Instrumentation and Measurements Dr. Mohammad Kilani Analog Electrical Devices and Voltage Dividing Circuits

Instrumentation and Measurements

Dr. Mohammad Kilani

Analog Electrical Devices and Voltage Dividing Circuits

Electric Signal in Measuring Instruments

Measured Variable

Sensor

Electric signal measurement and conditioning. Common components in a large variety of

Instruments

Variable Conversion

ElementSignal Processor

Use of Measurement at Remote

Location

Signal Transmission

Presentation / Recording

Unit

Output

Advantages of Electrical signal forms over Non-Electrical Forms

Many detecting elements provide an output in the form of varying resistance, capacitance or inductance, voltage

or current. Compared to mechanical, pneumatic or hydraulic forms, electric signal forms provide the following

advantages::

Fast transient response. Inertia effect are negligible compared with hydraulic and pneumatic systems.

Light weight and low power consumption in the measuring instrument

Possibility of obtaining large amplification for the signal, with good linearity.

Lower hysteresis effect. Hysteresis is present in mechanical devices due to friction, backlash, and

windup effects.

Availability a multitude of proven analog and digital signal conditioning circuits for filtering,

compensation, error-rejection, recording and other signal processing operations.

Galvanometer Instruments

Galvanometer Instruments

A galvanometer is an analog device that produces a

deflection proportional to the current in passing in its coil. It

utilizes the principle that a current-carrying conductor

placed in a magnetic field is acted upon by a force that is

proportional to the current passing through the conductor

This force can be used as a measure of the flow of current in

a conductor by moving a pointer on a display.

Galvanometer Instruments:Force on a straight current conductor placed on magnetic field

Consider a straight length l of a conductor through

which a current I flows. The force on this conductor

due to a magnetic field strength B is:

The vector k is a unit vector along the direction of the

current flow, and the force F and the magnetic field B

are also vector quantities.

BIlF

k

Galvanometer Instruments: Force on a Current Loop in a Magnetic Field

A current loop in a magnetic field experiences a torque if the loop is not aligned

with the magnetic field. The torque is given by

where

N Number of turns in the loop

I: current

A: cross-sectional area defined by the perimeter of the current loop

B: Magnitude of magnetic field strength

α angle between the normal cross-sectional area of the current loop and

the magnetic field

sinNIABT

Galvanometer Instruments: Analog DC Current Measurement

One approach to utilizing the previous equation in current

measurement is in the D’Arsonval arrangement. In this

arrangement, the uniform radial magnetic field and torsional

spring result in a steady angular deflection of the coil that

corresponds to the coil current.

The coil and fixed permanent magnet are arranged in the

normal direction to the current loop. The problem with this

arrangement is that the deflection is nonlinear with the

current. (torque proportional to sin α) The torque becomes

zero when α = 0 irrespective of the amount of current in the

coil.


To produce a deflection proportional to the current, the coil is usually wrapped

around cylindrical iron core with the two poles form circular arcs around the core.

The presence of the iron core causes the magnetic field lines to direct themselves

radially toward the center of the core.

As the field lines act in the radial direction, they always act normal to the inwards

and outwards current paths irrespective over the range of motion. The angle α is

always 90⁰ and

where

r radius of the cylindrical iron core

l: length of the cylindrical coreNIrlBT 2


In a typical ammeter the deflection of the pointer

indicates the magnitude of the current flow. The

range of current that can be measured is

determined by selection of the combination of

shunt resistor and the internal resistance of the

meter movement.

The shunt resistor provides a bypass for current

flow, reducing the current that flows through the

movement. A break switch prevents current

overload through the meter coil.


A galvanometer is a highly sensitive

D’Arsonval movement device calibrated

about zero current; the indicator can deflect

to the plus or minus direction.

The pointer can be used to adjust a circuit to

a zero current state. This mode of operation

is called the null mode. The highest

sensitivity of commercial galvanometers is

about 0.1 µA/division.

Galvanometer Instruments: Analog DC Voltage and Resistance Measurement

DC voltage can be measured through a D’Arsonval

configuration placed in series with a resistor. The

arrangement is basically a current measuring device,

but can be calibrated in terms of voltage by using an

appropriate known fixed resistor and through Ohm’s

law relating it to the measured current.

