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Lab. 8 주파수 응답(Frequency Response)

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이 실험의 목표

Transfer Function의 이해

Frequency Response이란 무엇인가?

Filter란 무엇인가? Low-Pass Filter의 특성 High-Pass Filter의 특성 Band-Pass Filter의 특성 Band-Stop Filter의 특성

RC 회로의 frequency response

RL 회로의 frequency response

Dept. of Information and Communication Eng.

Starting with the general differential equation

we may propose the input signal and the output signal

Assuming zero initial conditions, and substituting and into the equation above, it becomes

Canceling common terms and arranging gives

1 0 1 0( ) ( ) ( ) ( )( ) ( )

M N

NM N

d y t dy t d x t dx tb b y t a a a x tdt dt dt dt

( ) stx t Xe( ) sty t Ye

( )x t

1 0 1 0M st st st N st st st

Ns Ye b sYe b Ye a s Xe a sXe a Xe ste

1 0

1 0

( )N

NM

a s a s aYH sX s b s b

( )y t

Transfer Function

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Dept. of Information and Communication Eng.

, which is called Transfer Function presenting by complex frequency because it describes how the input is transferred to the output in a transform domain

The description is said to be in s-domain, or frequency domain, or phasor domain

Example: Determine the transfer function for

Solution:( ) 3 ( ) 2 ( ) 3 ( ) ( )y t y t y t x t x t

2

3 1( )3 2

3 1/ 31 2

Y sH sX s s

ss s

Pole-ZeroDiagram

Transfer Function

s j

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Dept. of Information and Communication Eng.

Transfer Function

Relationship between system stability and pole-zero diagram

-1

-0.5

0

0.5

1

0 2 4 6 8 10

s j

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Dept. of Information and Communication Eng.

Example: Find the transfer function of the RL circuit

All the initial conditions are zero !

Solution: Applying KVL to the circuit yields

The transfer function is

( ) ( ) ( )R Ry t y t x tL L

/ 1( )/ 1

R RsY Y XL L

R LH ss R L s

Transfer Function

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Dept. of Information and Communication Eng.

It is a very efficient way to characterize an LTI system (likeanalogy filters) for sinusoidal input Let’s consider an input

signal of the form

Then, the output sinusoidal signal is

( ) ,j j tx t Ae e t

( ) ( ) ( ) ( ) ( )j tj j j tjy t h t x t h Ae e d Ae eh e d

( ) ( )j j ty t Ae eH j H j x t

, which is called the Frequency Response for an LTI system

Can obtain directly from the Transfer Function of s-domain !

( )s j

H s

Is the same concept as Fourier Transform

Frequency Response

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Dept. of Information and Communication Eng.

Example:Suppose that the impulse response is

Sketch the frequency response of the LTI system

Solution: The transfer function is

Thus, the magnitude and phase functions become

2( ) 2 ( )th t e u t

2( )2

H ss

2( ) ( )2s j

H s H jj

2

1

2( ) ,4

( ) tan / 2

H j

H j

Frequency Response

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Dept. of Information and Communication Eng.

To measure , we turn on the oscillator and wait till steady state is established and measure the input and output sinusoid

At each frequency, the ratio of the magnitude of the output to the magnitude of the input sinusoid defines the magnitude of the frequency response

The angle of the output minus that of the input defines the angle of the frequency response

( )H j

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Frequency Response

Dept. of Information and Communication Eng.

[ Relation of time waveforms, and frequency response ]

0

0.5

2

10

Frequency Response

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The input signal vin(t) consists of a 1-KHz sine wave plus high-frequency noise

By passing vin(t) through an ideal low-pass filter, the sine wave is passed and the noise is rejected, resulting in a clean output signal

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Filtering Property

[ Practical application example: Low-Pass filter ]

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Filtering Property

Lowpass Filter Highpass Filter

Bandpass Filter Bandstop Filter

(Cutoff frequency)

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Frequency Response of RC Circuit

Low Pass Filter Consider the circuit below.

1

2

11 1 tan1 1 1

o

s

v j j CH j RCv j j RC RCR

j C

Low pass filter circuit

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Frequency Response of RC Circuit

High Pass Filter Consider the circuit below.

1

2tan1 1 21

o

i

v j R j RC RCH j RCv j j RC RCR

j C

High pass filter circuit

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Frequency Response of RL Circuit

1

2

1 tan

1

oL

i

V j R LH jV j R j L RL

R

L+–

Vi

R

Low Pass

High Pass

1

2

1 tan

1

oH

i

V j j L RH jV j R j L LR

L

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Frequency Response of RL Circuit

[ Cutoff Frequency of RC and RL circuits ]A frequency value which passes all frequencies up to the value, and

blocks all frequencies above that one.The amplitude of frequency response is equal to at cutoff

frequency.1/ 2

Is called -3 dB bandwidth

10120log 3 dB2

(Cutoff frequency)

12 1

cRCRL

for RC circuit

for RL circuit

from 1( )2

H j

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Inductors in AC Circuit

Filter for woofer Low frequencies pass

through High frequencies get

blocked.

FrequencyWoofer

VR

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Capacitors in AC Circuit

Filter for tweeter High frequencies pass

through Low frequencies get

blocked.

FrequencyTweeters

VR