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A
Project Report
On
“CURRENT MODE BIQUAD FILTER”
Submitted to
The Department of Electronics Engineering
In partial fulfilment of requirements for the award of the Degree of
Bachelor of Technology
In Electronics and Communication Engineering

By:
Vishad Saxena (1405231054)
Saurabh Kumar (1405231042)
Under the guidance of
Dr. V. K. Singh
Department of Electronics Engineering
Institute of Engineering and Technology, Lucknow
Dr. A.P.J. Abdul Kalam Technical University
2017-2018
TABLE OF CONTENTS
CERTIFICATE…………………………………………………………..4
ACKNOWLEDGEMENT……………………………………………….5
ABSTRACT………………………………………………………………6
LIST OF FIGURES………………………………………………………7
INTRODUCTION
FUNDAMENTALS………………………………………………..8
TYPES OF FILTERS………………………………………………9
CLASSIFICATION ON THE BASIS OF WORKING…………..10
1.4 WHY WE USE ANALOG FILTERS…………………………….11
OPERATIONAL AMPLIFIER
INTRODUCTION…………………………………………………12
OPERATIONAL AMPLIFIER MODELS………………………..12
SLEW RATE ……………………………………………………..15
OPERATION AMPLIFIER WITH RESISTIVE FEEDBACK……………………………………………………….16
FIRST ORDER FILTERS
INTRODUCTION…………………………………………………18
BILINEAR TRANSFER FUNCTION……………………………18
REALIZATION WITH PASSIVE ELEMENTS…………………20
ACTIVE REALIZATION……………………………………….21
THE EFFECT OF A(s)…………………………………………..23
CASCADE DESIGN…………………………………………….23
SECOND ORDER LOW PASS AND BANDPASS FILTERS
INTRODUCTION………………………………………………..25
SECOND ORDER CRICUIT……………………………………25
FREQUENCY RESPONSE OF LOW PASS AND BAND PASS CIRCUITS……………………………………………………….26
INTEGRATORS: THE EFFECT OF A(s) ……………………..28
THE EFFECT OF A(s) IN BIQUAD……………………………29
PROPOSED KHN BIQUAD CIRCUIT
KHN-EQUIVALENT BIQUAD USING CURRENT CONVEYOR…………………………………………………….31
INTRODUCTION……………………………………………….31
PROPOSED KHN-EQUIVALENT BIQUAD………………….31
ADVANTAGES…………………………………………………34
KHN-EQUIVALENT PSPICE CIRCUIT SIMULATION……..35
KHN-EQUIVALENT SIMULATION OUTPUT………………35
CONCLUSION…………………………………………………36
REFERENCES…………………………………………………………37
CERTIFICATE
This is to certify that Project Report entitled “Current Mode Biquad Filter” which is submitted by Vishad Saxena and Saurabh Kumar, students of Electronics Engineering Department, Institute of Engineering and Technology Lucknow, in partial fulfilment of the requirement for the degree of B.Tech in Department of Electronics, of A.P.J. Abdul Kalam University is a record of the candidate own work carried by the, under my/our supervision. The matter embodied in this thesis is original and has not been submitted for the award of any other degree.

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Date: Project Guide
Dr. V.K. Singh
(Professor)
Department of Electronics Engineering
Institute of Engineering and Technology
Dr. A.P.J. Abdul Kalam Technical University, Lucknow
ACKNOWLEDGEMENT
In the department, Presentation and compilation of our work, we are highly indebted to all those people who provided their able guidance and support throughout. We would like to express our gratefulness towards our project guide Dr. V.K. Singh for his continuous encouragement and support, especially through all the obstacles and setback. His dedication, profound insight and attention to details have been true inspiration to my research and my personal life. His technical guidance has facilitated a rich educational experience to us by numerous discussion, valuable advice and direction, encouragement and fair criticism.

We would like to express special thanks to Prof. Subodh Wairya, Head of Department (Electronics Engineering) for giving us the opportunity to undertake project of such class.

We also do not want to miss the opportunity to acknowledge the contribution of all lab assistants of the department for their kind assistance and cooperation during the developments of our project. The guidance and support received from our classmates, friends and our M.Tech seniors who contribute to this project, was vital for our successful completion of the report. We are grateful for their constant support and help.

Date:
Signature:
Name:
Roll No: 1405231054
Signature:
Name:
Roll No: 1405231042
ABSTRACT
The proposed KHN biquad filter is better implementation of original KHN biquad filter. It provides better controllability over various parameters. Output fields can be independently controlled by different components. Some resistors can be replaced by FET based voltage control resistor. It also offers ideally infinite input impedance. Based on its state variable structure, several other CC based and OTA based biquads of varying complexity are derivable by exploiting the degrees of freedom in devising and combining the required building blocks.

