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**INTEGRATED CIRCUITS (EEC-501)**

INTEGRATED CIRCUITS (EEC-501) UNIT-2 FILTERS (ACTIVE) By: Mr. RAJAN VERMA

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Filters A filter is a system that processes a signal in some desired fashion. A continuous-time signal or continuous signal of x(t) is a function of the continuous variable t. A continuous-time signal is often called an analog signal. A discrete-time signal or discrete signal x(kT) is defined only at discrete instances t=kT, where k is an integer and T is the uniform spacing or period between samples By: Mr. RAJAN VERMA By: Mr. RAJAN VERMA

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**Types of Filters There are two broad categories of filters:**

Types of Filters There are two broad categories of filters: An analog filter processes continuous-time signals A digital filter processes discrete-time signals. The analog or digital filters can be subdivided into four categories: Lowpass Filters Highpass Filters Bandstop Filters Bandpass Filters By: Mr. RAJAN VERMA By: Mr. RAJAN VERMA

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**Analog Filter Responses**

Analog Filter Responses H(f) H(f) f f fc fc Ideal “brick wall” filter Practical filter By: Mr. RAJAN VERMA By: Mr. RAJAN VERMA

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**Ideal Filters Lowpass Filter Highpass Filter Bandstop Filter**

Ideal Filters Lowpass Filter Highpass Filter M(w) Stopband Passband Passband Stopband w c w w c w Bandstop Filter Bandpass Filter M(w) Passband Stopband Passband Stopband Passband Stopband w c 1 w c 2 w w c 1 w c 2 w By: Mr. RAJAN VERMA By: Mr. RAJAN VERMA

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There are a number of ways to build filters and of these passive and active filters are the most commonly used in voice and data communications. By: Mr. RAJAN VERMA By: Mr. RAJAN VERMA

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Passive filters Passive filters use resistors, capacitors, and inductors (RLC networks). To minimize distortion in the filter characteristic, it is desirable to use inductors with high quality factors (remember the model of a practical inductor includes a series resistance), however these are difficult to implement at frequencies below 1 kHz. They are particularly non-ideal (lossy) They are bulky and expensive By: Mr. RAJAN VERMA By: Mr. RAJAN VERMA

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Active filters overcome these drawbacks and are realized using resistors, capacitors, and active devices (usually op-amps) which can all be integrated: Active filters replace inductors using op-amp based equivalent circuits. By: Mr. RAJAN VERMA

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**Op Amp Advantages Advantages of active RC filters include:**

Op Amp Advantages Advantages of active RC filters include: reduced size and weight, and therefore parasitics increased reliability and improved performance simpler design than for passive filters and can realize a wider range of functions as well as providing voltage gain in large quantities, the cost of an IC is less than its passive counterpart By: Mr. RAJAN VERMA

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**Op Amp Disadvantages Active RC filters also have some disadvantages:**

Op Amp Disadvantages Active RC filters also have some disadvantages: limited bandwidth of active devices limits the highest attainable pole frequency and therefore applications above 100 kHz (passive RLCfilters can be used up to 500 MHz) the achievable quality factor is also limited require power supplies (unlike passive filters) increased sensitivity to variations in circuit parameters caused by environmental changes compared to passive filters For many applications, particularly in voice and data communications, the economic and performance advantages of active RC filters far outweigh their disadvantages. By: Mr. RAJAN VERMA

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Bode Plots Bode plots are important when considering the frequency response characteristics of amplifiers. They plot the magnitude or phase of a transfer function in dB versus frequency. By: Mr. RAJAN VERMA

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**1 bel = 10 decibels (dB) The decibel (dB)**

The decibel (dB) Two levels of power can be compared using a unit of measure called the bel. The decibel is defined as: 1 bel = 10 decibels (dB) By: Mr. RAJAN VERMA

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**A common dB term is the half power point **

A common dB term is the half power point which is the dB value when the P2 is one- half P1. By: Mr. RAJAN VERMA

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Logarithms A logarithm is a linear transformation used to simplify mathematical and graphical operations. A logarithm is a one-to-one correspondence. By: Mr. RAJAN VERMA

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**The value power (x) can be determined by **

Any number (N) can be represented as a base number (b) raised to a power (x). The value power (x) can be determined by taking the logarithm of the number (N) to base (b). By: Mr. RAJAN VERMA

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**Any base can be found in terms of the common logarithm by:**

Although there is no limitation on the numerical value of the base, calculators are designed to handle either base 10 (the common logarithm) or base e (the natural logarithm). Any base can be found in terms of the common logarithm by: By: Mr. RAJAN VERMA

