EE292 Ch06 Filter Part1 Signals Part 2 Filters Em v2

8/18/2019 EE292 Ch06 Filter Part1 Signals Part 2 Filters Em v2

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Electrical Engineering: Principles and Applications, 6eAllan R. Hambley

Copyright ©2014 by Pearson Education, Inc.All rights reserved.

EE292

CHAPTER 6FREQUENCY RESPONSE,BODE PLOTS, ANDRESONANCE


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EE292

00

1 t idt t v L

t i

t

t

dt di

Lt v

Inductor

90)2

exp( L j L L j Z L

LLL I Z V

0cosrmsrmsavg LY I V P


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EE292 5.3 Complex Impedance

9011 C C j

Z

C

90 L L j Z L

0 R R Z L R

C

L


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t vvdt t iC t Ri

dt t di L sC

t

010

Second Order Circuit:RLC serial circuits

Second Order Circuit:RLC parallel circuits

t iidt t v Ldt

t dvC R

t v s

t

01

0


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SECOND-ORDER CIRCUIT EQUATION ANALYSIS

t vvdt t iC t Ridt

t di

L sC

t

01

0

L

R

2

LC

10

t f t idt

t di

dt

t id

2

02

2

2

0


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t f t i

dt

t di

dt

t id 2

02

2

2

2

0

2

1 s 2022 s

t st sc e K e K t x 21 21

nn j s j s 21 and

220 n

t e K t e K t x nt

n

t

c sincos 21


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EE292

The short segment of a music waveform shown in (a) is thesum of the sinusoidal components shown in (b).

6.1 Fourier Analysis, Filters andTransfer functions


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EE292Figure 6.2 A square wave and some of itscomponents.

All real-worldsignals are sums

of sinusoidal

componentshaving various

frequencies,

amplitudes, andphases.


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Filters and Transfer Functions Filters process the sinusoid components of an input signal differently

depending of the frequency of each component. Often, the goal of the filter

is to retain the components in certain frequency ranges and to rejectcomponents in other ranges.

The transfer function H(f ) of the two-port filter is defined to be the ratio

of the phasor output voltage to the phasor input voltage as a function of

frequency:

in

out

V

V f H

The magnitude of the transfer function shows how the amplitude of each

frequency component is affected by the filter. Similarly, the phase of the

transfer function shows how the phase of each

frequency component is affected by the filter.


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EE292

When an input signal vin(t ) is applied to the input port of a filter, some componentsare passed to the output port while others are not, depending on their frequencies.

Thus, vout(t ) contains some of the components of vin(t ) , but not others. Usually, theamplitudes and phases of the components are altered in passing through the filter.

Two-Port Network


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EE292

Filters behave as if they separate theinput into components, modify theamplitudes and phases of thecomponents, and add the alteredcomponents to produce the output.

The transfer functionof a filter.


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EE292

To measure the transfer function we apply a sinusoidal input signal, measure theamplitudes and phases of input and output in steady state, and then divide the phasoroutput by the phasor input. The procedure is repeated for each frequency of interest.

1. Determine the frequency and phasor representation for each input component.

2. Determine the (complex) value of the transfer function for each component.3. Obtain the phasor for each output component by multiplying the phasor for each inputcomponent by the corresponding transfer-function value.4. Convert the phasors for the output components into time functions of various frequencies. Addthese time functions to produce the output.

Determining the output of a filter for an input withmultiple components:


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EE292 Figure PA6.1 Active Noise Cancellation


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EE292

A first-order lowpass filter.

6.2 First-Order Lowpass Filters


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EE292

Figure 6.8 Magnitude and phase of

the first-order lowpass transfer function versus frequency.

RC

f B

2

1

B f f j

f H

1

1

21

1

B

f f f H

B f

f f H arctan

Characteristics of First-Order Lowpass Filters


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EE292Figure 6.10 Another first-order lowpass filter;see Exercise 6.4.


