Introduction to Electrical Systems

7/26/2019 Introduction to Electrical Systems

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SCHOOLOFENGINEERING

IntroductiontoElectricalandElectronic

Engineering

Part7 BodePlots,FrequencyResponse

Grant A. Ellis, PhD

1


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Frequency Response, Bode Plots,

and Resonance Our concern here is the information-bearing currents andvoltages that we call now signals

Example 1, sensors or transducers on an internalcombustion engine provide electrical signals that representtemperature, speed, throttle position and the rotational

position of the crankshaft. These signals are processed todetermine the optimum firing instant of the cylinder

Example 2, signal processing related toelectrocardiogram (plot of the electrical signals generated

by the human heart). The information extracted from thesignals is used to analyse the behaviour of a patientsheart.

Grant A. Ellis


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and Resonance Signal processing is concerned with manipulating signals to extractinformation and using that information to generate other useful electrical signals Even though many signals are not sinusoidal, we can

construct any waveform by adding sinusoids (sometimes thousands of them)that have the proper amplitude, frequencies and phases.

Grant A. Ellis

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


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and Resonance Fourier analysis is a mathematical technique for finding theamplitudes, frequencies and phases of a given waveform

The range of the frequencies of the components depends onthe type of signal under consideration (table below)

Grant A. Ellis

Frequency Ranges of Selected Signals


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and Resonance Example, Fourier series of asquare wave is a combination of

sinusoidal components :

where 0 = 2/T is called thefundamental angular frequency ofthe wave

Figure shows the sum its first 5components only Approximation will get better asmore components are added

The frequency, amplitude andphases can be determined bymeasurement using a spectrumanalyzer

Grant A. Ellis

A square wave and some

of its components.


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and Resonance Filters are electrical circuits that are used to retain components in a givenrange of frequencies and discard the components in another range. Example,

television antenna produces a voltage composed of signals from many transmitters.

Filters select only frequencies of a particular channel and reject others

Grant A. Ellis

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.


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and Resonance Transfer function H(f) of the two-port filter is defined to be the ratio ofthe phasor output voltage to the phasor input voltage as a function of

frequency

Because phasors are complex, the transfer function is a complex

quantity having both magnitude and phase

The magnitude of the transfer function is the ratio of the output

amplitude to the input amplitude

The phase of the transfer function is the output phase minus the input

phase

In steady-state, the output signal is sinusoidal and has the same

frequency as the input signal

in

out

V

V

fH

Grant A. Ellis


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and ResonanceExample 6.1: Given an input signaland transfer function of the filter.Find the output.

By inspectionthe frequencyf= 0/2=1000Hz From the Figurefor his frequency:

Magnitude =3 andPhase =30

Therefore: and

or

Grant A. Ellis

The transfer function of a filter. See Examples 6.1 and 6.2.


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and Resonance

Input signals with multiple frequency Components (procedurefor determining the output)

1. Determine the frequency and phasor representation for eachinput components

2. Determine the (complex) value of the transfer function for eachcomponent

3. Obtain the phasor for each output component by multiplyingthe phasor for each input component by the correspondingtransfer-function value

4. Convert the phasor for the output components into time

function of various frequencies. Add these time functions toproduce the output

Grant A. Ellis


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Example 6.2 Given an input signaland transfer function of the filter, find the output:

Break the input into components:

By inspection: frequencies f are 0, 1000, 2000 from Figure:

Outputs: dc output phasors:


and Resonance

Grant A. Ellis


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Frequency Response, Bode Plots,and Resonance

Experimental Determination of the Transfer Function

1. Connect a sinusoidal source to the input port of the circuit2. Measure the amplitude and phase of both the input signal and the

resulting output signal

3. Divide the output phasor by the input phasor

4. Repeat 1 to 3 for various frequencies of interest

The measurement can be done with instruments such as: Voltmeters,

oscilloscopes, network analyzers, signal generators & spectrum

analyzers

Grant A. Ellis


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First-order Lowpass Filter

First-order lowpass filter tends to pass low-frequency componentsof input signal and reject high-frequency components. In other words, for low frequencies, the output amplitude is nearlythe same as the input and for high frequencies, the output is muchless than the inputInput current is

Output voltage is

Substitute the current

Transfer function is

If thenRC

fB21

Bffj

fH

1

1

Grant A. Ellis

A first-order lowpass filter.

Another first-order lowpass filter


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First-order Lowpass Filter

Bffj

fH 1

1

21

1

BfffH

BfffH arctan

Grant A. Ellis

Magnitude and phase of the first-order lowpass

transfer function versus frequency.

