3_Analog_SP

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3 Analog Signal Processing- Data Acquisition

3.1 Analog Measurement Procedures

3.3 Measurement of Impedances

3.2 Measurement of Resistances

Three- and Four-Wire-Method, Self-Heating

¼-Bridge, ½-Bridge, and Full-Bridge

3.3.1 AC-Current-Bridges

3.3.2 Measurement with Oscillator

P. 3-1Prof. Dr.-Ing. O. KanounChair for Measurement and Sensor Technology

3.3.4 Power Measurement Method

3.4 Measurement of Capacitances

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Why analog measurement procedures?

Most sensors have an analog output signal

No quantization-error and no conversion time

Current, voltage, resistance

Analog measurement systems can be faster!

Low-cost signal processing and realization


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Impedance

R

C

L

jX R Z

R Z

L j Z L

C j Z C

1

L j R Z Coil real L ,

C j G

jBGY

losses

lossesreal

L R Coil

00 C

G j

C

B

j r r r

C

Glosses

cabel cabel C j R R Z ||

R

cabel C

R cabel

ideal real



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4-Wires-Method


2-Wires-Method

Error by• Temperature dependence of

wires

Cable length RL

4-Wires-Method

Total separation between current-

and voltage-wires

Rx

RL

RL

U

I

UL

UL

Utarget valueRx

RL

RL

I

RL

UM

RL

I

IU≈0

I

U R M x

IU≈0


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Compensation of cable resistance and itstemperature dependence

3-Wires-Method

Prerequisit

• 3 wires have the identical length and

temperature

Rx

RL

RL

IRL

IU≈0 IRR2U xL3L

x M

L R I

U R

I R R U L x M

I R U U x M L 23

I

U U R LM x

32

3-Wires-Method

2-Wires-Method

Error by• Temperature dependence of

wires

Cable length RL

Rx

RL

RL

U

I

UL

UL

Uwahr

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Which current schould be used?



I

I

U R

Signal to noise

Ratio

ISNRIresolution limit

ILow Energy Consumption

Self-Heating

I SH

Under this limitsensor signal

disappears in

the noise

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x R I P 2

Converted electric power

P R T W th

Self-heating through accumulated heat Tth

Rw: Heat resistance is dependent on:

- Packaging

- Surrounding media, its temperature and velocity!

Self-Heating



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Comparison with a reference resistance (reference principle)

Us

Ur

Ux

Rr

Rx

r

r

x x R

U

U R

Rr and Rx should have the same order of

magnitude

Reduction of uncertainty through

common mode rejection

Typical values Rr = Rx, max

Rr : Reference resistance

Rx: Resistance to be measured



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Voltage Devider

Suitable for high resistance values

r

r

r x

U

UURR

0

0U

RR

RU

x r

r

r

R r

U 0

R x

U r



Realization with op-amps

Rx >> Rr

Ur

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P. 3-10

Supply with a constant voltage

Supply with a constant current

4 –Wires-connection and constant

current supply

Inverting Amplifier

x a R R

U U

0

0

x a R I U 0

I4=0

I3=0

x a R I U 0

Offset-currents and offset-voltages are

problematic if resistances have small

changes!

U0 =


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Wheatstone-Bridge

Bridge Voltage

Sensitivity

Proportional to supply voltage

Dependence on Rx is nonlinear!



R R R R 321

x

d R R

R U U

2

10

02U

R R

R

R

U E

x x

d

R1 R2

R3Rx

Ud

U0

0321312

3

3

21

20

U R R R R

R R R R R R

R

R R

R U U

x

x

x

d

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12P. 3-12Prof. Dr.-Ing. O. KanounProfessur für Mess- und Sensortechnik

R1 R2

R3Rx

Ud

U0

R R R x 0

x x

x d

R R R R

R R R R

R R

R

R R

R

U U

321

312

3

3

21

2

0

0

0

0

0

0

0000

0000

424U

R

R U

R R

R

U R R R R R

R R R R R U d

0

04U R

R U d

0321 R R R R

Wheatstone-Bridge


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P. 3-13

))(( 4321

41320

R R R R

R R R R U U d

[Schrüfer]

4

4

3

3

1

1

2

20

4 R

R

R

R

R

R

R

R U U d

For small R

+ R0+R

- R0-R

R0

U d near a certain working point

¼-brücke

½-brücke

Fullbridge

1/4-, 1/2- and Fullbridge


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AC-Bridge

Z2 Z1

Ud

U0

Z4 Z3

0

34

4

21

2

U Z Z

Z

Z Z

Z U d

3.3 Measurement of Impedances

2341 Z Z Z Z Balancing for :

k j

k k eZ Z

2341

2341 j j j j

eZ eZ eZ eZ

23412341

j j eZ Z eZ Z

With

Necessary conditions for Balancing:

2341 Z Z Z Z

2341

2. Modulus condition

1. Phase condition

!P. 3-16Prof. Dr.-Ing. O. Kanoun

Chair for Measurement and Sensor Technology

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Capacitance bridge by Wien(measurement of lossy capacitances)

