Booming BassEPO-1 ProjectGroup C1Friday 15th January, 2016

Athul Perumpillichira, 4448588Jayme Freeke, 4484045Koen Peelen, 4457692Koen Rodewijk, 4453638Lars Dijkstra, 4490150Michael Hoogenboom, 4497449RyanHoekman, 4453530Simon Verkleij, 4431812Sjoerd Cnossen, 4492064Swier Garst, 4484320Thomas Roos, 4492242TijsMoree, 4449479

Contents

Foreword 1Introduction 21 Power supply analysis 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Calculating the values of the power supply circuit. . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Simulating the power supply using PSpice . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Building and testing the power supply . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Loudspeaker analysis 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Loudspeaker Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Passive filter design 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Impedance Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Separation filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.3 Phase shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.4 Volume adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.5 Final schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Power amplifier design 214.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.2 Objective and requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.3 Global description of design steps . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Amplifier Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.1 Discussion schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.2 Derived transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.3 Role of unknown components . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.4 Calculation of component values . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.5 Recalculation for available components . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.1 Simulation in Matlab and PSpice. . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.2 Final schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.5 Volume control knob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5.2 Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

ii

Contents iii

5 Booming bass design 285.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2.1 Needed behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.2 The circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.3 Final schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.5 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Acoustic characterization of the total system 317 Conclusions and recommendations 34

7.1 Power supply analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.2 Loudspeaker analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3 Passive filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4 Power amplifier design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.5 Booming bass design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Bibliography 36Appendix 37

A Derivations of the equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.1 Loudspeaker analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.2 Passive filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

B Measurement results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.1 Loudspeaker analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.2 Passive filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49B.3 Power amplifier design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

C MATLAB codes and simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51C.1 Loudspeaker analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51C.2 Passive filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53C.3 Power amplifier design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

D Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59D.1 Used components for the passive filters . . . . . . . . . . . . . . . . . . . . . . 59

E Assignments of power amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61E.1 Tutorial 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61E.2 Tutorial 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63E.3 Tutorial 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65E.4 Tutorial 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67E.5 Tutorial 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

F Datasheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71F.1 LM3886 Overture Audio Power Amplifier. . . . . . . . . . . . . . . . . . . . . 71F.2 RDE VideoMic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Foreword

Over the course of 10 weeks a group of 12 students have designed, simulated and built the electrical com-ponents necessary to get a predetermined audio system working optimally. To accomplish this, the studentshave also build a booming bass extension to widen the dynamic range of the speaker.

In this project, the students have tackled the basics of electrical engineering. The theoretical lecture courseshave been used to understand the concepts of the loudspeaker. Furthermore, the system must comply to thefollowing specifications:

The requirement best possible reproduction of an input signal may be interpreted as: the loudness ofa tone measured at a distance of one meter only depends on the amplitude of the electrical input signal,and not (or as little as possible) on the frequency. In other words, within the displayed frequency range,the transfer of the system is as flat as possible.

The frequency of the audio system ranges from 20 Hz to 20 kHz (-3dB bandwidths).

When the signal source produces an electrical signal with an amplitude of maximum 0.4 V, the maxi-mum amplitude of the signal presented to the speakers must be 10 V.

The system is built in the EPO-1 chassis, which can be seen in Appendix B of the EPO book. [3]

1

Introduction

In the EPO-1 project the main goal is to create an audio system which reproduces an input signal as best aspossible. The system has to be built for a speaker which is provided by the TU-Delft. The audio system thatmust be built has to include the following parts: [3]

A power supply

A power amplifier

A three-way speaker filter

A booming bass extension

The speaker has to be analysed as well which means that there were 5 main parts of the project.

The project has been approached by dividing the group in 4 parts, with each group working on a differentcomponent of the system. As certain parts of the project got finished, the group members joined other groupsor started working on the final part, the booming bass. When all 4 parts were finished everybody went to helpwith the booming bass or making the finishing touches to the parts and the reports.

In the further chapters of this report all the components of the system will be discussed and analysed. Thecomplete system will also be discussed and analysed and conclusions will be drawn afterwards to see if thesystem satisfies all of the specifications.

2

1Power supply analysis

1.1. IntroductionWhether it is from a battery, or from the power outlet, every electronic amplifier needs power. For this am-plifier, the obvious choice would be the power outlet. However, the power from the main outlet cant be feddirectly into the amplifier. So in order for us to supply the amplifier with the right power, a power supplyunit (PSU) will be needed that converts the mains voltage into a voltage that is usable by the amplifier. Theschematic of our power supply unit is given in figure 1.1.

Figure 1.1: A schematic version of the power supply we built

There were a couple of requirements for the PSU given in the EPO book [3]:

1. Without any load, both the V+ and the V rails should be at least 22 V .2. When the amplifier would demand a current of 1 A, the ripple voltage should only be 5% of 19 V .

3. When the PSU is not loaded, the capacitors should be discharged within 2.5 minutes (5).

4. The PSU should be a double-sided rectifier with a positive and negative output voltage.

Please note that these are minimum requirements. If in any way it is possible to lower the ripple voltage evenbelow the earlier mentioned 5%, this should be aimed to do so.

The main challenges of the PSU are finding the values of the components. The capacitors should be capableof keeping the ripple voltage below 5%, diodes with a reverse voltage higher than the current through oursystem. The bleeder resistors should be able to keep the discharge time below the 2.5 minutes.

3

1.2. Calculating the values of the power supply circuit 4

1.2. Calculating the values of the power supply circuitBefore building the PSU, the components should be chosen and their values should be calculated.

1.2.1. Capacitors

To obtain and reduce the ripple voltage above the 19 V , capacitors will be added at both rails of the PSU.To know what value to use for the capacitors, an equation is needed to calculate this value, starting off withequation 1.1.

Figure 1.2: V+ when adding a capacitor

q =Cv (1.1)

If both sides are divided by the time, the following equations are obtained using the fact that I qt

.

q

t= Cvt

(1.2)

I = Cvt

(1.3)

Before going any further, the ripple voltage Vr should be defined. Vr is the potential difference between theaverage voltage VRMS and the maximum voltage. The difference between the maximum and the minimumvoltage v is thus twice the Vr .

I = 2CVrt

(1.4)

The t is half the period of the original power outlet, which equals T = f 1. Then it comes down to:

t = 12 f

(1.5)

Then the value of the capacitors can be calculated in the following way:

C = I4 f Vr

(1.6)

Using equation 1.6 the minimum value can be calculated to keep the ripple voltage below 5%. The ripplevoltage Vr is 5% of the 19 V and the frequency is f = 50 Hz.

1.3. Simulating the power supply using PSpice 5

Cmin = 14 50 0.05 19 = 5300 F (1.7)

This exact capacitor wasnt available. However, a bigger capacitor should reduce the the ripple even more.That is why a 6800 F 40 V model capacitor is used in the PSU

1.2.2. Resistors

When switching off the amplifier, the capacitors theoretically will not discharge. Because charged capacitorscan hold enough energy to damage components when handled incorrectly, bleeder resistors were added todischarge the capacitors in a safe manner to keep the discharge time tmax below 2.5 minutes. Because thisdischarging process follows the exponential behavior of an RC-circuit, mathematically it is impossible to fullydischarge the capacitors. 5 is considered empty enough and it should therefore equal the maximum timetmax allowed, which is the mentioned 2.5 minutes.

5= tmax (1.8)

With =RC in a parallel RC circuit, the maximum resistance can be calculated via the formula:

Rmax = tmax5C

(1.9)

This means that a smaller resistance always means a shorter discharge time. However, having shorter dis-charge times is not always desirable. A shorter discharge time needs an smaller bleeder resistor resistor, andthat resistor will also load the PSU down unnecessarily.

With the actually used capacitance being 6800 F and the maximum time being 2.5 minutes, the maximumresistance can be calculated using equation 1.9:

Rmax = 2.5 605 6800 106 = 4.4 k (1.10)

A pair of 3.9 k resistors were used, because they were the largest resistors available under 4.4 k. They willkeep the discharge time below 2.5 minutes and the load on the PSU the smallest.

1.3. Simulating the power supply using PSpice

Figure 1.3: The simulation model of the double sided rectifier

Orcad PSpice was used to simulate the schematic of the PSU that can be seen in figure 1.3. To mimic thecenter-tapped transformer, two AC voltage sources in phase were used. Their amplitudes were 22 V and the

1.4. Building and testing the power supply 6

frequency was set to be 50 Hz, to make them resemble the output voltages of the transformer box of the TUDelft. On the right side of the figure, there are two 20 resistors to simulate a load of 1 A. The diodes used arenot the ones that will be used to build the PSU, but are able to handle the voltage and the current delivered,so they are no problem in the simulation.

This setup resulted in the graph that can be seen in figure 1.4. The difference between the minimum andmaximum voltage never exceeds 1.8 volts and therefore never exceeds the 10% of the 19 V . We can concludefrom this simulation that the circuit meets all the requirements.

