Operation Of Different Band Pass Filter Designs

Published Date: 02 Nov 2017

Politechnika Łódzka

Wydział Elektrotechniki, Elektroniki, Informatyki i Automatyki

Instytut Elektroenergetyki

BACHELOR THESIS

Filtr pasmowoprzepustowy o regulowanej częstotliwości środkowej

Band-pass filter with adjustable midband frequency

Author: Piotr Kamiński

Student number: 158338

Supervisor: dr inż. Michał Kaczmarek

Łódź, December 2012

Executive summary

Table of Contents

Introduction

Theoretical background

Parallel resonant filter

Passive band-pass Chebyshev filter

Multiple feedback band-pass active filter

Choice of the filter design based on simulations

Investigation of parallel resonant filter

Investigation of passive band-pass Chebyshev filters of various orders

Investigation of multiple feedback band-pass active filter

Construction of the filter

Selection of the components

Issues that occurred during the filter construction

Discussion of results

Summary

Bibliography

List of figures

List of tables

List of acronyms

Introduction

Theoretical background

This section provides an insight into principles of operation of different band-pass filter designs. Although, only designs that were considered the most applicable in the project are presented. Therefore, this is a fundamental part for a thorough understanding of the topic. This chapter is divided into three subchapters. First one describes one of the passive filters, which is a parallel resonant filter. In the second subchapter emphasis is put on Chebyshev filters of different orders. While the third subchapter elaborates the topic of multiple feedback band-pass active filters, which is an active filter. Reader has an opportunity to get acquainted with reasons why exactly those filter designs are considered. Furthermore, differences between mentioned filter designs are brightly illuminated. All the relations needed to manipulate the centre frequency as well as a bandwidth, signal-to-noise ratio and attenuation of the signal are clearly presented. The following chapters outline the most important aspects of the filter design taking into account the aim of the project, which is the filter with adjustable mid-band frequency.

Four basic filter types can be distinguished (band-pass, low-pass, high-pass and notch). They are illustrated in figure 2.1, 2.2, 2.3, and 2.4. Fifth type can also be distinguished. It is called all-pass filter. It has no effect on the amplitude of the signal. However, its function is to change the phase. That is why it is also called phase-shift filter.

Figure 2.1. Low-pass filter.

Figure 2.2. High-pass filter.

Figure 2.3. Band-pass filter

Figure 2.4. Notch filter.

Several examples of amplitude response curves for various filter types are presented above. Ideally, transfer function has a rectangular shape, indicating that the boundary between the pass-band and the stop-band was instant and that the roll-off slope was infinitely steep. Unfortunately, such an amplitude response curve is not physically realizable. Nonetheless, different filter functions are already developed. Each one optimizes some filter property. In the following part, some of them will be briefly described.

The first filter approximation is the Butterworth one. It is also known as a maximally-flat response. It exhibits a nearly flat pass-band with no ripple. The roll-off is smooth and monotonic.

Another approximation to the ideal filter is called Chebyshev filter or equal ripple one. As the second name implies, this kind of filter has ripple in the pass-band amplitude response and its amount is one of the parameters describing a Chebyshev filter. The Chebyshev characteristic has a significantly steeper roll-off near the cut-off frequency compared to the Butterworth one. However, its monotonicity in the pass-band and group delay response are much worse.

Finally, the third filter approximation is called Bessel-Thomson filter or linear phase one. Most of filters introduce phase shift varying with frequency. Nonetheless, it is a standard feature of filters, it may cause some issues in certain instances. If the phase increases linearly with frequency, its effect is simply to delay the output signal by a constant time period. However, if the phase shift is not directly proportional to frequency, components of the input signal at one frequency will appear at the output shifted in phase or time with respect to other frequencies. This can lead to ringing and overshoot build up on the waveform because of the shift in time of square wave’s component frequencies. In order to avoid it, Bessel-Thomson filter is used. As its second name states it exhibits approximately linear phase shift with frequency. The higher the filter order, the more linear the Bessel-Thomson’s phase response. The amplitude response of the Bessel filter is monotonic and smooth, although its roll-off near the cut-off frequency is quite slow compared to the Butterworth and Chebyshev filters. Comparison of the amplitude response of filter designs mentioned above is presented in figure 2.5.

Figure 2.5. Comparison of amplitude response of the mentioned filter designs.