In this project, Filters and its basic principles and implementation is being studied. Types of filters and their replacement and realisation is being studied. State variables representation and conversion is studied. And with the help of this knowledge, KHN biquad filter is analysed and this circuit is simulated on pSpice software and the result obtained from the practical is verified with its theoretical value.

LIST OF FIGURES
Figure 1.1 : Two Port Network………………………………………….8
Figure 1.2 : Types of Filters……………………………………………10
Figure 2.1 : Integrator Model…………………………………………..13
Figure 2.2 : Ideal Operational Amplifier………………………………14
Figure 2.3 : Non Inverting Amplifier………………………………….16
Figure 2.4 : Inverting Amplifier……………………………………….17
Figure 3.1 : Passive Voltage Divider to realize Bilinear Function…….21
Figure 3.2 : Inverting Operational Amplifier with impedance in passive feedback network……………………………………………………….22
Figure 3.3 : Active circuit realizing the Bilinear Function…………….23
Figure 3.4 : Cascade Connection on “N” Sections…………………….24
Figure 4.1: Magnitude Response of Lowpass Filter…………………..27
Figure 4.2: Phase Response of Lowpass Filter………………………..27
Figure 4.3: Magnitude Response of Bandpass Filter………………….28
Figure 4.4 Phase Response of Bandpass Filter…………………………28
Figure 5.1: KHN-Equivalent biquad using Opamp and State variable diagram………………………………………………………………….32
Figure 5.2: Proposed KHN- Equivalent Biquad using Current Conveyor………………………………………………………………..34
Figure 5.3: KHN Equivalent Simulation Circuit……………………….35
Figure 5.4: Simulation Output of KHN Equivalent…………………….35
CHAPTER 1
INTRODUCTION
1.1 FUNDAMENTALS
A filter is a circuit capable of passing (or amplifying) certain frequencies while attenuating other frequencies. Thus, a filter can extract important frequencies from signals that also contain undesirable or irrelevant frequencies.

In the field of electronics, there are many practical applications for filters. Examples include:
Radio communications: Filters enable radio receivers to only “see” the desired signal while rejecting all other signals (assuming that the other signals have different frequency content).

DC power supplies: Filters are used to eliminate undesired high frequencies (i.e., noise) that are present on AC input lines. Additionally, filters are used on a power supply’s output to reduce ripple.

Audio electronics: A crossover network is a network of filters used to channel low-frequency audio to woofers, mid-range frequencies to midrange speakers, and high-frequency sounds to tweeters.

Analog-to-digital conversion: Filters are placed in front of an ADC input to minimize aliasing.

The fundamentals of analogue filter design can be described from a two port network as shown in figure.

Figure1.1: Two port Network
In this, an input voltage V(1) is connected on one side and output voltage V(2) is connected on the other side.The transfer function of this two port network is defined as
(1.1)
The general derivation of transfer function is derived to be as
(1.2)
Where Theta is the phase angle of the I/O voltages.

The magnitude of the transfer function is equal to
(1.3)
1.2 TYPES OF FILTERS
Filters can be placed in one of two categories: passive or active.

Passive filters include only passive components—resistors, capacitors, and inductors. In contrast, active filters use active components, such as op-amps, in addition to resistors and capacitors, but not inductors.

Passive filters are most responsive to a frequency range from roughly 100 Hz to 300 MHz. The limitation on the lower end is a result of the fact that at low frequencies the inductance or capacitance would have to be quite large. The upper-frequency limit is due to the effect of parasitic capacitances and inductances. Careful design practices can extend the use of passive circuits well into the gigahertz range.

Active filters are capable of dealing with very low frequencies (approaching 0 Hz), and they can provide voltage gain (passive filters cannot). Active filters can be used to design high-order filters without the use of inductors; this is important because inductors are problematic in the context of integrated-circuit manufacturing techniques. However, active filters are less suitable for very-high-frequency applications because of amplifier bandwidth limitations. Radio-frequency circuits must often utilize passive filters.

1.3 CLASSIFICATION ON THE BASIS OF WORKING
Filters are classified on the basis of functions they are to perform. Over the frequency of filter where |T| = 1 and alpha = 0, that is, signals are transmitted from input to output without attenuation or gain. In a stop band |T| = 0 and alpha = -Infinite, which means that transmission is blocked completely. The patterns of stop band and pass bands that give rise to the most common filters which are shown below

Figure 1.2: Types of Filters
Some of the concepts that define these patterns are:
A low pass filter characteristics one in which the pass band extends from w=0 to w = wc. Where wc is known as the cut-off frequency.

