Frequency and Bode Plots

8/18/2019 Frequency and Bode Plots

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ENGINEERINGPrinciples and

ApplicationsChapter 6

Frequency Response, Bode Plots, and Resonance

CHAPTER 6

Frequency Response, BodePlots, and Resonance

1. State the fundamental concepts of Fourier

analysis.

2. Determine the output of a filter for a given

input consisting of sinusoidal componentsusing the filter’s transfer function.


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3. Use circuit analysis to determine thetransfer functions of simple circuits.

4. Draw first-order lowpass or highpass filter

circuits and sketch their transfer functions.

5. Understand decibels, logarithmic frequency

scales, and Bode plots.


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6. Draw the Bode plots for transfer functions of

first-order filters.


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Fourier Analysis

All real-world signals are sums of sinusoidalcomponents having various frequencies,

amplitudes, and phases.


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Filters

Filters process the sinusoid components of an inputsignal 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 reject components in other ranges.


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Transfer Functions

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:

( ) inout

V

V

= f H


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The magnitude of the transfer function shows

how the amplitude of each frequency componentis affected by the filter. Similarly, the phase of

the transfer function shows how the phase of eachfrequency component is affected by the filter.

( )in

out

V

V= f H


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Determining the output of a filterfor an input with multiple

components:

1. Determine the frequency and phasor

representation for each input component.

2. Determine the (complex) value of the transfer

function for each component.


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3. Obtain the phasor for each output

component by multiplying the phasor for each

input component by the corresponding

transfer-function value.

4. Convert the phasors for the output

components into time functions of variousfrequencies. Add these time functions to

produce the output.


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Linear circuits behave as if they:

1. Separate the input signal into components

having various frequencies.

2. Alter the amplitude and phase of each

component depending on its frequency.

3. Add the altered components to producethe output signal.


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FIRST ORDER LOWPASS


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FIRST-ORDER LOWPASS

FILTERS

RC f B π 2

1

=

( ) ( ) B f f j f H

+= 11

( )( )21

1

B f f f H

+=

( )

−=∠

B f f f H arctan


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( ) ( )21

1

B f f f H +=


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C S CASCA


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DECIBELS, THE CASCADE

CONNECTION, AND LOGARITHMICFREQUENCY SCALES

( ) ( ) f H f H log20dB =


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C d d T P t N t k


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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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Logarithmic Frequency ScalesOn a logarithmic scale, the variable is

multiplied by a given factor for equalincrements of length along the axis.

A decade is a range of frequencies for which


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A decade is a range of frequencies for which

the ratio of the highest frequency to the

lowest is 10.

=

1

2logdecadesof number

f

f

An octave is a two-to-one change in frequency.

( )

( )

=

= 2loglog

logoctavesof number

12

1

2

2

f f

f

f


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BODE PLOTS


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BODE PLOTS

A Bode plot shows the magnitude of a network

function in decibels versus frequency using a

logarithmic scale for frequency.

( )( ) B f f j

f H

+=

1

1

( )

+−=

2

dB1log10

B f f f H


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1. A horizontal line at zero for f < f B / 10.

2. A sloping line from zero phase at f B / 10 to –90° at 10 f B.

3. A horizontal line at –90° for f > 10 f B.


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FIRST ORDER HIGHPASS


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FIRST-ORDER HIGHPASS

FILTERS

( ) ( )( ) B

B

f f j

f f j f H

+==

1in

out

V

V

RC f B π 2

1

=


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SERIES RESONANCE

Resonance is a phenomenon that can beobserved in mechanical systems and

electrical circuits.

f1

Q1


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LC

f

π 2

0 =

R L f Q s02π =

CR f

Q s02π

=

( )

−+=

f f

f f jQ R f Z s s

0

0

1


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Series Resonant Circuit as a


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Bandpass Filter

( ) f f f f jQ s s R

001

1

−+=V

V


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L H f f B −=

sQ

f

B

0

=

20

B f f H +≅

20 B

f f L −≅


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PARALLEL RESONANCE


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( ) ( ) fL j fC j R Z p

π π 21211 −+=

LC f

π 2

10 = L f

RQ p02π

= CR f Q p 02π =

( ) f f f f jQ R Z

p

p

001 −+=


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Ideal Filters


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Ideal Filters


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Second-Order Lowpass Filter


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Second-Order Lowpass Filter

( ) ( )( ) f f f f jQ

f f jQ f H s

s

00

0

in

out

1 −+−==

VV


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DIGITAL SIGNAL


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G S GN

PROCESSING


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Conversion of Signals from


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g

Analog to Digital Form

If a signal contains no components with

frequencies higher than f H

, the signal can be

exactly reconstructed from its samples, provided

that the sampling rate f s is selected to be more

than twice f H .


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Digital Lowpass Filter


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g p

( ) ( ) ( ) ( )n xanayn y −+−=11

T

T a

τ

τ

+= 1


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Frequency and Bode Plots

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