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CHAPTER 6Frequency Response, Bode Plots, and

Resonance

Fourier Analysis, Filters, and Transfer

Functions First-Order Lowpass Filters Decibels,

the Cascade Connection, and Logarithmic Frequency

Scales Bode Plots First-Order Highpass

Filters Series Resonance Parallel Resonance Ideal

and Second-Order Filters Digital Signal Processing

CHAPTER 6Frequency Response, Bode Plots, and

Resonance

- State the fundamental concepts of Fourier

analysis. - 2. Determine the output of a filter for a given

input consisting of sinusoidal components using

the filters transfer function.

3. Use circuit analysis to determine the

transfer 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.

6. Draw the Bode plots for transfer functions of

first-order filters. 7. Use software to

produce Bode plots for more complex RLC

filters. 8. Calculate parameters for series

and parallel resonant circuits.

9. Select and design simple filter

circuits. 10. Design simple digital

signal-processing systems.

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Fourier Analysis

All real-world signals are sums of sinusoidal

components having various frequencies,

amplitudes, and phases.

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Filters

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 reject components in

other ranges.

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

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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Determining the output of a filter for 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.

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 various frequencies. Add

these time functions to produce the output.

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Linear circuits behave as if they

- 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 produce the

output signal.

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FIRST-ORDER LOWPASS FILTERS

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DECIBELS, THE CASCADE CONNECTION, AND LOGARITHMIC

FREQUENCY SCALES

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Cascaded Two-Port Networks

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Logarithmic Frequency Scales

On a logarithmic scale, the variable is

multiplied by a given factor for equal increments

of length along the axis.

A decade is a range of frequencies for which the

ratio of the highest frequency to the lowest is

10.

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

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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.

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- A horizontal line at zero for f lt fB /10.
- 2. A sloping line from zero phase at fB /10 to

90 at 10fB. - 3. A horizontal line at 90 for f gt 10fB.

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FIRST-ORDER HIGHPASS FILTERS

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Computer-Generated Bode Plot

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SERIES RESONANCE

Resonance is a phenomenon that can be observed in

mechanical systems and electrical circuits.

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Series Resonant Circuit as a Bandpass Filter

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PARALLEL RESONANCE

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Ideal Filters

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Second-Order Lowpass Filter

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DIGITAL SIGNAL PROCESSING

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Conversion of Signals from Analog to Digital Form

If a signal contains no components with

frequencies higher than fH, the signal can be

exactly reconstructed from its samples,

provided that the sampling rate fs is selected to

be more than twice fH.

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Digital Lowpass Filter

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