The magnitude of the complex function is

The phase is the phase of the numerator (which is zero) minus the

phase of the denominator, which is

From the plot:- For low frequencies, the magnitude is approximately unity and thephase is nearly zero (they are passed)- For high frequencies, the magnitude approaches zero (they arerejected) When f = fB themagnitude of transfer

function is 1/ . Sinceit causes the poweroutput to cut half (V2/z),fB is called half-powerfrequency


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Example 6.3 Given an input signaland the filter find the output:

Half power frequency is:

For the first input component:

-The amplitude almost retained

For the second (reduced by ); For the third (significant reduction)


and Resonance

Grant A. Ellis


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The transfer function can be defined in decibels as follows:

Note from the Table, that decibels are positive formagnitudes greater than 1 and negative formagnitudes less than 1

A filter designed to eliminate components ina narrow range of frequencies, is called notchfilter (example, filter to eliminate 60 Hz ofpower line noise from the audio signals, and pass component of at otherfrequencies, called passband). Both results clearly seen inonly in decibelplot but not inlinear plot

Decibels

fHfH log20dB

Grant A. EllisTransfer-function magnitude of a notch filter

used to reduce hum in audio signals

Transfer-Function Magnitudesand their Decibel Equivalents


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When we connect the output of one two-port circuit to the inputterminals of another two-port circuit, we have a cascade connection

For cascade connection the following is trueor (multiplied) Expressing both sides in decibelsor

or

The overall transfer function of a cascade connection in decibels isthe sum of individual transfer functions in decibel (or dB).

Cascaded Two-port Network

fHfHfH 21

dB2dB1dB

fHfHfH

Grant A. Ellis

Cascade connection of two two-port circuits


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Logarithmic scale is often used for frequency when plotting the transferfunction to clearly see its variation in large range of values

On a logarithmic scale, the variable is multiplied by a given factor on toobtain equal increments of length along the axis, whereas on linear

scale, equal lengths correspond to adding a given amount to thevariable

A decade is the range of frequencies for which the ratio of the highestfrequency to the lowest one is 10 eg 2 to 20 is one decade; 50 to 5000are two decades, from 50 to 500 and from 500 to 5000Hz

f2

>f1

(range f1

to f2

)

An octave is a two-to-one change in frequency eg10 to 20 Hz is oneoctave

Logarithmic Frequency Scales

Grant A. Ellis


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A Bode plot uses a logarithmic scale for frequency to show the

magnitude of a network function in decibels versus frequency.

The Bode plots can be closely approximated by straight-line

segments that are relatively easy to draw and then make quick

estimate of transfer functions

Given a transfer function magnitude

Convert to decibel

Substitute and modify

since log(1)=0

Since then

Bode Plots

2

dB 1log10

Bf

ffH

Grant A. Ellis


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Plot of

If f>fB (high frequency range)

then

and the amplitude is approximated by the straight slope line on the right-hand side of the graph These two straight-line asymptotes intersect at the half-power frequency

fB, called the corneror break frequency The slope of the high-frequency asymptote is -20 dB per decade The asymptotes are in error at the corner by only

(3 dB frequency).

Bode Plots (amplitude)

2

dB 1log10BfffH

Grant A. Ellis

Magnitude Bode plot for first-order lowpass filter.

20 dB / decade= 6 dB / octave


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The phase of transfer function

By observation, the phase

approaches zero at very low

frequencies, equals -45 at the

break frequency, andapproaches -90 at high

Frequencies

The curve can be approximated by the following straight-linesegments:

Bode Plots (phase)

Grant A. Ellis

Phase Bode plot for thefirst-order lowpass filter.


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The analysis of the circuit for thefirst-order highpass filter similar tothat of the lowpass filter

if

then

The magnitude is

The phase is

Amplitude goes to 0 fordc signal

First- Order Highpass Filter

RCfB

2

1

B

B

ffj

ffjfH

1in

out

V

V

Grant A. Ellis

Magnitude and phase for the first-order highpass

transfer function.

First-order highpass filter.

RCf2j1

RCf2j

V

VfH

in

out


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For Bode Plot, the magnitude is converted to decibels (dB) and alogarithmic frequency scale is used

If the second term in amplitude expression is almost zero (f >fB the resulting magnitude is almost 0 (horizontal asymptote

Intersection is at fB

Bode Plots First-Order Highpass Filter

Grant A. EllisBode plots for the first-order highpass filter.


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Example 6.4: Highpass filter with magnitude -30 dB at frequency

f = 60 Hz. Define the break frequency of this filter.

Low-frequency asymptote slopes at the rate of 20db / decade

Then the number of decades from -30 dB to 0 dB is:

According to the definition of the decade:

Solving the equation, we obtain:

Highpass filter (example)

Grant A. Ellis

31.6 * 60 = 1896


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Series Resonance

LCf

2

10

Grant A. Ellis

The RLC circuit (resonant circuit) have better performance (inpassing desired signals and rejecting undesired signals that are

relatively close in frequency) than first-order filters

In a resonant circuit, when a sinusoidal source of a proper

frequency is applied, voltages much larger than the source voltage

can appear in the circuit

The impedance seen by the source

in this circuit is

The resonant frequency f0 is defined to

be the one at which the impedance is purely

resistive (i.e. the total reactance is zero), that means

or


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The quality factor QS is defined to be ratio of the reactance of theinductance at the resonant frequency to the resistance:

Excluding L in this equation (L is from the resonant frequencyequation), we obtain:

Note:

Substituting L and C from the equations above into theimpedance formula we obtain:

Series Resonance

R

LfQs

02

CRfQs

02

1

f

f

f

fjQRfZ ss

0

0

1

Grant A. Ellis

CRf2

1

R

Lf2Q

0

0s


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Plots of the normalized magnitude and the phase of the impedanceversus normalized frequency f / f0 shows that the impedancemagnitude is minimum at the resonant frequency (f = f0) (equal to

what?) As the quality factor Q increases, the minimum becomes sharper.

