Design of a 4-Way Passive Cross-Over Network

UNIVERSITY OF NAIROBI

SCHOOL OF ENGINEERING DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING

FINAL YEAR PROJECT REPORT

DESIGN OF A 4-WAY PASSIVE CROSS-OVER NETWORK

BY ODHIAMBO TONNY SILVANCE

REGISTRATION NUMBER: F17/1453/2011

SUPERVISOR: MR. S. L. OGABA

EXAMINER: PROF.ABUNG’U

This project was submitted as a partial fulfillment of the requirement for the award of Bachelor of Science Degree in Electrical and Information Engineering from University of Nairobi

DECLARATION OF ORIGINALITY

FACULTY/SCHOOL/INSTITUTE: ENGINEERING

DEPARTMENT: ELECTRICAL AND INFORMATION ENGINEERING

COURSE NAME: BACHELOR OF SCIENCE IN ELECTRICAL AND INFORMATION ENGINEERING

NAME: ODHIAMBO TONNY SILVANCE

REGISTRATION NUMBER: F17/1453/2011

COLLEGE: ARCHITECTURE AND ENGINEERING

PROJECT: DESIGN OF A 4-WAY PASSIVE CROSS-OVER NETWORK

PROJECT NUMBER: 112

1) I understand what plagiarism is and I am aware of the University policy on this regard.

2) I declare that this final year project is my original work and has not been submitted elsewhere for examination, award of degree or publication. Where other people’s work or my own work has been used, this has properly been acknowledged and referenced in accordance with University of Nairobi’s requirements.

3) I have not sought or used the services of any professional agencies to produce this work.

4) I have not allowed and shall not allow anyone to copy my work with the intention of passing it off as his/her own work.

5) I understand that any false claim in respect of this work shall result in disciplinary action, in accordance with University anti-plagiarism policy.

Signature:

……………………………………..……………………………………......

Date:

……………………………………..…………………………………………………. i

DECLARATION AND CERTIFICATION

This is my original work and has not been presented for any degree award in this or any other university. Information from other sources has been duly acknowledged.

…………………………………………………………………………

ODHIAMBO TONNY SILVANCE

F17/1453/2011

This report has been submitted to the Department of Electrical and Information Engineering, University of Nairobi with my approval as supervisor:

……………………………………………………………………………..

MR. S. L. OGABA

Date: ………………………….

DEDICATION I would like to dedicate this project to my parents, my uncle Dr.Jared Oule, my entire family and my supervisor for their support during the period of my project. You have been the drive and inspiration that has kept me on course and track in pursuit of this very interesting but rather demanding and challenging career path

iii

ACKNOWLEDGEMENTS I would like to thank the Almighty God for giving me a good health, strength and the ability to carry out this project. I would also like to thank my supervisor, Mr. S. L. Ogaba for the continuous guidance he has shown from when the project started all through to its completion. My sincere appreciation goes to my classmates and close friends for their views, opinions on various aspects of the project and their constructive criticism that enabled me deliver the final piece of work to the best of my ability.

ABBREVIATIONS SPL- Sound Pressure Levels AWG -American Wire Gauge OPAMP- Operational Amplifier mmf -Magneto motive force emf -Electromotive Force dB -Decibel Hz -Hertz Rms -Root mean square

Table of Contents

1 CHAPTER ONE- INTRODUCTION...... 1 1.1 BACKGROUND INFORMATION ...... 1 1.2 PROBLEM DEFINITION ...... 3 1.3 PROJECT JUSTIFICATION ...... 3 1.4 OVERALL OBJECTIVE ...... 4 1.5 SPECIFIC OBJECTIVES ...... 4 2 CHAPTER TWO- LITERATURE REVIEW ...... 5 2.1 Filter networks...... 5 2.1.1 Low-pass filters ...... 6 2.1.2 High pass filters ...... 6 2.1.3 Band-pass filters...... 7 2.1.4 Band-stop filters ...... 7 2.1.5 Butterworth response ...... 8 2.1.6 Chebyshev response ...... 9 2.1.7 Maximally flat time delay response (Bessel) ...... 10 2.1.8 COMPARISON ...... 11 2.2 Speakers ...... 12 2.2.1 SPEAKER PARAMETERS ...... 14 2.2.2 USABLE FREQUENCY RANGE ...... 17 2.2.3 POWER HANDLING ...... 18 2.2.4 SENSITIVITY ...... 18 2.2.5 SIGNAL-TO-NOISE RATIO (SNR) ...... 19 2.2.6 DRIVER SIZES [7] ...... 19 2.2.7 SPEAKER MODEL [8] [2] ...... 21 2.3 AUDIO CROSSOVER NETWORKS ...... 23 2.3.1 Active crossover...... 23 2.3.2 Passive crossover ...... 24 2.4 Components (inductors and capacitors) ...... 25 vi

2.4.1 Inductors ...... 25 2.4.2 CAPACITORS ...... 27 2.5 ADDITIONAL USEFUL CIRCUITS ...... 29 2.5.1 1.Zobel network [11] ...... 29 2.5.2 L-pad ...... 33 2.5.3 Series-notch filter ...... 33 2.5.4 Parallel notch (trap) filter ...... 33 2.6 WORKING MECHANISM OF A CROSSOVER NETWORK ...... 34 2.6.1 12dB crossover...... 34 3 CHAPTER 3- DESIGN ...... 36 3.1 Passive Crossover Circuit Design ...... 36 3.1.1 Selection of crossover frequency ...... 36 3.2 Component values determination ...... 38 3.2.1 Crossover points...... 38 3.2.2 Woofer (100W): ...... 39 3.2.3 Midrange1 (60W)...... 40 3.2.4 Midrange2 (30W)...... 40 3.2.5 Tweeter (10W) ...... 41 3.3 IMPEDANCE CURVES AND ZOBEL NETWORK DESIGN ...... 45 3.3.1 1. woofer (wf090wa02) ...... 46 3.3.2 specifications...... 46 3.3.3 midrange1(rs52an-8) ...... 47 3.3.4 specifications...... 47 3.3.5 midrange2 (nd105-8) ...... 48 3.3.6 specifications...... 48 3.3.7 tweeter (tw030wa14) ...... 49 3.4 THE FILTER NETWORKS FOR THE INDIVIDUAL SPEAKER DRIVERS ...... 51 3.4.1 woofer ...... 52 3.4.2 midrange1 ...... 52 3.4.3 midrange2 ...... 53 3.4.4 tweeter ...... 53 4 CHAPTER 4 - OBSERVATION AND RESULTS ...... 55 vii

4.1 impedance curves ...... 55 4.1.1 1.woofer ...... 55 4.1.2 midrange1 ...... 56 4.1.3 midrange2 ...... 57 4.1.4 tweeter ...... 58 4.2 FILTER NETWORKS SIMULATED RESPONSE ...... 59 4.2.1 woofer ...... 59 4.2.2 Midrange1 ...... 60 4.2.3 Midrange2 ...... 61 4.2.4 Tweeter ...... 62 4.3 SIMULATED RESULTS ...... 64 4.3.1 1.Woofer (baseband) ...... 64 4.3.2 2. Midrange1(lower midrange) ...... 65 4.3.3 Midrange2(higher midrange) ...... 66 4.3.4 4.Tweeter ...... 67 4.4 PRACTICAL DESIGN ...... 68 4.5 Practical results ...... 72 4.5.1 Woofer ...... 72 4.5.2 Lower midrange (midrange1) ...... 73 4.5.3 Higher midrange (midrange2)...... 74 4.5.4 Tweeter ...... 75 4.6 Analysis ...... 77 5 CHAPTER 5 ...... 81 5.1 Conclusion ...... 81 5.2 Recommendation ...... 81

viii

Table of Figures

Figure 1 low pass filter ...... 6 Figure 2 High pass filter ...... 7 Figure 3 Band pass filter ...... 7 Figure 4: Butterworth amplitude response ...... 9 Figure 5:Chebyshev amplitude response ...... 10 Figure 6:Bessel magnitude response ...... 11 Figure 7:comprarison of the magnitude response of Butterworth,Chebyshev and Bessel ...... 12 Figure 8:Speaker driver parts ...... 13 Figure 9:Speaker driver parts assembled in 3D ...... 14 Figure 10:Speaker impedance model ...... 21 Figure 11:Speaker driver electrical model ...... 22 Figure 12:Zobel network...... 30 Figure 13:Impedance curve with and without zobel network ...... 32 Figure 14:Series notch filter ...... 33 Figure 15:Parallel notch trap filter ...... 34 Figure 16:simulated circuit for impedance curves ...... 51 Figure 17:Woofer circuit...... 52 Figure 18:Midrange1 circuit ...... 52 Figure 19:midrange2 circuit ...... 53 Figure 20:Tweeter circuit ...... 53 Figure 21:Fabricated circuit ...... 54 Figure 22:Simulated woofer impedance curve ...... 55 Figure 23:simulated midrange1 impedance curve: ...... 56 Figure 24:Simulated midrange2 impedance curve ...... 57 Figure 25:simulated tweeter impedance curve ...... 58 Figure 26:simulated woofer frequency response ...... 59 Figure 27:simulated midrange1 frequency response ...... 60 Figure 28:simulated midrange2 response ...... 61 Figure 29:simulated tweeter frequency response ...... 62 ix

Figure 30:full network frequency response ...... 63 Figure 31:stage1 etching process ...... 68 Figure 32:stage2 etching process ...... 69 Figure 33:stage 3 etching process ...... 70 Figure 34:Final stage etching process ...... 71 Figure 35:complete soldered circuit...... 71 Figure 36:speaker cabinets ...... 76 Figure 37:practical woofer frequency response ...... 77 Figure 38:practical midrange1frequency response ...... 77 Figure 39:practical midrange2 frequency response ...... 78 Figure 40:practical tweeter frequency response ...... 78 Figure 41:full network frequency response ...... 80

List of tables

Table 1:Filter characteristics ...... 11 Table 2:Crossover frequency range for various systems ...... 36 Table 3:crossover frequency and speaker power proportions ...... 37 Table 4:Capacitor calculated,measured and standard values ...... 42 Table 5: Inductor calculated and measured values ...... 44 Table 6: Electrical and Mechanical parameters of the simulated speaker drivers ...... 50 Table 7:Simulated woofer frequency response ...... 64 Table 8:simulated midrange1 frequency response ...... 65 Table 9:Simulated midrange2 frequency response ...... 66 Table 10:simulated tweeter frequency response ...... 67 Table 11:Practical woofer frequency response results ...... 72 Table 12:Practical midrange1 frequency response results ...... 73 Table 13:practical midrange2 frequency response results ...... 74 Table 14:practical tweeter frequency response results ...... 75

ABSTRACT

The project involves the design of a 4-way cross-over network that employs four second order Butterworth filter networks to split the audible frequency range into four separate frequency sound bands which include the low frequency bass, which operates between 20Hz and 400Hz,two midranges between 500Hz and 5000Hz for both and the high frequency treble above 5000Hz .The separate frequency bands are then directed to the speaker drivers optimized to handle them and they are woofer, lower mid-range(midrange1),higher mid-range(midrange 2)and tweeter respectively. The most common driver nominal impedances are 4ohms and 8 ohms for the speaker drivers but in this case 8 ohm speakers were used in the design. Resistors were used at the output to represent the speaker drivers in the simulation but in the practical design and demonstration, actual speaker drivers in their cabinets were used. Commercially available capacitor values which slightly differed from the actual design values were used in the simulation and final fabrication. The inductors were successfully manually coiled after their values had been determined from design calculations.

Designed circuits are simulated using Microcap and TINA TI soft wares, and then implemented on PCB.