A high pass filter is the complement of the low pass filter in that the frequency range from 0 to wc is the stop band and from wc to infinity is the pass band.

A band pass filter is one in which the frequencies extending from w1 to w2 are passed while signals at all other frequencies are stopped.

The band stop filter is the complement of the band pass filter where signal components at the frequencies from w1 to w2 are stopped and all others are passed. These filters are also sometimes referred to as notch filters because of the “notch” in their transmission characteristics.

In practice, it is not possible to realize the ideal transfer functions shown by solid lines with real filters consisting of a finite number of elements. We shall see throughout our study of filters that for real rational functions of the complex frequency s. A real rational function is a ratio of polynomial in s as shown
(1.4)
1.4 WHY WE USE ANALOG FILTERS
The basic concept of electric filters were developed in 1915 independently by Wagner in Germany and Campbell in United States. In the years since that invention, filter theory have been developed to a high degree of perfection. Implementing economically an active filter, a filter that uses gain, became possible with the invention of the vacuum tube, and the development of feedback theory by Black, Bode and others in early 1930s. The present era of wide use of high quality low-cost discrete analog active filters is due to the development of the inexpensive monolithic IC, filter designers felt pressure to devise techniques that allow the integration of analog filters onto the IC along with digital circuitry .The solution was found in the more recent developments of switched capacitor filters.

Analog active filters always use gain and capacitors .In practical discrete active filters resistors are also used and gain is obtained from the op-amp. In integrated active filters we obtain gain by making use of op-amps or trans-conductance amplifiers, and we utilize capacitors, resistors and at the highest frequencies integrated inductors. To be able to decide which components to use, we must consider factors such as:
The technology desired for the system implementation.

Availability of dc supplies for the active devices, and power consumption.

Cost.

The range of frequency of operation.

The sensitivity to parameter changes and stability.

Weight and size of the implemented circuit.

Noise and dynamic range of the realized filter.

The meaning of several of these criteria will become clear as we progress in our discussion of active filters.
CHAPTER 2
OPERATIONAL AMPLIFIERS
2.1 INTRODUCTION
In this chapter we introduce the operational amplifier (opamp), the main device used to provide gain in the design of active filters. The other gain device is the trans-conductor or operational trans-conductance amplifier (OTA). It is used mainly in filters for very high frequencies. Since the performance of the active filters depends critically on the opamp, it is very important that we fully understand the behaviour of opamps to be able to undertake successful filter designs. Opamps are relatively complicated electronic circuits, consisting of transistors, resistors and capacitors.

To analyse the behaviour of each filter with complete transistor-level electronic circuit for the vast majority of applications studied. We will also define some simple models like integrator model, the ideal opamp that will allow us to develop circuits very rapidly and to gain preliminary insight into their behaviour under carefully observed conditions.

2.2 OPERATIONAL AMPLIFIER MODELS
Opamps are differential amplifiers, familiar in modern electronics. They differ from ordinary amplifiers by having two inputs. Their operation is such that the output voltage Vo is the difference between two input voltages multiplied by an overall gain. In terms of voltage defined we have,
(2.1)
Where A is the gain of the opamp. An important property of differential amplifiers is that signals that are common to both inputs are not amplified; the amplifier rejects them. Signal that are common to both inputs are called common mode signals and are rejected by the opamp.

2.1.1 THE INTEGRATOR MODEL
An understanding of the opamp behaviour can be gained from the simple block diagram of the opamp circuit, excluding the bias circuitry and, as we agreed, excluding the power supply. We have included the resistor R to represent the large but finite output resistance of the trans-conductance stage gm. Without having to concern ourselves with the details of the operation of the trans- conductance and amplifier stage A2. The integrator model is shown as follows

Figure 2.1: Integrator Model
Where current is given by
(2.2)
Further note from figure that –A2V1 = Vo. Combining these equations and reordering the terms results in the open loop transfer function of the opamp circuit in previous figure.

(2.3)
And here the frequency is given by
(2.4)
2.1.2 THE IDEAL OPERATIONAL AMPLIFIER
We have developed a simple model that will be useful for our work in active filter design. We have seen that the gain is a function of frequency, is very large at dc, and that the opamp has a large input resistance and a small output resistance. Specifically, we define a model with
A = infinity, Ri = Infinity, Ro = Zero

Figure 2.2: Ideal Operational AmpifierThe idealizations of equations rarely cause problems in filter design, and are used quite liberally in most circumstances. It leads to the conclusion that the net input voltage into and ideal opamp must be zero. Property No.4: Zero Noise Contribution n In the ideal op-amp, zero noise voltage is produced internally g This is, any noise at the output must have been at the input as well n Practical op-amp are affected by several noise sources, such as resistive and semiconductor noise g These effects can have considerable effects in low signal-level applications.

