Designing of different kinds of filter

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Filter Circuits

Pa s.

Th = 1/RC.

o approaches +90 .

Th 1/RC.

o approaches −90 .

C---vin

jωRCjωRC 1+-----------------------vin=

2R

2C

2

ω2R

2C

2-----------------------

°

C---vin

1jωRC 1+-----------------------vin=

1

ω2R

2C

2-----------------------

°

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ATORY ELECTRONICS II

ssive filters with a single resistor and capacitor are called one-pole filter

e high-pass filter selects frequencies above a breakpoint frequency ωB.

r small ω, A goes as ω or 6 dB/octave; φ = tan-1(1/ωRC); for small ω, φ

e low-pass filter selects frequencies below a breakpoint frequency ωB. =

r large ω, A goes as 1/ω or 6 dB/octave; φ = tan-1(−ωRC); for large ω, φ

vin

RC

vout voutR

R 1 jω⁄+------------------------=

Avout

vin----------

ω

1 +

-----------= =

vin

CR

vout vout1 jωC⁄

R 1 jω⁄+------------------------=

Avout

vin----------

1 +

-----------= =

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Transfer Function

Th

Fo

wi

Th

Ag

al

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e complex gain for a filter is the transfer function.

r a high-pass filter it is,

th the breakpoint frequency ωB = 1/RC.

e transfer function describes behavior as a function of frequency.

ain for the high-pass filter, the real gain G(ω) = |H(jω)|

ls off below ωB at 20 dB/decade or 6 dB/octave.

vout

vin---------- R

R 1 jωC⁄+---------------------------

jω ωB⁄1 jω ωB⁄+--------------------------- H jω( )≡= =

G ω( )ω ωB⁄

1 ω ωB⁄( )2+

------------------------------------=

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Speed-up Capacitor

Co short circuit at high f .

h capacitor.

h

Fo

Hi

n

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nsider that a capacitor looks like an open connection to low f and like a

e circuit is a resistor divider with R1 replaced with Z1 which includes a

e expectation is that at high f, the divider has Z1 = 0.

r ω << 1/R1C, A = R2/(R1 + R2); ω >> 1/R1C, A = 1.

gh frequencies are enhanced, so a pulse edge becomes sharper

C

R1vin R2 voutvout

R2R2 Z1+-------------------vin=

Z1R jωC⁄

R 1 jωC⁄+--------------------------- R

jωRC 1+-----------------------= =

vout

R2R2 R1 jωR1C 1+( )⁄+------------------------------------------------------vin

jωR1R2C R2+

jωR1R2C R2 R1+ +-------------------------------------------------vi= =

AωR1R2C( )2

R2( )2+

ωR1R2C( )2R2 R1+( )2

+---------------------------------------------------------------=

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Two-Pole Filters

Tw

Th

Th

M

2--

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o RC high-pass filters can be placed in series.

e gain varies as ω2.

is is a second-order filter.

ore poles further increase the rapidity of fall off and add phase shifts.

RC

vin

RC

vout

H jω( ) RR

2R 1 jωC⁄+------------------------------⎝ ⎠⎛ ⎞ 1

1 jωC⁄ R R 1 jωC⁄+( )2R 1 jωC⁄+

------------------------------------+------------------------------------------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

=

H jω( ) ω2R

2C

2–

1 3jωRC ω2R

2C

2–+

------------------------------------------------------ω ωB⁄( )2

–

1 3j ω ωB⁄( ) ω ωB⁄( )–+--------------------------------------------------------------= =

G ω( )ω ωB⁄( )2

1 7 ω ωB⁄( )2 ω ωB⁄( )4+ +

---------------------------------------------------------------------=

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RLC Filter

A

wh

h

h

As

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second-order low-pass filter can be made with a resistor and capacitor.

ere ω02 = 1/LC and Q = ω0L/R.

e circuit is equivalent to a damped driven harmonic oscillator.

ere is a damping factor d0 = 1/Q = R/ω0L.

a second-order filter, the gain varies as ω2 above ω0.

RLvin

C

vout

H jω( ) 1 jωC⁄jωL R 1 jωC⁄+ +------------------------------------------- 1

1 jω Qω0⁄ ω ω0⁄( )2–+

-----------------------------------------------------------= =

H jω( ) 1

1 jd0 ω ω0⁄( ) ω ω0⁄( )2–+

------------------------------------------------------------------=

G ω( ) 1

d02 ω ω0⁄( )2

1 ω ω0⁄( )–2[ ]

2+

------------------------------------------------------------------------------=

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Series RLC Circuit

An frequency.

Th

Th

A

ω2LC–

jωC--------------------

vout

vin---------- 1 ω2

LC–

jωRC 1 ω2LC–+

--------------------------------------------=

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RLC circuit can form a notch filter that only negates a narrow band of

e series impedance can be calculated and inserted to find the gain.

e width of the filtered region is the Q value.

graph of the behavior shows the notch.