Subsequently, passive filters will be described since two of three considered filters do not compose of active elements. A passive filter is based on combinations of resistors, inductors and capacitors. It does not depend upon an external power supply or contain active components such as transistors. In this respect, it is the simplest (in terms of the number of necessary components) implementation of a given transfer function. Passive filters have other advantages as well. Because of the fact that they are not restricted by the bandwidth limitations of operational amplifiers, they can be used at very high frequencies. Moreover, they can be used in applications involving larger current or voltage levels than can be handled by active devices. Passive filters also generate less noise than circuits using active gain elements. However, they have also several important disadvantages in certain applications. Since they use no amplifying elements, they cannot provide signal gain. Input impedances might be lower than required and output impedances can be higher than optimum for some applications. Inductors are indispensable part of passive filters and these can be expensive if large value, small physical size or high accuracy are required. Finally, complex passive filters, higher than 2nd-order, can be difficult and time consuming to design. Consequently, theory concerning parallel resonant filter and Chebyshev filters of various orders will be presented. As these are types of passive filters considered in the design of band-pass filter with adjustable mid-band frequency.

Parallel resonant filter

It was considered in the design of band-pass filter with adjustable mid-band frequency since it is one of the simplest band-pass filters with easily adjustable centre frequency. In order to change centre frequency capacitance or inductance need to be changed. Parallel resonant filter is a 3-element network that contains two reactive components making it a second-order circuit. It is composed of resistance, inductance and capacitance. Because of the energy of oscillations large current will be circulating between the capacitor and the inductor at resonance. Circuit energy is stored in the capacitor’s electric field and the inductor’s magnetic field. It is constantly being exchanged between the capacitor and the inductor resulting in zero current being drawn from the supply. It is caused by the fact that corresponding instantaneous values of IL and IC are equal and opposite. Hence, the current taken from the supply is the sum of the current flowing in IR and two currents in the inductor and the capacitor. Therefore, one may also say that the LC circuit acts like an open circuit and the current is determined only by the resistor as the total impedance of the circuit at resonance is only the value of resistance. The admittance of parallel circuit is given as:

2.1.1

Where, is an admittance of the resistor, is called an inductive susceptance, BL, and is called a capacitive susceptance, BC. It might be noticed that inductive susceptance is inversely proportional to the frequency, while capacitive susceptance is directly proportional to it. For frequencies lower than the resonant frequency, ‘lagging’ power factor is produced by the inductive susceptance dominating the circuit, however if the frequency is higher than the resonant one, ‘leading’ power factor is produced by the capacitive susceptance dominating the circuit. Resonance frequency can be found by solving a following equation:

2.1.2

2.1.3

Where XL is the reactance of an inductor, XC is the reactance of the capacitor, f is the frequency, L is the inductance, and C is the capacitance, for the frequency:

2.1.4

In figure 2.6, sample parallel resonant filter is presented. Using the equations presented above, its centre frequency can be computed and it is equal to fresonant=151.8Hz.

RLC.jpg

Figure 2.6. Parallel resonant filter.

Frequency response of the circuit shown in figure 2.6 is presented in figure 2.7. It is clearly visible that the peak occurs at the resonant frequency of the circuit and all the other frequencies are attenuated. However, there is a band of frequencies that get through. The width of the band is usually measured at points where the height of the curve is equal to of the amplitude at the centre frequency, maximum height. It can also be said that it occurs 3dB below the maximum amplitude since:

In frequency response presented below in figure 2.7, -3dB point occurs at 150.4Hz, which is lower cut-off frequency, and 153.2Hz, which higher cut-off frequency. Therefore the bandwidth is equal to:

2.1.5

Figure 2.7. Frequency response of the parallel resonant filter presented in figure 2.6.

The narrower the bandwidth, the greater quality factor is. Quality factor can be defined as a ratio of centre frequency and the bandwidth. Hence quality factor of the parallel resonant filter presented in figure 2.6 can be found in the following way:

2.1.6

If slope of the frequency response is steeper then quality factor also goes up. The greater the Q is, the more selective filter is. Quality factor can be obtained by dividing circulating branch currents by the supply current and can be expressed in the following manner:

2.1.7

Based on the equation presented above it is clear that keeping centre frequency fixed, quality factor can be manipulated by changing inductance L, capacitance C or resistance R. Therefore in order to acquire high quality factor resistance and capacitance should be as large as possible and inductance needs to stay as small as possible.