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**Properties of Logarithms**

Properties of Logarithms The common or natural logarithm of the number 1 is 0. The log of any number less than 1 is a negative number. The log of the product of two numbers is the sum of the logs of the numbers. The log of the quotient of two numbers is the log of the numerator minus the denominator. The log a number taken to a power is equal to the product of the power and the log of the number. By: Mr. RAJAN VERMA

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**Poles & Zeros of the transfer function**

Poles & Zeros of the transfer function pole—value of s where the denominator goes to zero. zero—value of s where the numerator goes to zero. By: Mr. RAJAN VERMA

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**Single-Pole Passive Filter**

Single-Pole Passive Filter vin vout C R First order low pass filter Cut-off frequency = 1/RC rad/s Problem : Any load (or source) impedance will change frequency response. By: Mr. RAJAN VERMA

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**Single-Pole Active Filter**

Single-Pole Active Filter vin vout C R Same frequency response as passive filter. Buffer amplifier does not load RC network. Output impedance is now zero. By: Mr. RAJAN VERMA

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**Low-Pass and High-Pass Designs**

Low-Pass and High-Pass Designs High Pass Low Pass By: Mr. RAJAN VERMA

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**To understand Bode plots, you need to use Laplace transforms!**

To understand Bode plots, you need to use Laplace transforms! R Vin(s) The transfer function of the circuit is: By: Mr. RAJAN VERMA

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**Break Frequencies Replace s with jw in the transfer function:**

Break Frequencies Replace s with jw in the transfer function: where fc is called the break frequency, or corner frequency, and is given by: By: Mr. RAJAN VERMA

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Corner Frequency The significance of the break frequency is that it represents the frequency where Av(f) = 0.707-45. This is where the output of the transfer function has an amplitude 3-dB below the input amplitude, and the output phase is shifted by -45 relative to the input. Therefore, fc is also known as the 3-dB frequency or the corner frequency. By: Mr. RAJAN VERMA

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By: Mr. RAJAN VERMA

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**Bode plots use a logarithmic scale for frequency.**

Bode plots use a logarithmic scale for frequency. where a decade is defined as a range of frequencies where the highest and lowest frequencies differ by a factor of 10. One decade By: Mr. RAJAN VERMA

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**Consider the magnitude of the transfer function:**

Consider the magnitude of the transfer function: Expressed in dB, the expression is By: Mr. RAJAN VERMA

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**Look how the previous expression changes with frequency:**

Look how the previous expression changes with frequency: at low frequencies f<< fb, |Av|dB = 0 dB low frequency asymptote at high frequencies f>>fb, |Av(f)|dB = -20log f/ fb high frequency asymptote By: Mr. RAJAN VERMA

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**Note that the two asymptotes intersect at fb where**

Note that the two asymptotes intersect at fb where |Av(fb )|dB = -20log f/ fb Low frequency asymptote 3 dB Actual response curve High frequency asymptote By: Mr. RAJAN VERMA

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The technique for approximating a filter function based on Bode plots is useful for low order, simple filter designs More complex filter characteristics are more easily approximated by using some well-described rational functions, the roots of which have already been tabulated and are well-known. By: Mr. RAJAN VERMA

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**Real Filters The approximations to the ideal filter are the:**

Real Filters The approximations to the ideal filter are the: Butterworth filter Chebyshev filter Cauer (Elliptic) filter Bessel filter By: Mr. RAJAN VERMA

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**Standard Transfer Functions**

Standard Transfer Functions Butterworth Flat Pass-band. 20n dB per decade roll-off. Chebyshev Pass-band ripple. Sharper cut-off than Butterworth. Elliptic Pass-band and stop-band ripple. Even sharper cut-off. Bessel Linear phase response – i.e. no signal distortion in pass-band. By: Mr. RAJAN VERMA

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By: Mr. RAJAN VERMA

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**Butterworth Filter The Butterworth filter magnitude is defined by:**

Butterworth Filter The Butterworth filter magnitude is defined by: where n is the order of the filter. By: Mr. RAJAN VERMA

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**From the previous slide:**

From the previous slide: for all values of n For large w: By: Mr. RAJAN VERMA

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**implying the M(w) falls off at 20n db/decade for large values of w.**

And implying the M(w) falls off at 20n db/decade for large values of w. By: Mr. RAJAN VERMA

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**www.uptunotes.com By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com**