The keywords: Woofer, Tweeter, Mid-range1, Mid-range2, Pass-band, Stop-band and Butterworth

xii

1 CHAPTER ONE- INTRODUCTION 1.1 BACKGROUND INFORMATION The audible frequency range is generally taken to be between 20 Hz for the lower limit and 20 kHz for the upper limits. This basically means that human ear can only perceive sound within this bandwidth and anything above and below is not possible. This audible range is defined by the Fletcher-Munson curves. [1]

Ideally, a single loudspeaker should reproduce the full audible frequency range without any detectable distortion, but this is unfortunately not easily possible and although good full-range loudspeakers do exist, that is those that reproduce the full audible frequency range, the frequency range of a full-range loudspeaker is limited with weak bass and unsatisfactory treble, the frequency response is irregular or at least compromised by the directivity at high frequencies and it is difficult to keep distortion low when the same diaphragm is used for bass and treble. The one and only way of distortion reduction is decreasing diaphragm excursion, but this requires an increase of diaphragm area to compensate for the lost sound pressure; and enlarging loudspeaker size worsens high frequency reproduction. It all boils down to a requirement of loudspeakers optimized for reproduction of a limited frequency range and thus the need for a frequency dividing network, and this is the genesis of crossover networks. [2]

In the simplest sense, a crossover is a device that separates the audio spectrum into different ranges and sends them to specific drivers. The crossover is responsible for sending bass information to the woofer, midrange information to the midrange, and treble to the tweeter. In a world where a single driver could easily and faithfully reproduce the entire audio spectrum, the use of a crossover would not be necessary. However there are a number of real-world problems that make the use of crossovers requisite. The prime reason is that multiple drivers are usually needed to cover the entire audio spectrum. It is very difficult to manufacture a driver that is capable of producing both high and low frequencies simultaneously. Various types of drivers are designed to perform well in different ranges, once outside of their optimum range they begin to operate poorly and erroneously. The use of a crossover can prevent corrupt information from being produced outside of a driver's operating range. Most tweeters and many smaller drivers would actually be physically damaged or destroyed by sending low frequency information to

1 them and woofers cannot move fast enough to generate higher frequencies. A crossover network can be used in a limitless number of ways to help tailor frequency response. There are notch filters which can remove peaks in response, conjugate networks to flatten impedance curves, and a myriad of other possible filters. All of these various crossover circuits can be used to help reach the goal of a flat frequency response. [2]

Crossovers use a combination of electrical high-pass and low-pass filters to separate the frequency band. A low-pass filter allows low frequency signals to pass without attenuation, but will attenuate signals above a certain frequency. A high-pass filter will allow high-frequency signals to pass without attenuation, but will attenuate signals below a certain frequency. When a low-pass filter on a woofer and a high-pass filter on a tweeter are combined, a smooth transition from woofer to tweeter can be accomplished. Unfortunately, a passive crossover filter cannot act with an infinitely steep slope; it produces a gradual roll-off. The high-pass and low-pass crossover points and slopes must be carefully combined to produce a flat response between drivers.

In the broadcast sense, crossovers can be classified by the number of bands into which the audio spectrum is divided. A two-way crossover separates the audio spectrum into two portions and sends the information to two different types of drivers. A three-way crossover separates the audio spectrum into three portion, a four- way crossover separates the audio spectrum into four portions and so on. [3]

In terms of components used and construction architecture, there two types of crossover networks and they include:

1. Active crossover network

2. Passive crossover network

In active crossover networks, the power amplifier which is used to drive the network is located between the network and the speaker driver and this basically means that each driver is be fed by its own power amplifier and so in a nutshell, active crossovers contain active circuit elements for instance op-amps. In passive crossover networks, the power amplifier driving the network is located before the network itself and so just one power amplifier is necessary unlike the latter.

Passive crossover networks are made of passive circuit’s elements which mainly are inductors and capacitors that make up the filter network.

The passive crossover network is currently the most used approach but the active crossover network is expected to be increasingly popular in the near future since high quality power amplifiers are becoming a serious alternative to the linear power amplifiers of today.

Terms often used to describe the slope of a crossover include 6dB/octave, 12dB/octave, 18dB/octave, or 24dB/octave. The crossover slope that these terms refer to is just as you would imagine. With a change of one octave, a 6dB/octave crossover will have an output that is 6 dB down from the beginning point; 12 dB/octave will have an output that is 12dB down.

Another set of terms that are often used to describe a crossover slope are 1st order, 2nd order, 3rd order, and 4th order. These terms are derived from the number of components that are needed to produce the described slope. A 1st order crossover uses 1 component, and will yield roughly a 6 dB/octave cutoff. A 2nd order crossover uses 2 components, and will yield roughly a 12 dB/octave cutoff, etc.

1.2 PROBLEM DEFINITION Loudspeaker systems as discussed above, cannot efficiently operate without crossover networks and hence their proper design cannot be overlooked. Commercially available are 2-way and 3- way crossover networks which basically employ the use of two (woofer and tweeter) and three drivers (woofer, midrange and tweeter) respectively. It is realized that the more the speaker drivers optimized to handle certain frequencies within the audible frequency band, the better the sound quality and thus the need to design a four way crossover network.

1.3 PROJECT JUSTIFICATION A 4- way crossover network means that four speaker drivers (woofer, midrange1, midrange2 and tweeter) optimized to reproduce sound within the audible frequency range are used and this eventually leads to a more efficient and higher quality sound production. The use of a passive network means that circuit complexity is reduced and thus less costly to produce and noise is extremely minimal since the circuit has no active elements where noise is inherent.

1.4 OVERALL OBJECTIVE This project focuses on the design of a 4-way passive cross-over network

1.5 SPECIFIC OBJECTIVES 1. To determine the crossover frequency based on the output power versus frequency relation and the sound frequency bandwidth of particular speaker drivers

2. To choose the speaker driver’s nominal impedance to use in the design

3. To evaluate, after derivation of the relevant equations, the network’s component values that is, the capacitors and inductors based on the crossover frequency, speaker driver nominal impedance and finally the speaker driver’s power ratings.

4. To research on and determine the formula used to evaluate inductance based on the physical parameters and thereafter manually coil the inductors to be used based on this formula.

2 CHAPTER TWO- LITERATURE REVIEW 2.1 Filter networks A filter is a network designed to pass signals having frequencies within certain bands (called passbands) with little attenuation, but greatly attenuates signals within other bands (called attenuation bands or stopbands). Between the pass band of a filter, where ideally the attenuation is zero, and the attenuation band, where ideally the attenuation is infinite, is the cut-off frequency, this being the frequency at which the attenuation changes from zero to some finite value. Filters are classified in a number of ways and the first to be considered is classification based on the circuit components which leads to active and passive filters. Passive filters are those that consist of combinations of resistance, capacitance and inductance and so basically they consist of passive circuit elements and do not contain any source of power. Capacitors block low-frequency signals and conduct high frequency signals and inductors do the reverse. Resistors on their own have no frequency –selective properties, but are added to inductors and capacitors to determine the time-constants of the circuit, and therefore frequencies to which they respond. Passive RLC structures are capable of achieving relatively good filter characteristics in applications ranging from the audio frequency range to the upper limit of the lumped parameter range. Problem occurs with passive RLC filters at the lower end of the audio frequency range since inductance values increase as the required frequency decreases creating several problems. Firstly, inductors are somewhat imperfect devices due to internal losses and these losses increased markedly in the very large inductance range required at low frequencies. These losses terribly degrade the quality factor for each coil and the associated filter responses have large deviations from the desired form. Second, the actual physical sizes of the large inductance values limit their usefulness and lastly their cost are certainly not trivial. Active filters are theoretically capable of achieving the same response as passive RLC filters and since inductors are not required, the problems associated with inductance at low frequency are eliminated but they do have a few problems of their own. Since they are active, power is required to operate them, they add noise to the response and are highly susceptible to instability since they employ feedback which consist of combinations of resistors and capacitors and one or more active

5 devices such as op amps employing feedback. They contain power sources since they have active devices in their circuitry. Our focus is however on the passive filters and not active. [4] [5] Filters can be categorized according to frequency ranges as: 2.1.1 Low-pass filters A low-pass filter is one designed to pass signals at frequencies below a specified cut-off frequency. A typical one consists of an inductor in series with the signal and a capacitor shunting the signal to the ground. At high frequencies, the inductive impedance increases while the converse is true for capacitive impedance and as a result, depending on the designed cut of frequency, frequencies higher that this value experience high impedance and are thus blocked or attenuated by the inductor and those that manage to pass are shunted to the ground by the capacitor thus overally, the capacitor, inductor combination realize the functionality of this type

Figure 1: low pass filter

2.1.2 High pass filters This is a filter designed to pass signals at frequencies beyond a certain point and attenuate frequencies below that point(cut off frequency).A typical one consists of a capacitor in series with the signal path and an inductor shunting the signal path to the ground. Frequencies below the cut off frequency experience a very high capacitive impedance which increases as frequency decreases and those that manage to pass are further directed to the ground through the shunting inductor whose impedance is low at low frequencies and hence only frequencies beyond the cut off are allowed to pass. A typical high pass filter is shown in figure2 below.

Figure 2: High pass filter

2.1.3 Band-pass filters A band-pass filter is one designed to pass signals with frequencies between

Two specified cut-off frequencies. Such a filter may be formed by cascading a high-pass and a low-pass filter. 푓퐶퐻 Is the cut-off frequency of the high-pass filter and푓퐶퐿 is the

Cut-off frequency of the low-pass filter. 푓퐶퐿 Is greater than 푓퐶퐻since it’s the high pass filter that first meets the signal to reject frequencies below the cut off frequency then the low pass to reject frequencies beyond the cut off frequency for the low pass filter so basically the band pass is defined by the difference between the low pass and high pass cut off frequencies.

Figure 3 :Band pass filter

2.1.4 Band-stop filters A band stop filter is one designed to block or attenuate signals with frequencies between two specified cut off frequencies. Such a filter is formed by cascading a low pass and a high pass filter as shown below. The cut off frequency of the high pass filter is higher than that of the low pass filter since the signal first encounters the low pass filter which allows frequency up to its cut off frequency to pass and blocks the rest then the high pass filter that allows frequencies above

7 its cut off frequencies o pass and blocks the rest and so basically the stop band is defined by the difference between the high pass and low pass cut of frequencies. In the view of the non-ideal nature of filter responses, one method of classifying filters is according to the type of approximation to the block characteristic employed [4].the amplitude response forms for a few of the major types using low pass characteristics are: 2.1.5 Butterworth response The form is illustrated in the diagram below. Butterworth amplitude response is also referred to as maximally flat amplitude response because of the mathematical structure of its development. Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband, very useful for audio signals that easily get affected by distortions that occur due to ripples that are almost always never present with this type of filter. The Butterworth utilize a convenient reference frequency at which the amplitude response drops to 1/√2 of its maximum pass band level corresponding to the response being down 3dB this frequency is the cut off frequency and it should be understood that it is not an abrupt cut off. Let fc represent the cut off frequency and n represent the order (number of poles) of the approximation. The amplitude response of the M (w) of the Butterworth low pass function is given by

푴(풘) = ퟏ ÷ √(ퟏ + (풇/풇풄)^ퟐ풏)

The maximum value of M (w) occurs at f=0 and it has been established as unity for convenience. The amplitude response for this high pass filter has essentially the same cut off frequency as for low pass filters except of course that it is at the low end of the pass band and the response is also 3db down at this point. The amplitude response m (w) of the Butterworth high pass function is given by:

푴(풘) = ퟏ ÷ √((풇풄/풇)^ퟐ풏)

Figure 4: Butterworth amplitude response

2.1.6 Chebyshev response The form of one particular Chebyshev amplitude characteristic is shown below. Chebyshev response is referred to as an equiripple response because the pass band is characterized by a series of ripples that have equal maximum levels and equal minimum levels. The number of ripples is a function of the number of reactive elements in the design. Chebyshev filters have a sharper slope than Butterworth filters and are thus capable of achieving more attenuation at the stop band for a given number of reactive elements. However, there time delay and phase characteristic are less ideal than those of the Butterworth filters and they tend to exhibit a ringing effect with transient signals which is not very good for audio signal processing.

Figure 5:Chebyshev amplitude response

2.1.7 Maximally flat time delay response (Bessel) A different approach to the approximation problem is that of the maximally flat time delay (MFTD) filter. With the mftd filter, the phase response is optimized so that all frequency components have nearly constant time delay through the filter.at first glance, this response resembles that of the Butterworth as response decreases as the frequency increases but compared closely to the Butterworth, the pass band amplitude response is not as constant and the attenuation is not as high in the stop band. MFTD filters are used in phase sensitive applications where constant time delay is very important but the attenuation requirements are moderate.