Ideal Opamp Properties
1. Infinite Open-Loop Gain: Open-Loop Gain Avol is the gain of the op-amp without positive or negative feedback n In the ideal op-amp Avol is infinite g Typical values range from 20,000 to 200,000 in real devices g
2. Infinite Input Impedance: Input impedance is the ratio of input voltage to input current
Zin = Vin/IinWhen Zin is infinite, the input current Iin=0. High-grade op-amps can have input impedance in the T? range. Some low-grade op-amps, on the other hand, can have mA input currents.

3. Zero Output Impedance: The ideal op-amp acts as a perfect internal voltage source with no internal resistance g This internal resistance is in series with the load, reducing the output voltage available to the load g Real op-amps have output-impedance in the 100-20? range.

4. Zero Noise Contribution: In the ideal op-amp, zero noise voltage is produced internally. This is, any noise at the output must have been at the input as well. Practical op-amp are affected by several noise sources, such as resistive and semiconductor noise. These effects can have considerable effects in low signal-level applications.

5. Zero output Offset: The output offset is the output voltage of an amplifier when both inputs are grounded. The ideal op-amp has zero output offset, but real op-amps have some amount of output offset voltage.

6. Infinite Bandwidth: The ideal op-amp will amplify all signals from DC to the highest AC frequencies. In real opamps, the bandwidth is rather limited. This limitation is specified by the Gain-Bandwidth product (GB), which is equal to the frequency where the amplifier gain becomes unity. Some op-amps, such as the 741 family, have very limited bandwidth of up to a few KHz
7. Differential Inputs Stick Together: In the ideal op-amp, a voltage applied to one input also appears at the other input.

2.3 SLEW RATE
The slew rate of an operational amplifier may be important in many applications.

The op amp slew rate is particularly important parameter in applications where the output is required to switch from one level to another quickly. In these applications the rate at which the op amp can change between the two levels is important.

The slew rate of an op amp or any amplifier circuit is the rate of change in the output voltage caused by a step change on the input. A typical general purpose device may have a slew rate of 10 V / microsecond. This means that when a large step change is placed on the input, the device would be able to provide an output 10 volt change in one microsecond.

The figures for slew rate change are dependent upon the type of operational amplifier being used. Low power op-amps may only have figures of a volt per microsecond, whereas there are fast operational amplifiers capable to providing rates of 1000 V / microsecond.

Op amps may have different slew rates for positive and negative going transitions because of the circuit configuration. They have a complementary output to pull the signal up and down and this means the two sides of the circuit cannot be exactly the same. However it is often assumed that they have reasonably symmetrical performance levels.

Slew Rate = 2*3.14*f*V
Where f = the highest signal frequency of the signal
V = the maximum peak voltage of the signal
It is measured as a voltage change in a given time – typically V / µs or V / ms.2.4 OPERATIONAL AMPLIFIERS WITH RESISTIVE FEEDBACK: Inverting and Non-Inverting Amplifiers
To help us gain an understanding of opamp behaviour, and to appreciate the use of the models we introduced, we study the inverting and the non-inverting gain amplifiers built with opamps. Amplifiers are used to increase the amplitude or the power of a signal. Ideally, this amplification should be performed without the signal source, and the amplified signal should be able to drive arbitrary loads. This implies that no current should be drawn by the amplifier from the source, i.e. the amplifier’s input resistance should be very large, ideally infinite. Being able to drive arbitrary loads in turn implies that the output of the amplifier should appear like an ideal voltage source, i.e. it should have a very small, ideally zero output resistance.
2.4.1 THE NON-INVERTING AMPLIFIER
Figure shows the opamp placed into a harness made up of 2 resistors. To emphasize that the following discussion is valid for arbitrary signals, lower-case symbols are used. The input V1 is applied to the non-inverting input terminal and a fraction of output voltage is applied (“fed back”) to the inverting input.

Figure 2.3: Non Inverting Amplifier
Solving the equation for V2/V1 gives the gain of the non-inverting amplifier as
(2.5)
Now, if the opamp gain is very large, in the limit infinite (A= infinity) for an ideal opamp, we obtain from previous equation
(2.6)
Thus, we see that the circuit in figure is an amplifier whose positive (non-inverting) gain Kp is determined by two resistors, and is – to the first order, for large gain A – independent of the opamp gain A. By choosing the proper values of the 2 resistors, the gain is determined.

