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Bridge Circuits & Variable Impedance Devices

• Strain Gages

• Bridge Circuits

• Variable Impedance Devices

Slide 1

Strain Gages

• Strain Gage– The change in resistance of the device is a reasonably linear

function of its deformation.

– The elastic element to which the strain gage is attached deforms in response to the physical phenomenon (input) that is of interest, for example, force, torque, pressure,

Slide 2

deforms in response to the physical phenomenon (input) that is of interest, for example, force, torque, pressure, displacement, acceleration, …etc.

Elastic Element Strain GageInput

Force; Torque;

Pressure; ...

DeformationVariable Resistance

Q: How does one measure resistance variation?

Expand resistance R into a Taylor series and ignore higher order terms:

Strain Gage

Active Axis

Pas

siv

e A

xis

R R R

RR

LL

R

AA

R

AL

L

AA

L

A

-

0

2

D

D D D D

D D D

where

Slide 3

Define Gage Factor Gf :

Resistance of wire:

Resistivity

Length of Wire

Cross Sectional Area

RL

A

L

A

:

:

:

AL

AA

A

R

R

L

L

A

A

-

-

2

2

1 2

D D D

D D D D D ( )

e e

e

GR R R

RGf f

D D D

e

e e1 2

1

Strain Gages

• Metallic Strain Gages:

– Gage Factor: Gf = 2.0 ~ 2.2

– R0 = 120 W ± 1 W .

– DR = - 2.4 W ~ 4.8 W .

• Semiconductor Strain Gage

– Silicon (Si) “doped” with phosphorus (P), arsenic (As) or boron (B).

– Gage Factor: G = 100 ~ 175

Slide 4

– DR = - 2.4 W ~ 4.8 W .

– Maximum gage current:

15 mA to 100 mA .

– Relatively low sensitivity to temperature variation.

– Gage Factor: Gf = 100 ~ 175

– R0 = 120 W ± 5 W .

– High sensitivity to temperature change.

• Measure Resistance (Impedance) Variation– Use constant voltage supply and measure current variation.

Strain Gages

R0 + DR

Vs

i

Amp Meter

iV

R R

RR

s

0

01 1

D

D

1/ i

Slide 5

– Use constant current source and measure voltage variation.

Meter

i

R

V VR

s s

01 1D

t

R0 + DRis

is

V i R R

V i R i R

o s

o s s

=

=

( )0

0

D

D

t

Vo

Vo

• Measure Resistance (Impedance) Variation– Use constant voltage supply and measure voltage variation.

Strain Gages

VR R

R R RVo s=

( )

( )0

1 0

D

D

R0 + DRVsVo

R1

Slide 6

VR

R RV

Vs

R RR R Ro s = if 0

1 0 1 00

D D

t

Vo

R0 + DRVsVo

Strain Gages

Ex: In the previous case, let Vs = 5 V, R0 = R1 = 100 W , and |DR| < 2 W,calculate the “nominal” output voltage and the maximum output voltage variation.

Slide 7

If a 10-bit ADC with an input range from 0 to 5 V is used to digitize the output voltage, how many effective bits of resolution are available to measure the variation in resistance?

Find the relationship between V and V :

Bridge Circuits

Vs

Vo

+

-

AC

Slide 8

Find the relationship between VS and VO:

• The bridge is balanced when VO = 0, i.e.,

V e eZ

Z ZV

Z

Z ZV

VZ Z Z Z

Z Z Z ZV

O A C S S

O S

-

-

-

3

2 3

4

1 4

1 3 2 4

1 4 2 3

( )( )

Z

Z

Z

Z

Z

Z

Z

Z1

4

2

3

1

2

4

3

or

Bridge Circuits

Ex: For a strain gage with R0 = 100 W and |DR| < 2 W; calculate the maximum output voltage with Vs = 5 V, if the strain gage is placed in Z1, and Z2

through Z4 are replaced by resistors with resistance R0 .

Vs

Vo

+

-

Slide 9

with resistance R0 .

If the maximum current through the strain gage is 50 mA, determine the

maximum permissible supply voltage Vs.