Figure 6:Bessel magnitude response

2.1.8 COMPARISON There are various other approximations some of which include Cauer-elliptic, Gaussian, Thompson, Equi-ripple Group Delay but these three types are among the most widely employed. Within the group of this filter characteristics, the Chebyshev amplitude response has the sharpest rate of attenuation increase above the cut off but its phase and time delay characteristics are the poorest. In contrast, the MFTD filter has the most ideal time delay and phase characteristics but its amplitude response is the poorest. The Butterworth filter is a reasonable compromise between these extremes and is as a result a very popular choice and since in addition to this they can achieve any basic form of filtering, it was used in the project. Table 1 below summarizes the above characteristics

Table 1: Filter characteristics

Filter Main Other Q-factor characteristics characteristics

Butterworth Maximally flat _ 0.707 amplitude

Bessel Maximally flat Fastest settling 0.5 to 0.707* phase time

Chebyshev/Chebys Fastest roll-off Slight peaks/dips 0.8 to 1.2 hev

Figure 7: comprarison of the magnitude response of Butterworth, Chebyshev and Bessel

2.2 Speakers Speaker driver or speaker as commonly known is an electrical transducer that converts electrical energy to sound energy. They generally consist of the following parts [6] [7]:

Permanent magnet-This is a magnet that acts as the motor of the loudspeaker.it provides the energy that’s needed by the voice coil to move

Pole pieces-these are used to focus the magnetic field so that it is strongest around the voice coil

Basket-this is the chassis of the driver unit all elements are attached and which itself bolts into the cabinet.

Suspension Spider and voice coil-the spider holds the voice coil centrally within the magnet and acts as a spring to bring it back after each pulse. The speaker cone is attached to the voice coil which sits in a magnetic field and moves when an electric pulse passes through it according to Fleming’s left hand rule or motor rule. Variations in the signal make the coil vibrate in the drive

12 in a pistonic motion which produces sound by resonating airwaves in the room much like how a drum operates

Drive cone and surround-many different materials are used here for mid and low frequencies.Varlar, aluminium, paper, and polypropylene are very popular choices.

Mounting ring-this cosmetic device hides the raw alloy of the basket when it’s mounted in the cabinet.

Phase plug-this is not common to all drivers but for those drivers where present, it is designed to avoid phase charges. A dust cap is also included at this point to prevent particles from entering it.

Figure 8: Speaker driver parts

Figure 9: Speaker driver parts assembled in 3D

2.2.1 SPEAKER PARAMETERS Fundamental small signal mechanical parameters

•SD – projected area of driver diaphragm (m2)

• Mms – mass of diaphragm (kg)

• Cms – compliance of driver’s suspension (m/N)

• Rms – mechanical resistance of driver’s suspension

(N•s/m)

• L – voice coil Le inductance (mH)

• Re – DC resistance of voice coil (Ω)

• Bl – product of magnetic field strength in voice coil gap and length of wire in magnetic field (T•m)

2.2.1.1 FS [8] This parameter is the free-air resonant frequency of a speaker. Simply stated, it is the point at which the weight of the moving parts of the speaker becomes balanced with the force of the speaker suspension when in motion. If you’ve ever seen a piece of string start humming uncontrollably in the wind, you have seen the effect of reaching a resonant frequency. It is important to know this information so that you can prevent your enclosure from ‘ringing’. With a loudspeaker, the mass of the moving parts, and the stiffness of the suspension (surround and spider) are the key elements that affect the resonant frequency. As a general rule of thumb, a lower Fs indicates a woofer that would be better for low-frequency reproduction than a woofer with a higher Fs. This is not always the case though, because other parameters affect the ultimate performance as well.

2.2.1.2 RE This is the DC resistance of the driver measured with an ohm meter and it is often referred to as the ‘DCR’. This measurement will almost always be less than the driver’s nominal impedance. Consumers sometimes get concerned the Re is less than the published impedance and fear that amplifiers will be overloaded. Due to the fact that the inductance of a speaker rises with a rise in frequency, it is unlikely that the amplifier will often see the DC resistance as its load.

2.2.1.3 LE This is the voice coil inductance measured in millihenries (mH). The industry standard is to measure inductance at 1,000 Hz. As frequencies get higher there will be a rise in impedance above Re. This is because the voice coil is acting as an inductor. Consequently, the impedance of a speaker is not a fixed resistance, but can be represented as a curve that changes as the input frequency changes. Maximum impedance (Zmax) occurs at Fs.

2.2.1.4 Q PARAMETERS Qms, Qes, and Qts are measurements related to the control of a transducer’s suspension when it reaches the resonant frequency (Fs). The suspension must prevent any lateral motion that might allow the voice coil and pole to touch (this would destroy the loudspeaker). The suspension must also act like a shock absorber. Qms is a measurement of the control coming from the speaker’s mechanical suspension system (the surround and spider). View these components like springs. Qes is a measurement of the control coming from the speaker’s electrical suspension system (the voice coil and magnet). Opposing forces from the mechanical and electrical suspensions act to absorb shock.Qts is called the ‘Total Q’ of the driver and is derived from an equation where Qes is multiplied by Qms and the result is divided by the sum of the same.

2.2.1.5 VAS/CMS Vas represents the volume of air that when compressed to one cubic meter exerts the same force as the compliance (Cms) of the suspension in a particular speaker. Vas is one of the trickiest parameters to measure because air pressure changes relative to humidity and temperature — a precisely controlled lab environment is essential. Cms is measured in meters per Newton. Cms is the force exerted by the mechanical suspension of the speaker. It is simply a measurement of its stiffness. Considering stiffness (Cms), in conjunction with the Q parameters gives rise to the kind of subjective decisions made by car manufacturers when tuning cars between comfort to carry

15 the president and precision to go racing. Think of the peaks and valleys of audio signals like a road surface then consider that the ideal speaker suspension is like car suspension that can traverse the rockiest terrain with race-car precision and sensitivity at the speed of a fighter plane. It’s quite a challenge because focusing on any one discipline tends to have a detrimental effect on the others.

2.2.1.6 VD This parameter is the Peak Diaphragm Displacement Volume — in other words the volume of air the cone will move. It is calculated by multiplying Xmax (Voice Coil Overhang of the driver) by Sd (Surface area of the cone). Vd is noted in cc. The highest Vd figure is desirable for a sub-bass transducer.

2.2.1.7 BL Expressed in Tesla meters, this is a measurement of the motor strength of a speaker. Think of this as how good a weightlifter the transducer is. A measured mass is applied to the cone forcing it back while the current required for the motor to force the mass back is measured. The formula is mass in grams divided by the current in amperes. A high BL figure indicates a very strong transducer that moves the cone with authority!

2.2.1.8 MMS This parameter is the combination of the weight of the cone assembly plus the ‘driver radiation mass load’. The weight of the cone assembly is easy: it’s just the sum of the weight of the cone assembly components. The driver radiation mass load is the confusing part. In simple terminology, it is the weight of the air (the amount calculated in Vd) that the cone will have to push.

2.2.1.9 EBP This measurement is calculated by dividing Fs by Qes. The EBP figure is used in many enclosure design formulas to determine if a speaker is more suitable for a closed or vented design. An EBP close to 100 usually indicates a speaker that is best suited for a vented enclosure. On the contrary, an EBP closer to 50 usually indicates a speaker best suited for a closed box design. This is merely a starting point. Many well-designed systems have violated this rule of thumb! Qts should also be considered.

2.2.1.10 XMAX/XLIM Short for Maximum Linear Excursion. Speaker output becomes non-linear when the voice coil begins to leave the magnetic gap. Although suspensions can create non-linearity in output, the point at which the number of turns in the gap (see BL) begins to decrease is when distortion starts to increase. Eminence has historically been very conservative with this measurement and indicated only the voice coil overhang (Xmax: Voice coil height minus top plate thickness, divided by 2). The Xmax figures on this website are expressed as the greater of the result of the formula above or the excursion point of the woofer where THD reaches 10%. This method results in a more real world expression of the usable excursion limit for the transducer. Xlim is expressed by Eminence as the lowest of four potential failure condition measurements: spider crashing on top plate; Voice coil bottoming on back plate; Voice coil coming out of gap above core; or the physical limitation of cone. A transducer exceeding the Xlim is certain to fail from one of these conditions. High pass filters, limiters, and enclosure modeling software programs are valuable tools in protecting your woofers from mechanical failure.

2.2.1.11 SD This is the actual surface area of the cone, normally given in square cm.

2.2.2 USABLE FREQUENCY RANGE This is the frequency range for which Eminence feels the transducer will prove useful. Manufacturers use different techniques for determining ‘Usable Frequency Range’. Most methods are recognized as acceptable in the industry, but can arrive at different results. Technically, many loudspeakers are used to produce frequencies in ranges where they would theoretically be of little use. As frequencies increase, the off-axis coverage of a transducer decreases relative to its diameter. At a certain point, the coverage becomes ‘beamy’ or narrow like the beam of a flashlight. If you’ve ever stood in front of a guitar amplifier or speaker cabinet, then moved slightly to one side or the other and noticed a different sound, you have experienced this phenomenon and are now aware of why it occurs. Clearly, most two-way enclosures ignore the theory and still perform quite well. The same is true for many guitar amplifiers, but it is useful to know at what point you can expect a compromise in coverage.

2.2.3 POWER HANDLING This specification is very important to transducer selection. Obviously, you need to choose a loudspeaker that is capable of handling the input power you are going to provide. By the same token, you can destroy a loudspeaker by using too little power. The ideal situation is to choose a loudspeaker that has the capability of handling more power than you can provide lending some headroom and insurance against thermal failure. To use an automobile as an analogy; you would not buy a car that could only go 55mph if that were the speed you always intended to drive. Generally speaking, the number one contributor to a transducer’s power rating is its ability to release thermal energy. This is affected by several design choices, but most notably voice coil size, magnet size, venting, and the adhesives used in voice coil construction. Larger coil and magnet sizes provide more area for heat to dissipate, while venting allows thermal energy to escape and cooler air to enter the motor structure. Equally important is the ability of the voice coil to handle thermal energy. Eminence is renowned for its use of proprietary adhesives and components that maximize the voice coil’s ability to handle extreme temperatures. Mechanical factors must also be considered when determining power handling. A transducer might be able to handle 1,000W from a thermal perspective, but would fail long before that level was reached from a mechanical issue such as the coil hitting the back plate, the coil coming out of the gap, the cone buckling from too much outward movement, or the spider bottoming on the top plate. The most common cause of such a failure would be asking the speaker to produce more low frequencies than it could mechanically produce at the rated power. Be sure to consider the suggested usable frequency range and the Xlim parameter in conjunction with the power rating to avoid such failures. The Eminence power rating is derived using an EIA 426A noise source and test standard. All tests are conducted for eight hours in a free-air, non-temperature controlled environment. Eminence tests samples from each of three different production runs and each sample must pass a test exceeding the rated power by 50 to 100W. The Eminence music program is double that of our standard Watts rating.

2.2.4 SENSITIVITY This data represents one of the most useful specifications published for any transducer. It is a representation of the efficiency and volume you can expect from a device relative to the input power. Loudspeaker manufacturers follow different rules when obtaining this information —

18 there is not an exact standard accepted by the industry. As a result, it is often the case that loudspeaker buyers are unable to compare ‘apples to apples’ when looking at the sensitivities of different manufacturers’ products. Eminence sensitivities are expressed as the average output across the usable frequency when applying 1W/1M into the nominal impedance. ie: 2.83V/8 ohms, 4V/16 ohms.

2.2.5 SIGNAL-TO-NOISE RATIO (SNR) The sound that a speaker produces includes some level of noise.

In other words, audio signals are sent to a speaker which are then converted into the sound (via internal driver movements) that we hear. But the sound that we hear are not purely audio signals that a speaker gets, in fact, it also includes some level of noise. This noise is added by internal components of the speaker/device.

Therefore, this spec describes how much noise is there in the output (sound that we hear) of a device in relation to the signal level. It is also expressed in decibels (dB).

