What is a filter Passive filters Some common filters Lecture 23. Filters I 1

2 What are filters? Filters are electronic circuits which perform signal processing functions, specifically intended to remove unwanted signal components and/or enhance wanted ones. Common types of filters: –Low-pass: deliver low frequencies and eliminate high frequencies –High-pass: send on high frequencies and reject low frequencies –Band-pass: pass some particular range of frequencies, discard other frequencies outside that band –Band-rejection: stop a range of frequencies and pass all other frequencies (e.g., a special case is a notch filter)

3 Bode Plots of Common Filters Frequency Low Pass Frequency Band Pass Frequency Band Reject Gain High Pass

4 Passive vs. Active filters –Passive filters: RLC components only, but gain < 1 –Active filters: op-amps with RC elements, and gain > 1

5 Passive Filters Passive filters use R, L, C elements to achieve the desired filter Some Technical Terms: The half-power frequency is the same as the break frequency (or corner frequency) and is located at the frequency where the magnitude is 1/  2 of its maximum value The resonance frequency,  0, is also referred to as the center frequency

6 First-Order Filter Circuits L +–+– VSVS C R Low Pass High Pass H R = R / (R + sL) H L = sL / (R + sL) +–+– VSVS R High Pass Low Pass G R = R / (R + 1/sC) G C = (1/sC) / (R + 1/sC)

7 Second-Order Filter Circuits C +–+– VSVS R Band Pass Low Pass L High Pass Band Reject Z = R + 1/sC + sL H BP = R / Z H LP = (1/sC) / Z H HP = sL / Z H BR = H LP + H HP

8 Higher Order Filters We can use our knowledge of circuits, transfer functions and Bode plots to determine how to create higher order filters For example, let’s outline the design of a third-order low- pass filter

9 Frequency & Time Domain Connections First order circuit break frequency:  break = 1/  Second order circuit characteristic equation s  0 s +  0 2 [  = 1/(2Q) ] (j  )  (j  ) + 1[  = 1/  0 ] s 2 + BW s +  0 2 s 2 + R/L s + 1/(LC)[series RLC] Q value also determines damping and pole types Q 1) overdamped, real & unequal roots Q = ½ (  = 1) critically damped, real & equal roots Q > ½ (  < 1) underdamped, complex conjugate pair

10 Time Domain Filter Response It is straightforward to note the frequency domain behavior of the filter networks, but what is the response of these circuits in the time domain? For example, how does a second-order band-pass filter respond to a step input?

11 Other types of filters Butterworth – flat response in the passband and acceptable roll-off Chebyshev – steeper roll-off but exhibits passband ripple (making it unsuitable for audio systems) Bessel – yields a constant propagation delay Elliptical – much more complicated

12 Butterworth filters Butterworth – The Butterworth filter is designed to have a frequency response which is as flat as mathematically possible in the passband. Another name for them is 'maximally flat magnitude' filters. Example: A 3 rd order Butterworth low pass filter. C 2 = 4/3 farad, R 4 = 1ohm, L 1 = 3/2 and L 3 =1/2 H.

Butterworth filters n th order Butterworth filter. where n = order of filter ω c = cutoff frequency (approximately the - 3dB frequency) G 0 is the DC gain (gain at zero frequency As n approaches infinity, it becomes a rectangle function The poles of this expression occur on a circle of radius ω c at equally spaced points

14 Class Examples Example 10-1 and 10-2 Drill Problem 10-1