Figure 1.4: The voltage time diagram of the power supply

1.4. Building and testing the power supplyThe PSU was built on the PCB provided by the university and analysed to check if the simulations were cor-rect. The measurements were done between the + and instead of + and ground , because this way boththe negative rail and positive rail were tested in one measurement. To measure at the 1 A current at whichthere should be only 5% ripple voltage, a 40 resistor was required, but such a resistance was not available.Thus, two measurements were executed, one with a 50 and one with a 60. These were the results:

H resistance 50 60maximum voltage 41.6 V 42.0 Vaverage voltage 40.6 V 41.2 Vminimum voltage 39.6 V 40.4 Vripple voltage 1.0 V 0.8 Vcurrent (DC) 812 mA 675 mA

As expected, the current is lower than the 1 A at which the ripple is specified, because 50 and 60 resis-tances were used instead of the calculated 40 that wasnt available.

The ripple voltage as percentage of the total voltage is calculated using the following equation:

Vr ipple (%)=Vmax Vavg

Vavg100% (1.11)

Filling in the results of the first measurement gives us:

Vr ipple (%)=41.640.6

40.6100%= 2.46% (1.12)

And for the second measurement:

Vr ipple (%)=4241.2

41.2100%= 1.94% (1.13)

1.4. Building and testing the power supply 7

Using PSpice, the first measurement was executed with a 50 load connected. This simulation is shown infigure 1.5. In the simulation, the average current iavg = 810 mA and the average voltage Vavg = 42 V . Theseresults match with our real measurements within the tolerances of the used resistors.

Figure 1.5: The simulated voltage and current with a 50 resistor, with a ripple voltage of 1 V

The results of the simulation with a 60 load are show in figure 1.6. The average current iavg = 680 mA,the average voltage Vavg = 41 V and other results also match within tiny tolerances to our real measure-ments.

Figure 1.6: The simulated voltage and current with a 60 resistor, with a ripple voltage of 1 V

2Loudspeaker analysis

2.1. IntroductionA dynamic loudspeaker is a device that converts electric signals into sound. In order to produce sound, a cer-tain pressure variation in the medium (air) is needed. For low frequencies, relatively large but slowly-vibratingvolumes of air are required and for high frequencies relatively small but quickly-vibrating air volumes are re-quired. Separate types of speakers, optimized for certain frequency ranges, are needed to accurately producethe desired frequencies. A speaker element usually consists of a metal frame: the basket, in which a movablecone is attached (see figure 2.1 [4] ).

Figure 2.1: Loudspeaker intersection

Depending on the type of speaker, the cone can be made of paper, plastic, or other materials such as Kevlar,carbon, or other light metals. The rear of the cone ends in a circular bus around which a coil: the voice coil, iswound. The voice coil is the electrical input of the speaker, with which it will be connected to the filter of thespeaker.

The voice coil is located in a strong permanent magnetic field and is centered in here by the spring-mountedspider, which allows the voice coil to be flexibly mounted to the speaker frame.

At the front, the cone is mounted by means of an edge made of rubber or foam in order to smoothly moveup and down. With a tweeter, intended to sound the treble, the cone must be very light, and is thereforeoften made of silk, or a very thin metal foil that is glued directly onto the voice coil. For a loudspeaker it isimportant to analyze the behaviour of the different internal components at different frequencies, becausesound consists of different frequencies. These components have their own resistances, inductances and ca-pacitances. The loudspeakers electrical characteristics have to be analysed in order to properly understand

8

2.2. Theory 9

its behaviour when playing audio through it, which consists of signals with varying frequencies. These sig-nals are essentially alternating current, which means the response of the circuitry inside the loudspeakerwill be very different because it contains components such as inductances and capacitances, which behavedifferently for different frequencies.

In this report a basic explanation of impedance, an explanation of the electrical model of a loudspeaker andits behaviour under alternating current will be given. Measurements of the impedance of the loudspeakerswill be included, the corresponding models and the determined component values based on the measure-ments will also be included. To accomplish that, the impedance of the speakers will be measured for thefrequency range of 0 Hz to 24 kHz. By using the impedance measurements a model can be made whichincludes:

Resistance of the voice coil Self inductance of the voice coil The values of the electrical equivalent of mass-spring system representing the cone

Based on the measurements, the values of the model parameters can be determined in such a way thatthe calculated impedance corresponds as much as possible with the measured impedance. The simulationresults will be compared with the measured impedance and the causes of the differences will be investi-gated.

Furthermore, the behaviour of the loudspeaker at different frequencies, measured by the actual sound out-put of the loudspeaker in decibels will be measured. The purpose of these measurements is to investigateon which frequency a switch of output between the different speakers can best be made, to make sure theeffectiveness of the entire system is maximized.

2.2. TheoryA circuit operating in steady-state with alternating current has impedance as an intrinsic property instead ofresistivity. Impedance can be seen as the equivalent of resistance for AC circuits. This means that impedance,just like resistance, is the ratio of voltage over current (equation 1 in appendix A.1).

The impedance of certain components (such as inductors and capacitors) is dependent on the frequency ofthe alternating current flowing through the circuit. Because the loudspeakers that are used are working underAC, a proper understanding of impedance is needed.

The impedance of a resistor is given in equation 3 in appendix A.1. As is apparent from equation 3 theimpedance of a resistance is independent of the frequency.

The impedance of an inductance is given in equation 4 and the impedance of a capacitor is given in equation5 in appendix A.1. These equations show that the impedance is frequency dependant in these cases.

The formulas for the resonant frequency for an RLC circuit and the formula for the bandwith of an RLC circuitare needed too. These are given in equations 6 & 7.

2.3. Loudspeaker Analysis

Figure 2.2: Simplified electric model of a loudspeaker [3]

To analyse the loudspeaker a simplified electric model of thespeaker is needed. This model is shown in figure 2.2. In thismodel the resistance (Re ) and the inductance (Le ) in seriesrepresent the voice coil. The parallel RLC (Rp ,Lp ,Cp ) circuitrepresents a cone. These two parts joined in series form theelectric model of a loudspeaker. The simplified model comesfrom a more extensive model. The more extensive model in-cludes two extra resistances in series with Re . These resis-tances are Ra , which represents the power losses to air, andRv , which represents the power lost to mechanical friction.These losses however are so low that they can be ignoredwhich gives us the model in figure 2.2 to work with during the analysis the speaker.

2.4. Measurements 10

The value of the resistance (Re ) of the voice coil is equal to the DC resistance of the loudspeaker. This is be-cause the frequency of DC is 0, which means ZL(0)= 0, which means ZLe = (0) and ZLp (0)= 0. The impedanceof the cone is also zero, so all the impedance from the loudspeaker comes from the resistance of the voicecoil.

At high frequencies, the impedance of a capacitor is very low, to the point where it is negligible. So a RLC cir-cuit at high frequencies will have an impedance which is practically equivalent to zero. As shown by equation4 , the impedance of an inductance will be high at high frequencies. This means that at high frequencies, theimpedance of the circuit of the loudspeaker is equal to Re +ZLe ().The inductance of the voice coil can be calculated using this and equation 4, this is shown in equation 8 inappendix A.1.

When a parallel LC circuit resonates, the inductance and capacitor resonate and cancel each other out, cre-ating an open circuit. This is because together, their admittance(the reciprocal of the impedance) approaches0. When the admittance is 0, the circuit becomes to a set of open terminals. This is shown in equations 6through 18 in appendix A.1.

This means that at the resonant frequency, a parallel RLC circuits impedance is equal to the resistance of theresistor in the RLC. The resonant frequency is usually quite low, so that means that the added impedancefrom the inductance of the voice coil is low, to the point where it is negligible. This means that at the resonantfrequency, the impedance of the loudspeaker is equal to Re +Rp .The resistance of the cone can be calculated with this, as is shown in equation 20 in appendix A.1.

After calculating the resistance of the cone and measuring the bandwidth, the formula for the bandwidth(equation 7) will be used to calculate the value Cp , the inductance of the cone. This is shown in equation21.

Now that the value of Cp had been calculated, the value of Lp can been calculated by using the formula forthe resonant frequency. This is shown in equation 22.

Using all of the formulas the values of the components have been calculated for the low, mid and high speak-ers:

Low speaker

Re= 7.1

Le= 0.38 mH

Rp= 9.9

Lp= 0.011 H

Cp= 0.61 mF

Mid speaker

Re= 4.13

Le= 0.17 mH

Rp= 5.23

Lp= 1.9 mH

Cp= 1.8 mF

High speaker

Re= 4

Le= 53 H

Rp= 0.91

Lp= 96 H

Cp= 0.16 mF

2.4. MeasurementsFor measuring the complex frequency-dependent impedance of a speaker, the MATLAB program LS_Measurewas used, provided by the TU Delft. LS_Measure uses the sound card of the computer to generate a whitenoise signal containing different frequencies of a chosen range. It measures the response of the impedanceon the frequencies and plots three graphs containing:

The impedance amplitude Z in ohms. The phase shift of the returning signal in degrees. The calculated electrical frequency response in decibels.

The measurements are shown in Appendix B.1. In figure 8 the measurement setup had been tested by mea-suring the impedance values for some components. The impedance values of these components can be cal-culated. The measurements can be compared to these calculations to determine if the setup is valid. They arethe same so the setup is valid and can be used for further measurements. It can be seen that there is alwaysa impedance peak at a certain frequency. In figure 5 the impedance amplitude and the and the impedance

2.5. Simulations 11

phase shift is shown. The peaks are caused by the frequency of the voice coil. As for figures 6 and 7 thesame argumentation is applied. In 9, the amplitude of the frequency response of the loudspeakers is given indB.