So if a speaker has 120dB of Signal-to-Noise Ratio, it means that that the level of the audio signal is 120dB higher than the level of the noise. The higher the number, the better it is

2.2.6 DRIVER SIZES [7]

2.2.6.1 1. Tweeters Tweeters are high frequency drivers that are designed to reproduce the highest octaves of the frequency spectrum. Typically, this is from 2,000Hz to 20,000Hz, while some tweeters will reproduce frequencies as low as 1500Hz. These drivers are smaller and usually dome-shaped. The membrane on dome tweeters usually measures 0.5″-1.25″ in diameter. There are several other different types of tweeters that are used, including ribbon tweeters, planar tweeters, horn loaded tweeters, and so forth, all of which have a slightly different sound. Dome tweeters are the most popular, and come in a variety of materials, including poly (such as Mylar), silk (and other treated fabrics), aluminum, and ceramic. Each of these also has a different sound but hey generally serve the same purpose. Due to limited size and very limited excursion, tweeters

19 cannot play frequencies below their frequency range without sounding fatigued, distorted, or harsh.

2.2.6.2 2. Midrange Midrange drivers are designed to reproduce frequencies between a woofer and a tweeter. However, not all speaker systems have them, as they are often not required. Mids. Will pick up where the woofer drops off, and drop off where the tweeter picks up. This frequency range is typically from 500Hz to 2000Hz. These are typically 3″-6″ in diameter, and are most commonly typical cones, although some dome midrange drivers do exist.

2.2.6.3 3. Woofers Woofers are drivers that are designed to reproduce a variety of frequency ranges. In a 3-way system with a midrange, they will play below what the midrange plays, but in a 2-way system, they will play below what the tweeter plays. This depends heavily on the design and purpose of a given woofer. These are sometimes also known as mid-woofers or mid-bass drivers. How high and low in the frequency range a woofer will play depends heavily on the design and material used. Some woofers are capable of playing down to as low as 30 Hz musically, effectively eliminating the need for a subwoofer where high output isn’t needed. These are typically between 5″ and 15″ in size, with the larger varieties found in pro audio applications.

2.2.6.4 4. Subwoofers Subwoofers are specifically designed to reproduce bass frequencies. This is typically between 20Hz and 125Hz. Due to their size and design, they are rarely able to play above mid-bass frequencies, with some exceptions existing in pro audio. Subwoofers are designed with higher excursion capabilities in order to create enough sound pressure and move enough air to reproduce frequencies down to 20Hz.

2.2.6.5 5. Full Range Speaker Drivers Full range drivers are drivers that are designed to reproduce a large frequency range, although their ability to reproduce a wide range of frequencies is much compromised. These range from anywhere between 8″ and 3″ in diameter. Their ability to reproduce high frequency sound is often lacking, and many of them suffer from a “beaming” effect. While dome tweeters are able to reproduce high frequencies (which are very directional) over a significant area, full range drivers

20 function more like a flashlight that focuses the frequency response of that beam directly at the area they are pointing to. For you as the listener, this means that you need to be sitting in a specific location and have the speakers pointed directly at you in order to get the best sound you can, while 2-way and 3-way speaker systems are much more forgiving.

2.2.7 SPEAKER MODEL [8] [2] The single most dominant branch of the model is the voice coil DC resistance, re. It's going to be in series with everything else we will look at (you mentioned "stray capacitance". Yes, there is some, but its magnitude is absolutely miniscule compared to all other components so it can be ignored).Next we have the voice coil inductance (we'll call it Lvc). Now, it, too, is in series with everything else, but it's no simple inductance. So far, we have the two real electrical components, as shown in figure 10 below

Figure 10: Speaker impedance model

Now, the next major set of components are the electrical equivalents of the major mechanical components of suspension compliance, cone mass and suspension losses. The suspension compliance is modelled as an inductor, Lces. The cone mass is modelled as a capacitance, Cmes, and the suspension losses are modelled as a resistor, Res. These three are in parallel and form a damped, parallel resonant branch called the driver mechanical branch. Finally, in series with that is the radiation impedance. No single lumped-parameter synthesis comes close to approximating this. Also the magnitude of the impedance of this branch is small compared to the others, so for simulating the electrical characteristics, it can be safely eliminated.

The driver electrical model, then, is as shown in figure 11 below

Figure 11: Speaker driver electrical model

Now, the relative values of these components depends upon the magnitudes of the physical values times a transformation factor. That transformation factor is the electromagnetic transduction

Factor, proportional to the Bl product (the product of the length of the wire l immersed in the magnetic field B), measured in N/A (or T/M, if you will). So, if we know the magnitudes of the physical components, we can easily calculate their electrical equivalents:

1.Lces -Depends upon the suspension compliance: Lces = Cms ∗ (Bl)2 Where Cms is the mechanical compliance in m/N, and the resulting Inductance is in Henries.

2.Cmes -Depends upon the cone mass Cmes = Mms/(Bl)2 Where Mms is the mechanical compliance in kg, and the resulting Capacitance is in farads

3.Res -Depends upon the suspension losses and its equivalent to (Bl)2/Rms Where Rms is the mechanical losses in 1/s, and the resulting Resistance is in ohms.

4.Xrs- Depends upon the air, the driver diameter, the baffle dimensions, position of the driver on the baffle, etc., but has little effect on the electrical impedance

2.3 AUDIO CROSSOVER NETWORKS As mentioned in [2], The loudspeakers are driven from power amplifiers, which can either be located before the crossover network; the conventional approach using passive crossover networks, or the power amplifier can be located between the crossover network and the loudspeaker; thus requiring an amplifier for each loudspeaker. Each has its own advantages and disadvantages [10]

2.3.1 Active crossover

2.3.1.1 Advantages 1) Direct control of each driver by its own amplifier.

2) Easier impedance load on the amp.

3) No loss of power or damping factor.

4) Reduced clipping. If clippings occurs, only one driver/amp is affected.

5) Crossover works at line level maintaining its design properties.

6) Each amp deals with only a specific bandwidth.

7) Reduced harmonic distortion.

8) Reduced intermodulation distortion

2.3.1.2 Disadvantages 1) Residual noise from X/O, less of an issue for digital processors.

2) Greater susceptibility to EMI (electromagnetic interference), and RFI (radio frequency interference).

3) Multiple amps & cables, more complex setup.

2.3.2 Passive crossover

2.3.2.1 Advantages 1) Plug & Play simplicity.

2) One amp, one cable, done.

3) Can handle large currents and voltages

4) No power supply required

5) They are not restricted by the bandwidth limitations of the OPAMPs; they can work well at very high frequencies

6) Very reliable

7) Requires least number of components for a given filter

8) produce less noise only thermal noise from the resistors

2.3.2.2 Disadvantages 1) Back EMF (electro motive force) goes back into the crossover, interferes with the input signal from the amplifier.

2) Passive crossover buffers the amplifier from the drivers resulting in loss of damping, loss of direct amplifier control over the drivers.

3) Loading effects, inductors, magnetic coupling, larger Cs, and parameters less adjustable.

4) Passive network wastes power, lowers efficiency, requires higher wattage amplifier to compensate.

5) Differing impedance of various drivers and the resulting phase shifts from the crossover present a difficult load for the amplifier, especially 1st order crossovers.

6) Crossover properties and accuracy varies with power and temperature resulting in shifting properties and inconsistent linear response.

7) Low order crossover reduces phase & time shifts but introduces other issues. Greater frequency sharing between drivers, and higher strain on drivers due to wider bandwidth demands increases distortion, both THD and intermodulation, induces interference patterns, amplitude irregularities, driver resonances, cone breakup, and hampers off-axis response.

8) High power draw in a specific frequency range, usually the bass, may cause amplifier clipping and possible damage to the woofer, midrange, or most likely, the tweeter. The amp has to deal with the combined complex impedance load and power draw.

2.4 Components (inductors and capacitors) 2.4.1 Inductors A component called an inductor is used when the property of inductance is required in a circuit. Inductance is the name given to the property of a circuit whereby there is an e.m.f. induced into the circuit by the change of flux linkages produced by a current change [5]. The basic form of an inductor is simply a coil of wire.

Factors which affect the inductance of an inductor include:

(i) The number of turns of wire—the more turns the higher the inductance

(ii) The cross-sectional area of the coil of wire—the greater the cross sectional area the higher the inductance

(iii) The presence of a magnetic core—when the coil is wound on an iron core the same current sets up a more concentrated magnetic field and the inductance is increased

(iv) The way the turns are arranged—a short thick coil of wire has a higher inductance than a long thin one

Inductors with iron core are called ‘iron core inductors’ while those without are called ‘air core inductors’. Both have the same basic characteristics. Windings of inductors are capacitively coupled to each other, thus introducing a parallel capacitor across the coil. The inductive reactance increases with frequency and so an ideal inductor behaves as a short circuit in dc but an open circuit in very high frequencies. This rise in impedance at higher frequencies allows us to use the inductor as a filter that passes low frequencies and chokes off high frequencies.

The inductive reactance is given by:

푿퐿=ퟐ훑퐟퐋

퐗 Hence,퐋 = 퐋 ퟐ훑퐟퐋

At crossover point, the reactance is equal to the speaker impedance, 퐙ퟎ,퐇ence,

풁 퐋 = ퟎ ퟐ흅풇푳

For second order crossover network,

풁 √ퟐ 푳 = ퟎ ퟐ흅풇푳

Where;

푿퐿 = inductive reactance f = frequency

L = Inductance in Henries

The inductance of a coil depends on its geometrical characteristics, the number of turns and the method of winding the coil. The larger the diameter, length, and the larger the number of winding turns, the greater its inductance.

If the coil is tightly wound, turn to turn, then it will have more inductance than a not tightly wound coil, with gaps between the turns. Sometimes you need to wind a coil with a given geometry, and you don't have a wire with required diameter, then if you use a thicker wire you should increase slightly number of turns, and if you use a thinner wire it takes to reduce the number of turns of the coil to get the required inductance.

They are different formulae derived depending on different factors, for example, depending on if the inductor is multilayer or single layer or if it has an air core or ferrite core. For air core single layer with length of core greater than half the diameter of conductor the inductance, L is given by:

푫ퟐ푵ퟐ 퐋 = ퟒퟓ푫+ퟏퟎퟎ풍

D – Diameter of the conductor in centimeters

L-length of core in centimeters

N-Number of turns

[11]In the case of toroidal coils on ferrite ring or rings of carbonyl iron powder is widely used in amateur radio designs. Their advantage is the high combined inductance with small stray field. Calculation of the coil can be carried out in various ways. The most used method for calculation is based on a special parameter AL. This parameter is usually included in the specification of the ferromagnetic rings Numerical parameter AL is the inductance in uH at 100 turns of the coil for the iron powder ring or the inductance in mH at 1000 turns of the coil for the ferrite ring. Knowing the AL parameter, the number of windings of the toroid can be calculated using the following expressions:

For the iron powder ring

퐿(µ퐻) 푁 푡푢푟푛푠 = 100 . √ 퐴퐿(µ퐻) 100 푡푢푟푛푠

For the case of the ferrite ring,

퐿(푚퐻) 푁 푡푢푟푛푠 = 1000 √ 퐴퐿(푚퐻) 1000 푡푢푟푛푠

2.4.2 CAPACITORS Every system of electrical conductors possesses capacitance which is the property of a pair of electrically charged plates which determines how much charge corresponds to a given potential difference between the plates. For example, there is capacitance between the conductors of overhead transmission lines and also between the wires of a telephone cable. In these examples

27 the capacitance is undesirable but has to be accepted, minimized or compensated for. There are other situations where capacitance is a desirable property. A Capacitor is a passive element that stores electric charge statistically and temporarily as a static electric field. It is composed of two parallel conducting plates separated by non-conducting region that is called dielectric, such as vacuum, ceramic, air, aluminum, etc.

The capacitance formula of the capacitor is represented by;

∈ 푨 푪 = 풅

Where C is the capacitance that is proportional to the area of the two conducting plates (A) and proportional with the permittivity ε of the dielectric medium. The capacitance decreases with the distance between plates (d). We get the greatest capacitance with a large area of plates separated by a small distance and located in a high permittivity material. The standard unit of capacitance is Farad which is defined as the capacitance when a potential difference of one volt appears across the plates when charged with one coulomb. most commonly it can be found in micro- 푸 farads, pico-farads and nano-farads. Hence capacitance can also be taken to be 푪 = in terms 푽 of charge stored (Q) and the potential difference between the two plates (V)

Hence it is cleared that, by varying ε, A or d we can easily change the value of C. If we require higher value of capacitance (C) we have to increase the cross-sectional area of dielectric or we have to reduce the distance of separation or we have to use dielectric material with stronger permittivity.

