The Design and Simulation of a Loudspeaker...

The Design and Simulation of a Loudspeaker System

ECE 4445 – Prof. Leach.

By:

Joshua Perkel Jason Clark

April 21, 2004

2

I. Introduction.

The purpose of this report is to describe the design and analysis of a Paradigm B145WFACTV020 6.5 inch driver from a Paradigm Reference Active 20 2-way loudspeaker system. The following tasks are described in subsequent sections:

Analysis of the off-box and on-box responses in order to obtain the driver parameters.

Simulation in an infinite baffle. Design and simulation of an optimum closed-box. Design and simulation of an optimum vented-box. Design of 1st, 2nd, and 3rd order crossover filters. Design of a Zobel (matching) network.

II. Driver Parameters.

Prior to any design work, it is necessary to determine the following driver parameters: RE, fs, QMS, QES, QTS, VAS, and lossy inductor parameters n and Le. The value of RE is determined by direct measurement of the dc voice-coil resistance using a multimeter. This was measured to be 3.6Ω. 2.1 Resonance Frequency (fs, fc).

The resonance frequency of the driver in both free air and the test enclosure is determined by observing the frequency at which the peak in the magnitude of the voice coil impedance occurs. This simple technique yields the following results:

HzfHzf

c

s

030.84414.64

==

2.2 Mechanical Quality Factor (QMS, QMCT).

The mechanical quality factors require additional calculations involving the impedance measurements of this driver. The first step is to determine the value of RE + RES. This sum is the real part of the driver impedance calculated at the resonance frequency. This will produce a value for both the on and off box measurements.

Table 2.2.1: Voice-coil impedance parameters.

Parameter Off-Box On-Box Zvc Ω°∠ 3181.28801.32 Ω°−∠ 2068.84076.30

( )VCESE ZRR Re=+ 30.0962Ω 32.8532Ω

( ) EESE RRRR +=1 10.87Ω 10.409Ω

3

Next, using the values obtained for R1 as shown above, I identify the frequencies to either side of the resonance frequency that exhibit this same impedance magnitude. These two frequencies will be called f1 and f2. Furthermore, as a check to verify the validity of these two frequencies the geometric mean of the two can be taken and compared to the resonance frequency, they should be the same or within 5% of each other.

Table 2.2.2: Frequency data from measurements.

Parameter Off-Box On-Box 1f 49.35Hz 66.875Hz

2f 84.025Hz 105.55Hz

21 ff 64.394Hz 84.016Hz From the above table, it is apparent that the values for f1 and f2 are valid in both cases since the geometric mean is consistent with the driver’s resonance frequency. I am now able to use Eqn.1 to determine QMS and QMCT.

2822.660.30962.30

875.6655.10503.84

6106.560.38532.32

35.49025.844.64

12

12

=−

⇒+

−=

=−

⇒+

−=

E

ESCE

CC

CMCT

E

ESEsMS

RRR

fffQ

RRR

fffQ

Eqn.1

2.3 Electrical Quality Factor (QES, QECT).

The electrical quality factors can be calculated using the value obtained for QMS in the previous section. The calculation is simple and is shown in Eqn.2.

8535.02822.660.30962.30

60.3

6905.06106.560.38532.32

60.3

=−

⇒=

=−

⇒=

MCTECT

EECT

MSES

EES

QRRQ

QRRQ

Eqn.2

It should be noted that the value for QES is Eqn.2 will be too high for designing the vented enclosure. Therefore, during that portion of the report we will consider a corrected value of QES that is obtained by increasing the Bl product described in section 3.4. 2.4 Total Quality Factor (QTS).

The total quality factor can now be calculated using Eqn.3.

4

6148.06106.58532.3260.3

=⇒+

= MSESE

ETS Q

RRRQ Eqn.3

2.5 Volume Compliance of the Driver Suspension (VAS).

The volume of the test enclosure was determined to be 0.4970 ft3 (or 3210407.1 m−× ). Equation 4 can then be used to determine VAS.