Figure 2.3: Measurement setup for measuring impedances [3]

To measure the impedance of the speaker, the program LS_Measure, the circuit shown in figure 2.3 and a PCfrom the Tellegen Hall in the EEMCS building of TU Delft have be used. The signal processing and -generationare being distributed among the PC and sound card. Two cables connected to the PC are needed to generateand process the signals. The cables are connected to the microphone- and the audio jack. The microphonejack at the rear of the PC is used and the audio jack at the front due to disruptive environmental factorsintroduced by the other ports when used. The cables connected with the PC are routed to the ConnectionBoxs line in and line out. The Vin and Vout ports are connected with the positive terminal of the high, midor low speaker. Only one ground port of the Vin and Vout is needed to connect the negative terminal of thespeaker to ground it. The frequency input of the measuring system has a certain limit. That is determined bythe background noise of the frequency. The higher the frequency, the more background noise it produces. At20 kHz the background noise is way to high to produce a normal sound. Furthermore, the soundcard of thecomputer used was not build to produce frequencies higher than 20 kHz.

2.5. SimulationsIn order to verify the measurements of the loudspeakers, a simulation of the loudspeaker has been created.The formulas for calculating the impedance of the components in the loudspeaker are already given. Bycalculating the impedance of each component at a certain range of frequencies and putting them together,the total impedance of the loudspeaker can be calculated.

There are two models of loudspeakers in the manual: Model 1, and model 2. Model 1 is the full representationof a the electrical circuit of a loudspeaker. Model 2 on the other hand, only represents the electric circuit of aloudspeaker for frequencies above those of the resonance peak.

The values of the components of the models are already given and are given in list C.1 in appendix C.1.

Using these values and the formulae given in the Introduction and Loudspeaker Analysis sections, the be-haviour of this model in response to different frequencies can be simulated.

First, the values are imported into MATLAB. Then, the impedance at a range of the frequencies has to becalculated. A range of frequencies that is logarithmically spaced will be chosen to get more samples in thelower frequency ranges, where the resonant frequency usually is. A frequency range of 20 kHz is chosen sincethat is aesthetically pleasing when plotted, for the simulations of the actual loudspeakers, different values willbe used. The impedance of a loudspeaker is given in equation 23 in appendix A.1.

2.5. Simulations 12

The equation is converted into MATLAB code and the impedance will be calculated. Now that the impedanceis calculated, it has to be converted into, as complex numbers dont have much meaning on their own. So theabsolute value of the impedance is taken to get the amplitude of the impedance. The angle is also taken sothat the phase shift caused by the loudspeaker can be obtained.

Then the phase shift and impedance amplitude can be plotted (figure 15 in C.1) to see if it matches the simu-lation from the tutorial.

Since this is identical to the results of the simulation on page 128 of the manual, it can be concluded that thecode for the simulation works as intended.

The code thats already written to simulate the behaviour of the loudspeakers can be extended.

By implementing the equations (2 through 7, A.1) that are used to calculate the values of the componentsin MATLAB, the process of creating a simulation of the speaker can be simplified. In the event of incorrectmeasurements, a new simulation can easily be created if this part will also be automated. First the values needto be added that can be (and need to be) measured. Then, equations 1 through 23 have to be implemented inMATLAB. Next, some of the code has to be rewritten so it works properly with vectors instead of scalars. Thisalso means that what was previously a vector, is now a matrix, so the code has to be adjusted for that as well.The plotting functions should be updated too so that multiple vectors can be plotted.

While there is a fairly noticeable difference in the phase shift of the simulation (figure 16 ) compared to thephase shift of the actual model (figure 7), there is little reason to worry. This is because the unpredictablebehaviour of the phase shift only starts to occur well after the frequencies where the signal transitions fromone speaker to the other. This means that any interference caused by phase shift can be accounted for withless difficulty.

Aside from that, there is a minor difference in the minimum impedance amplitude. This could come fromexternal sources such as the PC used to measure the loudspeakers impedance or background.

3Passive filter design

3.1. IntroductionAmplifiers take a signal and boost the amplitude. But in the case of an acoustic amplifier (a speaker) theelectric signal gets amplified and converted into sound waves. To make sure that all frequencies produce aslittle distortion as possible, specialized speakers are used: two bass speakers for the low domain, one midspeaker for the middle domain and a tweeter for the highest domain. To use these, they must only receive thefrequencies they were made to handle. The part of the frequency spectrum used for our system is equal tothe HiFi range (this ranges from 20 Hz to 20 kHz). [3]

Not every speaker is able to give an accurate sound representation over a wide frequency range, due to physi-cal limitations. To deliver the most accurate sound representation, specific frequency ranges have to be sendto different speakers. In this section, a filter will be build that separates the frequency range, and sends theranges to the best suited speaker. The best frequency ranges for our specific speaker have been analysed inthe Speaker Analysis section, and will be used here.

The speaker filter must satisfy the following requirements: [3]

Frequencies that are not in the frequency range of a speaker must be attenuated by that specific speaker. The overall transfer function of the frequency and phase must be as flat as possible. All frequencies of the HiFi spectrum must be diverted to at least one speaker element. The frequency ranges for each speaker element must be based on the recommended values from the

loudspeaker analysis section.

3.2. DesignIt is not easy to just start making the filters on the go. Therefore the components and schematics for everyfilter are designed in this section.

The measurement results of our loudspeakers are used as a baseline for the design procedure of our filters.The results of these measurements can be found in appendix B.1. The derivations of the equations used inthis section can be found in appendix A.2.

3.2.1. Impedance Correction

The results of the impedance measurements show that the impedance of a speaker increases at high frequen-cies. This is caused by the inductor in a speaker, which impedance is linearly dependent of the frequency.The increase of the impedance means that the sound pressure increases with the frequency. [3] The com-pensation of this increase can be done with a Zobel network, which consists of a resistor and a capacitor inseries. The Zobel network is placed in parallel with the speaker and since the frequency dependent part of theimpedance is different for each speaker element, a different Zobel network has to be made for each speaker

13

3.2. Design 14

element. The result of a Zobel network is that the speaker element can be written as an impedance with onlya real part.

The measured values of the resistor and inductor of the three different speakers are shown in table 3.1. Thecalculated values of the resistor and the capacitor of the three different Zobel networks are shown in table 3.2.The impedance of a speaker element (Zl s) will be, due to the Zobel network, equal to Re .

Table 3.1: The measured values of Re and Le

Speaker Re LeLow 7.1 380 HMid 4.13 170 HHigh 4 53 H

Table 3.2: The calculated values of Rz and Cz

Speaker Rz CzLow 7.1 7.54 FMid 4.13 9.97 FHigh 4 3.31 F

3.2.2. Separation filters

An easy way to keep the overall transfer function of the system as flat as possible, is to only use a speaker inthe frequency range where its transfer function is as flat as possible. This can be achieved by separating thefrequency spectrum into three regions, and passing each speaker with its matching region. In this sectionthree filters will be designed, one for each speaker. The filters will allow certain regions of the frequencyspectrum to pass through, while attenuating the others. The frequency regions of each filter should matchclosely with each other, to avoid dead bands or volume peaks in the spectrum.

The EPO book prescribes two different types of filters. [3] There is a choice between a first order and a secondorder low-pass filter. The differences between a first and a second order filter are: the maximum slope afterthe 3 dB frequency and the availability of a quality factor. For a first order low-pass filter, the maximumslope is 20 dB per decade. For a second order low-pass filter, the maximum slope is 40 dB per decade,double as steep as the slope of the first order filter. [3], [1] The advantage of a steeper slope is that less soundfrom outside the selected frequency region will leak to the wrong speaker. The quality factor gives controlover extra attenuation or resonance at the 3 dB frequency. Because of this, these systems only uses secondorder separation filters.

The 3 dB frequency of a filter is the frequency at which the power output is halved. It is important that the3 dB frequencies of the different filters match, because the two speaker will then both output half power,and make sure the total power remains 100%. On top of that it is important that the filter degree (first orsecond order) match, because the cut-off slope (20 dB or 40 dB) should match each other. The choicesfor the cut-off frequencies can be seen in table 3.3 and are based on the speaker analysis diagrams in ap-pendix B.1.

Table 3.3: The chosen cut-off frequencies

Switch point FrequencyLow to mid 170 HzMid to high 950 Hz

Woofer

Figure 3.1: A second order LPF

Results of the speaker analysis indicate that the woofer is ableto deliver the flattest acoustic transfer in the lowest part of thefrequency spectrum. For this reason only the lowest part of ourfrequency range should be sent to the woofer, while attenuat-ing higher frequency signals. This can be achieved by using alow-pass filter (LPF).