It should be mentioned that in equation 2.1.4 it is assumed that all the elements are purely inductive and capacitive with negligible resistance. Although, in reality inductors have some resistance and it should be taken into account while calculating centre frequency. Hence, the equation looks like:

2.1.8

2.2. Passive band-pass Chebyshev filters

Filter design considered as the next one was passive band-pass Chebyshev filters, mainly because of its superior attenuation. It is designed based on low-pass filter prototypes, which are normalized in terms of frequency and impedance. It is done so in order to simplify the design of filters for arbitrary impedance, frequency and type. Afterwards, those low-pass prototypes are scaled to the desired impedance and frequency.

The Chebyshev response is a mathematical strategy for achieving a faster roll-off by allowing ripple in the frequency response. Insertion loss of an N-order low-pass filter using Chebyshev polynomial can be expressed as:

2.2.1

Where ε is the ripple factor responsible for determining the pass-band ripple level, ω0 is the cut-off frequency and TN() is a Chebyshev polynomial of the n-th order. Since oscillates between ±1 for , the pass-band response will have ripples of amplitude 1+ε2. For , then the insertion loss can be expressed as:

2.2.2

It can be noticed that it increases at the rate of 20N dB/decade. The insertion loss for the Chebyshev case is (22N)/4 greater than the binomial response at the frequency much greater than the cut-off frequency.

Low-pass prototypes can be implemented using one of the two topologies. One begins with a shunt element and another one with the series. Afterwards, elements alternate between series and shunt connections. The order of both filter prototypes is determined by the number of reactive components. It should be mentioned that both topologies give the same responses. Sample topology starting with the component connected in series is shown in figure 2.8 and another one starting with the component connected in shunt is presented in figure 2.9. It is normalized low-pass filter design with source impedance being equal to 1 and the cut-off frequency .

Figure 2.8.Topology showing 6th order low-pass prototype starting with element connected in series.

Figure 2.9.Topology showing 6th order low-pass prototype starting with element connected in shunt.

As one may notice in figure 2.8, elements are numbered from g0 at the generator impedance to gN+1 at the load impedance, for a filter of the Nth order. G0 can be considered as a generator resistance if the topology starts with a series element and as a generator conductance if the topology start with the shunt element. Similarly, gN+1 is a load resistance if gN is a shunt capacitor or it is a load conductance if gN is a series inductor. Furthermore, all the other components from g1 to gN stand for inductance for series inductor and conductance for shunt capacitor.

The element values of circuit shown in figure 2.8 can be found in tables. Table 2.1 shows such a values for low-pass Chebyshev prototype( g0=1, ω0=1). Values are presented for filters of the 1st order up to the 10th. Actually, they are divided into two tables one presenting values for low-pass Chebyshev prototype with 0.5dB ripples and another one with 3dB ripples.

Table 2.1. Element values for Chebyshev low-pass filter prototypes (g0=1, ω0=1, N=1 to 10, 0.5dB and 3dB ripple)

0.5 dB Ripple

N

g1

g2

g3

g4

g5

g6

g7

g8

g9

g10

g11

1

0.6986

1.0000

 

 

 

 

 

 

 

 

 

2

1.4029

0.7071

1.9841

 

 

 

 

 

 

 

 

3

1.5963

1.0967

1.5963

1.0000

 

 

 

 

 

 

 

4

1.6703

1.1926

2.3661

0.8419

1.9841

 

 

 

 

 

 

5

1.7058

1.2296

2.5408

1.2296

1.7058

1.0000

 

 

 

 

 

6

1.7254

1.2479

2.6064

1.3137

2.4758

0.8696

1.9841

 

 

 

 

7

1.7372

1.2583

2.6381

1.3444

2.6381

1.2583

1.7372

1.0000

 

 

 

8

1.7451

1.2647

2.6564

1.3590

2.6964

1.3389

2.5093

0.8796

1.9841

 

 

9

1.7504

1.2690

2.6678

1.3673

2.7239

1.3673

2.6678

1.2690

1.7504

1.0000

 

10

1.7543

1.2721

2.6754

1.3725

2.7392

1.3806

2.7231

1.3485

2.5239

0.8842

1.9841

3 dB Ripple

N

g1

g2

g3

g4

g5

g6

g7

g8

g9

g10

g11

1

1.9953

1.0000

 