For f < f0 the impedance has a capacitive nature and for f > f0it has an inductive nature.

Series Resonance

f

f

f

fjQRfZ ss

0

0

1

Grant A. Ellis

Plots of normalized magnitude and phase for the impedance of the

series resonant circuit versus frequency.


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Define transfer function of the filter:

Plot of the magnitude of transfer function:

Series Resonant Circuit as a

Bandpass Filter

ffffjQ1

1fH

V

V

00ss

R

Grant A. Ellis


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Discussions: At lower frequencies f < f0 theimpedance is capacitive in nature, itreduces the circuit current and VRbecomes small compared to VS At higher frequencies f > f0 theimpedance is inductive in nature, itstill reduces the circuit current and

again VR becomes small compared to VS

At f=f0 the impedance becomes minimum (equal R) and VR becomesequal to VS

As a result, if a source has components ranging in frequencyabout the resonant frequency f0 the components of the source close tothat frequency will have only a small change, but other components willbe significantly reduced. So Bandpass filter passes the

components centered at the resonant frequency the

rest of components are partly rejected.

Series Resonant Circuit as aBandpass Filter

Grant A. Ellis

Transfer-function magnitude

|VR / Vs| for the series resonantbandpass-filter.


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Discussions (continued): There are two half-powerfrequencies for series resonant circuit

when the transfer function magnitudehas fallen from its maximum by aFactor 1/ =0.707, i.e. fL and fH

The bandwidth B of this filter is the

difference between the half-powerfrequencies:

It can be shown that

For QS >> 1, the following approximation is valid:

Series Resonant Circuit as a

Bandpass Filter

LH ffB

sQ

fB 0

20

BffH

20

BffL

Grant A. Ellis

The bandwidth B is equal to thedifference between the half-power

frequencies.


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Example 6.5Given a filter circuit Resonant frequency:

Quality factor and bandwidth:

Half-power frequencies: Reactances:

Voltages across at resonant frequencies:

Grant A. Ellis

Series resonant circuit.

Series Resonance


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Parallel Resonance

fLjfCjRZp

2121

1

LCf

2

10

Lf

RQ

p02

CRfQp 02

ffffjQ

RZ

pp

001

Parallel resonant circuit The impedance is:

At the resonant frequency the impedance is purely resistive (sameas the series circuit)

The quality factor is defines as (reciprocal of series)and can be written as

Then impedance can be written as:then V

out

= I ZP

Half-power frequencies

and the band width:

Grant A. Ellis


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Ideal Filters

Ideal filters pass components in the desired frequency rangewith no change in amplitude or phase and totally reject the

components in the undesired frequency range.

They are ideal and can be just approximated by real circuits. fLandfH are cutoff frequencies.

Grant A. Ellis

Transfer functions of ideal filters.


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Ideal Filters

The input signal vin consists of a 1-kHz sine wave plus high-frequency

noise. By passing vin through an ideal

lowpass filter with the proper cutoff

frequency, the sine wave is passedand the noise is rejected, resulting in

a clean output signal.

Grant A. Ellis


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Second-Order Lowpass Filter

ffffjQffjQ

fHs

s

00

0

in

out

1

V

V

LCf

2

10

R

LfQs

02

Grant A. Ellis

It is based on series resonant circuit

For QS >> 1 the highpeak is reached in thevicinity of f0 To cut the peak andmake it flat we choose

QS=1 (more preciselyQS = 0.707). The transfer functionat this quality factor is said tobe maximally flat and is calledButterworth function. The transfer function for thesecond order filter falls more rapidlythan for the first order filter (different slope) Lowpass filter circuits and their

transfer-function magnitudes versus

frequency.


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Second-Order Highpass Filter

Grant A. Ellis

Second-order highpass filter and its transfer-function magnitude

versus frequency for several values of Qs


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Filter design Example 6.7.

Grant A. Ellis

Second-Order Highpass Filter

Design a filter which passes components higher than 1 kHz and rejects components

Less than 1 kHz using a second order circuit with L = 50 mH.

F0.507

1050102

1

Lf2

1C

CL2

1f

3-620

0

22

1B

fQ 0s To provide a nearly constant transfer function up to 1 kHz

314.11

1050102Q

Lf2QCf2

1R -33

s

0

s0


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Second-Order Bandpass Filter

Grant A. Ellis

Second-order bandpass filter and its transfer-function magnitudeversus frequency for several values of Q

s


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Second-Order Band-Reject (Notch)

Filter

Grant A. Ellis

Second-order band-reject filter and its transfer-function magnitude versus

frequency for several values of Qs.

Documents

Introduction to Electrical Systems