2.4.2 THE INVERTING AMPLIFIER
In this case, both input and output signals are connected through resistors to the inverting opamp input terminal for negative feedback. The circuit is shown in figure below

Figure 2.4: Inverting Amplifier
By calculating the node analysis of the given figure, we can solve for the gain and it is given by
(2.7)
Interpreting this expression proceeds similarly to the discussion following equations. If the opamp gain is very large, in the limiting case infinite (A=infinity), we obtain the following equation
(2.8)
The circuit is an inverting amplifier of gain –Kn. As in the non-inverting amplifier, the gain is set by two chosen resistors. It is (at low frequencies) independent of A, and the minus sign indicates a signal inversion (positive input results in negative output). Again the assumption of an ideal opamp would have permitted us to obtain the result in previous equation immediately.
CHAPTER 3
 
FIRST ORDER FILTERS
 
 3.1 INTRODUCTION
In chapter 2, we considered operational amplifier circuits with purely resistive feedback. Any frequency dependence arose from the operational amplifiers itself. However, since opamps have poorly controlled parameters, we saw that it is generally not advisable to let filter performance be determined by these parameters. We must therefore rely on the opamp to have large gain and make the feedback characteristic H frequency dependent. To accomplish this goal, we will in this chapter extend our studies from chapter 2 by adding capacitor as component to the resistor in the feedback network. Also, since we shall be interested from now on mainly in the frequency response, we shall always assume sinusoidal sources and steady state operation. Although we will remind ourselves from time to time of domain response, the language of filter design is predominantly that of the frequency domain. This means that v(t) and i(t) will be represented by phasors or laplace transforms V(s) and I(s). Earlier, we labelled the ratio of output to input of a circuit the gain. In the following we shall refer to this ratio as the transfer function and design again the symbol T or T(s). We shall also characterize the circuit elements, individual ones or combinations, by their impedances Z(s) = V/I or admittance Y(s) = I/V. This was done on occasion before, when we needed to consider frequency domain concepts, such as the bandwidth of a design.

We pointed out in chapter 1 that the transfer function of a filter with a finite number of discrete components must be a real rational function. Which we repeat here as Eq. 3.1 for convenience:
(3.1)
3.2 BILINEAR TRANSFER FUNCTION
An analog filter is a linear system that has an input and output signal. This system’s primary purpose is to change the frequency response characteristics of the input signal as it moves through the filter. The characteristics of this filter system could be studied in the time domain or the frequency domain. From a systems point of view, the impulse response h(t) could be used to describe the system in the time domain. The impulse response of a system is the output of a system that has had an impulse applied to the input. Of course, many systems would not be able to sustain an infinite spike (the impulse) being applied to the input of the system, but there are ways to determine h(t) without actually applying the impulse.

A filter system can also be described in the frequency domain by using the transfer function H(s). The transfer function of the system can be determined by finding the Laplace transform of h(t). Figure 2.1 indicates that the filter system can be considered either in the time domain or in the frequency domain. However, the transfer function description is the predominant method used in filter design, and we will perform most of our filter design using it. The transfer function H(s) for a filter system can be characterized in a number of ways. As shown in (2.1), H(s) is typically represented as the ratio of two polynomials in s where in this case the numerator polynomial is order m and the denominator is a polynomial of order n. G represents an overall gain constant that can take on any value.

3.2.2 TRANSFER FUNCTION CHARACTERISTICS
The transfer function H(s) for a filter system can be characterized in a number of ways. As shown in (2.1), H(s) is typically represented as the ratio of two polynomials in s where in this case the numerator polynomial is order m and the denominator is a polynomial of order n. G represents an overall gain constant that can take on any value.

center3954300
(3.2)
Alternately, the polynomials can be factored to give a form as shown in (2.2). In this representation, the numerator and denominator polynomials have been separated into first-order factors. The Zs represent the roots of the numerator and are referred to as the zeros of the transfer function. Similarly, the Ps represent the roots of the denominator and are referred to as the poles of the transfer function.

center969800(3.3)
Most of the poles and zeros in filter design will be complex valued and will occur as complex conjugate pairs. In this case, it will be more convenient to represent the transfer function as a ratio of quadratic terms that combine the individual complex conjugate factors as shown in (2.3). The first-order factors that are included will be present only if the numerator or denominator polynomial orders are odd. We will be using this form for most of the analog filter design material.