Capacitive reactance is the impedance that a capacitor offers in a circuit to the flow of current. In an ideal case scenario, capacitive impedance is considered to be infinite in dc circuits where frequency is taken to be zero and decreases with increase in frequency to a very minimal value at very high frequencies. It’s given by:

ퟏ 푿 = 푪 ퟐ흅풇푪

Therefore,

ퟏ 푪 = ퟐ흅풇푿푪

At the crossover point, the capacitive reactance is equivalent to the speaker nominal impedance

풁푂 ,

Thus,

ퟏ 푪 = ퟐ흅풇풁ퟎ

For second order crossover network,

ퟏ 푪 = ퟐ흅풇풁ퟎ√ퟐ

XC= capacitive reactance f = frequency

C = capacitance in Farads

풁ퟎ=speaker nominal impedance 2.5 ADDITIONAL USEFUL CIRCUITS 2.5.1 1.Zobel network [11] The voice-coil impedance of a loudspeaker driver is not purely resistive. This can have a major perturbation on the performance of crossover networks. At high frequencies, the voice-coil impedance becomes inductive. For odd-order crossover networks, this inductance can be utilized as part of the crossover network. Because the inductance is lossy, some experimentation may be necessary. For even-order networks, the inductance can be canceled by using a simple RC matching network as discussed below.

The impedance rise near the resonance frequency of closed-box midrange and tweeter drivers can have a major perturbation on the performance of the high-pass crossover networks. The

29 effect is to cause a peak to appear in the pressure output of the driver at or near its resonance frequency. It can be very difficult to pull down this peak without causing a depression in the frequency response over a much wider band. To minimize the problem, the lower crossover frequency for the midrange and the tweeter should be greater than the fundamental resonance frequency of the drivers. The matching network described below can be used to cancel the impedance rise, but the element values may not be practical. A matching network, sometimes called a Zobel network, between the crossover network and the voice-coil terminals of a driver can be used to cause the effective load on the crossover network to be resistive as shown in the impedance curves in the figure below. Fig. 1 shows the network connected to the voice-coil equivalent circuit for a closed box driver. The high-frequency part of the network consists of R1, C1, R2, and C2. This network can be designed to correct for the lossy voice-coil inductance in an equal ripple sense between two specified frequencies in the band where the impedance is dominated by Ze (ω). At the fundamental resonance frequency of the driver, L1 and C3 resonate and put R3 in parallel with the voice coil. This cancels the rise in impedance at the fundamental resonance frequency푓푐.Figure 12 shows a typical zobel network configuration

Figure 12:Zobel network

Let the lossy voice-coil inductance have the impedance Ze (ω) = Le (jω)n [12]. Let the network consisting of푅1,퐶 1 ,푅2, and 퐶2correct for the lossy voice-coil inductance over the frequency band from f1 to f2. The frequency f1 might be chosen to be the frequency above the fundamental resonance frequency 푓푐where the voice-coil impedance exhibits a minimum before the high-

30 frequency rise caused by the voice-coil inductance. The frequency 푓2might be chosen to be 20 kHz. In order for the input impedance to the network plus the driver to be approximately equal to

푅퐸 at all frequencies, the matching network elements are given by:

푹ퟏ = 푹푬

푳풆 푪ퟏ = (ퟏ−풏) ퟏ−풏 ퟐ 풏 (ퟐ+풏) ퟐ(ퟏ+풏) (ퟐ흅) 푹푬 (풇ퟏ 풇ퟐ )

푳풆 푪ퟐ= (ퟏ−풏) -푪ퟏ ퟏ−풏 ퟐ (ퟐ+풏) 풏 ퟐ(ퟏ+풏) (ퟐ흅) 푹푬 (풇ퟏ 풇ퟐ )

ퟏ 푹 = ퟐ ퟏ 풏 (ퟏ+풏) ퟐ흅풇ퟏ 풇ퟐퟏ+풏푪ퟐ

푸푬푪 푹ퟑ = 푹푬(ퟏ + ) 푸푴푪

푹푬푸푬푪 푳ퟏ = ퟐ흅풇풄

ퟏ 풄ퟑ= ퟐ흅풇풄푹푬푸푬푪

Where:

푅퐸 Is the voice-coil resistance,

푓푐 is the closed-box resonance frequency,

푄퐸퐶 is the electrical quality factor

푄푀퐶 is the mechanical quality factor.

The above equations are derived under the assumption that C1 and C2 are open circuits in the low-frequency range where R3, C3, and L1 are active and that L1 is an open circuit in the high- frequency range where R1, C1, R2, and C2 are active. For a lossless inductor, n has the value n = 1,typically used values are either 0.6 and 0.7 [12].

In this case, C1 = Le/R2 E, and both R2 and C2 are open circuits.

Zobel networks however have the following setbacks to the circuit :

1. It leads to increased power dissipation due to additional dissipating components.

2. It can reduce the driver impedance below the nominal value.

3. Leads to a shift in crossover frequency.

Alternative and better equalization mechanisms without pronounced limitations include:

1. Use of Ferro fluid this has a better cooling characteristic in addition to impedance equalization but can cause sluggish response.

2. By design so that the resonance frequency is well above Fs or below where there is least amount of interference

Figure 13:Impedance curve with and without zobel network

2.5.2 L-pad A speaker L pad is a special configuration of rheostats used to control volume while maintaining a constant load impedance on the output of the audio amplifier.[1] It consists of a parallel and series resistor in an "L" configuration. As one increases in resistance, the other decreases, thus maintaining a constant impedance, at least in one direction. To maintain constant impedance in both directions, a "T" pad must be used. In loudspeakers it is only necessary to maintain impedance to the crossover; this avoids shifting the crossover point.

It is in other words a level control used in passive speaker systems to attenuate (reduce) power to the tweeter in a 2-way system as well as the mid speaker in a 3-way system or 4-way system. Most mid-range speakers and tweeters are approximately +6dB more efficient than woofers. Inside the L-Pad is 2 wire wound elements which are arranged to maintain a constant impedance of 8R to the amplifier.

2.5.3 Series-notch filter Change of nominal impedance occurs mostly at resonance frequency (Fs) of the speaker, thus impedance peaks make crossover points to shift. This filter prevents this. It is mainly employed in tweeters due to its cost and size.

Figure 14:Series notch filter

2.5.4 Parallel notch (trap) filter This is designed to remove broad peaks in frequency response of a driver

Figure 15:Parallel notch trap filter

2.6 WORKING MECHANISM OF A CROSSOVER NETWORK L Inductor (mH milli-Henry) approaches being a short circuit at low frequencies and an open circuit at high frequencies. C Capacitor (μF micro-Farad) approaches being an open circuit at low frequencies and a short circuit at high frequencies. The Impedance of L and C (expressed as resistance they represent) at any one frequency, is called Reactance, symbolized by the letter X. This Reactance changes by double or half for each double of half the frequency (6dB/octave).The Reactance XL and XC, reduces power by shifting the phase, between Volts and Amperes (of the signal) in opposite directions. The phase shifting of the signal at the crossover point has to be compensated by reversing connections to one of the speakers or by other means. A physical experience of phase shift is being in a motor vehicle that is accelerating or breaking, being thrust forward or backward.

2.6.1 12dB crossover Bass (low pass). The Inductor L1 in series with the bass speaker approaches being an open circuit at high frequencies (6dB/octave). The Capacitor C1 across the bass speaker approaches being a short circuit at high frequencies (6dB/octave). The Inductor and Capacitor combined limit high frequencies getting to the woofer at -12dB/octave.

Bass to Mid-range (band pass). The second Capacitor C1 in series with the mid-range and approaches being an open circuit at low frequencies (6dB/octave). The Inductor L1 across the mid-range approaches being a short circuit at low frequencies (6dB/octave). The Inductor and Capacitor combined limit low frequencies getting to the mid-range at -12dB/octave.

Mid-range (band pass).The Inductor L2 in series with the mid-range speaker, approaches being an open circuit at high frequencies (6dB/octave). The Capacitor C2 across the mid-range speaker approaches being a short circuit at high frequencies (6dB/octave). The Inductor and Capacitor combined, limit high frequencies getting to the mid-range speaker at -12dB/octave.

Tweeter (high pass). The Capacitor C2 in series with the tweeter, approaches being an open circuit at low frequencies (6dB/octave). The Inductor L2 across the tweeter, approaches being a short circuit at low frequencies (6dB/octave). The Inductor and Capacitor combined, limit low frequencies getting to the tweeter at -12dB/octave. The reactance (X) of L and C, shift phase between Volts and Amperes therefore reducing power (Watts). L and C are in series, and phase is shifted in opposite directions between them. This is called a 'series resonant' circuit. If the speaker is not connected to the crossover, or the speaker has been destroyed (open circuit), the LC 'series resonance' without a load behaves as short circuit at the crossover frequency only. The amplifier can easily be destroyed.

Passive crossovers of higher order than 12dB/octave can be made but are difficult to construct. Most are inefficient and inaccurate, regardless of the academic theory that describes them as being superior. The more complex a passive crossover, the more energy is required from the amplifier for it to function. This increases insertion loss which generates distortion that often outweighs the benefits. Early research, referred to 'transient distortion' as the major problem of passive crossovers greater than 12dB/octave. Early Audiophiles only accepted first order crossovers, claiming this has least effect on colouring the music. Their descriptions were, '1st and 2nd order crossovers allow the sound to be open whereas higher order crossovers cause the sound to be closed’. Recent audiophile trends are for very complex passive crossovers, greater than 12dB/octave that use magical Capacitors. The larger the number of magical Capacitors the more magical the sound becomes. These passive crossovers attempt to adjust for time alignment and Impedance variations within each speaker [13]

3 CHAPTER 3- DESIGN 3.1 Passive Crossover Circuit Design The nominal impedances of the speaker drivers were chosen to be 8 ohms for all of the four that is woofer, the two midranges and the tweeter.

3.1.1 Selection of crossover frequency Crossover frequency points are to some extent influenced by the enclosure design but are mainly determined by the characteristics and specifications of the speaker units employed. Usual crossover frequency ranges for the various system configurations are as follows [14]:

Table 2:Crossover frequency range for various systems

System type Speaker driver Crossover frequency range

2-way systems Bass-treble 3khz-5khz

3-way systems Bass-midrange 500hz-1.5khz

Midrange-treble 3khz-5khz

4-way systems Bass – low midrange 200hz-400hz

Low Midrange-Midrange 800-1.5khz

Midrange-treble 3khz-5khz

Super tweeter 5khz-10khz

Crossover frequency range for various systems

When using passive crossover networks, the power proportion to each bandwidth varies in accordance with the crossover frequency as illustrated in table3 below [14]:

Table 3:crossover frequency and speaker power proportions

Crossover frequency % power to bass unit %power to upper units

250Hz 40 60

350Hz 50 50

500Hz 60 40

1200Hz 65 35

3000Hz 85 15

5000Hz 90 10

Other guidelines used in the choice of crossover frequency included:

 Bass power should not be reduced below about the 40% level of the total power regardless of crossover frequency [15].This because although the average power might be quite low, it is usually of relatively high peak amplitude. The wide dynamics of the bass content require an amplifier capable of far more power than might be imagined if clipping is to be avoided. Clipping is something that one should avoid at all costs, because apart from sounding horrible, the average power level is increased, placing loudspeakers at risk. Having said that, some peak clipping in a subwoofer may be inaudible, provided the remainder of the signal is clean.  Generally ,the bass amplifier should have at least the same power as that used for the mid+high frequency  Theoretically, a woofer should be used to cover only those frequencies where the wavelength of the sound to be reproduced does not exceed the diameter of the driver.  Midranges or tweeters are crossed at frequencies greater than twice their natural frequency of resonance.

