm

www.ti.com
TA

Pin

Rtv

Tv

Tm
Rtm

Ctv

Ctm

voice coilmagnetFigure 12. Typical Thermal Model of Speaker
Using Iphone 7 speaker as an example, thermal resistance and capacitance (shown in Table 4) can becalculated from the characterization result shown in below curve in Figure 12 by learning board 2.

Figure 13. Thermal Characterization Method in PPc3 and Learning Board 2
Table 4. Main Thermal Parameters of Iphone 7 Speaker
Thermal parameter

Description

Unit

Value

Rtv

Thermal resistance of voice
coil

K/W

76.9

Ctv

Thermal capacitance of voice
coil

J/K

0.0549Rtm

Thermal resistance of magnet
pole

K/W

495

Ctm

Thermal capacitance ofmagnet pole

J/K

0.832

From Figure 11, Power flows to the voice coil and magnet system, each with a time constant, denoting thetemperature rising speed respectively. This time constant can be calculated from thermal models shown in
Table 4.
Time constant for voice coil is:Wv

R tv C tv

76.9 u 0.0549

4.22 s(46)

Time constant for magnet part is:Wm

R tmC tm

495 u 0.832

411.8 s(47)

This means, temperature of voice coil can rise much faster than the magnet as the power increases. Also
for this case, temperature of voice coil is our main concern. Algorithm take s advantage of the time
constant and decides how to allocate power properly to make system work nearly the thermal limit while
ensure safety.
In power protection module, signals are attenuated according to the power output from the thermal
controller. This keeps the voice coil temperature within the thermal limit.
12

SmartPA Speaker Protection Algorithm

SLAA857 – August 2018
Submit Documentation Feedback
Copyright © 2018, Texas Instruments Incorporated

References

www.ti.com

5

References•••

Thiele/Small Parameters
Smart Amp Quick Start Guide
W. Marshall Leach Jr, Introduction to electro-acoustics and audio amplifier design, published by
Kendall Hunt

SLAA857 – August 2018
Submit Documentation Feedback

SmartPA Speaker Protection Algorithm
Copyright © 2018, Texas Instruments Incorporated

13

IMPORTANT NOTICE FOR TI DESIGN INFORMATION AND RESOURCES
Texas Instruments Incorporated (‘TI”) technical, application or other design advice, services or information, including, but not limited to,reference designs and materials relating to evaluation modules, (collectively, “TI Resources”) are intended to assist designers who aredeveloping applications that incorporate TI products; by downloading, accessing or using any particular TI Resource in any way, you(individually or, if you are acting on behalf of a company, your company) agree to use it solely for this purpose and subject to the terms ofthis Notice.
TI’s provision of TI Resources does not expand or otherwise alter TI’s applicable published warranties or warranty disclaimers for TI
products, and no additional obligations or liabilities arise from TI providing such TI Resources. TI reserves the right to make corrections,enhancements, improvements and other changes to its TI Resources.
You understand and agree that you remain responsible for using your independent analysis, evaluation and judgment in designing yourapplications and that you have full and exclusive responsibility to assure the safety of your applications and compliance of your applications(and of all TI products used in or for your applications) with all applicable regulations, laws and other applicable requirements. You
represent that, with respect to your applications, you have all the necessary expertise to create and implement safeguards that (1)anticipate dangerous consequences of failures, (2) monitor failures and their consequences, and (3) lessen the likelihood of failures thatmight cause harm and take appropriate actions. You agree that prior to using or distributing any applications that include TI products, youwill thoroughly test such applications and the functionality of such TI products as used in such applications. TI has not conducted any
testing other than that specifically described in the published documentation for a particular TI Resource.
You are authorized to use, copy and modify any individual TI Resource only in connection with the development of applications that include
the TI product(s) identified in such TI Resource. NO OTHER LICENSE, EXPRESS OR IMPLIED, BY ESTOPPEL OR OTHERWISE TOANY OTHER TI INTELLECTUAL PROPERTY RIGHT, AND NO LICENSE TO ANY TECHNOLOGY OR INTELLECTUAL PROPERTYRIGHT OF TI OR ANY THIRD PARTY IS GRANTED HEREIN, including but not limited to any patent right, copyright, mask work right, orother intellectual property right relating to any combination, machine, or process in which TI products or services are used. Information
regarding or referencing third-party products or services does not constitute a license to use such products or services, or a warranty or
endorsement thereof. Use of TI Resources may require a license from a third party under the patents or other intellectual property of thethird party, or a license from TI under the patents or other intellectual property of TI.
TI RESOURCES ARE PROVIDED “AS IS” AND WITH ALL FAULTS. TI DISCLAIMS ALL OTHER WARRANTIES OR
REPRESENTATIONS, EXPRESS OR IMPLIED, REGARDING TI RESOURCES OR USE THEREOF, INCLUDING BUT NOT LIMITED TO
ACCURACY OR COMPLETENESS, TITLE, ANY EPIDEMIC FAILURE WARRANTY AND ANY IMPLIED WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, AND NON-INFRINGEMENT OF ANY THIRD PARTY INTELLECTUAL
PROPERTY RIGHTS.
TI SHALL NOT BE LIABLE FOR AND SHALL NOT DEFEND OR INDEMNIFY YOU AGAINST ANY CLAIM, INCLUDING BUT NOT
LIMITED TO ANY INFRINGEMENT CLAIM THAT RELATES TO OR IS BASED ON ANY COMBINATION OF PRODUCTS EVEN IF
DESCRIBED IN TI RESOURCES OR OTHERWISE. IN NO EVENT SHALL TI BE LIABLE FOR ANY ACTUAL, DIRECT, SPECIAL,COLLATERAL, INDIRECT, PUNITIVE, INCIDENTAL, CONSEQUENTIAL OR EXEMPLARY DAMAGES IN CONNECTION WITH OR
ARISING OUT OF TI RESOURCES OR USE THEREOF, AND REGARDLESS OF WHETHER TI HAS BEEN ADVISED OF THE
POSSIBILITY OF SUCH DAMAGES.
You agree to fully indemnify TI and its representatives against any damages, costs, losses, and/or liabilities arising out of your noncompliance with the terms and provisions of this Notice.
This Notice applies to TI Resources. Additional terms apply to the use and purchase of certain types of materials, TI products and services.
These include; without limitation, TI’s standard terms for semiconductor products http://www.ti.com/sc/docs/stdterms.htm), evaluation
modules, and samples (http://www.ti.com/sc/docs/sampterms.htm).

Mailing Address: Texas Instruments, Post Office Box 655303, Dallas, Texas 75265
Copyright © 2018, Texas Instruments Incorporated