3607381350800
(3.4)
3.3 REALIZATION WITH PASSIVE ELEMENTS
Let us now contemplate how the bilinear function and its various special cases can be realized with passive elements. To this end consider the voltage divider in figure. It is characterizes by impedances, phasor voltages and currents. This analysis follows well known elementary procedures. If I2 = 0 , we have
V1 = (Z1 + Z2)I and V2 = Z2I
center350520then with Y = 1/ Z,
(3.5)
An equation that could also have been written directly by voltage division. Remembering that we do not wish to use inductors, we can simple use RC circuit in figure with G1 = 1 / R1 we see that
Y1 = G1 + sC1 and Y2 = G2 + sC2
So that the transfer function becomes
259080000Figure 3.1: Passive Voltage divider to realize a bilinear function
centertop
(3.6)
Clearly, it is a bilinear expression with real coefficients as required. If we bring this equation, we see that the zero and pole location in the circuit. Observe that pole and zero have units of 1/time i.e., they are frequencies set by RC time constants. We will find that this will always the case. Also note that pole and zero are on the real negative axis as we predicted as the coefficients of equation are positive and further that we may have z1 ; p1 and z1 ; p1 depending on how the elements values are chosen. Equation given is therefore a relatively general bilinear function with only few restrictions on possible pole zero locations.

3.4 ACTIVE REALIZATIONS
The transfer function we proposed have number of problems that we proposed to solve with the aid of operational amplifiers. To determine the circuit that might provide a solution, we consult chapter 2 and notice that the inverting amplifier may be suitable if we replaced the resistors by impedances. The analysis of the circuit is no different from its resistive version in chapter 2. We sum the currents at the inverting input node of the opamp or for simplicity replace Ri by Zi. The result is
415290127000
(3.7)
center1944139On the right hand side we have used admittances. Y = 1 / Z because in the following as in the case in active filter work, the treatment will be simpler and more transparent if it is based on admittances. Also we have included in equation the effect on the transfer function of finite and frequency dependent opamp gain A(s) to remind ourselves that this is a concern we must address. Initially however we shall assume A = infinite sothat we can focus on the main issues involved in designing first order circuits. Thus we base our treatment on equation and contemplate how the bilinear function of equation can be realized.

Figure 3.2: Inverting Amplifier with impedances in the passive feedback network
The procedure we will follow require that some parts of the prescribed right hand side be assigned to Y1 and Y2. The assignments are not unique resulting in several different design strategies and circuits. Since inductors are excluded we must avoid making the identifications Y = 1 / (sL) or
Z = sL. One of several possibilities that suggests itself is to make the admittances linear functions of frequency.

Y 1= sC1 + G1and Y2 = sC2 + G2
center170064To give
(3.8)
From which we can identify
5793510(3.9)
center456276The circuit is shown in figure 3.3. It is easy to find the components values for a prescribed design.

Figure 3.3: Active circuit realizing the bilinear function
3.5 THE EFFECT OF A(s)
In section 3.4 we assumed that the opamp gain was so large that its influence on transfer function can be neglected. In this section we shall examine how far the gain and bandwidth assumptions are valid for first order opamp circuits. Equation contains explicitly the finite and frequency dependent opamp gain A(s) so that we may examine its effect on transfer function. Since we are dealing with bilinear functions in this chapter we obtain the expression
centertop
(3.10)
3.6 CASCADE DESIGN
The method proposes to connect n lower order building blocks with transfer function T(s) as shown in figure such that the total higher order transfer function is the product of the individual functions.

1962720-47160(3.11)
For this simple method to work, the lower order sections must satisfy certain conditions that we can understand readily by considering the circuit depicted in figure.

center000
Figure 3.4: Cascade connections of n sections
CHAPTER 4
 
SECOND ORDER LOWPASS AND BANDPASS FILTERS
 
4.1 INTRODUCTION
 Second order filters often referred to as biquads are among the most useful circuits when electrical engineers have to solve analog signal processing requirements. Sections of second order can be configured to be universal filters. Cascade connection of biquads are used most frequently when filters of higher order are to be designed. We will discuss in this chapter specifically the second order lowpass and bandpass filters and leave the more general biquads next. We shall start by defining a few basic terms and the notation used by filter designers and we shall than investigate how fundamentally a biquad can be designed to meet arbitrary requirements.

4.2 THE SECOND ORDER CIRCUIT
The transfer function for the lowpass filter derived as in equation was written in a normalized form such that T(j0) = 1 . A more general form for T(s) in active circuits will recognize the possibility of gain and also that the associated circuit may be inverting or non-inverting. Such a transfer function is
226692095760
(4.1)
Normalized frequency Sn = S/Wo.

Result is same as we set Wo = 1 (setting Wo=1 is scaling).

Rewriting the function
2324880135720
(4.2)
In time domain, s is replaced by differential operator, so Laplace transform of second order differential equation is
18669005080
(4.3)
Dividing by s is much easier by which we get the equation for BP, LP ; HP.

