Kent Bertilsson Muhammad Amir Yousafapachepersonal.miun.se/~amiyou/Lecture2_analog_070912.pdf · Muhammad Amir Yousaf . Block Diagram • Electronic systems is often described by

Kent Bertilsson

Muhammad Amir Yousaf

Today’s topics

• Analog System (Rev)

• Frequency Domain

• Signals in Frequency domain

• Frequency analysis of signals and systems

• Transfer Function

• Basic elements: R, C , L

• Filters

• RC Filters

• jw method (Complex impedance method for AC circuits)

• Bode plot

• Amplifier

Kent Bertilsson


Block Diagram

• Electronic systems is often described by block diagram

Antenna Amplifier Filter Analog to

digital conversion

Kent Bertilsson


Signals in Frequency domain

Every signal can be described both in the time domain and the

frequency domain.

A periodic signal is always a sine or cosine or the sum of sines

and cosines.

Frequency representation of periodic signal is:

V

fs 2 fs 3 fs 4 fs 5 fs f

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Signals in Frequency domain

A periodic signal (in the time domain) can in the frequency

domain be represented by:

A peak at the fundamental frequency for the signal, fs=1/T

And multiples of the fundamental f1,f2,f3,…=1xfs ,2xfs ,2

xfs

V

T=1/fs t

V

fs 2 fs 3 fs 4 fs 5 fs f

Kent Bertilsson


Time domain vs Frequency domain

A non periodic (varying) signal time domain is spread in the

frequency domain.

A completely random signal (white noise) have a uniform

frequency spectra

V Noise

f

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Why Frequency Representation?

All frequencies are not treated in same way by nature and

man-made systems.

In a rainbow, different parts of light spectrum are bent differently

as they pass through a drop of water or a prism.

An electronic component or system also gives frequency

dependent response.

Kent Bertilsson


Frequency domain

Frequency domain is a domain for analysis of mathematical functions

and signals with respect to frequency.

A time-domain graph shows how a signal changes over time,

whereas a frequency-domain graph shows how much of the signal

lies within each given frequency band over a range of frequencies.

A frequency-domain representation also include information on

the phase shift that must be applied to each sinusoid in order to be

able to recombine the frequency components to recover the original

time signal.

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Transfer function

The transfer function is the relation between the amplitude for the

output and input in the frequency domain.

H(20kHz)=10 means that for a 20kHz signal the output is ten times larger

than the input.

H(f) is of course continuous function.

fU

fUfH

in

Out

H

10

5

0

f

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Basic Elements and jω-method

Any combination of passive (R, L, and C) and/or active (transistors or

operational amplifiers) elements designed to select or reject a band of

frequencies is called a filter.

The jω-method is a very powerful tool making it possible performing

advanced frequency dependent (alternating current, AC) functions

using the same rules that applies for direct current (DC)

How these elements behave with alternating voltage and current.

Resistor Capacitor Inductor

Symbol

Reactance

RX LjX

CjX

1

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jω-method

Impedance calculations can be performed in the same

way as for normal resistances.

LjRLZRZZ

RCj

R

CjR

CjR

CZ

RZ

CZ

RZ

CZRZZ

11

1

//

R L

R

L

Kent Bertilsson


Filter

Any combination of passive (R, L, and C) and/or active (transistors or

operational amplifiers) elements designed to select or reject a band of

frequencies is called a filter.

In communication system, filters are used to pass frequencies

containing the desired information and to reject the remaining

frequencies.

In stereo systems, filters can isolate particular bands of frequencies

for increased or decreased emphasis by the output acoustical system.

Filters are used to eliminate any unwanted frequencies, called noise,

due to non linear characteristics of electronic deices or signal picked

up from surrounding medium.

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Filters

Two classifications of the filters:

Passive Filters (R,L,C)

Active Filters (R,L,C, transistor, op-amps)

Four categories of filters

Low Pass Filters

High Pass Filters

Band Pass Filters

Band Stop Filters

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Filters

Cut-off Frequency

Pass Band

Stop Band

Any Frequency (in pass band)

with at least 70.7% of max.

output voltage will pass

through the filter

For stop band, the output

voltage is 1/1000 of Vmax or

-60dB

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RC - filter

Calculate the transfer function H(ω)

using jw-method

What is the output voltage level and

power level at the cut-off frequency?

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RC - filter

Analysis:

At low frequencies how capacitor

behaves?

Xc = 1/ jwC = 1/ 2 pi f C

How capacitor behaves at high

frequencies?

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Normalized plot for filter

Normalization is a process whereby a quantity such as voltage is

divided by a quantity of the same unit of measure to establish a

dimensionless level of specific value or range.

A normalized plot in filter domain can be obtained by dividing the

plotted quantity Vo with applied voltage Vi for the frequency of

interest.

1

0

0.707

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Bode Diagram

• It is a technique for sketching the frequency response of systems (i.e.

filter, amplifiers etc) on dB scale . It provides an excellent way to

compare decibel levels at different frequencies.

• Absolute decibel value and phase of the transfer function is plotted

against a logarithmic frequency axis.

fHangle

fHdB

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Decibel, dB

decibel, dB is very useful measure to compare two levels

of power.