The same device can be used to measure resistance if

instead if an appropriate fixed known DC voltage is

applied at the input. Pointer movement can, in this

case be calibrated to reflect resistance.


Practical analog ohmmeters use circuits employing a

D’Arsonval mechanism with shunt resistors for measuring

a wide range of resistance while limiting the flow of

current through the meter movement.

The lower limit on the measured resistance is

determined by the upper limit of current flow through it.

A practical limit to the maximum current flow through a

resistance is imposed by the ability of the resistance

element to dissipate the power generated by the flow of

current (Joule heating). At a too high current, the sensor

resistance may melt.


This basic circuit is employed in the construction of analog voltage dials and

volt-ohmmeters (VOMs), which were commonly used for the measurement of

current, voltage, and resistance.

Galvanometer Instruments: Analog AC Current and Voltage Measurement

Deflection meters can be employed for measuring AC current by the

uses of diodes to form a rectifier that converts the AC current into a

DC current. This current then can be measured with a calibrated

D’Arsonval movement meter.

An electrodynamometer is basically a D’Arsonval movement modified

for use with AC current by replacing the permanent magnet with an

electromagnet in series with the current coil. These AC meters have

upper limits on the frequency of the alternating current that they can

effectively measure; most common instruments are calibrated for use

with standard line frequency.

Galvanometer Instruments: Analog AC Current and Voltage Measurement

The waveform output of the rectifier must be

considered if a steady meter reading is to be

obtained.

An AC meter indicates a true rms value for a

simple periodic signal only, but a true rms AC

voltmeter performs the signal integration

required to accurately determine the rms

value in a signal-conditioning stage and

indicates true signal rms regardless of

waveform

Galvanometer Instruments: Errors in D’Arsonval Movement

Errors in the D’Arsonval movement include hysteresis and

repeatability errors due to mechanical friction in the pointer-

bearing movement, and linearity errors in the spring that

provides the restoring force for equilibrium.

Also, in developing a torque, the D’Arsonval movement must

extract energy from the current flowing through it. This

draining of energy from the signal being measured changes the

measured signal. Such an effect is called a loading error.

This is a consequence of all instruments that operate in

deflection mode.

Dynamic Response of a Galvanometer

A galvanometer consists of N turns of a conductor wound about a

core of length l and radius r that is situated perpendicular to a

magnetic field of uniform flux density B. A DC current passes

through the conductor due to an applied potential, Ei(t). The

output of the device is the rotation of the core and pointer, u.

Develop a lumped parameter model relating pointer rotation and

current.


Applying Newton’s 2nd law on the iron core we have

The rotation of the current carrying coil in the magnetic field generates an opposing

electromotive force Em

, given by:

Applying Kirchhoff’s law to the resulting circuit gives,

NABITkdt

dc

dt

dJ

dt

dJk

dt

dcT

dt

dJT

2

2

2

2

2

2

dt

dNABEm

dt

dNABEEEIR

dt

dIL imigg


The equations are coupled. The current due to potential E produces torque T that moves the pointer.

This motion develops an opposing potential Em

.

igg EIRdt

dIL

dt

dNAB

NABIkdt

dc

dt

dJ

2

2


Using Laplace

igg

igg

igg

ERsLNABs

NABkcsJs

ERsLNABs

NABkcsJs

ERsLNABs

NABkcsJs

0

0

0

12

2

2

i

gg

gggggg EkcsJsNABs

NABRsL

kRsNABcRkLsJRcLsJL

012223


Using Laplace

i

gg

gggggg EkcsJsNrlBs

NrlBRsL

kRsNrlBcRkLsJRcLsJL

0

2

2

2

12223


HW 2:

Analyze the dynamic response of a galvanometer to step input and harmonic input. Discuss the effect of system parameters on the stability, settling

time, overshoot and other pertinent dynamic characteristics.

Use system analysis tools including:

1. Routh’s stability criterion

2. Root-locus plots

3. Bode diagrams

4. Polar plots

5. Log-Magnitude-vs.-Phase plots

6. Nyquist stability analysis

Verify the results of your analysis with computer simulations (e.g. simulink), or video recorded experiments.