2048040-60840(4.4)
(4.5)
(4.6)
4.3 FREQUENCY RESPONSE OF LOWPASS AND BANDPASS CICUITS
4.3.1 Lowpass Filter
Let us simplify the equation by scaling transfer function by H, as we assume H = 1 and removing the negative sign i.e. non inverting circuit.

226692010080
(4.7)
For this quantity we found that the magnitude is given by
2152800124560
(4.8)
And phase is given by
226764057240
(4.9)
Here,
1381679105480
(4.10)
Similarly for phase
145728057240
(4.11)
Magnitude function of lowpass filter
171132599695
Figure 4.1: Magnitude response of lowpass filter
Phase response of lowpass filter
18243556350
Figure 4.2: Phase response of lowpass filter
4.3.1 Bandpass Filter
The Bandpass function is
2107439104760
(4.12)
And we know
2610360114840
(4.13)
Putting K in Bandpass function
220103966600
(4.14)
We set Wo = 1, Bandpass Magnitude function
217188066600
(4.15)
And Phase
224784019080
(4.16)
Magnitude response of Bandpass function
14020807620
Figure 4.3: Magnitude response of Bandpass filter
Phase Response of Bandpass function
center73660
Figure 4.4: Phase response of Bandpass filter
4.4 INTEGRATORS: THE EFFECT OF A(s)
We know that the biquad was designed on the assumption that the op-amps were ideal. The only effect that frequency dependent op-amp gain can have is to change the performance of integrators from their ideal behaviour. In capacitor, losses are modelled via a resistor RC. A loss capacitor generates a voltage by integrating current in time domain.

224784067320
(4.17)
Or in Laplace domain
2210400104760
(4.18)
On jw axis equation becomes
220968047520
(4.19)
The integration is described as
center13271500

(4.20)
We define the general integrator function F(s) as
7620009461500
(4.21)
4.5 THE EFFECT OF A(s) ON BIQUAD
Effect of A(s) is to make integrator lossy.

244872038880
(4.22)
Q = Error caused by non-ideal op amp and Tau(J) = Integrator time constant.

Q is function of frequency and inversely proportional to op amp gain. q is proportional to square of W hence losses increase with increase in operating frequency.

From behaviour of two integrator loop biquad, we replace ideal integrators +-(1/s) in eqn V2/V1. From BP function,
center-318504(4.23)
Dividing by Tau1 and Tau2 and q=0 for lossless tau(i) normalized = 1, Equation reduces to
2066925151765
(4.24)
And Pole Frequency,
2372400133200
(4.25)
CHAPTER 5
PROPOSED KHN BIQUAD CIRCUIT
5.1 KHN EQUIVALENT BIQUAD USING CURRENT CONVEYOR
A new current conveyor based biquad, equivalent to the well-known KHN circuit is introduced. The proposed circuit employs exactly the same number (five) of Ccs and resistors (six/seven) along with two grounded capacitor as in the two CC biquads recently reported by Soliman. However, in contrast to Soliman’s circuit, the proposed biquad offer several other advantages.

5.2 INTRODUCTION
Recently, Soliman presented two current conveyor base circuits equivalent to the well known Kervin Huelsman Newcomb (KHN) biquad. Both of these circuits employ five Ccs six (or Seven) resistors and realise exactly the same three transfer functions (lowpass, bandpass and highpass) as in the KHN circuit. An advantage of these CC based circuits is the use of two grounded capacitors, a feature not available in the opamp based KHN circuit.

The purpose of this letter is to introduce another CC based KN equivalent biquad, which uses exactly the same number of active and passive components and employs two grounded capacitor as in Soliman’s circuits, but in contrast to these, offers some additional advantages.

5.3 PROPOSED KHN EQUIVALENT BIQUAD
The KHN biquad (reproduced here in figure 5a) is essentially an opamp implementations of the state variable structure shown in figure 5b. The proposed biquad (figure 2) evolves by implementing the inverting integrators by Ccs 1 and 2 (and associated RC elements) and implementing the summer (shown in figure 5 within dotted box) by Ccs 3-5 and resistors R3-R6. Our circuit differs from those of 1 in the manner in which the summer part of the structures of figure 5b has been implemented. It is this difference in the realisation which results in several additional advantages over its predecessors in 1.