• One arm active

Either Z1 or Z3 is active (Z=R0+DR) and all the other arms have equal resistance (R0):

Bridge Circuits

VR R

VO S

D

D

02

Vs

Vo

+

-

Slide 10

– If the active arm is a strain gage (DR / R0 = Gf e ), then: Q: How would one improve the sensitivity

of the strain measurement?

– will making 2 or all of the bridge arms active help?

VR R R

V

V VR

RR R

O S

O S

D

DD ( )

02

0

00

4 2

1

4

V V GO S f1

4e

• Two arms active (same sign)

Both (Z1 and Z3) are active (Z=R0+DR) and all the other arms have equal resistance (R0):

Bridge Circuits

VR R R

VO S2 0

2

2 2

D D

Vs

Vo

+

-

Slide 11

– If the active arms are strain gages (DR / R0 = Gf e ), then: Q: What if the two strain gages have

opposite strain (one in tension and the other in compression)?

VR R R R

V

V VR

RR R

O S

O S

4 4

1

2

0

02

02

00

D D

DD ( )

V V GO S f1

2 e

• Two arms active (Push-Pull type)

Both (Z1 and Z4) or (Z2 and Z3) are

active (Z1=R0+DR and Z4=R0 -DR) and all the other arms have equal resistance (R0):

Bridge Circuits

Arm1 Arm 4

Z1 F

Vs

Vo

+

-

Slide 12

– If the active arms are strain gages (DR / R0 = Gf e ), then:

Q: How do temperature changes that cause additional strains affect Vo for this arrangement of the strain gages?

Arm1 Arm 4

0 0 0 0

0 0

0

0

( ) ( )

(2 )(2 )

1 ( )

2

O S

O S

R R R R R RV V

R R

RV V R R

R

D - - D

D D

V V GO S f1

2 e

Z4

• Four arms active

All four arms Z1 , Z2 , Z3 and Z4 are active (Z1 = Z3 = R0+DR and Z2 = Z4

= R0 -DR):

Bridge Circuits

Arm3Arm1 Arm2 Arm4

( ) ( ) ( ) ( )R R R R R R R R D D - - D - D

Vs

Vo

+

-

Slide 13

– If the active arms are strain gages (DR / R0 = Gf e ), then:

Q: How do temperature-induced strains affect the output voltage Vo in this configuration?

0 0 0 0

0 0

0

0

( ) ( ) ( ) ( )

(2 )(2 )

( )

O S

O S

R R R R R R R RV V

R R

RV V R R

R

D D - - D - D

D D

V V GO S f

e

• Temp. Compensation for 1 arm active

Z1 is active (Z1=R0+DRS+DRT), use a “dummy” device at Z4 such that Z4=R0+DRT and all the other arms have equal resistance (R0):

Bridge Circuits

R R R R R R R - ( ) ( )D D D

Arm1 Arm4

Vs

Vo

+

-

Slide 14

– If the active arm is a strain gage (DR / R0 = Gf e ), then:

VR R R R R R R

R R R RV

R R

R R R R RV

R R R R

OS T T

T SS

S

T SS

S T

-

( ) ( )

( )0 0 0 0

0 0

0

02

0 0

0 0

2 2 2

4 4 2

D D D

D D

D

D D

D D ( and )

V V GO S f1

4e

• Balancing the Bridge

Recall that a bridge is

balanced when the output voltage is zero for zero impedance change:

Bridge Circuits

Slide 15

impedance change:

The condition for a balanced bridge is:

VZ Z Z Z

Z Z Z ZVO S

-

1 3 2 4

1 4 2 3

0

( )( )

Z

Z

Z

Z

Z

Z

Z

Z1

4

2

3

1

2

4

3

or

• Calibration

– Balance the bridge while the switch is opened.

– Use a calibration resistor RC to simulate a change in resistance DR in arm 1 due

Bridge Circuits

V V GO S f CAL1

4 e

Rc

Slide 16

change in resistance DRCAL in arm 1 due

to an equivalent strain eCAL.

0

20 0

0 0

0

1

CALCAL CAL

f

CCAL C

C CAL

RR

G R

R R RR R R R

R R R

e D D

- D - D

Ex: To calibrate a bridge circuit, what should the calibration resistance be to simulate the output of a single active arm bridge with a strain of 0.02 in/in? The gage factor is 2.2 and R0 = 100 W.