The Oscilloscope

The Oscilloscope

The oscilloscope is a practical graphical display device which provides an

analog representation of a measured signal. It is used to measure and to

visually display voltage magnitude versus time for dynamic signals over a

wide range of frequencies with a signal frequency extending into the

megahertz or gigahertz range.

The oscilloscope provides a visual output of signal magnitude, frequency,

distortion, and a delineation of the DC and AC components. The visual

image provides a direct means to detect the superposition of noise and

interference on a measured signal, something non visual metering devices

cannot do.

The Oscilloscope

In addition to signal versus time, a typical unit can

also display two or more signals X(t) and Y(t), perform

addition and subtraction of signals, and display

amplitude versus amplitude (XY) plots and other

features.

Some digital oscilloscopes have significant internal

storage so as to mimic a data-logging signal recorder.

Others have internal fast Fourier transform (FFT)

circuitry to provide for direct spectral analysis of a

signal.

The Oscilloscope

In the cathode ray oscilloscope, the cathode tube operates as a time-based voltage transducer. A beam of electrons emitted by a

heated cathode are guided vertically and horizontally by pairs of plates to control the location of the impact of the electron beam

on the screen

The output of the oscilloscope is a signal trace created by the impact of the electrons on a phosphorescent coating on the screen.

The Oscilloscope

Input voltages result in vertical deflections of the beam, and produce a trace of the voltage variations versus time (horizontal or x axis) on

the screen.

The vertical plates are excited by the electrical field to be measured. The horizontal sweep frequency can be varied over a wide range,

and the operation of the oscilloscope is such that high-frequency waveforms can be resolved.

Resistance Measurement Circuits

Vs

Rm

G

kRt

0≤k ≤1

The Simple Current-Sensitive Circuit

Many sensing element produce their output in the form of varying

resistance. The element may be a sliding contact, a thermistor, a

piezoresistive element or other, where the resistance varies between zero

and a maximum value, Rt.

The simple current-sensitive circuit may be used to determine the

resistance of a varying resistance element through measurement of

current magnitude.

Rm

represents the resistance of the remaining parts of the circuit,

including the meter resistance and the internal resistance of the voltage

source. The current is related to kRt through Ohm’s law.

mt

so RkR

Vi

Vs

Rm

G

kRt

0≤k ≤1

io

The Simple Current-Sensitive Circuit

The maximum current flow, imax

= Vs/R

m takes place when the resistance of the

sensing element is zero. The current ratio, io

/ imax

may be written as:

The output (current ratio) is nonlinear with the input (resistance change). The

sensitivity is small for small values of Rt

/Rm

. Greater sensitivity may be obtained

for larger values of Rt

/Rm

but it drops rapidly as the resistance approaches its upper

limit (k approaches 1).

Careful control of the driving voltage is also necessary if calibration relationship

between current and resistance is to be maintained.

mts

m

mt

so

RRkV

R

RkR

V

i

i

1

1

max

Vs

Rm

G

kRt

0≤k ≤1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1R

t / R

m = 0.5

1

2

4

10

k

i o/im

ax

Voltage Dividing Circuits

Vs

R1

R2

Vo

The Resistive Voltage Divider

A resistive voltage divider is obtained from a set

of resistors connected in series to divide a

voltage source.

The voltage across each resistor may be

calculated by multiplying the value of the current

flowing through the circuit by the resistance of

the resistor considered.

21

2

2122 RR

RV

RR

VRIRV s

so

Vs

R1

R2

Vo

The Ballast Circuit

The ballast circuit is simply a voltage divider with one of

the resistors representing the resistive sensing element.

The voltage across that resistor changes as its resistance

varies with the input to be measured. A voltage sensitive

device is placed across that resistance which can be

calibrated to indicated the input of interest.

Assuming a high-impedance meter, the relationship

between the output ratio (Vo

/Vs) to the input ratio

(R/Rt) can be found from Ohm’s law

Vs

Rb

Rt

Vo

Voltage Indicator

bt

bt

tb

t

s

o

tb

tsto

tb

s

RRk

RRk

kRR

kR

V

V

kRR

kRVkRiV

kRR

VI

1

The Ballast Circuit

The relationship between the output ratio (Vo

/Vs) and

the input ratio (R/Rt) is nonlinear and a percentage

variation in the supply voltage results in a greater

variation in the output than does a similar percentage

change in k. Voltage regulation must be strictly

controlled for successful operation.

