PRITESH VASOYA (130420105057)VISHVARAJ CHAUHAN(130420105058)

VIVEK MISTRY (130420105059)MILAN HIRAPARA (140423105004)

FREQUENCY RESPONSE ANALYSIS

Active Learning Assignment, Instrumentation & Process Control

BE SEM VChemical Engineering

INTRODUCTION

Frequency responses are generally derived by using the standard Laplace transform of sinusodial forcing functions.

We shall look at a convenient graphical technique for obtaining frequency response of linear systems.

AMPLITUDE RATIO AND PHASE ANGLE

AMPLITUDE RATIO AND PHASE ANGLE

AMPLITUDE RATIO AND PHASE ANGLE

AMPLITUDE RATIO AND PHASE ANGLE

After sufficient time elapses, the response of a first order system to a sinusodial input of frequency ω also a sinusoid of frequency ω.

Amplitude Ratio is defined as output amplitude upon input amplitude, and is denoted by |G (jω)|.

To obtain AR and phase angle, one merely substitutes jω instead of s in the transfer function and then finds the magnitude and angle of the resulting complex number.

CHARECTERISTICS OF A STEADY STATE SINUSODIAL RESPONSE

CHARECTERISTICS OF A STEADY STATE SINUSODIAL RESPONSE

The output is also a sine wave.Input frequency=output frequency=ω.In general, AR < 1, which means output

amplitude is greater than input amplitude.The output is shifted in time, that is it lags

the input by a phase angle of φ.Amplitude ratio (AR) and phase angle are

both functions of frequency.

BODE DIAGRAMS

There is a convenient graphical representation of AR and phase lag’s dependence on frequency.

This is called Bode Diagram.It consists of two graphs: logarithm of AR VS

logarithm of frequency and phase angle versus logarithm of frequency.

It is plotted on semilog papers.

BODE DIAGRAMS

BODE DIAGRAMS (FOR FIRST ORDER SYSTEMS)

BODE DIAGRAMS (FOR FIRST ORDER SYSTEMS)

BODE DIAGRAMS (FOR FIRST ORDER SYSTEMS)

Some asymptotic considerations can simplify the construction of this plot. As ωτ 0, we can see that AR1. This is indicated by the low frequency asymptote.

As ωτ∞ the equation becomes asymptotic to,log AR = - log(ωτ), which is a line of slope -1, passing through the point ωτ=1. This is indicated as the high frequency asymptote.

The frequency ω=1/τ, where the two asymptotes intersect, is known as the corner frequency.

BODE DIAGRAMS (FOR FIRST ORDER SYSTEMS)

In the second part of the Bode Diagram, the phase curve is given by φ= tan-1 (ωτ)= -tan-1 (ωτ).

φ approaches 0 at low frequencies and -90 at high frequencies. At corner frequency, φ= tan-1 (ωτ)= -tan-1 (ωτ)=-tan-1 (1)= -45.

It should be noted that AR is often reported in decibels. It is defined by, dB= 20 log(AR).

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