The schematic of a second order low-pass filter can be seen infigure 3.1. The following equations can be used to calculate thevalues of the capacitor and inductor of the second order low-pass filter:

3.2. Design 15

LLPF = Zl sQ 0

(3.1) CLPF = QZl s 0

(3.2)

Filling in what we know gives us the values of the capacitor and inductor:LLPF = 9.4 mH and CLPF = 93 FTweeter

Figure 3.2: A second order HPF

As expected, speaker analysis shows that the tweeter excels atdelivering high frequencies without too many variations. Thatis why a high-pass filter (HPF) is needed. This will filter out thelow and mid frequencies that would come out distorted andlets through only the higher ones.

The schematic of a second order high-pass filter can be seen infigure 3.2.

The values of the components of the HPF can be calculatedwith the same equations as used for the LPF. So the values ofthe capacitor and inductor are as follows:CHPF = 29.6 F and LHPF = 948 HMidtoner

Figure 3.3: A second order BPF

Since the amplifier will feed three speakers, a circuitis needed to filter out the high and low frequencies,so the midtoner will only receive the middle ones.To achieve this kind of behavior, a band-pass filter(BPF) is being used. If the two cut-off frequenciesare sufficiently far apart, the BPF can be treated astwo separate filters in sequence: a high-pass and alow-pass filter. Then the same equations can be ap-plied for the inductances and capacitors as we usedfor the LPF and HPF. The frequencies differ enoughwhen the highest frequency is at least five times bigger than the lowest frequency, which is the case with170 Hz and 950 Hz. The values of the components of the BPF are as follows:CHPF = 28.6 F , LHPF = 979 H , LLPF = 5.5 mH and CLPF = 160 F

3.2.3. Phase shift

When a current moves through an impedance with a complex part, it is delayed a little, which creates a currentthat is similar to the original, only shifted. The phase has been altered: phase shift. Normally this should notcause any trouble. The sound will only be delayed with such a tiny bit that it is humanly impossible to notice,but problems could arise when the delay gets too big. When using multiple speakers it is possible to createdestructive interference. In this project two speakers will produce the frequencies near the -3dB frequency,but with a different phase shift. The two speakers have different filters and a different internal impedance,which is likely to result in different delays. The closer the total phase differences are to 180 degrees, the morethese will cancel one another out due to destructive interference.

Figure 3.4: An extreme example of two signals that will result in no sound at all

Re-syncing the phases is a difficult if not impossible task, as the phases are also dependent on the distanceand angle from the speaker you measure it. So for now, as there arent any predetermined specifications ofthis nature, this phenomenon was kept in mind while choosing the transfer frequencies.

3.2. Design 16

3.2.4. Volume adjustment

Figure 3.5: Volume adjustment

The various speakers that are used, produce a differ-ent volume at the same input voltage. [3] For a planeacoustic frequency characteristic, these differencesin volume should be corrected. The speakers thatproduce a sound which is too loud need an addednetwork for damping. The equivalent impedanceof the damping network has to be in combinationwith Zl s to be equal to Zl s . This is to prevent a re-dimension of the filter, which is dimensioned for aconstant output impedance, Zl s . The circuit for thedamping network is shown in figure 3.5.

For a known value of the damping factor (in decibel) and the speaker impedance Zl s , the resistors R1 andR2 can be calculated with equations 3.3 and 3.4.

R1 = Zl s(110(/20)

)(3.3) R2 = Zl s

(1

10(/20)1)

(3.4)

The damping factor can be determined from the simulation of the filters, which can be seen in figure 18 inappendix C.2. A more detailed description of this simulation can be found in section 3.3. From this simulationthe required damping factor equals 1.75 dB for the mid speaker and 1.35 dB for the high speaker. Theimpedance of the speakers, with the added Zobel networks, is also known. Zl s,mid = 4.13 and Zl s,high = 4.With this information the values of R1 and R2 can be calculated, which are shown in table 3.4.

Table 3.4: The calculated values of R1 and R2 for the damping networks

Speaker R1 R2Mid 0.754 18.5High 0.576 23.8

These damping factors were based on the electric measurements. After the acoustic measurements and hu-man listening, all volume adjustments were removed. All members of the group found the mid- and high-tones a bit too weak when listening different kinds of music. After removing all volume adjustments from thefilters, the group concluded that the sound was more clear without the volume adjustments. On top of that,the acoustic measurements of the power transfer were flatter. These acoustic measurement results can befound in figure 6.3 in chapter 6.

3.2.5. Final schematics

The final schematics turned out to the following figures 3.6, 3.7 and 3.8, where the values of the actually usedcomponents are shown. The internal resistances of the inductors are displayed as well. These schematicscontain the filters themselves and the Zobel networks. At each circuit the input signal is presented to theterminals of Vin and the speaker element is connected to the terminals of Vout .

Low-pass filter

Figure 3.6: Low-pass filter schematic

Since the exact values for these components werenot available, multiple capacitors, inductors and re-sistors are used together to create the correct values.These can be found in the appendix D.1, next to theother components of this filter in the tables 1 and 2in appendix D.1.

Band-pass filter

3.3. Simulations 17

Figure 3.7: Band-pass filter schematic

As with the components in the LPF, not all valuesfor the BPF components were available either, so wecombined components to get the correct values. These and all the other components can be found in ap-pendix D.1 in figures 3 and 4

High-pass filter

Figure 3.8: High-pass filter schematic

As with the components in the LPF and BPF, not all values for the HPF components were available either,so we combined components to get the correct values. These components can be found in appendix D.1 infigures 5 and 6.

AdjustmentsConsidering the acoustic measurements a few changes were made to the values of the components. Thevolume adjustment circuits were changed as mentioned in the designated section 3.2.4. Furthermore, thevalue of CHPF used in the BPF was changed from 160 F to 190 F to get a flatter power transfer which canbe seen in section 3.4.

3.3. SimulationsThe simulations are done using Matlab, because it has more possibilities to work with than PSpice. The valuesof the capacitors and inductors were calculated first. To simulate the filters very accurately, the values of theactually used components were calculated with. The inductors have a small resistance too, so these valueswere taken in account as well. Unfortunately, these resistances dropped the power transfer of the low-passand high-pass filter, as can be seen in figure 3.9. Therefore the decision was made to use a volume adjustmentnetwork to get the band-pass and high-pass filters have the same maximum power transfer as the low-passfilter. Another simulation was made including the volume adjustment networks. The result of this simulationcan be seen in figure 3.10. For both simulations, the speaker elements were simulated as a resistor with theZl s values. Those Zl s values can be found in section 3.2.1. The Matlab codes used for this section can befound in appendix C.2.

3.4. Measurements 18

Figure 3.9: Simulation of the filters

Figure 3.10: Simulation of the filters including volume adjustment

3.4. MeasurementsFirst, the electric power transfer response and phase of the filters, with the speakers attached, are measured.This is done using the Matlab program ls_measure. This resulted in figure 3.11.

3.4. Measurements 19

Figure 3.11: Measurements of the power transfer to the speakers using the filters

The little dimple in the sum where the midtoner takes on the signal from the woofer, could be easily fixedby reducing the cut-off frequency of the high-pass side of the BPF. This resulted in changing the value of theCHPF in the BPF from 160 F to 190 F . The inductor didnt have to be changed according to the Matlabsimulation. This change resulted in figure 3.12.

Figure 3.12: Measurements of the power transfer to the speakers using the filters

After this change, the acoustic measurements were done. The power amplifier, the filters and the speakerwere attached. The acoustic transfer of the total system is measured using a microphone on a distance ofapproximately 1 meter. This measurement is also done using the Matlab program ls_measure. The resultscan be seen in figure 3.13 and 3.14.

3.4. Measurements 20

Figure 3.13: The acoustic frequency response for the complete speaker systemincluding the power amplifier and the three filters. Distance from the speaker to the microphone: 1 meter.

Figure 3.14: The acoustic phase response for the complete speaker system including the power amplifier and the three filters. Distancefrom the speaker to the microphone: 1 meter.

The result in figure 3.14 is not reliable, because the microphone-to-speaker distance was1 meter. At this dis-tance, some frequencies have made more periods then others, which changes the phase. Therefore the phaseresponse at a distance of 1 meter is not equal to the phase response at a close distance to the speaker.

4Power amplifier design

4.1. Introduction

4.1.1. Background

Now that the power supply is finished, it is time to build the power amplifier. The power amplifier has threepurposes: the amplification of the input signal, blocking a possible DC-signal, and blocking a range of fre-quencies; the amplifier acts as an active bandpass filter. This is perhaps one of the most complex units of thesystem, requiring advanced knowledge of linear circuits. But luckily a lot of the hard work is already given inthe EPO student manual [3], so its more a matter of understanding than designing a whole new circuit.

4.1.2. Objective and requirements

The end goals of the power amplifier are as following [3]:

The amplifier should have a non-inverting configuration

The passband of the amplifier is 20Hz-40kHz (which is the -3dB bandwith)

The voltage gain in the passband equals 25

A possible DC input signal should be blocked and may not be present at the output

The DC offset of the op-amp may have a maximal amplification of 1.