Vs

Rb

Rt

Vo

Voltage Indicator

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

k

Vo

/Vs

Rt/R

b = 4

2

1

0.5 bt

bt

s

o

RRk

RRk

V

V

1

The Sliding Contact Resistive Voltage Divider

A sliding contact voltage divider is a resistive voltage divider whereby the

resistance ratio is adjusted with a movable contact.

A voltage Vs is applied across the two ends A and B of a resistance element ,

RAB

, and the output voltage Vo

is measured between the point of contact C

of the sliding element and the end A of the resistance element.

The output voltage from the divider is controlled by the position of the slider,

taking the range 0 ≤ Vo

≤ Vs. as the resistance ratio is adjusted from zero to

1 with a movable contact. sABsACACo V

L

xRVRIRV

A

Vs

x

L

C B

Vo

so VL

xV

The Null Type Resistive Potentiometer

The null type potentiometer uses the sliding contact voltage divider to

measure DC voltage. It balances an unknown input voltage against a known

internal voltage until both sides are equal. The voltage to be measured, Vm

,

is applied at the divider’s output, and a current-sensing device, such as a

galvanometer, is used to detect current flow on the output branch.

The output voltage of the divider VAC

is adjusted by moving the sliding

contact A; any current flow through the galvanometer, G, would be a result of

an imbalance between the measured voltage, Vm

, and VAC

. The value of Vm

is determined from knowledge of VAC

.

A

Vs

x

L

C B

Vm

G

sACm VL

xVV

The Null Type Resistive Potentiometer

A null balance, corresponding to zero current flow through G, occurs

only when Vm

= VAC

.

With a known and constant supply voltage Vs , V

AC is related to the

position of the slider, which can be calibrated to indicate Vm

.

Potentiometers have been supplanted by digital voltmeters, which are

deflection mode devices but have very high input impedances so as to

keep loading errors small, even at low-level voltages.

A

Vs

x

L

C B

Vm

G

sACm VL

xVV

The Deflection Type Resistive Potentiometer

The resistive potentiometer is commonly used as a

displacement-measuring device.

Assuming high galvanometer impendence, a linear

relationship exists between the output voltage Vo

and

the distance x, for uniform resistivity for RAB

Displacement (x)Resistance change (ΔR)

Voltage change (Δv)

A

Vs

x

L

C B

Vo

G

ABACABAC

so

ssABACo

ABsACACo

kRRRRk

kVV

VLxVRRV

RVRIRV

,

The Resistive Potentiometer: Loading ErrorLoading error occurs when the resistance R

L is finite. It can be estimated by calculating

the voltage across RL for finite values of R

L/R

AB . Noting that R

AC = kR

AB and R

BC = (1-

k)RAB

The total resistance seen by the source Vs is:

The current through RBC

is thus

The voltage across RL is

A

Vs

x

L

C B

RL

G

LAB

LABABLACBC RkR

RkRkRRRRR

1//

LABAB

LABsR

V

RRkkR

RkRVi s

12

21

1

kRRkRR

k

V

V

kiRViRVVVV

LABLABs

o

ABsBCsBCso

The Resistive Potentiometer: Loading ErrorComparing the values of V

o for finite and infinite values of R

L/R

AB , the error in V

o for a finite

RL/R

AB is:

The percentage error is thus:

A

Vs

x

L

C B

RL

G

ABLs

LABs

actualoidealo

RRkk

kkVError

RRkk

kkVError

VVError

1

1

112

ABL

idealo

actualoidealo

RRkk

kkError

V

VVError

1

1%

%100%

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

k

Err

or

%

RL

/RAB

= 1

2

5

10

0 0.2 0.4 0.6 0.8 10

5

10

15

k

Err

or

%

The Resistive Potentiometer: Loading ErrorThe error ratio may alternatively be defined as the ratio of the error to the full scale output,

Vs, in this case,

A

Vs

x

L

C B

RL

G

ABL

s

actualoidealo

RRkk

kkError

V

VVError

1

1%

%100%

2

RL

/RAB

= 1

2

5

10

Direct Voltage Dividing Circuits for Measurement of Variable Resistance

A voltage divider may be used to measure the small changes in

resistance from variable resistance detectors such as RTDs, strain

gauges and photoresistive elements.

