INTEGRATED CIRCUITS (EEC-501) www.uptunotes.com INTEGRATED CIRCUITS (EEC-501) UNIT-2 FILTERS (ACTIVE) By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

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

Types of Filters There are two broad categories of filters: www.uptunotes.com 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 Email:engr.rajan@gmail.com By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Analog Filter Responses www.uptunotes.com Analog Filter Responses H(f) H(f) f f fc fc Ideal “brick wall” filter Practical filter By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Ideal Filters Lowpass Filter Highpass Filter Bandstop Filter www.uptunotes.com 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 Email:engr.rajan@gmail.com By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

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

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

www.uptunotes.com 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 Email:engr.rajan@gmail.com

Op Amp Advantages Advantages of active RC filters include: www.uptunotes.com 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 Email:engr.rajan@gmail.com

Op Amp Disadvantages Active RC filters also have some disadvantages: www.uptunotes.com 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 Email:engr.rajan@gmail.com

www.uptunotes.com 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 Email:engr.rajan@gmail.com

1 bel = 10 decibels (dB) The decibel (dB) www.uptunotes.com 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 Email:engr.rajan@gmail.com

A common dB term is the half power point www.uptunotes.com 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 Email:engr.rajan@gmail.com

www.uptunotes.com 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 Email:engr.rajan@gmail.com

The value power (x) can be determined by www.uptunotes.com 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 Email:engr.rajan@gmail.com

Any base can be found in terms of the common logarithm by: www.uptunotes.com 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 Email:engr.rajan@gmail.com

Properties of Logarithms www.uptunotes.com 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 Email:engr.rajan@gmail.com

Poles & Zeros of the transfer function www.uptunotes.com 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 Email:engr.rajan@gmail.com

Single-Pole Passive Filter www.uptunotes.com 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 Email:engr.rajan@gmail.com

Single-Pole Active Filter www.uptunotes.com 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 Email:engr.rajan@gmail.com

Low-Pass and High-Pass Designs www.uptunotes.com Low-Pass and High-Pass Designs High Pass Low Pass By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

To understand Bode plots, you need to use Laplace transforms! www.uptunotes.com To understand Bode plots, you need to use Laplace transforms! R Vin(s) The transfer function of the circuit is: By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Break Frequencies Replace s with jw in the transfer function: www.uptunotes.com 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 Email:engr.rajan@gmail.com

www.uptunotes.com 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 Email:engr.rajan@gmail.com

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Bode plots use a logarithmic scale for frequency. www.uptunotes.com 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 10 20 30 40 50 60 70 80 90 100 200 By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Consider the magnitude of the transfer function: www.uptunotes.com Consider the magnitude of the transfer function: Expressed in dB, the expression is By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Look how the previous expression changes with frequency: www.uptunotes.com 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 Email:engr.rajan@gmail.com

Note that the two asymptotes intersect at fb where www.uptunotes.com 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 Email:engr.rajan@gmail.com

www.uptunotes.com 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 Email:engr.rajan@gmail.com

Real Filters The approximations to the ideal filter are the: www.uptunotes.com Real Filters The approximations to the ideal filter are the: Butterworth filter Chebyshev filter Cauer (Elliptic) filter Bessel filter By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Standard Transfer Functions www.uptunotes.com 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 Email:engr.rajan@gmail.com

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Butterworth Filter The Butterworth filter magnitude is defined by: www.uptunotes.com Butterworth Filter The Butterworth filter magnitude is defined by: where n is the order of the filter. By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

From the previous slide: www.uptunotes.com From the previous slide: for all values of n For large w: By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

implying the M(w) falls off at 20n db/decade for large values of w. www.uptunotes.com And implying the M(w) falls off at 20n db/decade for large values of w. By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

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 Email:engr.rajan@gmail.com

To obtain the transfer function H(s) from the magnitude www.uptunotes.com To obtain the transfer function H(s) from the magnitude response, note that By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Because s = jw for the frequency response, we have s2 = - w2. www.uptunotes.com 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 Email:engr.rajan@gmail.com

Note that for any n, the poles of the normalized Butterworth www.uptunotes.com 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 Email:engr.rajan@gmail.com

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Recall from complex numbers that the rectangular form www.uptunotes.com 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 Email:engr.rajan@gmail.com

Definition: If z = x + jy, we define e z = e x+ jy to be the www.uptunotes.com 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 Email:engr.rajan@gmail.com

The following equation is known as Euler’s law. www.uptunotes.com The following equation is known as Euler’s law. Note that By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

This leads to two axioms: www.uptunotes.com This implies that This leads to two axioms: and By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

www.uptunotes.com Observe that e jq represents a unit vector which makes an angle q with the positivie x axis. By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

www.uptunotes.com 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 Email:engr.rajan@gmail.com

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

The roots are: www.uptunotes.com By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

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Using the left half-plane poles for H(s), we get www.uptunotes.com Using the left half-plane poles for H(s), we get which can be expanded to: By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

www.uptunotes.com The factored form of the normalized Butterworth polynomials for various order n are tabulated in filter design tables. By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

n Denominator of H(s) for Butterworth Filter 1 s + 1 2 s2 + 1.414s + 1 www.uptunotes.com n Denominator of H(s) for Butterworth Filter 1 s + 1 2 s2 + 1.414s + 1 3 (s2 + s + 1)(s + 1) 4 (s2 + 0.765 + 1)(s2 + 1.848s + 1) 5 (s + 1) (s2 + 0.618s + 1)(s2 + 1.618s + 1) 6 (s2 + 0.517s + 1)(s2 + 1.414s + 1 )(s2 + 1.932s + 1) 7 (s + 1)(s2 + 0.445s + 1)(s2 + 1.247s + 1 )(s2 + 1.802s + 1) 8 (s2 + 0.390s + 1)(s2 + 1.111s + 1 )(s2 + 1.663s + 1 )(s2 + 1.962s + 1) By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

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

So far we have looked at the Butterworth filter www.uptunotes.com 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 Email:engr.rajan@gmail.com

Lowpass with Cutoff Frequency wu www.uptunotes.com Lowpass with Cutoff Frequency wu Transformation: By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Highpass with Cutoff Frequency wl www.uptunotes.com Highpass with Cutoff Frequency wl Transformation: By: Mr. RAJAN VERMA Email:engr.rajan@gmail.com

Band pass Butterworth Filters Multistage Wide Band pass www.uptunotes.com 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 Email:engr.rajan@gmail.com

Band reject KRC Filters : www.uptunotes.com 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 Email:engr.rajan@gmail.com