3.2 Component values determination 3.2.1 Crossover points

st 3.2.1.1 1 crossover point =350푯풁 Woofer nominal impedance =8ohms

As previously determined, at the crossover point, both the capacitive and inductive impedances are equivalent to the speaker drivers nominal impedance (8 ohms), and hence for the second ퟏ order crossover network being designed, capacitance 푪 = .This formula can be ퟐ흅풇풁ퟎ√ퟐ rationalized by multiplying both the numerator and denominator by √ퟐ and this simplifies to

√ퟐ 풁 ퟐ 푪 = while the formula for inductance 푳 = ퟎ√ .Hence: ퟒ흅풇풁ퟎ ퟐ흅풇푳

√ퟐ 푪 = ퟏ ퟒ흅 ∗ ퟑퟓퟎ ∗ ퟖ

푪ퟏ=40.19흁푭

ퟖ ∗ √ퟐ 퐋 = 1 ퟐ흅 ∗ ퟑퟓퟎ

퐋1 = ퟓ. ퟏퟒ퐦퐇

푪ퟐ =40.19흁푭

퐋2 = ퟓ. ퟏퟒ퐦퐇

nd 3.2.1.2 2 crossover point =1150푯풁 √ퟐ 푪 = ퟑ ퟒ흅 ∗ ퟏퟏퟓퟎ ∗ ퟖ

푪ퟑ=12.233흁푭

ퟖ ∗ √ퟐ 퐋 = 3 ퟐ흅 ∗ ퟏퟏퟓퟎ

퐋3 = ퟏ. ퟓퟔퟔ퐦퐇

푪ퟒ =12.233흁푭

퐋4 = ퟏ. ퟓퟔퟔ퐦퐇

rd 3.2.1.3 3 crossover point =4000푯풁 √ퟐ 푪 = ퟓ ퟒ흅 ∗ ퟒퟎퟎퟎ ∗ ퟖ

푪ퟓ=3.517흁푭

ퟖ ∗ √ퟐ 퐋 = 5 ퟐ흅 ∗ ퟒퟎퟎퟎ

퐋5 = ퟎ. ퟒퟓ퐦퐇

푪ퟔ =3.517흁푭

퐋6 = ퟎ. ퟒퟓ퐦퐇

The above determined component values of capacitance are theoretical and for the practical implementation, the closest standard values commercially available were chosen. They were then measured to determine the actual capacitance as shown in the table below.

The power ratings of the speaker drivers used were:

 Woofer-100W  Midrange1 -60W  Midrange2- 30W  Tweeter -10W

The nominal impedance of the speaker drivers was 8 ohms.

3.2.2 Woofer (100W): From the formula퐏 = 퐈ퟐ퐑,

퐏 퐈 = √ 퐑

ퟏퟎퟎ 퐈 =√ =3.54A 퐑퐌퐒 ퟖ

퐈퐦퐚퐱 = ퟑ. ퟓퟒ ∗ √ퟐ = ퟓ퐀

푽ퟐ Also, 퐏 = hence 퐯 = √(퐏 ∗ 퐑) 푹

푽푹푴푺 = √(ퟏퟎퟎ ∗ ퟖ) =28.28V

푽푴푨푿 = 푽푹푴푺 ∗ √ퟐ=40V

3.2.3 Midrange1 (60W) 퐏 = 퐈ퟐ퐑,

퐏 퐈 = √ 퐑

ퟔퟎ 퐈 =√ =2.74A 퐑퐌퐒 ퟖ

퐈퐦퐚퐱 = ퟐ. ퟕퟒ ∗ √ퟐ = ퟑ. ퟖퟕퟓ퐀

푽ퟐ Also, 퐏 = hence 퐯 = √(퐏 ∗ 퐑) 푹

푽푹푴푺 = √(ퟔퟎ ∗ ퟖ) =22V

푽푴푨푿 = 푽푹푴푺 ∗ √ퟐ=31.11V

3.2.4 Midrange2 (30W) 퐏 = 퐈ퟐ퐑,

퐏 퐈 = √ 퐑

ퟑퟎ 퐈 =√ =1.913A 퐑퐌퐒 ퟖ.ퟐ

퐈퐦퐚퐱 = ퟏ. ퟗퟏퟑ ∗ √ퟐ = ퟐ. ퟕퟎퟓ퐀

푽ퟐ Also, 퐏 = hence 퐯 = √(퐏 ∗ 퐑) 푹

푽푹푴푺 = √(ퟑퟎ ∗ ퟖ. ퟐ) =15.68V

푽푴푨푿 = 푽푹푴푺 ∗ √ퟐ=22.18V

3.2.5 Tweeter (10W)

퐏 = 퐈ퟐ퐑,

퐏 퐈 = √ 퐑

ퟏퟎ 퐈 =√ =1.1043A 퐑퐌퐒 ퟖ.ퟐ

퐈퐦퐚퐱 = ퟏ. ퟏퟎퟒퟑ ∗ √ퟐ = ퟏ. ퟓퟔ퐀

푽ퟐ Also, 퐏 = hence 퐯 = √(퐏 ∗ 퐑) 푹

푽푹푴푺 = √(ퟏퟎ ∗ ퟖ. ퟐ) =9.055V

푽푴푨푿 = 푽푹푴푺 ∗ √ퟐ=12.81V

The components thus had to have a rating of a value higher than the maximum voltages and currents for the respective speaker drivers. Since the highest voltage rating for the woofer was 40V, 250V DC capacitors were used in the design.

Table 4:Capacitor calculated,measured and standard values

CAPACITORS CALCULATED STANDARD MEASURED AVAILABLE VALUE

푪ퟏ 40.19흁푭 47흁푭 45.4 흁푭

푪ퟐ 40.19흁푭 47흁푭 45.4 흁푭

푪ퟑ 12.233흁푭 12흁푭 11.58 흁푭

푪ퟒ 12.233흁푭 12흁푭 11.58 흁푭

푪ퟓ 3.517흁푭 3.3흁푭 3.2흁푭

푪ퟔ 3.517흁푭 3.3흁푭 3.2 흁푭

The inductors used were manually coiled using the online coil32 program [11] which is based on the two empirical formulae shown below. The calculation using these two formulae required knowledge of the dimensions of the core which basically included the outer diameter of the toroid(퐷1), the inner diameter (퐷2), the height of the toroid used (h) and the relative permeability (µ) to realize the value of inductance calculated above, which then led to the determination of the number of turns required to wind a particular value of inductance using a specified wire gauge

푫 푫 푳 = ퟎ. ퟎퟎퟎퟐµ풉푵ퟐ퐥퐧 ( ퟏ) For the case where ퟏ >1.75 푫ퟐ 푫ퟐ

푫 −푫 푫 Or 푳 = ퟎ. ퟎퟎퟎퟒµ풉푵ퟐ( ퟏ ퟐ) for the case where ퟏ <1.75 푫ퟐ+푫ퟏ 푫ퟐ

All dimensions are in millimeters, the inductance in µH. At the same time, by the numerical algorithm the program calculates the length of wire needed for the winding. Wire length is calculated with a margin of 10 cm on the "ends".

In this design, similar torus ferrite cores in terms of dimensions were used but their relative permeability were different. The dimensions were:

푫ퟏ = ퟐퟓ풎풎

푫ퟐ = ퟏퟓ풎풎

h= 4.5mm

퐷 25 Since in this case, 1 = = 1.667 < 1.75 the second formula for determining inductance was 퐷2 15 used in the online program for determining the number of turns to be used and the length of wire to be used.

For the first pair of inductance to be wound, 5140µ퐻;

The relative permeability of the core used was 2470 according to the data sheet and this using the online calculator, this yielded 68 turns for 20 AWG and this was arrived at as shown below

푫 −푫 푳 = ퟎ. ퟎퟎퟎퟒµ풉푵ퟐ( ퟏ ퟐ) 푫ퟐ+푫ퟏ

Substituting for the values of L, µ, ℎ, 퐷1 and퐷2;

ퟐퟓ−ퟏퟓ ퟓퟏퟒퟎ = ퟎ. ퟎퟎퟎퟒ ∗ ퟐퟒퟕퟎ ∗ ퟒ. ퟓ풎풎푵ퟐ( ) ퟐퟓ+ퟏퟓ

The value of N was determined to be 68 and with a wire length of 1.47m and hence was the inductor was then made and measured and the value tabulated as shown below.

For the second pair of inductors to be wound, 1566µ퐻;

The wire gauge used was 18AWG.The relative permeability of the core used was 3867 according to its data sheet and this yielded 30 turns, with a wire length of 0.67m this as previously was arrived at as shown below:

푫 −푫 푳 = ퟎ. ퟎퟎퟎퟒµ풉푵ퟐ( ퟏ ퟐ) 푫ퟐ+푫ퟏ

Substituting for the values of L, µ, ℎ, 퐷1 and 퐷2;

ퟐퟓ−ퟏퟓ ퟏퟓퟔퟔ = ퟎ. ퟎퟎퟎퟒ ∗ ퟑퟖퟔퟕ ∗ ퟒ. ퟓ풎풎푵ퟐ( ) ퟐퟓ+ퟏퟓ

Solving for N, we determine it to be 30.

For third and last pair of inductors to be wound, 450 µ퐻,

The wire gauge used were 18AWG and 19AWG for each of the two inductors. The relative permeability of the core used was 2770 according to its data sheet and this yielded 19 turns, with a wire length of 0.461m for both wires used and this as previously was arrived at as shown below:

푫 −푫 푳 = ퟎ. ퟎퟎퟎퟒµ풉푵ퟐ( ퟏ ퟐ) 푫ퟐ+푫ퟏ

Substituting for the values of L, µ, ℎ, 퐷1 and 퐷2;

ퟐퟓ−ퟏퟓ ퟒퟓퟎ = ퟎ. ퟎퟎퟎퟒ ∗ ퟐퟕퟕퟎ ∗ ퟒ. ퟓ풎풎푵ퟐ( ) ퟐퟓ+ퟏퟓ

Solving for N, it was determined it to be 19.

They were then measured to determine the actual practical values used and the values obtained are as shown in the table 5 below.

Table 5: Inductor calculated and measured values

INDUCTORS CALCULATED MEASURED

푳ퟏ 5.14mH 4.8mH

푳ퟐ 5.14mH 4.8.mH

푳ퟑ 1.566mH 1.45mH

푳ퟒ 1.566mH 1.45mH

푳ퟓ 0.45mH 0.41mH

푳ퟔ 0.45mH 0.41mH

3.3 IMPEDANCE CURVES AND ZOBEL NETWORK DESIGN In the crossover network simulation, resistors of impedance equivalent to the nominal impedance of the speaker drivers were used to represent the speaker drivers themselves. This however, does not give a very accurate representation of the speaker drivers as their equivalent circuits have an inductor and resistor in series to represent the electrical parameters and a parallel RLC circuit to represent the mechanical parameters as shown in the figure above. This manifests itself best when the impedance curve of a particular speaker driver is plotted where there is a rising impedance as frequency increases due the series inductance of the voice coil unlike a constant impedance curve that would be obtained when the speaker driver impedance is assumed to be purely resistive and a simulation with microcap clearly showed this below. The remedy for the impedance rise which would otherwise lead to a shift in the design crossover point thus making the circuit not operate as intended is a zobel network but two challenges were realized during the project ,with the design of the zobel network.Firstly,zobel networks are designed with a particular speaker drivers specifications in mind for instance the electrical and mechanical quality factors,the mechanical compliance and many more which vary from one speaker driver to another and since no particular driver was to be used,the zobel network could not be designed as the overall crossover network was meant to be versatile in the sense that it could work with a number of speaker drivers and not just one particular driver.Secondly,the use of many components in a crossover leads to a higher susceptibility to distortions both in magnitude and phase and this is not very desirable with audio signals and so the zobel network was omitted. However there are speaker drivers that could be used with a proper design of the crossover network, without the use of the zobel network and the system would still not suffer impedance rise due to the voice coil inductance or rather the effect of this impedance rise would be very minimal. The concept behind the design is to crossover the audio signal spectrum within the frequency range where the voice coil impedance of a particular speaker driver is sort of constant since the significant rise in inductive impedance occurs beyond some frequency as shown in the simulations below. The following commercially available speaker drivers and their specifications were used to demonstrate this concept.