It is used for expressing amplification (and attenuation)

InV

OutV

VAVdBA

InV

OutV

InV

OutV

R

InV

R

OutV

InP

OutP

PdBA

R

VIVP

InP

OutP

PAPdBA

log20log20

log20

2

log102

2

log10log10

2

log10log10

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dB AV AP

20 10 100

10 3.16 10

6 2 4

3 2

0 1 1

-3 0.5

-20 0.1 0.01

414.12

707.02

1

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Bode Plot for High-Pass RC Filter

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Sketching Bode Plot for High-Pass RC

Filter

High-Pass R-C Filter

Voltage gain of the system is:

In magnitude and phase form

A change in frequency by a 2:1 ratio results in a 6dB change in gain.

A change in frequency by a 10:1 ratio results in a 20dB change in gain.

)/(1/1 ffjA cv

)/(1tan^2)̂/(1/1 ffcffcAA vv

vvdB AA 10log20

For f << fc

cvdB

f

fA 10log20

Kent Bertilsson


Bode Plot Amplitude Response

Must remember rules for sketching Bode Plots:

Two frequencies separated by a 2:1 ratio are said to be an octave

apart. For Bode plots, a change in frequency by one octave will result

in a 6dB change in gain.

Two frequencies separated by a 10:1 ratio are said to be a decade

apart. For Bode plots, a change in frequency by one decade will result

in a 20dB change in gain.

True only for f << fc

Kent Bertilsson


Asymptotic Bode Plot amplitude response

Plotting eq below for higher frequencies:

For f= fc/10 AvdB = -20dB



For f= fc AvdB = 0dB

This gives an idealized bode plot.

Through the use of straight-line segments called idealized Bode plots,

the frequency response of a system can be found efficiently and

accurately.

cvdB

f

fA 10log20

Kent Bertilsson


Actual Bode Plot Amplitude Response

For actual plot using equation

For f >> fc , fc / f = 0 AvdB = -3dB

For f = 2fc AvdB = -1dB

For f = 1/2fc AvdB = -7dB

At f = fc the actual response curve is 3dB down from the idealized Bode

plot, whereas at f=2fc and f = fc/2 the acutual response is 1dB down

from the asymptotic response.

)2)̂/(1/1log(20 ffcAv

dBAv 3)2/1log(20)2)̂0(1/1log(20

Kent Bertilsson


Asymptotic Bode Plot Phase Response

Phase response can also be sketched using straight line asymptote

by considering few critical points in frequency spectrum.

Plotting above equation

For f << fc , phase aproaches 90

For f >> fc , phase aproches 0

At f = fc tan^-1 (1) = 45

)/(1tan^ ffc

Kent Bertilsson


Asymptotic Bode Plot Phase Response

Must remember rules for sketching Bode Plots:

An asymptote at theta = 90 for f << fc/10, an asymptote at theta = 0

for f >> 10fc and an asymptote from fc/10 to 10fc that passes through

theta = 45 at f= fc.

Kent Bertilsson


Actual Bode Plot Phase Response

At f = fc/10

90 – 84.29 = 5.7

At f = 10fc

At f= fc theta = 45 whereas at f=fc/10 and f=10fc, the difference the

actual and asymptotic phase response is 5.7 degrees

29.84)10(1tan^

)10//(1tan^)/(1tan^ fcfcffc

7.5)10/1(1tan^

)10*/(1tan^)10/(1tan^ fcfcfcfc

Kent Bertilsson


Bode Plot for RC low pass filter

Draw an asymptotic bode

diagram for the RC filter.

Kent Bertilsson


Bode diagram for multiple stage filter

According to logarithmic laws

dBA

dBA

dBA

dBtotA

AAAtotA

321

321

321

AangleAangleAangletot

Aangle

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Kent Bertilsson



Kent Bertilsson


Bode diagram

• Complicated expressions can be factorized into

sub-expressions as

Const

Differentiator Integrator

Zero Pole

0

1

j

j

C

0

1

1

1

j

j

Kent Bertilsson


Exercise

R

R2 C

VIn R3 VOut

Draw an asymptotic bode

diagram for the shown filter.

Kent Bertilsson


Amplifier

• Voltage amplification

• Current amplification

• Power amplification

IIN IOut

PIN VIn VOut POut

In

OutV

V

VA

In

OutI

I

IA

In

OutP

P

PA

Kent Bertilsson


Amplifier model

• RIn – Input impedance

• AV – Voltage gain

• ROut – Output impedance

ROut

VIn RIn AVVIn VOut

• The amplifier model is often sufficient describing

how an amplifier interacts with the environment

Kent Bertilsson


H(f)

AVmax

0.707AVmax

f1 f2 f

Bandwidth

The bandwidth is the frequency range where the

transferred power are more than 50%.

12

maxmax

max

707.02

5.0

ffB

AAA

AA

VVV

PP

Kent Bertilsson


A nonlinear function between UIn and UOut distorts the

signal

An amplifier that saturates at high voltages

A diode that conducts only in the forward direction

Distortion

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Noise

• Random fluctuation in the signal

• Theoretically random noise contains all possible

frequencies from DC to infinity

• Practical noise is often frequency limited to an upper

bandwidth by some filter

• A limited bandwidth from the noisy reduce the noise

power

Kent Bertilsson


RC Filters in Mindi

Design a RC filter in Mindi.

Simulate output for diffrent frequencies

Analyse the results.

dB

Bode Plots

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Changes in Timetable

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References

• Introductory Circuit Analysis By Boylestad

Kent Bertilsson Muhammad Amir Yousaf

Documents

Kent Bertilsson Muhammad Amir Yousafapachepersonal.miun.se/~amiyou/Lecture2_analog_070912.pdf · Muhammad Amir Yousaf . Block Diagram • Electronic systems is often described by