Also, the power amp should be experimentally tested with either MATLAB or PSpice. Futhermore, the re-strictions of the LM3886TF should be taken into account. The datasheet of the LM2886TF can be found inappendix F.1

4.1.3. Global description of design steps

The students were first required to follow tutorial 5 of the EPO Student Manual [3] to get acquainted with allthe possible active filter configurations, before designing the power amplifier. After the assignments we arerequired to come up with a design. However, it should be specified that the design of the power amplifierisnt fully done by the students. A PCB and a few possible designs without component values are alreadygiven. It is the students task to choose the design, calculate the correct component values to match the givenspecifications, and solder the calculated components to the PCB. The power amplifier itself needs its ownsupply circuit; this should be soldered to the PCB as well.

21

4.2. Amplifier Analysis 22

4.2. Amplifier Analysis

Figure 4.1: The schematic of the final power amp circuit.

4.2.1. Discussion schematic

This schematic is chosen instead of the other possible schematics, because all the specifications can be metusing this circuit.

For a clear explanation of this schematic, it can be can divided in two parts. Part one consists of the compo-nents R2, Ra ,Ca andC1; part two consists of the operational amplifier and the components Rb , Rc andC3.The components R1 and R2 were already given from the EPO student manual [3].

Part one is actually a low-pass and a high-pass filter connected in series, while part two is the actual amplifierof the circuit, consisting of the operational amplifier and the components used to specify the gain.

So when a signal enters the circuit it will:

1. First come across the capacitor C*1, filtering out a possible DC signal. If the signal is AC however, thesignal will go through the low-pass filter and the high-pass filter respectively.

2. After the filters a voltage potential is present at the positive pin of the op amp, causing the op amp toamplify the signal on such a way specified with the components Rb, Rc and C*3.

3. Finally; if a DC offset is present in the output of the circuit, because of the capacitor C*3, its maximumamplification will be 1.

4.2. Amplifier Analysis 23

4.2.2. Derived transfer functions

There are four transfer functions to be derived. The transfer function of the high-pass filter, the low-pass filter,the amplifier and of the whole system.

The transfer functions are as follows:

VoutVin lowpass

= R21jC1 +R2

(4.1)

VoutVin hi ghpass

= 11+RCa j

(4.2)

VoutVin ampli f i er

= 1+ (Rb+Rc) j f 2piC31+ j f 2piRbC3

(4.3)

VoutVin whole

= VoutVin lowpass

VoutVin hi ghpass

VoutVin ampli f i er

(4.4)

4.2.3. Role of unknown components

The role of the possible components varies. R1 functions as a pull down resistor. C1 and R2 are there toboth provide a high-pass filter and to block a possible DC input. Ra andCa are combined a low-pass filter. Rband Rc are components that determine the amplification by the opamp. C3 is necessary to make sure a DCcomponent is not amplified by the opamp, because DC cant flow through the capacitor. And last, Cb and Cccan be used to counter the effects of C3 or to implement another filter. However, this isnt done in our finalcircuit, because of all the hard calculations needed for the values, and the advantages probably wont up waythe disadvantages.

4.2.4. Calculation of component values

The calculation of the components was done using Matlab. We wrote a program that could calculate certaincomponents if values of other components where already given. This is done using the transfer functionsderived in section 4.2.2 and choosing a cut-off frequency. The Matlab program can be found in appendixC.3.

It is impossible to calculate all the values at once using a system of equations, because only the ratio of thevalues counts, not an absolute number. So there are an infinite possibilities of values. The calculated resultsare therefore only a possible configuration. A configuration with other values, doesnt mean immediately thatthe eventual system will comply less to the specifications.

4.2.5. Recalculation for available components

The Matlab program calculated the values based on values we thought were decent to use. However thecomponent values we initially came up with werent available. So a recalculation was needed to come upwith values that were actually available. This was done by calculating these new values based on the valuesthat are available. Eventually the component values listed in table 4.1 were calculated.

4.3. Simulations 24

Table 4.1: The initially calculated component values and the values weve finally chosen

Specified component Initial calculated value Best available component valueRa 100 1kRb 12.5k 20kRc 300k 470kC2 6.77F 4.7F +0.88FCa 15.9nF 1.5nFC3 4.7F 4.7F

4.3. Simulations

4.3.1. Simulation in Matlab and PSpice

The circuit is simulated twice, namely with the initially calculated values, and with the values calcu-lated for the available components. The schematics used for the PSpice simulation are shown in figure4.2 and 4.3. The code used for the simulation in Matlab is found in Appendix C.3.

Figure 4.2: The schematic of the power amplifier circuit with the initially calculated values in PSpice.

Figure 4.3: The schematic of the final power amplifier circuit with the calculated available component values in PSpice.

4.3. Simulations 25

4.3.2. Final schematic

The circuit that was eventually made is presented in figure 4.4 with all the component values clearly marked.

Figure 4.4: The schematic of the final power amp circuit with the used values clearly marked.

4.3.3. Results

The results of simulation of the final schematic is shown in figure 4.5 and 4.6.

The other results of the schematics with the initial values are found in appendix C.3.

Figure 4.5: The result of the PSpice simulation of the final schematic

Figure 4.6: The result of the Matlab simulation of the final schematic

4.4. Measurements 26

The calculated results agree with the simulated results, since the figures look practically the same. No realdifferences can be pointed out. As would be expected; in both cases the opamp is ideal and the values of thecomponents are the same.

4.4. MeasurementsThe final circuit is hooked up to a function generator and a oscilloscope to test the basic functions of thesystem. The results of the measurements are also found in appendix B.3.

4.4.1. Measurement setup

The measurements were performed using an oscilloscope and a function generator. The signal from thefunction generator is split and one part is put into the oscilloscope, while the other part is connected to theinput of the power amplifier. Finally the output of the power amplifier is connected to the oscilloscope as well.Then any frequency can be send out from the function generator through the power amplifier and comparedto the input on the oscilloscope.

4.4.2. Results

(a) Result with the frequency in thepassband

(b) Result with the frequency belowthe passband

(c) Result with the frequency abovethe passband

Figure 4.7: Results of the measurements with CH1 being the output from the power amplifier and CH2 being the reference signal fromthe signal generator

Note that the circuit indeed amplifies by 25 if the frequency is in the passband; while a phase shift is presentand the gain is less if the frequency is out of the passband. With this information the following conclusionscan be made:

Table 4.2: Conclusion of the results of the final circuit of each of the requirements

Measurable specifications MeasurementsA non-inverting configuration Satisfied

The passband is 20Hz - 40kHz f < 20Hz and f > 40kHz are suppressedVoltage gain in the passband is 25 The voltage gain equals 25

DC input should be blocked 0 V is measured with DC input

4.5. Volume control knob

4.5.1. Introduction

Although the design specification state that the amplifier can only amplify the 25 times, we also liked to havethe ability to adjust the volume on the amplifier itself, and not rely completely on the audio source. This hasseveral advantages; for example we can easily mute the amplifier when switching between audio sources toavoid the 50Hz mains hum.

4.5. Volume control knob 27

4.5.2. Schematic

We could have changed one of the 2 resistors near the power op-amp, but to keep things simple, we addedan potentiometer right after the input jack. The potentiometer we ended up adding to the circuit was anlogarithmic 10k potmeter. The resistor value of the potmeter is not that important, as everything before thepower amp is voltage-driven. We did use a logarithmic potentiometer because sound intensity is measuredin decibels, which is a logarithmic scale. The schematic can be seen in figure 4.8.

Figure 4.8: Schematic of the volume control knob

5Booming bass design

5.1. IntroductionOverall the sound from the system was quite nice at this stage, but the spectrum was not equally present.Especially the 20Hz to 80Hz range was lacking volume with respect to the other tones.

5.2. DesignTo achieve the better bass sound, a few solutions could be applied. The higher frequencies could be damped,which would get all tones to sound as good as the bass does. A better solution is to boost the bass tones to thelevel of the mid and high tones.

This implies that an extra op-amp is needed to boost a certain range of frequencies before amplifying withthe Power Amplifier.

5.2.1. Needed behavior

To keep the setup easy, it should boost the low frequencies and send the higher frequencies on to the poweramplifier without changing their amplitude. Thus the wanted dB output response should look like the oneshown in figure 5.1.

Figure 5.1: The wanted dB output for the Booming Bass Extension

5.2.2. The circuit

This behavior can be achieved using a non-inverting active shelving low-pass filter, which can be seen infigure 5.2a. This filter is not invented by us, but found on a website [2].

28

5.2. Design 29

(a) A non-inverting shelving low-pass filter(b) The corresponding graph

Figure 5.2: The circuit and its behavior

This filter consists of an op-amp and a passive shelving low pass filter underneath. This whole circuit followsthe next few equations. d and 0 are the same variable as can be seen in figure 5.2a. This site gives us thecircuit including the following two equations to calculate with.

d = 20 log10(1+ RBRA

) (5.1)

0 = 1C RB

(5.2)

Using these two equations the C value can be chosen and the RA and RB are dependent on that capacitanceaccording to the following equations.

RA = 10 C (10d/201)

(5.3)

RB = 10 C

(5.4)

5.2.3. Final schematic

In our case the bass tones should be boosted by 12 dB up until 60 Hz. This will cause a large section of themid to be boosted a little as well, but we agree with that. Thus d = 12 dB and 0 = 2pi 60 Hz. So we canchoose the value of the capacitance, therefore we choose the available value of C = 680 nF , the resistanceshave to be the following values.