32977.016965.08535.0

4.6403.844970.01 ft

QQ

ffVV

ES

ECT

s

CTTAS =

−⇒

−= Eqn.4

2.6 Lossy Voice Coil Inductance (Le, n).

For this driver, I used the method described in the course textbook that uses the voice coil impedance at two high frequencies to calculate the values of Le and n. The table below shows the values used in the upcoming calculation.

Table 2.6.1: Lossy inductance data.

Parameter f1 = 10.098kHz f2 = 20kHz VCZ Ω°∠ 6071.567521.17 Ω°∠ 9412.600207.29

EVC RZ − Ω°∠= 3975.670546.161Z Ω°∠= 5233.674531.272Z

The calculation of n is shown in Eqn.5.

( ) ( )[ ]

( )

75116.01009820000ln

3975.675233.673602

1009820000ln

0546.164531.27ln

ln

argarglnln

22

2

2

1

2

211

2

1

2

=

+

−

+

×

⇒

+

++

×

=

π

ππ

πωω

πωω ZZ

ZZ

n

Eqn.5

Finally, the value of Le can be determined using Eqn.6.

( ) ( )H

ZZL nne

375116.075116.0

21

21 10011.4200002100982

4531.270546.16 −×=×××

×⇒

×=

ππωω Eqn.6

5

To verify the validity of these calculations of the lossy inductance parameters, Figs.2.6.1 and 2.6.2 show the measured off-box impedance (without RE) and that predicted using Eqn.7.

( ) ( ) 75116.0310011.4 ωω jjLZ nee

−×⇒= Eqn.7

Figure 2.6.1: Magnitude of Ze both measured (blue) and calculated (red).

Figure 2.6.2: Phase of Ze for both measured (blue) and calculated (red).

6

From these plots, the estimated values of Le and n seem to reasonably predict the magnitude and phase of Ze.

7

III. Simulation in an Infinite Baffle.

This section deals with the simulation of our driver in an infinite baffle. This requires some additional calculations before implementing a PSpice simulation. This simulation will lead to plots of the SPL, voice-coil impedance, and diaphragm displacement. 3.1 Additional Parameter Calculations.

The following parameters are calculated in the next few sections: CMS, MMS, MMD, Bl, RMS, MA1, RA1, RA2, and CA1. 3.1.1 Mechanical Compliance of the Diaphragm Suspension (CMS).

The parameter CMS is calculated directly using Eqn.8 and the volume compliance of the driver suspension. Please note that VAS has been converted to m3 for this calculation.

( )( ) Nm

ScVC

min

mftinftV

D

ASMS

AS

4222

3

220

333

33

3

333

1041.3065.034518.1

10429.8

10429.81

0254.01

122977.0

−−

−

×=×××

×⇒=

×=

=

πρ

Eqn.8

3.1.2 Mechanical Mass of Diaphragm and Air Load (MMS).

The value of MMS can be calculated from the equation for resonance frequency as shown in Eqn.9.

( )

( )kgM

CfM

CMf

MS

MSSMS

MSMSs

2422

22

10793.11041.3394.642

12

12

1

−−

×=×××

=⇒

=⇒=

π

ππ Eqn.9

3.1.3 Mechanical Mass of Diaphragm (MMD).

The mechanical mass of the driver diaphragm can be calculated using Eqn.10.

( )kg

M

Sa

MM

MD

DMSMD

2

222

2

220

106202.1

065.0065.03

18.18210793.1

382

−

−

×=

×××

−×=⇒

−=

ππ

πρ

Eqn.10

8

3.1.4 Bl Product

Equation 11 outlines the calculation of the Bl product using QES, RE, MMS, and CMS.

mTBlCM

QRBl

MS

MS

ES

E ⋅=××

=⇒= −

−

1486.61041.310793.1

6905.060.3

4

2

Eqn.11

3.1.5 Mechanical Resistance of the Diaphragm Suspension (RMS).

This parameter can be calculated directly using Eqn.12.