These detectors show only a small change in their resistance in

response to the detected input. For example, the resistance of metal

strain gauges may vary only be 1 x 10-4 in use. The smallness of

resistance change has important consequences on the performance of

the voltage dividing circuit.

Vs

R1

Vo

R0

Direct Voltage Dividing Circuits for Measurement of Variable Resistance : Sensitivity

Suppose that a voltage divider is formed from a detector of an initial

resistance R0

and a second resistance with R = R0

. The initial output is:

Assuming no change in the supply voltage during measurement, dVs = 0,

0210

1

10

0

0210

0

1010

0

10

0

1

dRRR

RVdV

RR

RdV

dRRR

R

RRVdV

RR

RdV

VRR

RV

sso

sso

so

Vo

→ Vo

+ Δ Vo

Vs

R1

R0

→ R0

+ ΔR

0201

01

0

0201

201

0210

1

1

1

dRRR

RR

R

VdV

dRRR

RRVdV

dRRR

RVdV

so

so

so

oso

so

sR

Vs

dR

dV

RR

RR

R

V

dR

dV

00

201

01

00 1


The sensitivity of the potentiometer increases with Vs , s

o = r/(1+r2) and with

1/Ro

. Practical considerations place an upper limit on Vs . R

o can not be made

too small as δRo

is related to the value of Ro

. The ratio r = R1

/ Ro

, can be

chosen to maximize so

= r/(1+r2)

Vo

→ Vo

+ Δ Vo

Vs

R1

R0

→ R0

+ ΔR

1,12

1

2

1

1

01

2

1

1

1

33

32

2

rrr

r

r

r

r

r

r

rdr

ds

r

rs

o

o

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

01 RRr

21 r

rso

oss

so

sR

V

r

r

R

Vs

RR

RR

R

V

dR

dVs

02

0

201

01

00

1

1


The sensitivity coefficient so

is maximum when r = R1

/ Ro

= 1. The maximum value of so

occurs when r = R1

/ Ro

= 1,

and its value is 1/4. Thus R0

is usually chosen to be equal

to R1

, and the sensitivity is

Vo

→ Vo

+ Δ Vo

Vs

R1

R0

→ R0

+ ΔR

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

01 RRr

21 r

rso

00

00

4

4

RR

VV

R

V

dR

dV

so

so

Example: Strain Gauge in Ballast Circuit Configuration

A strain gauge with a nominal resistance R0

= 120 Ω is

used in a ballast circuit designed to produce maximum

sensitivity. The maximum expected change in the

resistance of the gauge when maximum load is applied

is ΔR = 240 x 10-6 Ω .

What is the percentage change in the

measured voltage when maximum load is

applied.

What is the percentage change in the

measured voltage due to a 0.1 % drift in the

supply voltage Vs

Vo

→ Vo

+ Δ Vo

Vs

R0

R0

→ R0

+ ΔR


Solution: percentage change in the measured voltage when maximum load is

applied. For a ballast circuit with maximum sensitivity, we have

Vo

→ Vo

+ Δ Vo

Vs

R0

R0

→ R0

+ ΔR

%0001.010

2

2

4

4

4

6

0

0

00

00

o

o

o

o

s

s

o

o

so

so

V

V

RRV

V

V

VRR

V

V

RR

VV

R

V

dR

dV

A meter with a resolution of better than one part in a millionth is needed in order to see any change in

Vo

. This excludes most common voltmeters, which may resolve to only 0.01%, (10-4)

R0

= 120 Ω

ΔR = 240 x 10-6 Ω

Maximum sensitivity configuration

Δ Vo

/ Vo

?