RA = 160 2pi 680 109 (1012/201) = 1.2k (5.5)

RB = 160 2pi 680 109 = 3.9 k (5.6)

With these values the full schematic can be drawn, it can be seen in figure 5.3. There are two voltage dividersin there as well to get the both the positive and negative voltages V = 20 V of the power supply down to themaximum allowed voltage Vmax = 15 V of the op-amp.

5.3. Simulations 30

Figure 5.3: The full booming bass extension circuit

5.3. SimulationsWhen simulating the expected acoustic measurement, the following graph is acquired. The power dBoutput of the booming bass is summed up with the original acoustic measurements of the speakerwithout the booming bass. This results in the graph in figure 5.4.

Figure 5.4: The simulated acoustic measurements including the booming bass extension

5.4. MeasurementsThe low frequencies are boosted enough and the mid tones look still good as well, therefore we agreeto build this booming bass. With this booming bass added, another two acoustic measurements aredone, they can be seen in chapter 6. These acoustic measurements look very good in our opinion. Butthe high tones will need some tweaking according to this graph.

5.5. AdditionThe booming bass extension works great, but while listening to music the bass can sometimes over-power the other tones. Although we found 12 dB as an ideal number during the measurements andsimulations, we wanted the option to reduce the bass. So we added an option to reduce the boost ofthe booming bass by 4 dB . This was done by adding a switch and a 3.9 k in parallel to the 3.9 kresistor that exists in the circuit already. Thus there are two switches, one boosts the bass by 8 dB andthen another switch can be pulled to boost it with another 4 dB .

6Acoustic characterization of the total

system

In Figure 6.1 the entire circuit is shown, with the booming bass extension, power amplifier and filter systemfrom left to right.

Figure 6.1: The whole circuit

By simulating this circuit we get the results in figure 6.2, with the simulation of the system without the boom-ing bass on the top and the simulation with booming bass on the bottom. These simulations are verygood as the transfer is as flat as possible. According to these simulations, the system should perform verywell.

31

32

Figure 6.2: The Bode-plot of the whole circuit

The acoustic measurements were done and the results are shown in figure 6.3, with the first measurementbeing the one without booming bass, the second one with the damped-booming bass, and the last one beingthe one with the booming bass.

These measurements were done at 5 cm distance for the high, mid, and low speakers individually and ameasurement at 1 meter distance for all speakers at once. At first we used a microphone provided by the TUDelft but for these measurements we used a better microphone. The datasheet for this microphone is shownin appendix F.2. The graph in figure 6.3 is summation of the measured acoustic responses of all speakersseparately at a a distance of 5 cm. The measurements done at a distance of 1 meter were affected by sound ofother sources than our speaker. That is why the other measurement results were used.

33

Figure 6.3: Acoustic measurement without booming bass

7Conclusions and recommendations

7.1. Power supply analysisBecause a 40 resistor was not available, it was not possible to load the PSU with specified 1 A, but it waspossible to check the validity of the pSpice simulations with the measurements from the 50 and 40 load.Both simulations are within the 5% tolerances of our load resistors to the measurements, thus we can con-clude that the simulations with the 1 A-load are valid and therefore our PSU will meet all the conditions in realmeasurements. To optimize the PSU even more, we could have used bigger capacitors to reduce the ripplevoltage even more. However, these were not available.

To conclude, the requirements enumerated in the introduction were reached. The ripple voltage stays underthe 5% required, the discharge time after shutting off the circuit stays under the 2.5 minutes and the powersupply is two-sided.

7.2. Loudspeaker analysisThe final purpose of analyzing the loudspeakers, is to give recommendations on how certain filters shouldbe made and placed. At first, the transition from the LOW-speaker to the MID-speaker will be discussed. Itis desirable for the quality of the sound that the used range of the speaker does not lie in the range of theimpedance peak. This is, because the at this frequency, the whole speaker resonates, generating a variety ofundesirable reflections and other resonances. Regarding the LOW, speaker, it is unavoidable that the usedrange of the LOW-speaker lies in the range of the impedance peak. This has to be solved with a filter. Re-garding the MID-speaker, the peak of the impedance has more or less no effect at approximately 120 Hertz.Therefore, when the MID-speaker has an acceptable frequency response a switch can be made. This is atapproximately 170 Hertz.

Second, the transition from the MID-speaker to the HIGH-speaker will be discussed. The impedance peak ofthe MID-speaker has no effect around the range of the switch. The impedance peak of the HIGH-speaker liesaround the 1200 Hz. After approximately 1000 Hertz the frequency response of the MID-speaker fluctuates alot, resulting in disruptions in the eventual sound quality. Therefore, the choice has been made to solve theeffects of the HIGH-speaker impedance peak using a filter, and use a transition frequency of 950 Hertz.

Also, because the frequency response after the 1000 Hertz is increasing significantly, this needs to be elevated,so that this is on the same level. Furthermore, after the 9000 Hertz the HIGH speaker decreases significantly,so this needs to be fixed as well.

7.3. Passive filter designThe goal of the passive filter design was to make the overall transfer function as flat as possible. The usedspeaker consists of three different speaker elements, for the low, mid and high frequencies. Therefore three

34

7.4. Power amplifier design 35

different passive filters are designed and built. Parallel with each different speaker element, a Zobel networkis placed, which makes the impedance of the speaker element frequency independent (for frequencies higherthan its resonance frequency).

After designing the Zobel networks, the low-pass, high-pass and band-pass filters are designed. For thesefilters, the cut-off frequencies are based on the measurement results of the loudspeaker.

According to the simulations of the filters, the high-pass and band-pass filters needed an added damping net-work to get a simulated plane electric frequency characteristic. These desired damping networks are achievedwith an volume adjustment network.

During the simulations of the filters, first it was impossible to create a flat line when adding the speakers up.This was due to a fault in the simulation, not the design. To simulate a speaker, a parallel RLC circuit wasdrawn, but because of the Zobel network that was added, the speaker should be simulated as a resistance.When this was altered, the line became flat like it was supposed to.

While measuring the filters, the filters were connected to the speakers, so that the Zobel network would becombined with the speaker impedance. The electronic signal coming off the output of the filters was mea-sured. The alternative, measuring with a microphone, would be more inaccurate, since the microphone isntas accurate as the electronic measurements.

The measurements did not correspond completely with the simulations, mainly because of physical proper-ties of the coils. The inductors started to interfere with each other, as these were rather large and were notplaced perpendicular to each other and relatively close to each other. This problem was solved by placing theinductances away from each other, but the measurements were still a bit off. Further tweaking and testingwith slightly different inductances and capacitors ultimately lead the results as seen in section 3.4.

7.4. Power amplifier designThe goal of the power amplifier is to amplify the input audio signal with a gain of 25 while suppressing thesignal out of the bandwidth. To achieve this goal, first we got acquainted with active filters by doing theassignments in the student manual [3]. Then a schematic was chosen and the component values were cal-culated to comply with the specifications given also in the student manual. Finally simulations were made,before building the actual circuit.

When the circuit was build, measurements were made to make sure the circuit complied to all the given spec-ifications. The results of these measurements agreed with the results of the simulations and the calculations.All the requirements are met and when the power amplifier was connected to the rest of the system finallysound came out of the speakers, which means the power amplifier works successfully.

It became clear during the design process that a lot of time was wasted in the project hall figuring out what todo. So it is recommended to read up on the subject before the project time and to spend in the overall processless time on the assignments and begin sooner with the design of the actual power amplifier.

7.5. Booming bass designThe aim of the booming bass extension is to boost the lower range of frequencies, because the signals withthese frequencies are acoustically not as powerful as the others. There is more than just one possibility toachieve this. In our case, an active shelving low-pass filter is used, which boosts the low frequencies by 12 dBor 8 dB and lets the higher frequencies through without boosting. This is a big advantage, because it meansthat the extension can easily be added to the system, without any summing amplifiers for example.

Our cut-off frequencies are quite low, which has as result that the woofer does not carry a big range of fre-quencies. The booming bass extension does therefore boost the mid-toner a little as well, which could beconsidered a disadvantage. However, the difference between the sound with and without the extension isbig. The extension gives the low frequencies a nice saturated sound and hence it can be considered a suc-cessful circuit.

Bibliography

[1] C. K. Alexander and M. N. O. Sadiku. Fundamentals of Electric Circuits, international edition.McGraw-Hill Education, New York, 5 edition, 2013.

[2] ElectroSmash. (2015). Boss CE-2 Analysis. ElectroSmash. [Online]. Available:http://www.electrosmash.com/boss-ce-2-analysis. Consulted: 2015-12-16.

[3] G. J. M. Janssen, J. F. Creemer, D. Djairam, M. Gibescu, I. E. Lager, N. P. van der Meijs, S. Vollebregt,B. Roodenburg, J. Hoekstra, and X. van Rijnsoever. Lab Courses EE Semester 1 Student Manual. CourseLabs of EE1C11, EE1P11, and EE1M11. TU Delft, 2015-2016.

[4] Rohitbd. (2014). Cross-section of a subwoofer drive unit. Wikipedia. [Online]. Available:https://upload.wikimedia.org/wikipedia/commons/c/cd/Speaker-cross-section.svg. Consulted:2015-11-20.

