• What is a filter• Passive filters• Some common filters

Lecture 23. Filters I

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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)

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Bode Plots of Common Filters

FrequencyFrequency

Low Pass

Frequency

Band Pass

Frequency

Band Reject

Gai

nG

ain

Gai

nG

ain

High Pass

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Passive vs. Active filters

– Passive filters: RLC components only, but gain < 1

– Active filters: op-amps with RC elements, and gain > 1

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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

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First-Order Filter Circuits

L+–

VSC

R

Low Pass

High Pass

HR = R / (R + sL)

HL = sL / (R + sL)

+–

VSR

High Pass

Low Pass

GR = R / (R + 1/sC)

GC = (1/sC) / (R + 1/sC)

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Second-Order Filter Circuits

C+–

VS

R

Band Pass

Low Pass

LHigh Pass

Band Reject

Z = R + 1/sC + sL

HBP = R / Z

HLP = (1/sC) / Z

HHP = sL / Z

HBR = HLP + HHP

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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

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Frequency & Time Domain Connections

• First order circuit break frequency: break = 1/• Second order circuit characteristic equation

s2 + 20 s + 02 [ = 1/(2Q) ]

(j)2 + 2(j) + 1 [ = 1/0 ]

s2 + BW s + 02

s2 + R/L s + 1/(LC) [series RLC]Q value also determines damping and pole types

Q < ½ ( > 1) overdamped, real & unequal rootsQ = ½ ( = 1) critically damped, real & equal roots

Q > ½ ( < 1) underdamped, complex conjugate pair

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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?

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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

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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 3rd order Butterworth low pass filter.

C2 = 4/3 farad, R4 = 1ohm, L1 = 3/2 and L3=1/2 H.

Butterworth filtersnth order Butterworth filter.

where n = order of filter ωc = cutoff frequency (approximately the -3dB frequency) G0 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 ωc at equally spaced points

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Class Examples

• Example 10-1 and 10-2• Drill Problem 10-1