3.3.1 1. woofer (wf090wa02) 3.3.2 specifications

푹푫푪 = ퟓ. ퟓ 풐풉풎풔

푭푺 = ퟏퟏퟎ푯풁

Cms=0.6mm/N

Rms =0.39NS/m

풍풆 = ퟎ. ퟐퟐ풎푯 풂풕 ퟏퟎ풌푯풁

ퟏ 푭푺 = ퟐ흅√푪푴푺. 푴푴푺

ퟏ 110= ퟐ흅√(ퟎ.ퟔퟐ∗ퟏퟎ−ퟑ.푴푴푺)

ퟏ ퟐ MMS= (ퟐ흅∗ퟏퟏퟎ) ퟎ.ퟔퟐ∗ퟏퟎ−ퟑ

MMS=3.376g

ퟐ 풍풄풆풔 = 풄풎풔 ∗ (풃풍)

−ퟑ ퟐ 풍풄풆풔 = ퟎ. ퟔퟐ ∗ ퟏퟎ ∗ (ퟑ. ퟓ)

풍풄풆풔 = ퟕ. ퟓퟗퟓ풎푯

푴푴푺 ퟑ.ퟑퟕퟔ∗ퟏퟎ−ퟑ 푪 = = = ퟐ. ퟕퟔ ∗ ퟏퟎ−ퟒ 푴푬푺 (푩풍)ퟐ (ퟑ.ퟓ)ퟐ

푹 (풃풍)ퟐ (ퟑ.ퟓ)ퟐ 풆풔= = 푹풎풔 ퟎ.ퟑퟗ

=31.41ohms

3.3.3 midrange1(rs52an-8) 3.3.4 specifications

푹풆 = ퟔ풐풉풎풔

풍풆 = ퟎ. ퟏퟑ풎푯

푸푴푺 = ퟑ. ퟏퟔ

푸푬푺 = ퟏ. ퟏퟖ

푪푴푺 = ퟎ. ퟏퟑ풎풎/푵

풃풍 = ퟒ. ퟐퟔ푻푴

푴푴푺 = ퟏ. ퟕ품

푭푺 = ퟑퟏퟒ푯풛

ퟐ흅 ∗ 푭풔 ∗ 푴풎풔 푸 = 푴푺 푹풎풔

ퟐ ∗ 흅 ∗ ퟑퟒퟏ ∗ ퟏ. ퟕ ∗ ퟏퟎ−ퟑ ퟑ. ퟏퟔ = 푹풎풔

ퟐ ∗ 흅 ∗ ퟑퟒퟏ ∗ ퟏ. ퟕ ∗ ퟏퟎ−ퟑ 푹풎풔 = = ퟏ. ퟏퟓퟐퟔ ퟑ. ퟏퟔ

ퟐ −ퟑ ퟐ 풍풄풆풔 = 푪풎풔*(푩풍) = ퟎ. ퟏퟑ ∗ ퟏퟎ ∗ ퟒ. ퟐퟔ = ퟐ. ퟑퟔ풎푯

푴풎풔 ퟏ.ퟕ∗ퟏퟎ−ퟑ Cmes= = = ퟗퟑ. ퟔퟖ흁푭 (푩풍)ퟐ ퟒ.ퟐퟔퟐ

(푩풍)ퟐ ퟒ.ퟐퟔퟐ Res= = = ퟏퟓ. ퟕퟓ풐풉풎풔 푹풎풔 ퟏ.ퟏퟓퟐퟔ

3.3.5 midrange2 (nd105-8) 3.3.6 specifications Power handling=30w

Maximum=60w

Impedance=8ohms

Response=60Hz-10,000Hz

Re=7.6ohms

Le=1.37mH

Fs=65.3Hz

Qms=7.61

Qes=0.73

Qts=0.66

Cms=0.95mm/N

Mms=6.3g

Bl=4.9TM

ퟐ −ퟑ ퟐ 풍풄풆풔 = 푪풎풔*(푩풍) = ퟎ. ퟗퟓ ∗ ퟏퟎ ∗ ퟒ. ퟗ = ퟎ. ퟎퟐퟐퟖ푯

푴풎풔 ퟔ.ퟑ∗ퟏퟎ−ퟑ Cmes= = = ퟐ. ퟔퟐퟒ ∗ ퟏퟎ−ퟒ (푩풍)ퟐ ퟒ.ퟗퟐ

ퟐ흅 ∗ 푭풔 ∗ 푴풎풔 푸 = 푴푺 푹풎풔

ퟐ ∗ 흅 ∗ ퟔퟓ. ퟑ ∗ ퟔ. ퟑ ∗ ퟏퟎ−ퟑ ퟕ. ퟔퟏ = 푹풎풔

ퟐ ∗ 흅 ∗ ퟔퟓ. ퟑ ∗ ퟔ. ퟑ ∗ ퟏퟎ−ퟑ 푹풎풔 = = ퟎ. ퟑퟑퟗퟔퟔ ퟕ. ퟔퟏ

(푩풍)ퟐ ퟒ.ퟗퟐ Res= = = ퟕퟎ. ퟔퟗ풐풉풎풔 푹풎풔 ퟎ.ퟑퟑퟗퟔퟔ

3.3.7 tweeter (tw030wa14) Power handling =35w

Frequency range =1500=30000Hz

Fs=715Hz

Nominal impedance =8 ohms

Re=6.5ohms

Bl=2.25N/A

Le=0.059mh

Mms=0.4g

Qms=2.37

Qes=2.31

Qts=1.17

ퟏ 푭푺 = ퟐ흅√푪푴푺. 푴푴푺

ퟏ 715= ퟐ흅√(푪풎풔.ퟎ.ퟒ∗ퟏퟎ−ퟑ)

ퟏ cms= ퟎ.ퟒ∗ퟏퟎ−ퟑ∗(ퟕퟏퟓ∗ퟐ흅)ퟐ cms=1.24*ퟏퟎ−ퟒ

ퟐ −ퟒ ퟐ −ퟒ 풍풄풆풔 = 푪풎풔*(푩풍) = ퟏ. ퟐퟒ ∗ ퟏퟎ ∗ ퟐ. ퟐퟓ = ퟔ. ퟐퟖ ∗ ퟏퟎ 푯

푴풎풔 ퟎ.ퟒ∗ퟏퟎ−ퟑ Cmes= = = ퟕ. ퟗ ∗ ퟏퟎ−ퟓ푭 (푩풍)ퟐ ퟐ.ퟐퟓퟐ

ퟐ흅 ∗ 푭풔 ∗ 푴풎풔 푸 = 푴푺 푹풎풔

ퟐ ∗ 흅 ∗ ퟕퟏퟓ ∗ ퟎ. ퟒ ∗ ퟏퟎ−ퟑ ퟐ. ퟑퟕ = 푹풎풔

ퟐ ∗ 흅 ∗ ퟕퟏퟓ ∗ ퟎ. ퟒ ∗ ퟏퟎ−ퟑ 푹풎풔 = = ퟕ. ퟓퟖퟐퟐ ∗ ퟏퟎ−ퟑ ퟐ. ퟑퟕ

(푩풍)ퟐ ퟐ.ퟐퟓퟐ Res= = = ퟔퟔퟕ. ퟔퟖ풐풉풎풔 푹풎풔 ퟕ.ퟓퟖퟐퟐ∗ퟏퟎ−ퟑ

The table 6 below shows the values of the electrical and mechanical parameters of the simulated speaker drivers

Table 6: Electrical and Mechanical parameters of the simulated speaker drivers

ELECTRICAL ELECTRICAL EQUIVALENCE OF PARAMETERS MECHANICAL PARAMETERS

SPEAKER 푹풅풄 푳풆 푹풆풔 푪풎풆풔 푳풄풆풔 DRIVER

1.WOOFER 5.5ohms 0.22mH 31.41ohms 276µF 7.595mH

2.MIDRANGE1 6.0ohms 0.13mH 15.75ohms 93.68µF 2.36mH

3.MIDRANGE2 7.6ohms 1.37mH 70.69ohms 262.4µF 22.8mH

4.TWEETER 6.5ohms 0.059mH 667.68ohms 79µF 0.628mH

The circuit simulated to realize the impedance curves is as shown in the figure 16 below

Figure 16:simulated circuit for impedance curves

3.4 THE FILTER NETWORKS FOR THE INDIVIDUAL SPEAKER DRIVERS They are as shown in figures 17,18,19,20 below:

3.4.1 woofer

Figure 17:Woofer circuit

3.4.2 midrange1

Figure 18:Midrange1 circuit

3.4.3 midrange2

Figure 19:midrange2 circuit

3.4.4 tweeter

Figure 20:Tweeter circuit

The overall passive four way crossover network that was simulated and fabricated is as shown in figure 21 below

Figure 21:Fabricated circuit

4 CHAPTER 4 - OBSERVATION AND RESULTS 4.1 impedance curves The impedance curves obtained via simulation from the circuit above were as shown in figures 22,23,24,25 below:

4.1.1 1.woofer

Figure 22: Simulated woofer impedance curve

4.1.2 midrange1

Figure 23:simulated midrange1 impedance curve:

4.1.3 midrange2

Figure 24:Simulated midrange2 impedance curve

4.1.4 tweeter

Figure 25:simulated tweeter impedance curve

4.2 FILTER NETWORKS SIMULATED RESPONSE The simulated response of the filter networks for the particular speaker drivers were as shown in the figures 26,27,28 and 29 shown below

4.2.1 woofer

Figure 26:simulated woofer frequency response

4.2.2 Midrange1

Figure 27: simulated midrange1 frequency response

4.2.3 Midrange2

Figure 28:simulated midrange2 response

4.2.4 Tweeter

Figure 29:simulated tweeter frequency response

The response for the entire 2nd order 4-way crossover network is as shown in figure 30 below

Figure 30:full network frequency response

4.3 SIMULATED RESULTS The simulated results at varied frequencies for all the four speaker drivers were as shown in the tables 7,8,9,10 below:

4.3.1 1.Woofer (baseband) Table 7:Simulated woofer frequency response

FREQUENCY(푯풛) MAGNITUDE(dB)

20 183.486m

50 183.486m

100 183.486m

200 -275.229m

250 -733.945m

300 -1.193

400 -4.16

500 -7.615

600 -10.367

700 -13.119

800 -15.413

900 -16.789

1000 -19.803

4.3.2 2. Midrange1(lower midrange) Table 8:simulated midrange1 frequency response

FREQUENCY(푯풁) MAGNITUDE (dB)

50 -31.5

100 -20.344

200 -8.57

300 -3.542

400 -321.1m

600 2.76

800 2.933

1000 1.537

1200 -114.619m

1500 -3.21

2000 -8.578

2500 -12.706

3000 -16.216

3500 -19.16

4000 -21.78

5000 -24.64

4.3.3 Midrange2(higher midrange) Table 9:Simulated midrange2 frequency response

FREQUENCY (푯풁) MAGNITUDE (dB)

500 -14.61

600 -12.12

700 -9.36

800 -7.61

900 -5.58

1000 -4.45

2000 1.531

4000 504.587m

5000 -2.592

6000 -5.894

7000 -8.372

8000 -10.642

9000 -13.01

10000 -14.77

20000 -27.569

4.3.4 4.Tweeter Table 10:simulated tweeter frequency response

FREQUENCY (푯풁) MAGNITUDE (dB)

1000 -24.68

2000 -12.706

3000 -6.927

4000 -3.83

5000 -1.766

6000 -733.945m

7000 -733.945m

8000 -321.101m

9000 -321.101m

10000 -114.679m

20000 91.743m

30000 91.743m

40000 91.743m

4.4 PRACTICAL DESIGN The fabricated circuit was designed using express PCB software as shown in figure 31below:

Figure 31:stage1 etching process

The etching process took place in the three stages in the images in the figures 32, 33, 34 below

Figure 32:stage2 etching process

Figure 33:stage 3 etching process

Figure 34:Final stage etching process

And the finally soldered circuit was as shown below in the image in figure35 below

Figure 35:complete soldered circuit

4.5 Practical results A signal generator was connected to the input of the circuit shown above. A 10 ohm dropper with a 5watt rating was connected across the input to the crossover network and the outputs connected to the various speaker drivers.an oscilloscope was used to observe both the input and output signals using both of the available channels. The input signal was a sinusoidal signal whose amplitude and frequency were varied from the signal generator.as the amplitudes and frequency of the input signal were varied, the corresponding values of the magnitude of the signal at the output were the speaker drivers were connected were also measured using both the oscilloscope and the digital multimeter.the values of input and output voltages were then used to calculate the gain and hence the frequency response of the network was determined. The results obtained are as shown in the tables 11,12,13 and 14 below:

4.5.1 Woofer Table 11:Practical woofer frequency response results

Frequency(Hz) Input Output Practical Simulated

voltage(VIN) voltage(VOUT) Magnitude(dB) Magnitude(dB)