Solution: percentage change in the measured voltage due to a 0.1 %

drift in the supply voltage Vs

The percentage change in the measured voltage due to a 0.1 % the

drift in Vs is 0.1 % which is a 1000 times greater than the strain

induced change in voltage, 0.0001%

Vo

→ Vo

+ Δ Vo

Vs

R0

R0

→ R0

+ ΔR

R0

= 120 Ω

ΔR = 240 x 10-6 Ω

Maximum sensitivity configuration

Δ Vo

/ Vo

?

s

s

o

o

ssso

V

dV

V

dV

dVdVR

RdV

RR

RdV

2

1

2 10

0

10

0

Voltage Dividing Circuits for Measurements of Variable Resistance Detectors

A solution to the voltage divider problems may be obtained by a circuit having an output

voltage proportional to ΔVo

itself, without the large offset voltage, Vo

. This can be done

by introducing another voltage divider with fixed resistors Ro, which has a midpoint

voltage Vo

. We now measure the difference between the midpoint voltages of the two

dividers as the output voltage of the circuit

This eliminates the problem of caused by the offset voltage Vo

.

The arrangement of two voltage dividers is in fact identical to the Wheatstone bridge

circuit to be discussed next.

A

so

ooooABout VR

RVVVVVV

4

B

Vs

R0

R0

R0

R0

Vout

Resistance Bridges

Bridges offer an attractive alternative for measuring small resistance

changes accurately. The basic Wheatstone bridge consists of four

resistors connected to form a quadrilateral, a source of excitation

(voltage or current) connected across one of the diagonals, and a

voltage detector connected across the other diagonal. The detector

measures the difference between the outputs of two voltage dividers

connected across the excitation.

So

sABo

sBsA

VRRRR

RRRRV

RR

R

RR

RVVV

RR

RVV

RR

RVV

3241

3241

32

2

41

1

32

2

41

1

11

,

At balance, Vo

= 0

Vs

R3

R4

R1 R

2

Vo

+ -

3241 RRRR

VB

VA

B

VS

R2

R3

R4

R1

AV

o

Resistance Bridges: Null Type Measurement

A bridge measures resistance indirectly by comparison with a similar

resistance. The two principle ways of operating a bridge are as a null

detector or as a device that reads a difference directly as voltage.

When R1/R4 = R2/R3, the resistance bridge is at a null, irrespective of the

mode of excitation (current or voltage, AC or DC), the magnitude of

excitation, the mode of readout (current or voltage), or the impedance of

the detector.

Therefore, if the ratio of R2/R3 is fixed at K, a null is achieved when R1 =

K·R4. If R1 is unknown and R4 is an accurately determined variable

resistance, the magnitude of R1 can be found by adjusting R4 until null is

achieved.

B

VS

R2

R3

R4

R1

AV

o

At balance, Vo

= 0

Vs

R3

R4

R1 R

2

Vo

+ -

3241 RRRR

VB

VA

Resistance Bridges: Null Type Measurement

Similarly, in sensor-type measurements, R4 may be a fixed reference,

and a null occurs when the magnitude of the external variable (strain,

temperature, etc.) is such that R1 = K·R4.

Null measurements are principally used in feedback systems

involving electromechanical and/or human elements. Such systems

seek to force the active element (strain gage, RTD, thermistor, etc.) to

balance the bridge by influencing the parameter being measured.

3241 RRRR

B

VS

R2

R3

R4

R1

AV

o

At balance, Vo

= 0

Vs

R3

R4

R1 R

2

Vo

+ -

3241 RRRR

VB

VA

Application: Null Type Measurement in Feedback SystemsExample 1: Room Temperature Controller

At balance, Vo

= 0

3241 RRRR

Vs

R3

R4

RTD

R2

Vo

+-

Heater

Temperature Controlled Room

R1

The resistor R4 in this system represents the reference input which may be adjusted by the user.

Zero output voltage occurs when the magnitude of the external variable (strain, temperature,

etc.) is such that R1 = (R2/R3)·R4.

The relationship between the bridge output voltage and ΔR is not linear. The system is good only

in ‘heating’ mode, and may become unstable if the room temperature becomes above the set

temperature.

Set temperature through R4

Room temperature through RTD

Bridge Output

Voltage

Heater Room

RTD

User adjustable

resistance

Application: Null Type Measurement in Feedback SystemsExample 2: Position Servo

At balance, Vo

= 0

3241 RRRR

VB

R3

R4

Magnet

R2

Vo

+-

R1

The resistor R4 in this system represents the reference input which may be adjusted by the user.

Zero output voltage occurs when the magnitude of the external variable (strain, temperature,

etc.) is such that R1 = (R2/R3)·R4.