RL

vout

ZLC

R ZLC+--------------------vin=vout

vin

C ZLC 1 j⁄ ωC jωL+1---= =

A =

Qω0Δω--------

Lω0R

---------- R LC----= = =

vout

vin----------

ω0ω

Δω Δω R= L⁄

ω0 1= LC⁄

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Twin-T Filter

A is low pass, and one is hig

Th

At

o

Hi

--⎠⎞ 2

vin

C----⎠⎞ 2

vin

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notch filter can be built with Combines two 2-pole passive filters. One h pass.

e combined effect of the two filters is:

ω = 1/RC, the gain is 0.

w frequencies are shifted by −90

gh frequencies are shifted by +90

C

Rvin

R/2

vout

vout LP–1 jωC⁄

R 1 jωC⁄+-------------------------⎝⎛=

C

2CR

vout HP–R

R 1 jω⁄+-----------------------⎝⎛=

vout1

1 jωRC+-----------------------⎝ ⎠⎛ ⎞ 2 jωRC

1 jωRC+-----------------------⎝ ⎠⎛ ⎞ 2

+ vin=

vout1 ωRC( )2

–

1 jωRC+( )2------------------------------- vin=

°

°

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Parallel RLC Circuit

If

Th

Th

jωL

ω2LC–

-------------------

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the inductor and capacitor are in parallel there is a positive resonance.

e impedance can be calculated and inserted to find the gain.

e filter selects only a narrow range of frequencies.

RL

vout

ZLC

R ZLC+--------------------vin=

voutvin

C

ZLCjωL jωC⁄

1 j⁄ ωC jωL+---------------------------------

1----= =

vout

vin---------- jωL

R 1 ω2LC–( ) jωL+

------------------------------------------------- ω2L

2

R2

1 ω2LC–( )

2ω2

L2

+

----------------------------------------------------------= =

vout

vin----------

1 LC⁄ω

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Filter Jargon - Time Domain

Thbe

Ri

Ov

e

h oscillator.

Ov

Un

t

oot

time

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ere are a number of terms used to describe the havior of signals as a function of time.

setime: time to get to 90% of the signal value.

ershoot: percent signal passes signal value.

ttling time: time to stay within ε of signal value.

e effect of filter damping in the time domain is like a damped harmonic

erdamped ( ) rises slowly.

derdamped ( ) rises quickly, but there is a ringing overshoot.

V

90%

10%

risetime oversh

settling

V

t

underdamped

overdamped

d0 2>

d0 2<

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Filter Jargon - Frequency Domain

Fifre

Pa

u

i

k

to

t

a

i

v

fln

d

skirt

stopband

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lter bahaior is also studied as a function of quency.

ssband: Unattenuated region 0 to -3 dB.

toff frequency: edge of passband.

pple band: passband that is not flat in frequency.

irt: transistion region from -3 to -40 dB.

pband: frequencies with attenuation greater than -40 dB.

eeper skirts require more poles - higher order filter

mping has an effect in frequency as well as time.

gh frequency ringing shows up as extra gain at resonant frequency.

erdamped circuits have extra non-uniform gain in the passband.

Aln

-3 dB

-40 dB

passban

Aln

fln

underdamped

overdamped

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Butterworth Filter

Ce special names.

A , so there is a critically

da

Th ponse.

Th equency ωC:

A

Fo

3---

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rtain transfer functions give special properties to the behavior and have

Butterworth filter is designed to give maximum flattness in the passband

mped response (d02 = 2) in the frequency domain.

is creates ringing in time domain in exchange for uniform frequency res

e gain of a Butterworth filter is an approximation in terms of a cutoff fr

butterworth filter can be made as a passive 3-pole circuit.

r (L2/R)2 = 2L1C, ωC = R2/L12L2

2C2.

G jω( )2H jω( ) 2 A

2

1 ω2 ωC2⁄( )

n+

--------------------------------------= =

R

L1vin

C

voutL2

H jω( ) 2 1 jωC⁄jωL1 1 jωC⁄+------------------------------------⎝ ⎠⎛ ⎞ R

jωL2 R+----------------------⎝ ⎠⎛ ⎞ 2 1

1 ω2 ωC2⁄( )+

-----------------------------------= =

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Chebyshev Filter

A sh

Th

Cn

h

Th

fln

t

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Chebyshev fileter is designed to maximize the arpness at the edge of the passband.

e transfer function takes the following form.

is an n-th order Chebyshev polynomial:

is gives an underdamped response (d0 = 0.767)

ere is substantial ringing in the time domain.

Aln

H jω( ) 2 A2

1 ε2Cn

2 ω ω0⁄( )+---------------------------------------------=

Cn x( ) n xacos[ ]cos=

V

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Bessel Filter

Thof

Th

wh

Th

Th

t

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e Bessel filter gives an equal rise time independent pulse height.

e transfer function is as follows.

ere Bn is an n-th order Bessel function.

is gives an overdamped response (d0 = 1.736)

ere is the softer rise in the frequency domain.

V

H jω( ) 2 A2

1 ε2Bn

2 ω ω0⁄( )+--------------------------------------------=

Aln

fln

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Designing of different kinds of filter