( ) ( )

msNR

QRQBlR MSMSE

ESMS ⋅=

××

=⇒= 292.161057.560.3

6905.01486.6 22

Eqn.12

3.1.6 Acoustic Mass for Piston Air Load (MA1).

The value of MA1 is also calculated directly using Eqn.13.

42120

1 905.4065.03

18.1838

mkgM

aM AA =

××

=⇒=ππ

ρ Eqn.13

3.1.7 Acoustic Resistors for Piston Air Load (RA1, RA2).

As in the previous two sections, the following calculations are simple and straightforward as shown in Eqn.14.

52312230

1

52220

2

9.441968.30670065.09

34518.11289

128

8.30670065.034518.1

msNRR

acR

msNR

acR

AAA

AA

⋅=−××

××=⇒−=

⋅=××

=⇒=

ππρ

ππρ

Eqn.14

3.1.8 Acoustic Compliance for Piston Air Load (CA1).

Equation 15 shows the calculation of the final parameter.

kgsmC

caC AA

248

2

3

120

3

1 101615.134518.1065.094.594.5

⋅×=××

=⇒= −

ρ Eqn.15

This concludes the calculations required by the PSpice circuit simulation.

9

3.2 PSpice Simulation.

The PSpice netlist used in this project was modeled on the netlist in Table 6.1 of the course textbook1. The parameters were modified for this particular driver and the resulting netlist is shown below: ***** Infinite_Baffle.cir ***** VEG 1 0 AC 1V RE 1 2 3.60 GZE 2 3 LAPLACE V(2,3) = 1/(0.004011*PWR(S,0.75116)) HBLUB 3 4 VD2 6.1486 VD1 4 0 AC 0V LMMD 5 6 1.6202e-2 RMS 6 7 1.292 CMS 7 8 3.41e-4 HBLIC 5 0 VD1 6.1486 ESDPD 8 9 10 0 0.01327 VD2 9 0 AC 0V LMA1 10 12 9.810 RA1 10 11 88393.8 RA2 11 12 61341.6 CA1 10 11 5.8073e-9 VD3 12 0 AC 0V FSDUD 0 10 VD2 0.01327

.AC DEC 50 10 20k .probe .end 3.3 Simulation Results.

Figures 3.3.1 and 3.3.2 show the magnitude and phase of the voice coil impedance as measured in free air in the lab. This data was used to calculate the parameters specified above in the PSpice netlist. The resulting plots of the magnitude and phase of the voice coil impedance are shown in Figs. 3.3.3 and 3.3.4.

1 Leach, W. Marshall, Jr. Introduction to Electroacoustics and Audio Amplifier Design, 3rd Edition,

Kendall/Hunt Publishing, 2003. pg109.

10

Figure 3.3.1: Magnitude of ZVC versus frequency from off-box measurements.

Figure 3.3.2: Phase of ZVC versus frequency from off-box measurements.

11

Figure 3.3.3: Magnitude of ZVC as calculated by PSpice.

Figure 3.3.4: Phase of ZVC as calculated by PSpice.

Figure 3.3.5 shows the on-axis SPL with 1VRMS input. This figure clearly illustrates the band-pass behavior of this driver. Finally, Fig. 3.3.6 shows the peak diaphragm displacement.

12

Figure 3.3.5: SPL at 1 meter with 1VRMS input.

Figure 3.3.6: Peak diaphragm displacement versus frequency.

13

IV. Simulation in an Optimum Closed Box.

This section describes the design and simulation of an optimum closed box for our particular driver. The definition of optimum in this case is the flattest frequency response possible. This definition leads to using a Butterworth alignment where the

total quality factor (QTC) is equal to2

1 . As before, there are several parameters that

must be calculated before the simulation can be created. 4.1 Basic Design Parameters.