Figure 5.1: KHN Biquad and its state variable representation
a KHN biquad
b State-variable representation
The three transfer function realised are given by:
T(LP)(s) = (b1/T1T2)/D(s)
T(LP)(s) = -(sb1/T1)/D(s)
T(HP)(s) = s2b1/D(s)
Where D(s0 = s2 + s(b2/T1) + (b3/T1T2)
The three transfer functions realised by the proposed circuit, which employs all second generation Ccs (CCII), characterised by i(y) = =0, v(x) = v(y), i(z) = +-i, are given by
(5.1) ; (5.2)
And
(5.3)
Where
(5.4)
Note that Soliman achieves equations exactly identical to those of the KHN circuit, by selecting R5 = R, R6 = 2R for the first circuit of 1 and R5 = R6 = R7 = R for the second circuit of 1. Thus without taking these conditions, the resulting transfer function will not be exactly the same as those of the KHN circuit; nevertheless, the circuits of 1 are still valid analogues of the KHN circuit because of the exact duplication of the mechanism of the state variable structure of figure 5b as is our structure of figure 2. The proposed circuit has been formulated in such a manner that no component matching conditions are needed whatsoever, while employing exactly the same number of Ccs, resistors and the two grounded capacitors as in the Soliman circuits 1.

Figure 5.2: Proposed KHN equivalent using CC
5.4 ADAVANTAGES
Advantages over KHN- Equivalent CC biquads of Soliman: The proposed circuit offers the following additional advantages over the structures of Soliman:
The first significant advantage is in respect of the controllability of the various parameters of the realised responses. A careful inspection of the expressions of the structures of Soliman (and hence also those of the circuit of Figure of opamp equivalent) reveals that independent control of the parameters Ho, Qo and Wo is not possible in any of the three responses; at most, a sequential tuning of the parameters Ho, BW and w is feasible only in the case of bandpass response. In comparison, in the biquad presented here, independent control of Ho and Qo is possible in the case of lowpass and highpass responses through separate resistors R5 and R3, respectively, whereas in the case of the bandpass response, Wo can be tuned independently by R2, BW is independently controllable by R3 and finally, the desired midband gain Ho can be independently adjusted by R5.

The structures of Soliman employ 2 grounded capacitors as preferred for integrated circuit implementation. Our circuit also has all the resistors grounded. This is particularly advantageous in facilitating electronic control of the pertinent parameters by replacing appropriate resistors by FET-based voltage control-resistor(s).

Neither structures of Soliman nor the original KHN biquad have infinite input impedance. Our circuit offers ideally infinite input impedance by virtue of the input terminal being the y-terminal of a CC.

5.5 PSpice SIMULATION
The proposed KHN biquad equivalent circuit is implemented and simulated on PSpice software. Theoretical value is calculated and verified with the practical values observed from the simulation. The circuit Diagram of KHN equivalent is shown in figure.

Figure 5.3: KHN Equivalent Simulation Circuit
5.6 PSpice SIMULATION OUTPUT
The circuit above in figure 5.3 is simulated in PSpice software and response of the circuit is shown in figure 5.4. The practical observed value from the circuit response is verified with the theoretical value. Output response of above circuit is given below in figure 5.4.

Figure 5.4: Simulation Output of KHN equivalent
5.7 CONCLUSION
In conclusion, a new formulation of a KHN- equivalent CC biquad is introduced which employs exactly same number of CCs and RC elements and employs 2 grounded capacitors as in circuits of Soliman, but in addition offers three more advantages as outlined above.

Lastly, it should be mentioned that based on the state variable structure of the Fig 5.2 and its other possible variants, several other CC-based or OTA-based biquads of varying complexity are derivable by exploiting the degrees of freedom in devising and combining the required blocks. However, all such alternative structures may not necessarily be equivalent to the KHN Biquad.

REFERENCES
1 SOLIMAN, A.M: ‘Kerwin-Huelsman-Newcomb circuit using current conveyors’, Electron. Lett., 1994, 30, pp. 2019-2020
2 KERWIN. W., HUELSMAN, L., AND NEWCOMB, R.: ‘State variable synthesis for insensitive integrated circuit transfer functions’ , IEEE J. Solid-State Circuits, 1967, SC-2, pp. 87-92.

3 R. Senani and V.K. Singh: ‘KHN-equivalent Biquad using Current Conveyors’ ELECTRONICS LETTERS 13th April 1995 Vol.31 No.8
4 Rolf Schaumann and Mac E. Van Valkenburg: ‘DESIGN OF ANALOG FILTERS’, Oxford University Press(2001), ISBN- 0-19-511877-4(cloth)
5 Adel S. Sedra and Kenneth C. Smith and Adapted by Arun N. Chadorkar:’ Microelectronic CIRCUITS Theory and Applications 6th Edition ‘, Oxford University Press INC, ISBN- 13:978-0-19-808913-1

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