Today

Dynamic Transducer Calibration

Theory 1

Terminology pinpointTerminology pinpoint

“response characteristic”:

relation between stimulus and response (when the stimulus varies as function of time, the response characteristic is the transfer function)

“sensitivity”: prime derivative of the change of the response of a measuring instrument on the change of the stimulus

“static sensitivity”: sensitivity of a measurement instrument whereas stimulus and response are not function of time.

“offset” – “bias”: constant value of reading whereas stimulus is null (if not corrected leads to systematic error)

“response time” – “settling time”: time interval between the instant when a stimulus changes and the instant when the response reaches and remains within specified boundaries

Dynamic Calibration

The black box model:

Suitable for every transducer regardless of the physical principles used for measuring.

Allows for a quick categorization of transducers.

1

0

n

i

ii xxy

TRANSDUCER input output

STATIC MODEL DYNAMIC MODEL

j

j

in

j

j

ii

out

i

idt

tqdb

dt

tqda

x, qin y, qout

Dynamic Calibration

Common transducer black box model:

Neglects the offset part as it can usally be corrected.

TRANSDUCER input output

DYNAMIC MODEL (limited to second order)

tqbtqa

dt

tdqa

dt

tqda inout

outout0012

2

2

x, qin y, qout

Dynamic Calibration

0th order model:

More theoretical than real, always an approximation

Usally means “the measurement chain (transducer AND conditioning) is faster than the signal”

Usually suitable for:

Estensimetric measurements

Optical measurements

Displacement measurements

DYNAMIC MODEL

tqbtqa inout 00

STATIC SENSITIVITY

0

0

ab

ks

Dynamic Calibration

1st order model:

Has a dynamic parameter which express the time delay between input and output: the time constant

Usually suitable for:

Thermal measurements

Speed measurements

DYNAMIC MODEL

tqbtqatqa inoutout 001 STATIC SENSITIVITY

0

0

ab

ks

TIME CONSTANT

0

1

aa

2nd order model:

Has a dynamic parameter which express the maximum bandwidth: the natural pulse

Usually suitable for:

Force measurements

Acceleration measurements

Dynamic Calibration

DYNAMIC MODEL

tqbtqatqatqa inoutoutout 0012

STATIC SENSITIVITY

0

0

ab

ks

NATURAL PULSE

2

0

aa

n

DAMPING 20

1

2 aa

a

Why is the frequency response function so important?

Every periodic function can be described by the composition of sine waves of given amplitude, phase and frequency

Frequency Response

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4

-3

-2

-1

0

1

2

3

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4

-3

-2

-1

0

1

2

3

4

Frequency Response:

Tells us the relationship between input and output in the frequency domain

Useful for handling period and quasi-period signals

Needed to estimate transducer readiness

Dynamic Calibration

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30

qo

ut/

(ks*q

in)

ω [rad/s]

0th order 1st order 2nd order

Frequency Response:

Given a tolerance boundary can give us the (passband) bandwidth

Dynamic Calibration

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

qo

ut/

(ks*q

in)

ω [rad/s]

0th order 1st order 2nd order

Tolerance boundaries

(e.g. 5%)

Bandwidth (5%)

Bandwidth (5%)

Frequency Response

0th order model:

1ins

out

qk

q

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16 18 20

qo

ut/

(ks*q

in)

τω [rad]

magnitude

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20

qo

ut/

(ks*q

in)

τω [rad]

phase

Frequency Response

1st order model:

1

22tan

1

1

ins

out

qk

q

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16 18 20

qo

ut/

(ks*q

in)

τω [rad]

magnitude

-100

-50

0

0 2 4 6 8 10 12 14 16 18 20

qo

ut/

(ks*q

in)

τω [rad]

phase

Frequency Response

2nd order model:

EACH FREQUENCY RESPONSE CURVE DEPENDS ON THE DAMPING PARAMETER

n

n

nn

ins

out

qk

q 2tan

21

1 1

22

2

Frequency Response

2nd order model:

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

qo

ut/

(ks*q

in)

ω/ωn

magnitude

0.2 0.3 0.6 0.8

-200

-150

-100

-50

0

0 0.5 1 1.5 2 2.5 3

qo

ut/

(ks*q

in)

ω/ωn

phase

Frequency Response

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

qo

ut/

(ks*q

in)

ω/ωn

magnitude

BANDWIDTH

Frequency response chart can be used to estimate

the response components of a given stimulus

Dynamic Calibration

How can I assess my transducer order

and quantify its dynamical properties? 1. Assess the frequency range of interest

2. Perform an excitation task in that range:

Studying IMPULSE RESPONSE

Studying STEP RESPONSE

Recording a DISCRETE SINE SWEEP

Recording a CONTINOUS SINE SWEEP

3. Compare with known models responses

4. Use a fitting method (such as LS)

Impulse Response:

Tells us what to expect from impulses and sudden variations of stimulus

Useful for identifying measurement systems dynamical characteristics

Useful for estimating settling time

Impulse Response

0 5 10 15-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time [s]

qo/k

sqi []

0th order

1st order

2nd order

Step Response:

Tells us what to expect from impulses and sudden variations of stimulus

Useful for identifying measurement systems dynamical characteristics

Useful for estimating settling time

Step Response

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

time [s]

qo/k

isqi []

0th order

1st order

2nd order

Exercise 9: Dynamic Calibration

95.03

1ln1

00

000

01

0

0

0

1

00001

0

iS

iS

tt

iS

iSoo

ioo

Sioo

qk

qtt

tt

qk

qeqkq

qkqq

aaqa

bqq

a

a

abkqbqaqa

Step Response (first order instrument) Instrument

dynamical

properties

eqq

qq

te

qkq

qkqqq

aa

a

aa

qa

bqq

a

aq

a

a

abkqbqaqaqa

oo

oo

n

t

iS

iSoo

nn

o

n

iooo

Siooo

n

min

max

22

20

2

20

1

20

0

0

0

1

0

2

000012

1arcsin1sin1

1

2

2

Addendum: Dynamic Calibration

Step Response (second order inst.) Instrument

dynamical

properties

Addendum: Dynamic Calibration

Step Response (second order inst.)