 

 

 

 

 

 

 

 

2

3.1013

0.5339

5.8095

 

 

 

 

 

 

 

 

3

3.3487

0.7117

3.3487

1.0000

 

 

 

 

 

 

 

4

3.4389

0.7483

4.3471

0.5920

5.8095

 

 

 

 

 

 

5

3.4817

0.7618

4.5381

0.7618

3.4817

1.0000

 

 

 

 

 

6

3.5045

0.7685

4.6061

0.7929

4.4641

0.6033

5.8095

 

 

 

 

7

3.5182

0.7723

4.6386

0.8039

4.6386

0.7723

3.5182

1.0000

 

 

 

8

3.5277

0.7745

4.6575

0.8089

4.6990

0.8018

4.4990

0.6073

5.8095

 

 

9

3.5340

0.7760

4.6692

0.8118

4.7272

0.8118

4.6692

0.7760

3.5340

1.0000

 

10

3.5384

0.7771

4.6768

0.8164

4.7425

0.8164

4.7260

0.8051

4.5142

0.6091

5.8095

Having all those values for different filter prototypes, it is necessary to realize filter order that is needed to satisfy the requirements. While choosing Chebyshev filter design attenuation in the stop-band is usually the most important parameter. In figures 2.10 and 2.11, attenuation characteristics of various N versus normalized frequency are presented. Where ω is the frequency at which specified attenuation is required.

Figure 2.10. Attenuation as a function of the normalized frequency for Chebyshev prototypes having 0.5dB ripple.

Figure 2.11. Attenuation as a function of the normalized frequency for Chebyshev prototypes having 3dB ripple.

Finally, in order to design band-pass Chebyshev filter low-pass prototype need to be transformed into band-pass response. As mentioned before attenuation characteristics can be used in order to determine the order of a band-pass filter. However, it need be taken into account that in that case ratio of bandwidths becomes more important than the ratio of frequencies. Therefore, ratio of frequencies should be replaced by ratio of the bandwidths BW/BW0. Having required order of the filter determined, values of the consecutive components can be found in table 2.1. Afterwards, each inductor in the low-pass prototype should be replaced with a series resonant circuit and each capacitor with a parallel resonant circuit. Although, it needs to be remembered that both components in each resonant circuit have the same normalized value. Furthermore, all the values need to be scaled to desired frequency and impedance. Firstly, impedance scaling is presented. In the prototype design source resistance is equal to the unity therefore after multiplication by R0 it is equal to R0. Rest of the scaled component values is expressed as:

2.2.3

2.2.4

2.2.5

2.2.6

Where gm, gk, gN+1, are component values of the original prototype and R0 is a normalization factor.

When it comes to the frequency scaling, assuming that ω1 and ω2 indicate edges of the pass-band, then in order to transform low-pass prototype into band-pass one following substitution is to be used:

2.2.7

where 2.2.8

Δ is called fractional bandwidth of the pass-band. Moreover, ω0 was chosen to be equal to the geometric mean of ω2 and ω1 because of the fact that calculations become significantly easier. Using 2.2.7 in expressions for series reactance (2.2.9) and shunt susceptance (2.2.12) it can be shown that a series inductor, Lk, is transformed to series LC circuit (2.2.9) and shunt capacitor, Ck, is transformed to parallel LC circuit (2.2.12) with the following values of the corresponding elements:

2.2.9

2.2.10

2.2.11

2.2.12

2.2.13

2.2.14

To conclude, low-pass prototype with the normalized values of the elements can be transformed to band-pass response by replacing series inductors with the series LC circuit and shunt capacitor with the parallel LC circuit as it is shown in figure 2.12. L1 and C1 are the normalized values of the low-pass prototype, which are taken from table 2.1 according to the requirements. In order to obtain values of the components of the band-pass filters they need to be frequency and impedance scaled in accordance with the formulas presented in figure 2.12.

Figure 2.12. Summary of low-pass filter transformation to the band-pass one.