To begin the design, we must calculate the basic parameters of the closed box system. The required parameters are: QEC, VAB, fc, and α. In the initial design we will assume a QMC of 3.5. If this were to actually be built we would need to verify this value of QMC but in this case we will assume it is correct. Equations 16 thru 19 show these calculations.

8861.0707.05.3707.05.3

=−×

⇒−×

=TCMC

TCMCEC QQ

QQQ Eqn.16

6468.016905.08861.01

22

=−

⇒−

=

ES

ECQQα Eqn.17

34603.0

6468.02977.0 ftVV AS

AB =⇒=α

Eqn.18

Hzff sc 661.826468.01414.641 =+⇒+= α Eqn.19

4.2 Additional Simulation Parameters.

The following calculations determine the remaining parameters needed to complete the PSpice simulation. Please note, in addition to the parameters in this section there are several from Sec. 3.1 that will also be used in this simulation. 4.2.1 Rear Air Load Mass (MAB). The value of MAB can be calculated using Eqn.20. As recommended by the text2, a typical value of B = 0.65 was used in this calculation.

48546.22065.0

18.765.0mkg

aBM AB ⇒

××

==ππ

ρ Eqn.20


Kendall/Hunt Publishing, 2003. pg114

14

4.2.2 Acoustic Resistance of the Box (RAB). This parameter is calculated directly using Eqn.21.

( )( ) 522

2

4.7752065.0

292.115.3

6106.56468.01

11

msNR

SR

QQR

AB

D

MS

MC

MSAB

⋅=×

×

−

×+=⇒

×

−

×+=

π

α

Eqn.21

4.2.3 Closed Box Compliance (CAB). This calculation is straightforward as illustrated in Eqn.22.

( ) ( )Nm

cpV

cpVC ASAB

AB

55

3

00102011.3

34518.16468.010429.8 −

−×=

××

⇒==α Eqn.22


The Spice netlist used in this project was modeled on the netlist in Table 7.3 of the course textbook3. The parameters were modified for this particular driver and the resulting netlist is shown below:

***** Closed_Box.cir ***** VEG 1 0 AC 1V RE 1 2 3.60 GZE 2 3 LAPLACE V(2,3) = 1/(0.004011*PWR(S,0.75116)) HBLUB 3 4 VD2 6.1486 VD1 4 0 AC 0V LMMD 5 6 1.6202e-2 RMS 6 7 1.292 CMS 7 8 3.41e-4 HBLIC 5 0 VD1 6.1486 ESDPD 8 9 10 0 0.01327 VD2 9 0 AC 0V FSDUD 13 10 VD2 0.01327 LMA1 10 12 9.810 RA1 10 11 88393.8 RA2 11 12 61341.6 CA1 10 11 5.8073e-9

`3 Leach, W. Marshall, Jr. Introduction to Electroacoustics and Audio Amplifier Design, 3rd Edition,

Kendall/Hunt Publishing, 2003. p124

15

LMAB 13 14 22.8546 RAB 14 15 7752.4 CAB 15 0 3.2011e-5 RAL 15 0 4e8 VD3 12 0 AC 0V EXD 16 0 LAPLACE I(VD2) = 1/S .AC DEC 50 10 20k .PROBE .END 4.4 Simulation Results.

The following results were obtained running the netlist shown in the previous section. Figures 4.4.1 and 4.4.2 show the voice-coil impedance magnitude and phase, respectively. Figure 4.4.3 shows the SPL of the closed box system while Fig. 4.4.4 shows the peak diaphragm displacement.

Figure 4.4.1: Voice-coil impedance magnitude versus frequency.

16

Figure 4.4.2: Voice-coil impedance phase versus frequency.

Figure 4.4.3: SPL at 1m with 1VRMS input versus frequency.

17

Figure 4.4.5: Peak diaphragm displacement versus frequency.