Ud

U0

R4 R3

C2 C1

R2 R1

4

1

1

1

1

3

2

2

2

2

11 R

C j R

C j R

R

C j R

C j R

1

3

42 R

R

R R

1

4

3

2 C R

RC

R 2,C 2 are unknownt

R 1,C 1 are modifiable

Condition for balancing

3.3.1 AC-Current-Bridge


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Inductivity-Bridge by Maxwell(Measurement of lossy Inductivities)

Inductivity-Bridge by Maxwell-Wien(Measurement of lossy Inductivities)

L2, R2 are unknown

L2, R2 are unknown



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U d

U dU d



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Checking the phase condition

03

04

0

02

03

04

01

02

03

04

01

02

04

023

041

023

041

23

Balancing is

possible

Balancing is

not possible

Balancing is

possible

beliebig

The judgement of the modulus condition is in the special case necessary!



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Frequency dependent

Balancing


Frequency dependence of the bridge balancing

2341 Z Z Z Z

111 L j R Z

22 R Z

33 R Z

444

1

C j R Z

234

4111

R R C

j R L j R

234

141 R R

C LR R

414

1 R LC

R


3 3 1 AC C B id

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2341 Z Z Z Z

Frequency independent balancing!


Frequency dependence of the bridge balancing

111 L j R Z

22 R Z

33 R Z

44

4 11

C j R

Z

4123 R R R R

32

14

R R

LC


3 3 2 Measurement with Oscillators

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G

C L j R X j R Z

1

C L

r r

1

Resonance frequency:

LC f r

1

2

1

L or C measurement

L

C

R

~~

Example

f = constant

L= constant

C

U R

C 0C 0-C C 0+C

3.3.2 Measurement with Oscillators


3 3 2 Measurement with Oscillators

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24

Parallel resonant circuit

series resonant circuit

Serie resonant circuit

Resonance: maximum currentminimum impedance

The resonance frequency: =1

2 =

1

4

3.3.2 Measurement with Oscillators


Parallel resonant circuit

Resonance: minimum current

maximum impedance

3 3 3 Power Measurement Method

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0 R

uu pw

2

ˆˆ10

Measurement of active power

R

)sin(ˆ)cos(ˆ)sin(ˆ

100 t ut C ut R u p

t uu sinˆ00

R x i i i

t uu sinˆ11 selectable

0

)sin()cos(ˆˆ)(sinˆˆ

10

201 t t C uut R

uu

)2sin(2

ˆˆ

)2cos(2

ˆˆ

2

ˆˆ100101

t C

uu

t R

uu

R

uu

y x y x y x sinsin21

)cos()sin(

)2cos(12

1)(sin2 x x

Case 1:

R

uudt t p

T p

T

w2

ˆˆ110

0

Low-passMultiplication


3 3 3 Power Measurement Method

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90 X

Measurement of the idle power

2

ˆˆ10uuC pB

t uu sinˆ00

R x i i i

t uu sinˆ11 selectable

90 Case 2:



)cos(ˆ)cos(ˆ)sin(ˆ

100 t ut C ut R u p

)(cosˆˆ)cos()sin(ˆˆ 2

1001 t C uut t

R

uu

)2cos(12

ˆˆ

)2sin(2

ˆˆ1001

t C

uu

t R

uu

y x y x y x sinsin2

1)cos()sin(

)2cos(121)(sin2 x x

C uu

dt t pT

pT

w 2

ˆˆ1 10

0

)2cos(12

1)(cos2 x x

Low-passMultiplication

3 4 Capacitance Measurement

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Practical capacitance has:

• Losses

• Stray field outside the electrodes

• Further stray capacitances to the environment:

Between the electrodes and the shielding screen

Between the electrodes and the measurement circuit

Cp1, Cp2: Stray capacitanceson circuit side

Cs1, Cs2: Stray capacitances

on sensor side

3.4 Capacitance Measurement


Practical aspects

Measured Capacitance


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P. 3-28


Prof. Dr.-Ing. O. KanounChair for Measurement and Sensor Technology

S1: On , S2: Off Charging

S1: Off , S2: On Discharging

= ( 1 −

) =

× (1

)

During the charging:

During the discharging:

= × −

=

× (

)

The measurement is affected by :

• Stray capacitance between the capacitance electrodes and earth

• Stray capacitance caused by the switches (S1, S2)

Charging/Discharging Method


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C x

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Up

U0

i

r x p

C

C C U U

0


Voltages F1 und F2 are switched together with the relai “Control”

During F1 is switched on U p :

- Relay “Control” is closed

- The (-)-Input of the OPV is on ground

- The capacitance C x is loaded

During F2 is switched on U p :

- The Relay “Control” is opened

- It flows a current from C x over C i

3.4 Capacitance MeasurementCharge-Transfer Method (2)


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Reference Method

V0 = ( +

´ + (

´ ))

Phase 1: Charging S1: On , S2: Off

Phase 2: Discharging S1: Off , S2: On

Assumptions: Stay capacitances are symmetrical

Quality of reference capacitance

Reference

CapacitanceMeasured

Capacitance

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