The relationship between the bridge output voltage and ΔR is not linear. The system is good only

in ‘heating’ mode, and may become unstable if the room temperature becomes above the set

temperature.

Set Position through R4

Current Position through R1

Bridge Output

Voltage

Solenoid Magnet

R1

User adjustable resistance

Solenoid Coil

Resistance Bridges: Deflection Type Measurement

For the majority of sensor applications employing bridges,

the deviation of one or more resistors in a bridge from an

initial value is measured as an indication of the magnitude

(or a change) in the measured variable. In this case, the

output voltage change is an indication of the resistance

change.

Because very small resistance changes are common, the

output voltage change may be as small as tens of

millivolts, even with VB = 10V (a typical excitation voltage

for a load cell application).

B

VB

R2

R3

R4

R1

AV

o

At balance, Vo

= 0

VB

R3

R4

R1 R

2

Vo

+ -

3241 RRRR

Deflection Type Resistance Bridges: Variable Resistance Configurations

In many bridge applications, there may be two, or even four elements which vary.

Note that since the bridge output is directly proportional to VB, the measurement accuracy

can be no better than that of the accuracy of the excitation voltage.

VB

R3

R4

R1 R

2

Vo

+ -

General Bridge

32

2

41

1

RR

R

RR

RVV Bo

VB

RR

RR + ΔR

Vo

+ -

24 RR

RVV Bo

VB

R+ΔRR

RR + ΔR

Vo

+ -

22 RR

RVV Bo

Single Varying Element Two Co-varying Elements

VB

R

RR + ΔR

Vo

+ -

R - ΔR

R

RVV Bo 2

Two Anti-varying Elements

VB

R + ΔR

Vo

+ -

R

RVV Bo 2

Four Varying Elements

R - ΔR

R - ΔR

R + ΔR

Deflection Type Resistance Bridges: Variable Resistance Configurations

In each case, the value of the fixed bridge resistor, R, is chosen to be equal to the nominal

value of the variable resistor(s). The deviation of the variable resistor(s) about the nominal

value is proportional to the quantity being measured, such as strain (in the case of a strain

gage) or temperature ( in the case of an RTD).

VB

R3

R4

R1 R

2

Vo

+ -

General Bridge

32

2

41

1

RR

R

RR

RVV Bo

VB

RR

RR + ΔR

Vo

+ -

24 RR

RVV Bo

Single Varying Element

VB

R+ΔRR

RR + ΔR

Vo

+ -

22 RR

RVV Bo

Two Co-varying Elements

VB

R

RR + ΔR

Vo

+ -

R - ΔR

R

RVV Bo 2


VB

R + ΔR

Vo

+ -

R

RVV Bo


R - ΔR

R - ΔR

R + ΔR

Deflection Type Resistance Bridges: Single Varying Element Configurations

The single-element varying bridge may be used for temperature sensing using

RTDs or thermistors. This configuration is also used with a single resistive strain

gage. All the resistances are nominally equal, but one of them (the sensor) is

variable by an amount ΔR.

The relationship between the bridge output and ΔR is not linear. Since there is a

fixed relationship between the bridge resistance change and its output,

software can be used to remove the linearity error in digital systems.

Alternative bridge configurations can also be used to linearize the bridge output

directly.

VB

RR

RR + ΔR

Vo

+ -

24 RR

RVV Bo

Single Varying Element

Deflection Type Resistance Bridges: Two Co-varying Elements Configurations

In this configuration, both elements change in the

same direction. The nonlinearity is the same as that of

the single-element varying bridge, however the gain is

twice that of the single-element varying bridge.

The two-element varying bridge is commonly found in

pressure sensors and flow meter systems.

VB

R+ΔRR

RR + ΔR

Vo

+ -

22 RR

RVV Bo

Two Co-varying Elements

Deflection Type Resistance Bridges: Two Anti-varying Elements Configurations

This configuration requires two identical elements that vary in

opposite directions. This could correspond to two identical strain

gages: one mounted on top of a flexing surface, and one on the

bottom.

Such a configuration could be used for measuring force, pressure,

stress, strain, etc. It produces a linear is output, and it has twice the

gain of the single-element configuration.