The above plots are all very similar to those obtained for the infinite baffle case. This is not unexpected since the formulas derived for this case generally differ by a constant multiplier. Therefore, this behavior is expected.

18

V. Simulation in an Optimum Vented Box.

We will now turn to the design and simulation of our driver in a vented-box. 5.1 Parameter Calculation.

Several parameters must be calculated before a simulation can be constructed and they depend largely on the driver parameters determined in Part 1. These values are reproduced below.

32977.0

6148.06905.06106.54.64

ftV

QQQ

Hzf

AS

TS

ES

MS

s

=

===

=

For this design, it is necessary to reduce the value of QTS since this value does not actually appear on the design charts. To accomplish this, the Bl product will be altered to allow for proper design using a Butterworth alignment. In order to obtain a reasonably sized enclosure, it is necessary to select α such that the value of VAB is larger than VAS. Therefore, the design presented here will use a QL of 3. This will require a QTS of 0.439. 5.1.1 Bl Product Revisited.

As already mentioned, it is necessary to reduce the total quality factor of the driver in order to use the design charts from the textbook. We wish to obtain a QTS of 0.439. Equations 1 and 2 show the calculation of the new Bl product.

4763.0439.06106.5439.06106.5' =

−×

⇒−

=TSMS

TSMSES QQ

QQQ Eqn.23

( ) ( ) mTQQ

BlBlES

ES ⋅=×⇒= 4032.74763.06905.01486.6'

' Eqn.24

5.1.2 Design Chart Parameters. Now that QTS has been set as 0.439, using the design chart for QL = 3, Table 5.1.1 shows the parameters from this chart.

19

Table 5.1.1: Design chart parameters.

Parameter Value Α 0.654 H 1 Q 1

These parameters lead to the following values for the Helmholtz frequency, lower -3dB frequency, and effective volume of the box.

33

455.0654.0

2977.0

4.644.64

ftftVV

HzqffHzhff

ASAB

Sl

SB

=⇒=

====

α

Eqn.25

5.2 Port Design

Equation 4 is used to calculate the dimensions of the port. For this design we will assume a port diameter of 2in.

( ) ( )

m

L

SVS

fcL

P

P

AB

P

BP

0772.0

0254.01463.10254.012455.0

0254.014.642

345

463.12

2

33

22

2

=

××−

××××

×=⇒

−

=

πππ

π

ππ

Eqn.26

5.3 Simulation Parameters.

The following equations are used to calculate the remaining parameters for the vented box simulation.

40 7561.3

065.018.165.0

mkg

aB

MW

AB =××

⇒=ππ

ρ Eqn.27

Nm

cVC AB

AB

58

2

33

20

101735.934518.1

0254.012455.0 −×=×××

⇒=ρ

Eqn.28

58 3.80820101735.94.642

32 m

sNCf

QRABB

LAL ⋅=

×××⇒= −ππ

Eqn.29

20

4220

1 551.120254.0318.18

38

mkg

aM

PPA =

××

⇒=ππ

ρ Eqn.30

( ) ( )

4

8212'

027.54

551.12101735.94.642

12

1

mkg

MCf

M PAABB

AP

=

−×××

⇒−=−ππ Eqn.31

5220

2 2008560254.0

34518.1msN

ac

RP

PA ⋅=×

×⇒=

ππρ

Eqn.32

5232230

1 885792008560254.09

34518.11289128

msNR

ac

R PAP

PA ⋅=−×

××⇒−=

ππρ

Eqn.33

Nm

caC P

PA

58

2

3

20

3

1 107286.234518.10254.094.594.5 −×=

××

⇒=ρ

Eqn.34

4220

1 905.4065.03

18.183

8mkg

aM

WWA =

××

⇒=ππ

ρ Eqn.35

5220

2 30670065.034518.1

msN

ac

RW

WA ⋅=×

×⇒=

ππρ

Eqn.36

5232230

1 1353030670065.09

34518.11289128

msNR

ac

R WAW

WA ⋅=−×

××⇒−=

ππρ

Eqn.37

Nm

ca

C WWA

58

2

3

20

3

1 10161.134518.1065.094.594.5 −×=

××

⇒=ρ

Eqn.38

21


The Spice netlist used in this project was modeled on the netlist in Table 8.2 of the course textbook4. The parameters were modified for this particular driver and the resulting netlist is shown below:

***** VentedBox.cir ***** VEG 1 0 AC 1V REW 1 2 3.60 GZE 2 3 LAPLACE V(2,3) = 1/(0.004011*PWR(S,0.75116)) HBLUW 3 4 V2W 7.4032 V1W 4 0 AC 0V HBLIW 5 0 V1W 7.4032 LMMDW 5 6 1.6202E-2 RMSW 6 7 1.292 CMSW 7 8 3.41E-4 ESWPW 8 9 17 10 0.01327 V2W 9 0 AC 0V V5W 16 0 AC 0V LMA1W 17 19 4.905 RA1W 17 18 1.353E4 CA1W 17 18 1.161E-8 RA2W 18 20 3.067E4 FSWUW 10 17 V2W 0.01327 LMABW 10 11 3.7561 CABW 11 12 9.1735E-8 V3W 0 12 AC 0V RALW 11 0 80820.3 LMAP 11 13 54.027 LMA1P 13 15 12.551 RA1P 13 14 88579 CA1P 13 14 2.7286E-8 RA2P 14 16 200856 V4W 15 16 AC 0V V6W 19 20 AC 0V V7W 20 0 AC 0V .AC DEC 50 10 10K .PROBE .END



22

5.5 Simulation Results.

Figures 5.5.1 and 5.5.2 show the voice-coil impedance magnitude and phase, respectively. While Fig. 5.5.3 shows the SPL of the port, diaphragm, and the total system output. Finally, Fig. 5.5.4 shows the peak displacement of the port and diaphragm. As usual, all plots are functions of frequency.

Figure 5.5.1: Voice-coil impedance magnitude versus frequency.

Figure 5.5.2: Voice-coil impedance phase versus frequency.

23

Figure 5.5.3: SPL of port (red), driver (green), and total system output (blue).

Figure 5.5.4: Peak displacement of diaphragm (blue) and port (red).

Each of the plots in this section differ substantially from their counterparts in the closed box and infinite baffle cases. The voice-coil impedance exhibits a much more complex behavior with a second peak in both the magnitude and phase responses. Furthermore,

the voice-coil impedance is roughly 31 less at the peaks than in either of the previous

cases.

24

5.6 Comparison of Enclosures.

Since we have now completed the design of all the required enclosures it is worth investigating their relative performance. Figures 5.6.1 and 5.6.2 show the voice-coil impedance magnitude and phase, respectively, for all three cases. Similarly, Fig. 5.6.3 shows the three SPL’s and Fig. 5.6.4 shows the peak diaphragm displacement for all cases.

Figure 5.6.1: Voice-coil impedance magnitude for infinite baffle (black), closed box

(blue), and vented box (red) systems.

Please note that in the Fig. 5.6.1, the infinite baffle case and closed box cases appear one on top of the other. This is also the case in Fig. 5.6.2.

25

Figure 5.6.2: Voice-coil impedance phase for infinite baffle (black), closed box (blue),

and vented box (red).

Figure 5.6.3: SPL for infinite baffle (black), closed box (blue), and vented box (red).

26

5.6.4: Peak diaphragm displacement for infinite baffle (black), closed box (blue), and

vented box (red).