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EE292

Table 6.2 Transfer-

Function Magnitudes andTheir Decibel Equivalents

f H f H log20dB

6.3 DECIBELS, THE CASCADE CONNECTION,AND LOGARITHMIC FREQUENCY SCALES


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EE292

Figure 6.12 Stop-band transfer-function magnitude ofa notch filter used to reduce hum in audio signals.

Notch Filters


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Cascaded Two-Port Networks

f H f H f H 21

dB2dB1dB

f H f H f H


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EE292Figure 6.13 Cascade connection of two two-port circuits.

f H f H f H 21

dB2dB1dB

f H f H f H


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A Bode plot shows the magnitude of a network function in

decibels versus frequency using a logarithmic scale forfrequency.

B f f j f H 1

1

2

dB 1log10 B f f f H

6.4 Bode Plot


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EE292 Logarithmic frequency scale.

1

2

logdecadesof number f

f

2log

loglogoctavesof number 12

1

22

f f

f

f


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EE292Figure 6.15 Magnitude Bode plot for the first-order lowpass filter.


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EE292Figure 6.16 Phase Bode plot for the first-orderlowpass filter.


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EE292Figure 6.17 Circuit for Exercise 6.11.


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EE292 Figure 6.18 Answers for Exercise 6.11.


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EE292 6.5 Highpass filter.

Ref: Lowpass filter

Highpass filter

B

B

f f j

f f j f H

1in

out

V

V

RC f B

2

1


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EE292Figure 6.20 Magnitude and phase for the first-order highpass transfer function.


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EE292

Table 6.4 Values of the

Approximate ExpressionGiven in Equation 6.26 forSelected Frequencies


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EE292Figure 6.21 Bode plots for the first-orderhighpass filter.


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EE292 Figure 6.22 Circuit for Exercise 6.13.


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EE292 6.6 The series resonant circuit.

C


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AC Power Review

cosrmsrms I V P

cosPF

iv sinrmsrms I V Q

Power“Effective Power”

“Real Power”

Power Factor

Power Angle

Reactive power

rmsrms I V P Apparent power [VA]

[W]

[VAR]


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Phasor / Power Analysis for Resonance

0 or,,0sinrmsrms I V Reactive power

Series Circuit Phasor diagram

0

1

C j L j Z Z C L

C L

1

LC

10 Resonance

Frequency

LC f

2

10


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EE292

Figure 6.24 Plots of normalized magnitudeand phase for the impedance of the seriesresonant circuit versus frequency.


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EE292Figure 6.25 Plots of the transfer-functionmagnitude |V R / Vs| for the series resonantbandpass-filter circuit.

LC f 2

10

f

f

f

f jQ R f Z s s

0

0

1

CR f Q s

021

R

L f Q s

02

Quality Factor

f f f f jQ s s R

0011

V

V


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EE292The bandwidth B is equal to the differencebetween the half-power frequencies.

L H f f B

20

B f f H

20

B f f L

sQ

f B 0


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EE292 6.7 The parallel resonant circuit.

Resonance is a phenomenon that can be

observed in mechanical systems andelectrical circuits.


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EE292 Phasor / Power Analysis for Resonance

Parallel Circuit Phasor diagram reactive

power = 0

0

111

C j L j Z Z C L

C L

1

LC 1

0 ResonanceFrequency

fL j fC j R Z p

2121

1

LC f

2

10

i 6 30 l h ll l


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EE292

Figure 6.30 Voltage across the parallelresonant circuit for a constant-amplitudevariable-frequency current source.

f f f f jQ

R Z

p

p

001

CR f Q p 02

L f

R

Q p02


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EE292 Summary

• Phasor approach suitable for steady state

AC circuit analysis

• Phasor approach simplifies circuit analysis

• Connection to AC Power formulation

• Mastering R, L, C complex impedances

facilitates further understanding of many

circuit characteristics

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EE292 Ch06 Filter Part1 Signals Part 2 Filters Em v2