Fourier Analysis, Low Pass Filters, Decibels

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Lecture 8 Furier Analysis, Lw Pass Filters, Decibels ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Chapter 6 Frequency Respnse, Bde Plts, and Resnance. State the undamental cncepts Furier analysis.. Determe the utput a ilter r a given put cnsistg susidal cmpnents usg the ilter s transer unct. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

3. Use circuit analysis t determe the transer uncts simple circuits. 4. Draw irst-rder lwpass r highpass ilter circuits and sketch their transer uncts. 5. Understand decibels, lgarithmic requency scales, and Bde plts. 6. Draw the Bde plts r transer uncts irst-rder ilters. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Furier Analysis ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Furier Analysis All real-wrld signals are sums susidal cmpnents havg varius requencies, amplitudes, and phases. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Furier Analysis v sq 4A 4A 4A ( t s( ω 0 t + s(3 ω 0 t + s(5 ω 0 t +... π 3π 5π ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Filters Filters prcess the susid cmpnents an put signal dierently dependg the requency each cmpnent. Oten, the gal the ilter is t reta the cmpnents certa requency ranges and t reject cmpnents ther ranges. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Filters ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Transer Functs The transer unct H( the tw-prt ilter is deed t be the rati the phasr utput vltage t the phasr put vltage as a unct requency: H ( ut ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Transer Functs The magnitude the transer unct shws hw the amplitude each requency cmpnent is aected by the ilter. Similarly, the phase the transer unct shws hw the phase each requency cmpnent is aected by the ilter. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Transer Functs Magnitude Phase ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Example 6. v 000π ( t cs(000πt + 40 000 Hz π H (000 v ut 40 3 30 ut 6cs(000πt + 70 ut (3 30 ( (3 30 ( 40 6 70 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6. v 4000π ( t cs(4000πt 000 Hz π H (000 v ut 0 60 ut 4cs(4000πt + 60 ut ( 60 ( ( 60 ( 0 4 60 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6. v ( t cs(6000πt 0 0 H (3000 0 90 ut 6000π 3000 Hz π v ut 0 cs(6000πt + 0 0 ut (0 90 ( (0 90 ( 0 0 0 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Graphic Equalizer Filter with an adjustable transer unct ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Determg the utput a ilter r an put with multiple cmpnents:. Determe the requency and phasr representat r each put cmpnent.. Determe the (cmplex value the transer unct r each cmpnent. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

3. Obta the phasr r each utput cmpnent by multiplyg the phasr r each put cmpnent by the crrespndg transer-unct value. 4. Cnvert the phasrs r the utput cmpnents t time uncts varius requencies. Add these time uncts t prduce the utput. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Example 6. v v v v ( t 3 + cs(000πt + cs(4000πt 70 ( t 3 3 0 ( t cs(000πt 33 ( t cs(4000πt 70 3 0 70 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Example 6. 0 70 ( 60 ( (000 60 (000 30 6 0 ( 30 (3 (000 30 3 (000 0 0 (3 0 (4 (0 4 (0 H H H H H H ut ut ut 0 (4000 30 6cs(000 0 70 ( 60 ( 000 ( 60 000 ( 3 3 + t v v ut ut ut π ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc. 0 cs(4000 30 6cs(000 0 cs(4000 3 3 + + + + + t t v v v v t v ut ut ut ut ut π π π

Lear circuits behave as i they:. Separate the put signal t cmpnents havg varius requencies.. Alter the amplitude and phase each cmpnent dependg n its requency. 3. Add the altered cmpnents t prduce the utput signal. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Determat the Transer Funct ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

First-Order Lw Pass Filter R + I C j R π + I C j R C j C j t ut π π π + I Hal pwer ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc. RC j RC j H B B ut π π / ( ( + + Hal pwer requency

First-Order Lw Pass Filter H ( + j ( + 0 B ( B arctan B H ( H ( arctan + ( B B ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

