Ch 4 Analog Measurand

7/23/2019 Ch 4 Analog Measurand

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Characteristics of time-dependent

measurand



Classification of time dependent

measurands



Examples of time-dependent functions



Simple harmonic relation• Any two variables can have a simple harmonic relation if the

second derivative of one w.r.t. the other is related to the

function itself.

• e.g. if the distance of a circular motion is described by

2

sin

first derivative (velocity) is

cos

and the second derivative (acceleration) is

sin

where is called the circular frequency ( rad/s)

while, the cyclic frequency (cycle p

x A t

the

xv A t

t

va A t t

er second, Hertz, f)is given by

=2 f



The scotch-Yoke mechanism

It illustrates the definition of both cyclic and

circular frequency

When the mass block rotates a complete

cycle in a time t

The number of cycles per time (cyclic

frequency) is calculated by

f=1/t

while the circular frequency is

=2f

Example:

given that the time of one cycle is 0.2 sec

what is the cyclic and circular frequency

F=1/t=5 cycle/sec (Hz)

= 2*3.14*5=31.4 rad/s=31.4*57.3 degree/sec

The mass is oscillating between two

peaks to form a harmonic motion.

One cycle t=1/f



Phase difference

In this case both of the mass block and the piston

have the same frequency, but they may have a

phase shift, i. e. there is a difference between the

start and end points for their cycles.



Different units for cyclic motion

Angular frequency ω (Ordinary) frequency

2π radians per second exactly 1 hertz (Hz)

1 radian per second approximately 0.159155 Hz

1 radian per secondapproximately 57.29578 degrees

per second

1 radian per second

approximately 9.5493 revolutions

per minute (rpm)

0.1047 radian per second approximately 1 rpm

https://en.wikipedia.org/wiki/Frequency

https://en.wikipedia.org/wiki/Hertz

https://en.wikipedia.org/wiki/Revolutions_per_minute




https://en.wikipedia.org/wiki/Hertz

https://en.wikipedia.org/wiki/Frequency



Fourier Analysis

• Putting any function in the form of the summation of its

trigonometric components (Sinusoidal functions).

• The special case is when the function is itself a periodic function.

Then the Fourier series is used to transform it to a summation of

infinity number of sines and cosines.

• The more general case is when the function is not periodic, then the

Fourier transform is the way to analyze it to its sinusoidal

constituents



The analogy of Fourier transform in

nature

A complex function can be represented as a summation of several more simple

sinusoidal functions. This is analogous to the analysis of the white light to many

simpler sinusoidal frequency-dependent components.



Mathematical representation of periodic

measurand

• Any periodic function can be represented in terms of

its sinusoidal components. This is the main purpose

of Fourier series analysis

01

0

cos sin

is the circular frequency

t is time

n is the number of the harmonic

is the constant component

is the variating component

,

n nn

n

f t A A n t B n t

where

A

A B

are the coefficients for each frequency component of the signal

n



Trigonometric identities

Using the relations between the

trigonometric functions, therepresentation of the periodic

functions can be simplified to:

0 1

0

1

*

0

1

2 2

1 * 1

cos sin

can be written as

cos( )

sin( )

tan , tan

n nn

n

n

n

n

n n n

n n

n n

f t A A n t B n t

f t A C n t

or

f t A C n t

where

C A B

B A

A B

2

0

2

2

2

2

2

1

2 cos( )

2 sin( )

1 21,2,3,...... and T is the periof of i.e. T=

T

T

T

n

T

T

n

T

A f t dt T

A f t n t dt

T

B f t n t dt T

where

n f t f



Even and odd functions

• The definition of even and odd functions:

• If f(-t) = f(t) …… it is an even function (e.g cos

• If f (-t)= - f(t) …… it is an odd function (e.g. sin

• It is helpful when you are going to calculate the coefficients of

Fourier series.• If the function is even then all sin’s part will be zero. On the other

hand, if the function is odd then all cosin’s part is zero.

1

1

So for even functions

( ) cos

for odd functions

( ) sin

n

n

nn

f t A n t

and

f t B n t

E l t fi d F i i ffi i t



Example to find Fourier series coefficients

for a periodic function

From the first look This is an odd function, why?

Then the function can be represented by

while the frequency and the period can be concluded from the

drawing as

The coefficients (1, , 3, ……. ) are calculated as follows:

1

( ) sinn

n

f t B n t

1 10.1 Hertz

10

and=2 2 *3.14 *0.1 0.628 rad/sec

f T

f

2

2

2 sin( )

T

n

T

B f t n t dt T



0 5

5 0

0 5

5 0

2 2 21 sin 1 sin

10 10 10

2 10 2 10 2 = cos cos

10 2 10 2 10

2 10 4 = 1 cos cos 110 2

the function can be

n

nt nt B dt dt

nt nt

n n

n nn n

then

written as

4 2 4 6 4 10sin sin sin .........

10 3 10 5 10 f t t t t



Periodic Function interpretation



Heterodyne and beat frequencies

This is a useful phenomenon that resulted when two signals that have the same

amplitude and close frequencies are mixed. Then two new frequencies are generated.

One important application based on this phenomenon is mixing the high frequency

from broadcast stations with a local oscillator signal that have a close frequency signal.

The output will contain a high and low frequency components. By filtering the high

frequency component, the lower frequency signal that is easier in dealing with can be

used to extract data.



Frequency spectrum



Steps for Fourier transform

the sampling frequency, Nyquist frequency and

resolution



Effect of sampling frequency and number of samples on

frequency analysis



ℎ > 2 ×



FT of common function

• Sine wave is represented by a impulse in the frequency domain at its own frequency



FT of common function• An impulse is represented by a constant value over all the frequency range (real part is a cosine, imaginary one

a sine)



FT of common function• Rectangular windows are represented by a sinc wave in the frequency domain (this

explains a lot in frequency analysis…)

Leakage and windowing



Leakage and windowing• To limit this effect a different set a windows can

be chosen, depending on what is required

• One commonly used is the “hanning” one

Ch 4 Analog Measurand

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