20 50mV 50.82 mV 162.39m 183.486m

50 100 mV 102 mV 161.41m 183.486m

100 200 mV 203.7 mV 161.346m 183.486m

200 300 mV 291 mV 268.365m -275.229m

250 400 mV 370 mV -698.439m -733.945m

300 500 mV 434 mV -1.23 -1.193

400 600 mV 383 mV -3.89 -4.16

500 700 mV 305 mV -7.21 -7.615

600 800 mV 269 mV -9.465 -10.367

700 900 mV 202 mV -12.69 -13.119

800 1V 180 mV -14.94 -15.413

900 2V 332 mV -15.61 -16.789

1000 3V 350mV -18.679 -19.093

4.5.2 Lower midrange (midrange1) Table 12:Practical midrange1 frequency response results

Frequency(Hz) Input Output Practical Simulated

voltage(VIN) voltage(VOUT) Magnitude(dB) Magnitude(dB)

50 50mV 2mV -28.69 -31.5

100 100Mv 10.5 mV -19.57 -20.344

200 200mV 82.7 mV -7.67 -8.57

300 300mV 204 mV -3.39 -3.542

400 400mV 386 mV -316.3m -321.1m

600 600mV 795 mV 0.46 0.76

800 650mV 904 mV 0.66 0.233

1000 700mV 825 mV 0.423 0.537

1200 750mV 740 mV -110.5m -114.679m

1500 800mV 570 mV -2.93 -3.21

2000 850mV 344 mV -7.86 -8.578

2500 900mV 222.5 mV -12.14 -12.706

2800 950mV 196 mV -13.69 -14.35

3000 1V 137 mV -17.26 -16.216

3500 2V 228 mV -18.86 -19.16

4000 3V 280 mV -20.61 -21.78

5000 4V 225 mV -25.69 -24.64

4.5.3 Higher midrange (midrange2) Table 13:practical midrange2 frequency response results

Frequency(Hz) Input Output Practical Simulated

voltage(VIN) voltage(VOUT) Magnitude(dB) Magnitude(dB)

500 50 10.4 -13.61 -14.61

600 50 13.76 -11.21 -12.12

700 100 38 -8.41 -9.36

800 200 89.5 -6.98 -7.61

900 300 161 -5.41 -5.58

1000 400 227 -4.91 -4.45

2000 500 603 1.62 1.537

4000 600 634 465.48 504.587

5000 700 530 -2.41 -2.592

6000 800 416 -5.68 -5.894

7000 900 362 -7.92 -8.372

8000 1 342 -9.31 -10.642

9000 2 540 -11.36 -13.01

10000 3 637 -13.46 -14.77

20000 4 233 -24.69 -27.569

4.5.4 Tweeter Table 14:practical tweeter frequency response results

Frequency(Hz) Input Output Practical Simulated

voltage(VIN) voltage(VOUT) Magnitude(dB) Magnitude(dB)

1000 50 mV 3.66 -22.7 -24.68

2000 100 mV 26.03 -11.69 -12.706

3000 200 mV 111 -5.11 -6.93

4000 300 mV 198mV -3.62 -3.83

5000 400 mV 338mV -1.46 -1.766

6000 500 mV 461.25mV -700.62m -733.945m

7000 600 mV 553.62 mV -698.78m -733.945m

8000 700 mV 675 mV -322.16m -321.101m

9000 800 mV 772 mV -318.54m -321.101m

10000 900 mV 889.35 mV -103.41m -114.679m

20000 1V 0.99V -86.23m 91.743m

30000 2V 1.96V -84.61m 91.743m

40000 3V 2.95V -88.36m 91.743m

Speaker cabinets were constructed from rough estimates of the diameters of the speaker drivers against the sizes of the speakers themselves. It was not possible to mathematically calculate accurate values of the measurements of the cabinets since the specifications of the speakers needed for this for instance the quality factors and the resonance frequencies were not known but even then, the rough estimates worked out really well as the perceived quality of sound produced was very good from music played. The constructed cabinets are as shown in the image in figure 35 below:

Figure 36:speaker cabinets

4.6 Analysis The practical response was found as indicated in the plots in figures 37,38,39,40 below.

woofer frequency response

2 0 0 100 200 300 400 500 600 700 800 -2 -4 -6

-8 Magnitude(dB) -10 -12 -14 Frequency(Hz)

Figure 37:practical woofer frequency response

lower midrange(midrange1) frequency response 5 0 -5 0 500 1000 1500 2000 2500 3000 3500 -10 -15 -20

magnitude(dB) -25 -30 -35 Frequency (Hz)

Figure 38:practical midrange1frequency response

Higher midrange(midrange2)frequency response 2 0 -2 0 2000 4000 6000 8000 10000 -4 -6 -8

-10 Magnitude(dB) -12 -14 -16 Frequency (Hz)

Figure 39:practical midrange2 frequency response

Tweeter frequency response 0 0 5000 10000 15000 20000 25000 30000 35000 40000 45000

-5

-10

-15 Magnitude()

-20

-25 Frequency(Hz)

Figure 40:practical tweeter frequency response

The matlab code shown below was used to determine frequency response of the full crossover network and it was as shown below: function[]=tonny9() ax=[-30,0,20,50000]; %sets axis limits axis(ax);%assigns the limits to axis function x=1:20;%produce values to be plotted on x-axis bassy=[0.162 0.161 0.161 -0.268 -0.698 -1.23 -3.89 -7.21 -9.465 -12.96 -14.94 -15.61 -18.679]; bassx=[20 50 100 200 250 300 400 500 600 700 800 900 1000]; mid1y=[-28.69 -19.56 -7.67 -3.369 -0.3161 0.26 0.86 0.423 -0.11 -2.93 -7.86 -12.14 -13.69 - 17.26 -18.86 -20.69 -25.69]; mid1x=[50 100 200 300 400 600 800 1000 1200 1500 2000 2500 2800 3000 3500 4000 5000]; mid2y=[-13.61 -11.21 -8.41 -6.89 -5.41 -4.91 0.32 0.504 -2.592 -5.894 -8.372 -10.642 -13.01 - 14.77 -27.569]; mid2x=[500 600 700 800 900 1000 2000 4000 5000 6000 7000 8000 9000 10000 20000]; highy=[-22.7 -11.69 -5.11 -3.62 -1.46 -0.7 -0.69 -0.32 -0.318 -0.103 -0.086 -0.084 0.084]; highx=[1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 40000]; plot(bassx,bassy,'-',mid1x,mid1y,'-',mid2x,mid2y,'-',highx,highy,'-'); legend('bass','mid1','mid2','bass');%defines the graphs grid;% sets the grid lines title('full response');% puts a title on the plot xlabel('Frequency(Hz)');% puts a label on the x axis ylabel('Magnitude(dB)')% puts a label on the y-axis end

The frequency response was as shown in figure 41 below:

Figure 41:full network frequency response

It was realized generally that as frequency increased, there was increased attenuation of the signals. The practical crossover frequencies realized from the matlab plots were 313Hz, 1515Hz and finally 4848Hz instead of the design crossover frequencies which were 350Hz, 1150Hz and 4000Hz.This was because the practical values of the capacitors and the inductors used in the design of the crossover network were slightly different from the values used in the design and this could led to a shift of the crossover frequency point. Also likely to have affected the crossover point was the fact that the impedance used in the determination of the crossover frequencies was 8 ohms but in a practical case scenario, speaker drivers with nominal impedances of 8 ohms have an impedance value slightly from t the 8 ohms specified in the design calculations. Also notable is the fact that speaker drivers are a reactive load and not resistive and due to the voice coil inductance, the impedance increases as frequency and even though there is a remedy for this by design of a zeal network, it was realized via simulation that inclusion of a zobel network led to a significant distortion of the output which overally affects the quality of sound produced and thus it was omitted.

5 CHAPTER 5 5.1 Conclusion The main objective of the project was to design and implement a 4-way passive crossover network. This aim has been achieved as the system has worked as expected with good quality sound at the output of the four speaker drivers. There were slight differences from the theoretical simulations and this was expected but it did not adversely affect the output and quality of sound produced. A system with several crossover points has increased distortions compared to that with fewer points of crossover. Comparison between this system and its active counterpart reveals that, this is cheaper and thus more desirable as most people desire to optimize the use of a device yet at a low price. Overally, the project was a success. 5.2 Recommendation The project generally involved a demonstration of the possibility of the design and use of a 4-way passive crossover network and since this concept is not fully explored and commercialized, for demonstration purposes, a mono instead of a stereo set up was used. Stereo systems produce the best quality sounds and hence for future works and demonstrations on this, I would recommend use and design of a stereo system.

The department should try and purchase ferrite cores in advance as they are not very easily available and the alternative use of air core inductors leads to very bulky crossover network circuits which are not practical incase it’s to be implemented in an actual system.

Lastly care should be taken when handling the speaker drivers and especially the domes. Any dent on the domes affect the quality of the sound produced.

REFERENCES

[1] C. D. Ambrose, Frequency Range of Human Hearing, 2003.

[2] T. A. Nielsen, Loudspeaker Crossover Networks, August,2005.

[3] www.diyaudioandvideo.com, "Speaker Crossover Wiring Guide," [Online].

[4] W. D. Stanley, Operational Amplifiers with Linear Integrated Circuits.

[5] J. Bird, Electrical Circuit Theory and Technology.

[6] D. Meyer, Loudspeaker Parameters.

[7] J. P. Bello, Loudspeakers.

[8] "http://www.eminence.com/support/understanding-loudspeaker-data/," [Online].

[9] Tomi_Engdahl, "Speaker Impedance".

[10] "http://audioundone.com/8-advantages-of-active-crossovers-douglas-self," [Online].

[11] "ferrite torroid core," 23 january 2015. [Online]. Available: http://coil32.net/ferrite-toroid- core.html.

[12] J. W. Marshall Leach, Introduction to Electroacoustics and Audio Amplifier Design, Second, Kendall/Hunt, 2001.

[13] "Loudspeaker Voice-Coil Inductance Losses: Circuit Models, Parameter Estimation, and Effect on Frequency Response".

[14] J. Bunett, "Crossover Basics Passive crossovers Active crossovers Time alignment".

[15] Fane, Loudspeaker Enclosure Design and Construction.

[16] R. Elliot, "http://sound.westhost.com/bi-amp.htm-power distribution and sound pressure levels," 1998-2009. [Online].

APPENDIX

A Amperes C Capacitance L Inductance M milli Μ Micro Ω Ohm X/o crossover

푋퐿- Inductive reactance

푋퐶-Capacitive reactance R Resistance V Volts F Frequency Fs Resonance frequency

Fc Cut-off frequency

Zo Nominal Impedance W Watts Q Quality factor

푄푀퐶 Mechanical quality factor

푄퐸퐶 Electrical quality factor

APPENDIX B

MATLAB CODE FOR THE FULL NETWORK FREQUENCY RESPONSE axis (ax);%assigns the limits to axis function x=1:20;%produce values to be plotted on x-axis bassy=[0.162 0.161 0.161 -0.268 -0.698 -1.23 -3.89 -7.21 -9.465 -12.96 -14.94 -15.61 -18.679];

83 bassx=[20 50 100 200 250 300 400 500 600 700 800 900 1000]; mid1y=[-28.69 -19.56 -7.67 -3.369 -0.3161 0.26 0.86 0.423 -0.11 -2.93 -7.86 -12.14 -13.69 -17.26 -18.86 -20.69 - 25.69]; mid1x=[50 100 200 300 400 600 800 1000 1200 1500 2000 2500 2800 3000 3500 4000 5000]; mid2y=[-13.61 -11.21 -8.41 -6.89 -5.41 -4.91 0.32 0.504 -2.592 -5.894 -8.372 -10.642 -13.01 -14.77 -27.569]; mid2x=[500 600 700 800 900 1000 2000 4000 5000 6000 7000 8000 9000 10000 20000]; highy=[-22.7 -11.69 -5.11 -3.62 -1.46 -0.7 -0.69 -0.32 -0.318 -0.103 -0.086 -0.084 0.084]; highx=[1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 40000]; plot(bassx,bassy,'-',mid1x,mid1y,'-',mid2x,mid2y,'-',highx,highy,'-'); legend('bass','mid1','mid2','bass');%defines the graphs grid;% sets the grid lines title('full response');% puts a title on the plot xlabel('Frequency(Hz)');% puts a label on the x axis ylabel('Magnitude(dB)')% puts a label on the y-axis end