2.3 Multiple feedback band-pass active filter

Multiple feedback band-pass filter is an active filter, therefore this type of filters will be briefly described at first. Active filters are composed not only from passive elements but they also use amplifying components such as operational amplifiers. Moreover, resistors and capacitors are parts of their feedback loop in order to acquire the desired filter characteristics. They can have unrestricted gain and low output impedance, high input impedance. Active filters might be easier to design than the passive ones. It is mainly caused by their lack of inductors leading to the reduction of the issues associated with those components. Inductors are usually the largest and most expensive parts used. They are subject to losses much more than capacitors. Furthermore, high quality polypropylene capacitor can be bought for few dollars, while high quality large value inductor is much more expensive. Although, capacitors have also problems with value spacing and their accuracy, it is significantly limited compared to the inductors. Operation of active filters can be limited at high frequencies by the gain-bandwidth product of the operational amplifier. However, very good accuracy can be achieved provided that it operates within frequency range and low-tolerance components are used. Another issue that might occur is deviation of filter’s response from the ideal transfer function. It can be a problem when the centre frequency multiplied by the filter’s quality factor is significant fraction of the gain-bandwidth product. It also depends on topology of the filter. Nonetheless, there exist certain topologies designed to decrease the impact of limited gain-bandwidths.

Multiple feedback band-pass filter is one of the most useful band-pass filters as it is able to produce an all-pole response. It is a circuit based around negative feedback active filter of a relatively high quality factor and steep roll-off on both sides of the centre frequency. Since its frequency response reminds the one of a resonance circuit, its centre frequency is referred to as the resonant frequency ( fr ).

The performance of the multiple feedback band-pass filter strongly depends on the gain-bandwidth of the operational amplifier as it should be high in comparison with the resonant frequency.

In order to obtain the frequency response of a second order band-pass filter transformation 2.3.1 needs to be applied to a first order low-pass transfer function (2.3.2 ), which is also presented below:

2.3.1

2.3.2

where , . Therefore, replacing s in 2.3.2 with 2.3.1 results in obtaining the general transfer function for a second order band-pass filter:

2.3.3

Multiple feedback band-pass filter schematic is presented in figure 2.13. It should be noted that both capacitors were chosen to be of the same value in order to simplify calculations. Based on the schematic transfer function of the multiple feedback band-pass filter can be expressed as:

2.3.4

After few transformations following expression can be obtained:

2.3.5

Figure 2.13. Multiple feedback band-pass filter.

As it can be noticed in figure 2.13, the two feedback paths are through R2 and C2, that is why it called multiple feedback filter. As most of the band-pass filters are composed of low-pass and high-pass filters, this one also exhibits those features, components R1 and C2 provide the low-pass response while C1 and R2 provide the high-pass response. Equation 2.3.4 was subjected to certain transformations in order to achieve expression of the same form as the general transfer function for a second order band-pass filter. This resulted in equation 2.3.5 which coefficients can be compared with the ones of equation 2.3.3. The comparison led to the following expressions:

Centre frequency 2.3.6

Gain at centre frequency 2.3.7

Quality factor 2.3.8

Bandwidth 2.3.9

Based on those equations values of components can be computed for certain design requirements. It should be also mentioned that multiple feedback band-pass filter allows to adjust quality factor, gain at centre frequency and centre frequency independently. Bandwidth and gain do not depend on . Hence, centre frequency can be modified by changing , without making any impact on the bandwidth or gain. However it should be noted that quality factor is changed then.

In order to design multiple feedback band-pass filter value of the capacitor should be chosen and bandwidth specified at first. Then value of can be obtained by transforming equation 2.3.9. Having , can be computed for the specified gain at the centre frequency based on the equation 2.3.7. Afterwards, having and , the value of resistor can be found out for the specified centre frequency. In order to perform those calculations following expressions are to be used:

2.3.10

2.3.11

2.3.12

It should be noticed that increasing capacitance, the resistance values are decreased. In that case loading on the input buffer of the operational amplifier should be maintained on the reasonable level. On the other hand, low values of capacitance may cause issues with a stray capacitance and resistor values becoming too high. Finally, multiple feedback band-pass filters with slightly spread centre frequencies can be cascaded in order to achieve steeper roll-off.

However, the emphasis is put on

The aim of the project is to design a highly selective bandpass filter with adjustable centre frequency. Therefore, the first design technique taken under consideration was parallel resonant filter. Arguments that were convincing enough to try it out were following. Firstly, parallel resonant filter is composed of small number of elements making it easier to construct. Schematic showing it is illustrated in figure 1. Secondly, only one element (inductor or capacitor) is required to vary in order to change centre frequency