Bridge Circuits

Rc

Slide 17

Ex: Suppose that R0 = 120 W and the gage factor is 2.1. What equivalent strain will be indicated if the calibration resistance RC = 100 kW.

Variable Impedance Devices

• Potentiometer (Displacement Sensor)

Without output loading RL , the potentiometer output voltage VTH is:

To measure V , a voltmeter with input

VxR

RV xV xTH

P

PS S

RL VOUT

RP

x

1-x

VTH

VS

+

-

Slide 18

To measure VTH , a voltmeter with input

impedance RL is used. The measured

output voltage VOUT is:

VZ

Z ZV

ZxR x R

Rx x R

VR

R x x RxV

x xxV

OUTL

L OUTTH

OUTP P

PP

OUTL

L PS R

R

SP

L

-

-

-

-

( )( )

( )( )

( )( )

11

1

1

1 1

VTH

ZOUT

ZL VOUT

OUTTH


• Potentiometer (Displacement Sensor)

Vx x

xVOUT RR

SP

L

-

1

1 1( )( )

RL VOUT

RP

x

1-x

VTH

VS

+

-OUT

Slide 19

Maximum nonlinearity occurs at:

MAX. Nonlinearity = % f.s.d

x

R

RP

L

2

3

400

27

OUTTH


• Variable Inductance Displacement Sensor

LL

d xL

L

d x1

02

0

1 1

-

a a( ) ( ) ,

For a typical device: d - x

Slide 20

d = 0.25 in

L0 = 25 mH

a = 1538.4 m-1

Q: How would one use this displacement sensor?

2d d +x

Place L1 and L2 at the 1st and 4th arms of a bridge circuit, respectively. Let the resistance in arms 2 and 3 be



LL

d xL

L

d x1

02

0

1 1

-

a a( ) ( ) ,

Vs

Vo

+

-

Slide 21

Let the resistance in arms 2 and 3 be the same:

The output voltage VO is:

Z Z Z R

Z1 2 3

4

;

1 4

1 2 1 2

1 2 1 2

1

( ) 2 2

ARM ARM

O S S

L j R R L j L LV V V

L j L j R L L

w w

w w

- -

Known Constant

1

2 1 O SV V x

d

a

a

Q: What kind of supply voltage (VS) should we use?

- Constant DC voltage, i.e. 6 V (DC)

- Oscillating voltage source?



• DC Supply Voltage (VS = constant)

The inductance is zero at DC. No inductance change will be sensed.

• AC Supply Voltage (V = V sin(w t))

Vs

Vo

+

-

Slide 22

Vd

V x

K x V t

A x t

O

K

S

C C

OUT

K V x

C

C

1

2 1

a

a

w

w

Known ConstantLet it be

Amplitude of the output

sin( )

( ) sin( )

• AC Supply Voltage (VS = VCsin(wCt))

The amplitude of the output voltage is modulated by the input signal x.



AC excited bridge circuit with variable inductance sensor:

Vd

V xO

K

S

1

2 1

a

a Known ConstantLet it be

Slide 23

( ) ( )

Let V V t

x X X t

V K x V t

K X X t V t

KV X t X t t

V KV X tX

tX

t

S C C

DC X

O C C

DC X C C

C DC C X C

O C DC C C X C X

-

sin( )

cos( )

sin( )

cos( ) sin( )

sin( ) cos( )sin( )

sin( ) sin ( ) sin ( )

w

w

w

w w

w w w

w w w w w2 2

Ex:

Let K = 2

VS = 5 sin(2p(10)t) [V]

x = 4 + 2 cos(2p(1)t) [mm]


Vd

V xO

K

S

1

2 1

a

a Known ConstantLet it be

-6

-4

-2

0

2

4

Inpu

t D

ista

nce

(um

)

Slide 24

x = 4 + 2 cos(2p(1)t) [mm]

0 0.5 1 1.5 2-60

-40

-20

0

20

40

Time (sec)

Outp

ut V

olta

ge (

V)

0 0.5 1 1.5 2-6

Time (sec)

Inpu

t D

ista

nce

(um

)

( ) ( )

V

KV X t

Xt

Xt

O

C DC C

C X C X

-

sin( )

sin ( ) sin ( )

w

w w w w2 2

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