36

Appendix

A. Derivations of the equations

A.1. Loudspeaker analysis

Following is a summary of the most important formulas used in this report.

Z ( f )= U ( f )I ( f )

(1)

= 2pi f (2)

ZR ()=R (3)The formula for the impedance of a resistance

ZC ()= L (4)The formula for the impedance of an inductance

ZC ()= 1C

(5)

The formula for the impedance of a capacitance

0 = 1pCL

(6)

The formula for the resonant frequency of an RLC circuit

B = 1RC

(7)

The formula for the bandwidth of an parallel RLC circuit

Following is the derivation of the formulas and answers used during the section of Loudspeaker Analysis. Forhigh values of :

Z ()=Re +ZLe ()= ZDC + Le Le = |Z ()Re

| (8)

ZL(0)= LpCL

(9)

ZC (0)= 1CpCL

=pCL

C(10)

37

A. Derivations of the equations 38

YRC ()= 1L

+ C (11)

= 22LC+L

(12)

= 2LC +1L

(13)

YRC (0)= LCLC +1

LpLC

(14)

= 1+1LpLC

(15)

= 0LpLC

(16)

= 0 (17)

Req = 11R +0

(18)

=R (19)

Z0 =Re +Rp =RDC +Rp Rp = Z0 RDC (20)

B = 1RpCp

Cp = 1BRp

(21)

0 = 1CpLp

Lp = 102Cp

(22)

Zloudspeaker = ZRe +ZLe +1

1ZRp

+ 1ZLp +1

ZCp

(23)

=Re + Le + 11Rp+ 1Lp

11

Cp

(24)

A.2. Passive filter design

Speaker impedanceTo obtain the equation of the speaker impedance, three general equations are written down. The impedancemodel for frequencies above the resonant frequency of the speaker consists of an resistor and an inductor inseries. The total impedance for the speaker equals the sum of the impedance of this resistor and inductor.This can be seen in equation 25. Furthermore the impedance of the resistor and the inductor are known.[1]

Zl s = ZR +ZL (25) ZR =R (26) ZL = jL (27)

The substitution of equation 26 and 27 in equation 25 gives us the equation of the speaker impedance.

Zl s =R+ jL (28)

A. Derivations of the equations 39

Zobel network

Figure 1: Speaker element with Zobel network

The values of the resistor and the capacitor of a Zobel network can be determined as following. The Zobelnetwork is in parallel with the speaker element. The total impedance of the Zobel network can be written as

Zzobel =Rzobel+1

jCzobel. The equivalent impedance of the total circuit will be Zeq = Zzobel Zl s . In the next

calculations Rzobel and Czobel will be written as Rz and Cz .

Zeq = Zzobel Zl sZzobel +Zl s

=

(Re + jLe

) (Rz + 1jCz

)Re + jLe +Rz + 1

jCz

(29)

Zeq =ReRz + Re

jCz+ jLeRz + jLe

jCz

Re +Rz + j(Le 1

Cz

) (30)

Zeq =ReRz + Le

Cz+ j(LeRz Re

Cz

)Re +Rz + j

(Le 1

Cz

) (31)The equivalent impedance has only a real part. So the imaginary part can be set to zero. Furthermore theequivalent impedance is equal to Re , because it is desirable that the equivalent impedance to equals the realpart of the speaker impedance just after its resonance frequency. The real part of the speaker impedancejust after its resonance frequency can be approximated by the value of Re . This can be done, because theinfluence of the inductor on the speaker impedance is relatively low at that frequency. This leads to equation32.

Zeq =ReRz + Le

CzRe +Rz

=Re (32)

Rz + LeReCz

=Re +Rz (33)

Cz = LeR2e

(34)

Equation 34 gives us the equation for the value of Czobel , which is dependent on the values of the speakerelement.

For the value of Rzobel following theory has been made. When the input frequency is very low, the branchwith the Zobel network will behave like an open circuit and the branch with the speaker will behave like it

A. Derivations of the equations 40

has only a resistor Re . That is because the inductor approximates an open circuit at these circumstances.So the equivalent impedance at very low frequencies equals Re . Furthermore, when the input frequency isvery high, the branch with the speaker element will behave like an open circuit. The branch with the Zobelnetwork will behave like is has only a resistor, so the equivalent impedance at very high frequencies equalsRz . The equivalent impedance has to be independent of frequency. In other words, this impedance has tohave at very low frequencies the same value as at very high frequencies. Because of that reason, Rz is equal toRe .

To conclude, the values of the components for the Zobel network can be calculated with equation 35 and 36.Where Re and Le are the measured values of the resistor and inductor of the speakers.

Rzobel =Re = Zl (35) Czobel =LeR2e

(36)

Low-pass filterTo find the output voltage in the second order low-pass filter, which is shown in figure 2, the inductor and

capacitor are rewritten to impedances. The impedance of a capacitor can be written as ZC = 1jCLPF

and the

impedance of an inductor can be written as ZL = jLLPF . [1] Nodal analysis in node 1 leads to equation 37.Further calculations on this equation gives us equation 40.

Figure 2: A second order LPF

iL = iC + il s (37)

Uin UoutZL

= UoutZC

+UoutZl s

(38)

UinZL

=Uout (

1

ZL+ 1ZC

+ 1Zl s

)(39)

Uout = 1ZLZL

+ ZLZC

+ ZLZl s

Uin (40)

Substitution of ZL = jLLPF and ZC = 1jCLPF

in equation 40 leads to equation 41. By simplifying this

equation, equation 42 is made.

Uout = 11+ jLLPF(

1

jCLPF

) + jLLPFZl s

Uin (41)

A. Derivations of the equations 41

Uout = 112LLPFCLPF + jLLPF

Zl s

Uin (42)

Equation 43 is provided in the EPO book [3].

Uout = 1

1 2

20+ j

Q0

Uin (43)

Equation 43 together with equation 42 leads to the equations for 0 and Q. 0 = 1pLLPFCLPF

, which is both

the resonance and 3 dB frequency (in radians per second), and Q = Zl s

CLPFLLPF

, which is the quality factor

of the filter. With these two equations, the equations for LLPF and CLPF can be derived.

Calculations on the equation for 0 leads to equation 45.

LLPFCLPF = 1

0(44)

CLPF = 1LLPF20

(45)

Calculations on the equation of Q leads to equation 48.

CLPFLLPF

= QZl s

(46)

CLPFLLPF

= Q2

Z 2l s(47)

CLPF = LLPFQ2

Z 2l s(48)

The combination of equation 45 and equation 48 leads to equation 51.

1

LLPF20= LLPFQ

2

Z 2l s(49)

L2LPFQ220 = Z 2l s (50)

LLPF = Zl sQ 0

(51)

Substitution of equation 51 in equation 45 leads to equation 53.

CLPF = 1( Zl sQ 0

)20

= 1(Zl s 0

Q

) (52)

CLPF = QZl s 0

(53)

A. Derivations of the equations 42

Here the Q equals 1p2

, because that gives a critically damped response. Filling in what we know gives us the

values of the capacitor and inductor:

LLPF = 7.11p2170 2pi = 9.4 mH (54) CLPF =

1p2

7.1 170 2pi = 93 F (55)

High-pass filter

Figure 3: A second order HPF

To find the output voltage in the second order high-pass filter, which is shown in figure 3, the inductor and

capacitor are rewritten to impedances. The impedance of a capacitor can be written as ZC = 1jCHPF

and

the impedance of an inductor can be written as ZL = jLHPF . [1] Nodal analysis in node 1 leads to equation56. Further calculations on this equation gives us equation 59.

iC = iL + il s (56)

Uin UoutZC

= UoutZL

+UoutZl s

(57)

UinZC

=Uout(

1

ZC+ 1ZL

+ 1Zl s

)(58)

Uout = 1ZCZC

+ ZCZL

+ ZCZl s

Uin (59)

Substitution of ZL = jLHPF and ZC = 1jCHPF

in equation 59 gives equation 60.

Uout = 11 1

2LHPFCHPF+ 1

jCHPF Zl s

Uin (60)

Multiplication of equation 60 with2LHPFCHPF2LHPFCHPF

leads to equation 61.

Uout = 2LHPFCHPF

12LHPFCHPF + jLHPFZl s

Uin (61)

Equation 62 is provided in the EPO book [3].

A. Derivations of the equations 43

Uout =

2

20

1 2

20+ j

Q0

Uin (62)

Equation 61 together with equation 62 leads to the equations for 0 andQ. 0 = 1pLHPFCHPF

, which is both

the resonance and 3 dB frequency (in radians per second), and Q = Zl s

CHPFLHPF

, which is the quality factor

of the filter.

These two equations are the same for the LPF and the HPF, so the equations for LHPF andCHPF are the sameas well. The derivation of LLPF and CLPF can be found in section Low pass filter of this appendix.

