ABE425 Engineering Measurement Systems

Filters

Dr. Tony E. Grift

Dept. of Agricultural & Biological EngineeringUniversity of Illinois

Agenda

Recap complex numbersRelationship Laplace, frequency (Fourier) domainRelationship time, s and frequency domainsdecibel notation (dB)RC circuit as a Low-Pass and High-Pass filterBode plotsCombination filters

Complex number in complex plane

jyxess j

s Argument of s

Absolute value of s (aka Modulus or Magnitude)

Operations on complex numbers cont.

1

2

1 1 1 1

2 2 2 2

j

j

s x jy s e

s x jy s e

2121212121 ** jjj essesesss

2121

2

12121 // jjj e

ss

esesss

Multiplication/divisionusing Euler’s notation

Operations on complex numbers cont.

111111

jj essess

211111

11 ** sesesss jj

Complex conjugate

Multiplying a complex number by its conjugate gives a real number

Relation Laplace and Fourier Transform

0

stL f t f t e dt F s

s-domain (Laplace Domain)

Time domain

j tF f t f t e dt F j

-domain (Frequency Domain)j

Time domain

js

Transient response (step, impulse, ramp)

Frequency response (filters)

Relation time, s and frequency ( ) domain

tUtUtU OOIN

OC UU INU

i

ssG

sUsU

sUsUssU

IN

O

OOIN

11

Time domain

Laplace (s)-domain

jjG

jUjU

IN

O

11

-domainjjs

j

Concept of impedance (Capacitor)

C CC C C

dQ dUQ CU C idt dt

CC CUQ

1C CsU s i s

C

1CZ j

j C

Volt Current

Impedance

1C CU j i j

j C

Laplace transform

js

Concept of impedance (Inductor (coil))

LL

diU Ldt

L LU s Lsi s

LZ j j L

ImpedanceVolt Current

L LU j j L i j

Laplace transform

js

dtdiLU L

L

Low-Pass filter using RC network

Derivation transfer function with impedance

OC UU INU

iiO U

RCjU

CjR

CjU

1

11

1

jjG

jUjU

IN

O

11

Decibel notation

Addition is much simpler than multiplication

Notation in Bel (after Alexander Graham Bell)

For Power

For Voltages (Power ~ Voltage2)

In deciBel (0.1 Bel)

Belin log10 P

log*2 log 10210 UU

(dB) deciBelin log*20Belin log*2 1010 UU

The transfer function of a RC circuit is a complex number

jjG

jUjU

IN

O

11

OC UU INU

i

2 2

1 1 11 1 1 1

jG j jj j

2 2 2

2

2

1 11 1 1

1arctan arctan1

1

G j

G j

First order system analysis in standard notation (laborious)

2

11

Re

Im

21

2

1

1

arctan

11

G jj

First order system analysis in standard notation (laborious)

First order system analysis in Euler’s notation

11

jG j ej

2

1

1

arctan

11 jj ee

11

G jj

Re

Im 21

1

j

arctan

11 jj ee

2

1

1

arctan

First order system analysis in Euler’s notation

RC circuit as a Low-Pass filter

Transfer function has anAbsolute value (Magnitude of complex number)Phase (argument of complex number)

Analyze three points:Very low frequencies

‘Corner’ frequency

Very high frequencies

1

1

1

jjG

jUjU

IN

O

11

Filter response at very low frequency

Magnitude

Magnitude in dB

Phase (argument)

dB01log*20 10 dB

jG

j

jG

1

1

deg0 jG

111 jG

Filter response at corner frequency

Magnitude

Magnitude in dB

Phase (argument)

dB32

1log*20 10

dBjG

j

jG

1

1

deg45 jG

j

jG

1

11

Filter response at very high frequency

Magnitude

Magnitude in dB

Phase (argument)

j

jG

1

1

deg90 jG

j

jG 11

jjjG

jjjG

jjG

dB

dB

dB

1log20dB2010

1log2010

1log20dB62

1log202

1log20

1010

1010

10

Summary 1st order low pass filter characteristics

G j dB

G j Phase

1 11

1020* log 1 0dB 1 0 0degj

1 11 j

10 120* log 32

dB

1 45deg1 j

1 1j

-6 dB / octave or -20 dB / decade

1 90degj

OC UU INU

Bode plot of a Low-Pass filter for = 1s

-40

-30

-20

-10

0

Mag

nitu

de (d

B)

10-2

10-1

100

101

102

-90

-45

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

MatLab: bode([0 1],[1 1])

High-pass filter using RC network

High-Pass filter characteristics

OC UU INU

1 1O i i

j RCRU U U

j RCRj C

1O

IN

U j jG jU j j

RC circuit as a High-Pass filter

Filter response has a Absolute value (Magnitude of complex number) andPhase (argument of complex number)

11

G j jj

1

1O

IN

U jG j j

U j j

11dBdB

dB

G j jj

1 21 21 2 1 2

jj je e e

1st order High Pass filter characteristics

G j dB

G j Phase

1 1j

+6 dB / octave or +20 dB / decade 1 90 0 90deg

1j

1

1jj

10 120* log 32

dB

1 90 45 45deg1

jj

1 jj

1020* log 1 0dB 1 90 90 0degjj

OC UU INU

1jG jj

-50

-40

-30

-20

-10

0

Mag

nitu

de (d

B)

10-2

10-1

100

101

102

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

MatLab: bode([1 0],[1 1])

Bode plot of a High-Pass filter for = 1s

Band-Pass filter through cascading

Cascade of High-Pass and Low-Pass filters to obtain a Band-Pass filter

Since the sections are separated by a buffer: Add absolute values in dB;s. Add phase angles

INU OC UU

Buffer

LOW HIGHdB dBdB

LOW HIGH

G j G G

G j G G

ABE425 Engineering Measurement Systems

Filters

The End

Dept. of Agricultural & Biological EngineeringUniversity of Illinois