09 rc filters

9-1

Lecture 9

RC Filters

9-2

Outlines of Filter Design

Filterinput output

Filtering:

Certain desirable features are retainedOther undesirable features are suppressed

9-3

FiltersFilters have the property of removing unwanted frequencies from our signal.

Classes: Passive (made of capacitors, resistors, inductors)

Active (involving an amplifier)

Types: Low-Pass (remove high frequencies)

High-Pass (remove low frequencies or DC)

Band-Pass (remove a range of frequencies on two sides)

Notch (removes frequencies in the middle)

9-4

Classification of Filters

Signal Filter

Analog Filter Digital Filter

Element Type Frequency Band

Active Passive Low-Pass

High-Pass

Band-Pass

Band-Reject

All-Pass

9-5

Filters – Type of filters

http://www.ece.eps.hw.ac.uk/~pmr/teaching/ae/lectures/circuits1.htm

Passive filters

9-6

Terminology in Filter Design

• Signal-To-Noise Ratio (S/N)

• Bandwidththe range of frequencies of |G(jw)|>0.707

• Cutoff Frequencythe end of pass-band frequency

• Break-point of a filterthe point with a gain of -3dB

dBW

WN

SN

S⎟⎟⎠

⎞⎜⎜⎝

⎛⋅= log10

9-7

RC FiltersIn combination with a resistor, a capacitor’s variation in reactance with frequency can beused to construct a simple low-pass or high-pass filter:

R

RC

C

Vin Vout Vin Vout

High-pass filter Low-pass filter

fCX

XRZ

Z

RVV

C

C

π21

22

inout

=

+=

⋅=

fCX

XRZ

Z

XVV

C

C

C

π21

22

inout

=

+=

⋅=

9-8

Passive Low-Pass Filter

• The pass-band is from 0 to some frequency wp.

• Its stop-band extends form some frequency ws, to infinity.

• In practical circuit design, engineers often choose amplitude gain of 0.95 for passive RC filters:

ωωp ωs

)( ωjH

Vout

Vin

C

R

VVoutoutVVinin RL

9-9

Passive High-Pass Filter

• Its stop-band is form 0 to some frequency ws

• The pass-band is from some frequency wp to infinity.

• In practical circuit design, engineers choose amplitude gain of 0.95 for passive CR filters:

ωωs ωp

)( ωjH

Vout

Vin

R

C

VVoutoutVVinin

9-10

Design of Passive Filters

( )1

1

+=

ωω

jRCjH

( )1

1

+=

RCssH

Transfer Function

( )21

1

ωRCV

V

in

out

+=

The amplitude response:

πτπ 2

1

2

13 ==

RCf dB

The 3dB break-point is at:

LF

L

ZZ

ZG

+=

The amplitude gain:

C

R

VVoutoutVVinin RL

9-11

Guideline of Pass Filter Design

R

( )1

1

+=

ssH

τ

Transfer Function

C VVoutoutVVinin RL

RC=τ

Time Constant

Select resistor based on amplitude gain:

95.0=+

=LF

L

ZZ

ZG

LLF RZRZ ⋅==≈ 053.095.0

05.0

Select capacitor based on cut-off freq:

dBRfRC

32

1

πτ==

9-12

Higher Order Filters

C

R

VVoutoutVVinin

First Order RC Low Pass Second Order RC Low Pass

C2 VVoutoutVVininC1

R1 R2

The higher the order of the filter, the closer it approaches ideal characteristics.

9-13

Active Filters

• Active filters employ Op-Amps to attenuate select frequencies and amplify signal during filtering process.

• Q factor of a filter is defined as the ratio of the center frequency fc to the bandwidth fH -fL :

( )LH

Cff

fQ −=

9-14

Active filters- cascading low pass filters

First order

Op Amp for everyone, Ron Mancini, Ed, Texas instrument, 2001.

Second order

3rd order

5th order

9-15

Low-Pass Active Filter

+

-R1

RF

R2

C1

C2

Passive filters take up lots of space in a circuit andcause signal to be lost. Combining a passive RCfilter with an op amp for amplification createswhat is known as an active filter. By “active”we mean that the filter requires powerto operate.

Here is an example of anactive low-pass filter. The signalis provided to the noninvertedinput through an RC low-pass filtermade up of R2 and C2. Feedback tolimit gain comes through C1 and RF. Theparallel combination of C1 and RF presentsan impedance which decreases with increasingfrequency, meaning that more negative feedback is provided to the inverting input athigher frequencies, reducing gain at those frequencies.

9-16

Design of Low Pass Active Filters

Example:Design a low pass filter with cut-off frequency of 5 kHz, and DC gain of 10:

Two equations, three unknowns

-

+

Vin

Vout

R1

RF

A

B

C2

Transfer Function:

0

0..ω

ω+

=s

KFT LP

( )221

CRfF

H π=

The -3 dB cut-off frequency:

1RRK F

LP −=

The DC gain:

9-17

High-Pass Active Filter

+

-R1

RF

R2

C1 C2

R3

Here is an example of an active high-passfilter. C2 and R2 make up an RC high-passfilter at the input of the op amp. R3 providesa path for the input when the frequency istoo low for C2 to freely conduct. When theinput signal passes through R3 instead ofinto the amplifier, the output is tied directly to the input and the gain is reduced. So, thisamplifier has low gain at low frequencies and higher gain at high frequencies. C1 preventany DC at the input from being coupled to the output.

9-18

Design of High Pass Active Filters

Vout

-

+

Vin

R1

RF

A

B

C1

The -3 dB cut-off frequency:

The DC gain:

Two equations, three unknowns

Select one component based on other conditions, and determine the values of the other two components.

( )1121

CRfH π=

1RRK F

HP =

Transfer Function:

0

..ω+

=s

sKFT HP

9-19

Filter Class

• A filter of a given order can be made to approximate to ideal characteristics in a number of ways, depending on the values of the filter components (or say: depending on the filter class.

• Two useful classes are Butterworth (maximally flat) and Chebyshev (equal-ripple) filters (n is the filter order)

n

C

in

out

ffV

V2

1

1

⎟⎠⎞⎜

⎝⎛+

=Butterworth Filter

Chebyshev Filter⎟⎠⎞⎜

⎝⎛+

=

Cn

in

out

ffCEV

V

221

1

9-20

Higher Order Active Filters

Vout-

+Vin

R2

Rb

C1

R1

Ra

C2

Gain=K

Filter Class R1 R2 C1 C2 K

Buterworth3.01 dB at ωH

1.00 1.00 1.00 1.00 1.59

Chebyshev1 dB ripple

1.00 1.00 0.94 0.97 2.00

The above list gives the gain and component valves for one of themany choices for ωH=1. You may find more combinations from filter design handbook(s).

9-21

Active Filters– High Pass Filters


High pass Low pass

9-22

Active Filters – Band Pass Filter


9-23

Active Filters – Band Reject Filter


Passive band reject filterActive band reject filter

9-24

References

• Op Amp for everyone, Ron Mancini, Ed, Texas instrument, 2001.