20 db/decade 40 db/decade 60 db/decade By: Mr. RAJAN VERMA

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**To obtain the transfer function H(s) from the magnitude **

To obtain the transfer function H(s) from the magnitude response, note that By: Mr. RAJAN VERMA

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**Because s = jw for the frequency response, we have s2 = - w2.**

Because s = jw for the frequency response, we have s2 = - w2. The poles of this function are given by the roots of By: Mr. RAJAN VERMA

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**Note that for any n, the poles of the normalized Butterworth **

The 2n pole are: e j[(2k-1)/2n]p n even, k = 1,2,...,2n sk = e j(k/n)p n odd, k = 0,1,2,...,2n-1 Note that for any n, the poles of the normalized Butterworth filter lie on the unit circle in the s-plane. The left half-plane poles are identified with H(s). The poles associated with H(-s) are mirror images. By: Mr. RAJAN VERMA

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By: Mr. RAJAN VERMA

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**Recall from complex numbers that the rectangular form **

Recall from complex numbers that the rectangular form of a complex can be represented as: Recalling that the previous equation is a phasor, we can represent the previous equation in polar form: where By: Mr. RAJAN VERMA

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**Definition: If z = x + jy, we define e z = e x+ jy to be the **

Definition: If z = x + jy, we define e z = e x+ jy to be the complex number Note: When z = 0 + jy, we have which we can represent by symbol: By: Mr. RAJAN VERMA

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**The following equation is known as Euler’s law.**

The following equation is known as Euler’s law. Note that By: Mr. RAJAN VERMA

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**This leads to two axioms:**

This implies that This leads to two axioms: and By: Mr. RAJAN VERMA

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Observe that e jq represents a unit vector which makes an angle q with the positivie x axis. By: Mr. RAJAN VERMA

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Find the transfer function that corresponds to a third-order (n = 3) Butterworth filter. Solution: From the previous discussion: sk = e jkp/3, k=0,1,2,3,4,5 By: Mr. RAJAN VERMA

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**Therefore, www.uptunotes.com**

By: Mr. RAJAN VERMA

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**The roots are: www.uptunotes.com**

By: Mr. RAJAN VERMA

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By: Mr. RAJAN VERMA

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**Using the left half-plane poles for H(s), we get**

Using the left half-plane poles for H(s), we get which can be expanded to: By: Mr. RAJAN VERMA

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The factored form of the normalized Butterworth polynomials for various order n are tabulated in filter design tables. By: Mr. RAJAN VERMA

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**n Denominator of H(s) for Butterworth Filter 1 s + 1 2 s2 + 1.414s + 1**

n Denominator of H(s) for Butterworth Filter 1 s + 1 2 s s + 1 3 (s2 + s + 1)(s + 1) 4 (s )(s s + 1) 5 (s + 1) (s s + 1)(s s + 1) 6 (s s + 1)(s s + 1 )(s s + 1) 7 (s + 1)(s s + 1)(s s + 1 )(s s + 1) 8 (s s + 1)(s s + 1 )(s s + 1 )(s s + 1) By: Mr. RAJAN VERMA

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**Frequency Transformations**

Frequency Transformations By: Mr. RAJAN VERMA

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**So far we have looked at the Butterworth filter **

So far we have looked at the Butterworth filter with a normalized cutoff frequency By means of a frequency transformation, we can obtain a lowpass, bandpass, bandstop, or highpass filter with specific cutoff frequencies. By: Mr. RAJAN VERMA

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**Lowpass with Cutoff Frequency wu**

Lowpass with Cutoff Frequency wu Transformation: By: Mr. RAJAN VERMA

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**Highpass with Cutoff Frequency wl**

Highpass with Cutoff Frequency wl Transformation: By: Mr. RAJAN VERMA

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**Band pass Butterworth Filters Multistage Wide Band pass**

Band pass Butterworth Filters Multistage Wide Band pass The center frequency fc or fo for the wide band pass filter is defined as, fc or fo = √ fH fL fL = 1 / 2π √ RC fH = 1 / 2π √ R’ C’ and Q (Quality Factor) = fC / (fH - fL) BW = (fH - fL) Q = fC / BW By: Mr. RAJAN VERMA

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**Band reject KRC Filters :**

Band reject KRC Filters : Vo / Vin = HON HN Where HON = DC gain and HN is given by HN = 1 – ( ω / ωO )2 / 1 – ( ω / ωO )2 + (j ω / ωO) / Q HON = K ωO = 1 / RC and Q = 1 / 4-2k By: Mr. RAJAN VERMA

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