Recap Filters ABE425 Engineering Tony Grift, PhD Dept. of Agricultural & Biological Engineering University of Illinois.

Recap Filters

ABE425 Engineering

Tony Grift, PhDDept. 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 sAbsolute 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

s

sesesss

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

sFdtetftfL st

0

s-domain (Laplace Domain)

Time domain

jFdtetftfF tj

-domain (Frequency Domain)j

Time domain

js

Transient response (step, impulse)

Frequency response (filters)

Relation time, s and frequency domain

tUtUtU OOIN

OC UU INU

i

ssG

sU

sU

sUsUssU

IN

O

OOIN

1

1

Time domain

Laplace (s)-domain

jjG

jU

jU

IN

O

1

1

-domainjjs

Concept of impedance (Capacitor)

cCC

CC idt

dUC

dt

dQCUQ

dtiC

U cC 1

CC CUQ

Current

ImpedanceVolt

2 11

jiCj

jUsisC

sU cCFL

cC

Cj

jZ

1

Concept of impedance (Inductor (coil))

dt

diLU L

L

CurrentImpedanceVolt

2 jiLjjUssLisU LLFL

LL

dt

diLU L

L

LjjZ

Low-Pass filter using RC network

Derivation transfer function with impedance

OC UU INU

iiO U

RCjU

CjR

CjU

1

11

1

jjG

jU

jU

IN

O

1

1

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

Transfer function of RC circuit is complex number

jjG

jU

jU

IN

O

1

1

OC UU INU

i

RC circuit as a Low-Pass filter

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

Analyze three points:Very low frequencies

‘Corner’ frequency

Very high frequencies

1

jjG

jU

jU

IN

O

1

1

1

1

RC Filter response at very low frequencies

Magnitude

Magnitude in dB

Phase (argument)

dB01log*20 10 dB

jG

j

jG

1

1

deg0 jG

1

11 jG

RC 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

RC Filter response at very high frequencies

Magnitude

Magnitude in dB

Phase (argument)

j

jG

1

1

deg90 jG

j

jG1

1

jjjG

jjjG

jjG

dB

dB

dB

1log20dB20

10

1log2010

1log20dB6

2

1log202

1log20

1010

1010

10

Summary 1st order low pass filter characteristics

G j dB

G j Phase

1 1

1 1020* log 1 0dB 1 0 0degj

1 1

1 j 10 1

20* log 32

dB

1

45deg1 j

1 1

j

-6 dB / octave or -20 dB / decade

190deg

j

OC UU INU

RC circuit as a Low-Pass filter: Bode plot

-40

-30

-20

-10

0

Mag

nitu

de (

dB)

10-2

10-1

100

101

102

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

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

High-pass filter using RC network

High-Pass filter characteristics

OC UU INU

1 1O i i

j RCR

U U Uj RCR

j C

1O

IN

U j jG j

U j j

RC circuit as a High-Pass filter

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

1

1G j j

j

1

1O

IN

U jG j j

U j j

1

1dBdBdB

G j jj

Summary 1st order High Pass filter characteristics

G j dB

G j Phase

1

1

j

+6 dB / octave or +20 dB / decade 1

90 0 90deg1

j

1

1

j

j 10 1

20* log 32

dB

190 45 45deg

1j

j

1 j

j

1020* log 1 0dB 1

90 90 0degjj

OC UU INU

RC circuit as a High-Pass filter: Bode plot

-50

-40

-30

-20

-10

0

Mag

nitu

de (

dB)

10-2

10-1

100

101

102

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

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

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

The End