Comparing the various enclosures simulated here, if we begin with the voice-coil impedance magnitude plot (Fig. 5.6.1) the difference between the infinite baffle and closed box system is nonexistent. These each produce a single resonance peak at approximately 64.4Hz whereas the vented box design produces two such peaks but lower in magnitude. With the vented box the two resonance peaks occur because of the two LC circuits which result when we construct the equivalent electrical, mechanical, and acoustical circuits. These observations are equally applicable to the voice-coil impedance phase plot shown in Fig. 5.6.2. There are two bumps in the vented box plot that again the result of the two resonances. The SPL plot shown in Fig. 5.6.3 offers a slightly distorted look at these three designs. As you might recall, it was necessary to change the Bl product in the vented box design in order to allow the system to be designed for a Butterworth alignment. As a result of this modification, the overall SPL will be slightly higher in relative amplitude. On the other hand, we can still make some useful observations as far as the shape is concerned. The vented system has a much steeper low frequency roll-off than either the closed or infinite baffle systems. Again, the infinite baffle and closed box are very similar with the exception of the high frequency roll-off where the closed system is considerably slower than the infinite baffle. In this regard, the vented box system responds almost identically to the closed box system.

27

In looking at the diaphragm displacement the closed box and infinite baffle systems are again identical. Comparing these to the vented-box system, we see that at low frequencies the vented box requires greater movement of the diaphragm but this reverses above 35Hz. Between 35 and 110Hz the vented system experiences a sharp dip in the diaphragm displacement but above 110Hz all three systems operate identically. VI. Crossover and Zobel Matching Networks.

The design of 1st, 2nd, and 3rd order crossovers, as well as the Zobel matching network rely heavily on values obtained in previous sections. Specifically, the value of RE used here is 3.6Ω and the cutoff frequency we will use is 1.5 kHz. Furthermore, the values of Le and n are 4.011mH and 0.75116, respectively. The calculations are shown in the next several sections. 6.1 Parameter Calculations.

6.1.1 First Order Crossover.

Here, we will be using a 1st order low-pass crossover consisting of an inductor in series with the voice-coil resistance. This inductance is calculated using Eqn.39.

mHf

RL

W

EWW 382.0

150026.3

2=

×⇒=

ππ Eqn.39

6.1.2 Second Order Crossover.

We now add a capacitor in parallel with the driver’s voice coil. The values of the inductor and capacitor are calculated using Eqns.40 and 41, respectively. For 2nd order networks, we will assume a value of QW equal to 0.5.

mHQf

RL

WW

EWW 764.0

5.0150026.3

2=

××⇒=

ππ Eqn.40

FRf

QC

EWW

WW µ

ππ74.14

6.3150025.0

2=

××⇒= Eqn.41

6.1.3 Third Order Crossover. With the third order network we again add an inductor in series with the voice-coil resistance and a second order network. We will denote this new inductance L2 and the other inductor will be L1. Eqns.42-44 show the calculation of these values.

mHf

RL

W

EW 191.0150046.3

42 =×

⇒=ππ

Eqn.42

28

mHmHLL 573.0191.033 21 =×⇒= Eqn.43

FRf

CEWW

µππ

30.396.315003

23

2=

××⇒= Eqn.44

6.1.4 Zobel Network

The circuit used here is shown in the textbook in Fig. 10.75. The circuit parameters are calculated using Eqns.45-51. Please note, f1 and f2 are chosen to be 331Hz and 20 kHz, respectively.

Ω⇒= 6.31 ERR Eqn.45

( ) [ ] ( )

( ) [ ]F

ffR

LC

nn

nnE

n

e

µπ

π

74.20200003316.32

10011.4

2

75116.02275116.01

75116.275116.0275116.01

3

121

221

211

=

×⇒

=

×+−

−

−

+−

+−

Eqn.46

( ) [ ] ( )