LHPF = Zl sQ 0

(63) CHPF = QZl s 0

(64)

Filling in what we know gives us the values of the capacitor and inductor:

CHPF =1p2

4 950 2pi = 29.6 F (65)LHPF = 41p

2950 2pi = 948 H (66)

Band-pass filterThe bandpass filter consists of a second order low-pass and high-pass filter. If the high-pass frequency f HPF3dBand the low-pass frequency f LPF3dB are sufficiently far away from one another, the values of their componentscan be calculated as if the filters were separate. The frequencies differ enough when f LPF3dB > 5 f HPF3dB accordingto the product requirements. In this case, f LPF3dB = 950 Hz and f HPF3dB = 170 Hz. So the values of the compo-nents can be calculated as if the filters were separate. And like in the second order low-pass and high-passfilters the following equations hold for both low-pass and high-pass part of the band-pass filter:

L = Zl sQ 0

(67) C = QZl s 0

(68)

With Q = 1p2

and 0 = 2pi f3dB , the next four equations have to hold:

CHPF = 1Zl sp

8pi f HPF3dB(69) LHPF = Zl sp

2pi f HPF3dB(70)

LLPF = Zl sp2pi f LPF3dB

(71) CLPF = 1Zl sp

8pi f LPF3dB(72)

The values of the components of the BPF can be calculated as follows:

CHPF =1p2

4.1 170 2pi = 28.6 F (73)LHPF = 4.11p

2170 2pi = 979 H (74)

LLPF = 4.11p2950 2pi = 5.5 mH (75)

CLPF =1p2

4.1 950 2pi = 160 F (76)

A. Derivations of the equations 44

Volume adjustment

Figure 4: Volume adjustment

To calculate the output voltage of the circuit in figure 4 as a function of the input voltage, the equivalent

impedance has to be determined. This equivalent impedance is Zeq = R1 + (R2 Zl s) = R1 +Zl sR2Zl s +R2

. The

voltage over the speaker element Zl s is calculated as following.

Ul s =R2 Zl sZeq

Uin =Zl sR2Zl s +R2

R1+ Zl sR2Zl s +R2

Uin (77)

Multiplication of equation 77 withZl s +R2Zl s +R2

gives equation 78.

Ul s =Zl sR2

R1(Zl s +R2)+Zl sR2Uin (78)

As a result of this voltage divider, the power has decreased by a factor of .

= U2l s

U2in= (Zl sR2)

2

(R1(Zl s +R2)+Zl sR2)2(79)

The factor expressed in dB is as following.

= 10 log10U2l sU2in

= 20 log10 Ul sUin

= 20 log10 Zl sR2R1(Zl s +R2)+Zl sR2

(80)

To make sure that the input impedance of this circuit is equal to the speaker impedance an expression can bederived for R1 and R2.

R1 =Z 2l s

R2+Zl s(81) R2 =

Zl s(Zl s R1)R1

(82)

To get the equations of R1 and R2 as functions of and Zl s , the equation 80 has to be solved for the part insidethe log10.

Zl sR2R1(Zl s +R2)+Zl sR2

= 10(/20) (83)

Next equation 82 is substituted into equation 83 to get equation 84.

A. Derivations of the equations 45

10(/20) =Zl s

(Zl s(Zl s R1)

R1

)R1

(Zl s +

Zl s(Zl s R1)R1

)+Zl s

(Zl s(Zl s R1)

R1

) (84)

10(/20) =

(Zl s(Zl s R1)

R1

)R1+Zl s +R1+

Z 2l sR1

+Zl s=

(Zl s(Zl s R1)

R1

)Z 2l sR1

= Zl s R1Zl s

(85)

R1 = Zl s Zl s 10(/20) = Zl s(110(/20)

)(86)

Substituting equation 86 in equation 82 leads to the equation for R2.

R2 =Zl s(Zl s

(Zl s(110(/20)

))Zl s(110(/20)) = Zl s

(Zl s(110(/20))

)110(/20) =

Zl s10(/20)1 (87)

R2 = Zl s(

1

10(/20)1)

(88)

B. Measurement results 46

B. Measurement results

B.1. Loudspeaker analysis

Following are the results and graphs of the section measurements

Figure 5: Measurement results for the low-speaker

B. Measurement results 47

Figure 6: Measurement results for the mid-speaker

Figure 7: Measurement results for the high-speaker

B. Measurement results 48

Figure 8: Impedances for a 1uF capacitor, a 1mH inductor and 12, 56 and 100 ohm resistors

B. Measurement results 49

Figure 9: Frequency response graphs for LOW, MID and HIGH speakers

B.2. Passive filter design

Figure 10: The electric frequency response for all separate filters, with the speakers attached

Figure 11: The electric phase response for all separate filters, with the speakers attached

B. Measurement results 50

B.3. Power amplifier design

The measurement results of the power amplifier are shown in figures 12, 13 and 14

Figure 12: Result with the frequency in the passband

Figure 13: Result with the frequency below the passband

Figure 14: Result with the frequency above the passband

C. MATLAB codes and simulations 51

C. MATLAB codes and simulations

C.1. Loudspeaker analysis

Following are the results and MATLAB codes obtained in the section Simulations.

Re = 6Le = 4mHRp = 30Lp = 20mHCp = 1mF

Figure 15: The result of simulating models 1 and 2

R_e = [6];L_e = [0.004];R_p = [30];C_p = [0.001];L_p = [0.020];

f_vec = logspace(0, log10 (20000) , 10000);

%For convenience 's sake:s = 1i .* 2 .* pi .* f_vec ;Z_vec = R_e + s .* L_e + 1./(1./ R_p + 1./(s .* L_p) + (s .* C_p ));

Z_amplitude_vec = abs(Z_vec);Z_phase_vec = rad2deg(angle(Z_vec ));

subplot (2,1,1);semilogx(f_vec , Z_amplitude_vec );axis([0, 20000, 0, 30]);ylabel('Impedance amplitude [\ Omega]');xlabel('Frequency [Hz]');

subplot (2,1,2);semilogx(f_vec , Z_phase_vec );axis(0, 20000, -90, 90);ylabel('Impedance phase [\circ]');

C. MATLAB codes and simulations 52

xlabel('Frequency [Hz]');

Simulating model 1

graph_name = ['Woofer ';'Midrange '; 'Tweeter '];f_resonant = [62.3 , 86.61 , 1272];f_max = [12200 , 20000 , 20000];Z_f_resonant = [17.0 , 9.36, 4.908];Z_f_max = [30.0 , 22.0, 7.765];Z_dc = [7.1, 4.13, 4.0];graph_colour = ['r', 'b', 'g'];f_B_omega = [49.20 , 77.66 , 892.3; 75.44 , 94.40 , 1966];

B_omega = (f_B_omega (2,:) - f_B_omega (1,:)) .* 2 .* pi;R_e = Z_dc;L_e = abs((( ( (Z_f_max ).^2 - (R_e ).^2) ).^(1/2) )./( f_max .* 2 * pi));R_p = Z_f_resonant - Z_dc;C_p = 1./( B_omega .* R_p);L_p = 1./((( f_resonant .* 2 .* pi ).^2) .* C_p);

f_mat = zeros(length(R_e), 10000);for n = 1:( length(R_e))f_mat(n,:) = logspace(0, log10 (24000) , 10000);ends = 1i .* 2 .* pi .* f_mat;Z_mat = zeros(size(f_mat ));for n = 1: length(R_e)Z_mat(n,:) =R_e(n)+s(n ,:).* L_e(n)+1./((1./ R_p(n))+(1./(s(n ,:).* L_p(n)))+s(n,:).* C_p(n));endZ_ampl_mat = abs(Z_mat);Z_pha_mat = rad2deg(angle(Z_mat ));

subplot(2, 1, 1)for n = 1:( length(R\_e))semilogx(f\_mat(n,:), Z\_ampl\_mat(n,:), graph\_colour(n), 'Linewidth ', 2);hold onendhold offlegend(graph\_name , 'Location ', 'Best');set(gca , 'Fontsize ', 13);grid ongrid minorxlabel('Frequency (Hz)');ylabel('Impedance (\ textbackslash \{\} Omega)');title('Impedance as function of frequency ');axis([0, 24000, 0, 30]);

subplot(2, 1, 2)for n = 1:( length(R\_e))semilogx(f\_mat(n,:), Z\_pha\_mat(n,:), graph\_colour(n), 'Linewidth ', 2);hold onendhold offlegend(graph\_name , 'Location ', 'Best');;set(gca , 'Fontsize ', 13);grid ongrid minorxlabel('Frequency (Hz)');ylabel('Impedance phase (\ textbackslash \{\} circ)');title('Impedance phase as function of frequency ');axis([0, 24000, -90, 90]);

Calculating the component values of and simulating the speakers

C. MATLAB codes and simulations 53

Figure 16: The simulation of the actual loudspeaker

C.2. Passive filter design

The Matlab code used for the simulations of the filters can be seen below:

clear;

j = 1i;

f = 20 : 40000;w = f .* 2 .* pi;

f_l = 170;f_h = 950;

b_b = -1.75;b_h = -1.35;

Q = 1 ./ sqrt (2);

% Impedances speakersZ_sl = 7.1;Z_sm = 4.13;Z_sh = 4;

C. MATLAB codes and simulations 54

% LPFC_l = 92.7e-6;L_l = 9500e-6;R_Ll = 1.1;

Z_Cl = 1 ./ (j .* w .* C_l);Z_Ll = j .* w .* L_l + R_Ll;

V_l = Z_Cl .* Z_