First-Order Lw Pass Filter Fr lw requency signals the magnitude the transer unct is unity and the phase is 0. Lw requency signals are passed while high requency signals are attenuated and phase shited. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Calculat RC Lw-Pass Output v 5 cs(0πt + 5cs(00πt + 5cs(000πt B 00Hz πrc 000 6 π (0 x 0 π ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Calculat RC Lw-Pass Output H ( H ( arctan + ( 00 00 v 5cs(0 πt v v 3 5cs(00πt 5cs(000 π t 3 5 0 5 0 5 0 H (0 + H (00 + H (000 + 0.9950 5.7 j (0 /00 0.707 45 j(00 /00 j(000 /00 0.09950995 84.9 ut ut ut 3 H (0 H (00 H (000 (0.9950 5.7 (5 0 4.975 5.7 (0.707 45 (5 0 3.535 45 3 (0.0995 84.9 (5 0 0.4975 84.9 v v v ut ut ut 3 4.975cs(0πt 5.7 3.535cs(00πt 45 0.4975cs(000πt 84.9 v ut ( t v ( t + v ( t + v ( t 4.975cs(0πt 5.7 + 3.535cs(00πt 45 + 0.4975cs(000πt 84.9 ut ut ut 3 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6.4 j L R + ( I R j L R R j L R ut π π ( ( + + I L R j L j L j R R H j L R B B ut π π π π / ( ( ( + + + + ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc. R j +

Exercise 6.5 v ( t 0 cs(40πt + 5cs(000πt + 5cs(π0 t H ( + ( / B j v ( t 0 cs(40πt v ( t 5cs(000πt v ( t 5cs(π0 t 3 4 + ( / 00 3 j 0 0 5 0 5 0 4 H (0 + 0. j H (500 +.5 j B H (0,000 + 50 j πrc 0.995 5.7 0.374 68. 0x0 500 π 0x0 π 3 88.85 6 3 ut ut ut 5x0 3 0.x0 00Hz (0 0 (0.995 5.7 9.95 5.7 (5 0 (0.374 68..857 68. (5 0 (0x0 3 88.85 0. 88.85 v ut 4 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc. 9.95cs(40πt 5.7 +.85cs(000πt 68. + 0.0 cs(π0 t 88.9

Decibels, the Cascade Cnnect, and Lgarithmic Frequency Scales H ( lg H ( db 0 db decibels ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Numbers greater than are psitive Numbers smaller than are negative ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

ut ut ut ut ut H ( H( H ( ut ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Cascaded Tw-Prt Netwrks ( H ( H ( H ( H ( H ( db db db H + ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

One decade: One ctave: h l h l 0 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Lgarithmic Frequency Scales On a lgarithmic scale, the variable is multiplied by a given actr r equal crements length alng the axis. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

A decade is a range requencies r which the rati the highest requency t the lwest is 0. Number decades lg An ctave is a tw-t-ne change requency. ( g( lg Number ctaves lg lg ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6.6 Cnvert the magnitude the transer unct H( t db s: H ( 50 ( 50 34dB H ( 0lg db ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6.7 ( a H ( 5dB 0lg H ( H ( 0 0 db 5 5.63 30 ( b H ( 30dB 0lg H ( H ( 0 0 db 3.6 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6.8 (a What requency is tw ctaves higher than 000 Hz? lg 000 lg 000 000 lg( 0 lg( 000x0 lg( lg 000 lg( 4,000 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6.8 (b What requency is three ctaves lwer than 000 Hz? 3 lg 000 lg 3lg( 000 0 000 000 3lg( 3lg( 0 5 000 lg lg( ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6.8 (c What requency is ne decade lwer than 000 Hz? 000 000 lg 00Hz 0 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6.9 (a What requency is halway between 00 and 000 Hz n a lgarithmic requency scale? lg(00 lg(000 3 3 + average. 5.5 lg(.5 0 36. Hz ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

Exercise 6.9 (b What requency is halway between 00 and 000 Hz n a lear requency scale? 00Hz 000Hz 000Hz + 00Hz 550Hz ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Furth Edit, by Allan R. Hambley, 008 Pearsn Educat, Inc.

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