( ) [ ]F

F

CffR

LC

nn

nnE

n

e

µ

µπ

π

41.16

74.20200003316.32

10011.4

2

75116.02275116.01

75116.075116.2275116.01

3

112

1

22

121

2

=

−×

⇒

−=

×+−

−

−

+−

+−

Eqn.47

Ω=××××

⇒=−++

045.51041.16200003312

1

2

1

675116.175116.0

75116.11

21

21

1

1

2

ππ CffR

nn

n

Eqn.48

Ω=

+⇒

+= 04.4

6106.56905.016.313

MS

ESE Q

QRR Eqn.49

mHf

QRL

S

ESE 143.64.642

6905.06.321 =

××

⇒=ππ

Eqn.50



29

FQRf

CESES

µππ

2.9946905.06.34.642

12

13 =

×××⇒= Eqn.51


The netlist for each of the above networks has been included as sub-circuits in the infinite baffle netlist shown in Sec. 3.2. ***** Infinite_Baffle.cir ***** VEG 100 0 AC 1V Xcrossover 100 1 1stOrder XZobel 1 Zobel RE 1 2 3.60 GZE 2 3 LAPLACE V(2,3) = 1/(0.004011*PWR(S,0.75116)) HBLUB 3 4 VD2 6.1486 VD1 4 0 AC 0V LMMD 5 6 1.6202e-2 RMS 6 7 1.292 CMS 7 8 3.41e-4 HBLIC 5 0 VD1 6.1486 ESDPD 8 9 10 0 0.01327 VD2 9 0 AC 0V LMA1 10 12 9.810 RA1 10 11 88393.8 RA2 11 12 61341.6 CA1 10 11 5.8073e-9 VD3 12 0 AC 0V FSDUD 0 10 VD2 0.01327 ***** Zobel Network ***** .subckt Zobel Vin R1 Vin 1 3.6 C1 1 0 20.74u R2 1 2 5.045 C2 2 0 16.41u R3 Vin 3 4.04 L1 3 4 6.143m C3 4 0 994.2u .ends Zobel ***** 1st Order Crossover ***** .subckt 1stOrder Vin Vout Lw Vin Vout 0.382m

30

.ends 1stOrder ***** 2nd Order Crossover ***** .subckt 2ndOrder Vin Vout Lw Vin Vout 0.764m Cw Vout 0 14.74u .ends 2ndOrder ***** 3rd Order Crossover ***** .subckt 3rdOrder Vin Vout L1 Vin 1 0.573m C1 1 0 39.30u L2 1 Vout 0.191m .ends 3rdOrder 6.3 Simulation Results.

Figures 6.3.1 through 6.3.3 show the response of the 1st, 2nd, and 3rd order crossovers with and without the Zobel network.

Figure 6.3.1: Gain of First Order Crossover with (green) and without (blue) Zobel

Network.

31

Figure 6.3.2: Gain of Second Order Crossover with (green) and without (blue) Zobel

Network.

Figure 6.3.3: Gain of Third Order Crossover with (green) and without (blue) Zobel

Network.

32

From the above plots, it is clear that the Zobel network greatly improves the operation of the crossover networks. Moreover, the cutoff frequencies appear to lie exactly at 1500 Hz. as they were designed. VII. Conclusion.

This report has described in detail the analysis and design of several loudspeaker enclosures and crossover networks. We began this project by measuring a 6.5 inch woofer from a Paradigm Reference Active 20 loudspeaker. From these measurements we were able to determine the driver parameters that would be used in designing appropriate enclosures. This enabled us to design a closed and vented box system for this particular driver. Although we did find it necessary to change one driver parameter during these phases, it was nevertheless, a relatively straightforward procedure. The final phase required the design of several crossover networks using 1st, 2nd, and 3rd order transfer functions. Furthermore, we then designed a Zobel matching network that allows the loudspeaker to be seen by other electronics as a pure resistor. This simple technique greatly enhances the operation of the crossovers especially considering they are designed with the belief that the loudspeaker is a pure resistor. The next step in this process would be to actually construct a prototype loudspeaker but this is beyond the scope of this project.

The Design and Simulation of a Loudspeaker...

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