VB

R

RR + ΔR

Vo

+ -

R - ΔR

R

RVV Bo 2


R - ΔR

R + ΔR

F

Deflection Type Resistance Bridges: Four Varying Elements Configurations

The four varying elements bridge produces

the most signal for a given resistance change

and is inherently linear.

It is an industry-standard configuration for

load cells which are constructed from four

identical strain gages.

VB

R + ΔR

Vo

+ -

R

RVV Bo


R - ΔR

R - ΔR

R + ΔR

Case Study II:Diaphragms Pressure Sensors using Strain Gauge

The diaphragm, is an elastic element pressure transducer used for low pressure measurement (up to 2000 bar absolute pressure). A

diaphragm can also be used to measure differential pressure (up to 2.5 bar) by applying the two pressures to the two sides of the diaphragm.

A diaphragm pressure sensors consists of a thin membrane (the diaphragm) attached to the pressure measurement chamber. Applied

pressure causes displacement of the diaphragm and this movement is measured by a displacement transducer.


The diaphragm can be either plastic, metal alloy, stainless steel or ceramic.

Plastic diaphragms are cheapest, but metal diaphragms give better accuracy.

Stainless steel is normally used in high temperature or corrosive environments.

Ceramic diaphragms are resistant even to strong acids and alkalis, and are used

when the operating environment is particularly harsh.

The typical magnitude of diaphragm displacement is 0.1 mm, which is well

suited to a strain-gauge type of displacement-measuring transducer, although

other forms of displacement measurement are also used in some kinds of

diaphragm-based sensors.


If the displacement is measured with strain

gauges, it is normal to use four strain

gauges arranged in a bridge circuit

configuration. The output voltage from the

bridge is a function of the resistance

change due to the strain in the diaphragm.

This arrangement automatically provides

compensation for environmental

temperature changes.

42

2

31

1

RR

R

RR

RVV bridgeout

//

//2

out

bridge

R R R RV

V R R R R


Older diaphragm pressure transducers used metallic strain gauges bonded to a diaphragm typically made of stainless steel. Apart from

manufacturing difficulties arising from bonding the gauges, metallic strain gauges have a low gauge factor, which means that the low output

from the strain gauge bridge has to be amplified by an expensive d.c. amplifier.

The development of semiconductor (piezoresistive) strain gauges provided a solution to the low-output problem, as they have gauge factors

up to one hundred times greater than metallic gauges. However, the difficulty of bonding gauges to the diaphragm remained and a new

problem emerged regarding the highly non-linear characteristic of the strain–output relationship.

42

2

31

1

RR

R

RR

RVV bridgeout

//

//2

out

bridge

R R R RV

V R R R R


The problem of strain-gauge bonding was solved with the emergence of monolithic piezoresistive pressure transducers. These have a typical

measurement uncertainty of ±0.5% and are now the most commonly used type of diaphragm pressure transducer.

The monolithic cell consists of a diaphragm made of a silicon sheet into which resistors are diffused during the manufacturing process. Such

pressure transducers can be made to be very small and are often known as micro-sensors.

42

2

31

1

RR

R

RR

RVV bridgeout

//

//2

out

bridge

R R R RV

V R R R R


Silicon measuring cells have the advantage of being very cheap to

manufacture in large quantities. Non-linear characteristic is normally

overcome by processing the output signal with an active linearization circuit

or incorporating the cell into a microprocessor based intelligent measuring

transducer. Such instruments can also offer automatic temperature

compensation, built-in diagnostics and simple calibration procedures. These

features allow measurement inaccuracy to be reduced to a figure as low as

±0.1% of full-scale reading.

By varying the diameter and thickness of the silicon diaphragms, silicon

diaphragam sensors in the range of 0 to 2000 bar have been made.

Example: Errors Produced by Wiring Resistancein Remote Resistive Bridge Sensor

A 350Ω nominal resistance strain gage is used as a sensing element in a single varying element Wheatstone bridge. The gauge is connected to the rest

of the bridge circuit by 100 feet of 30 gage twisted pair copper wire with a resistance of 0.105 Ω /ft, at 25°C and a temperature coefficient of 0.385%/°C.

The full scale variation of the strain gage resistance (with flex) above its nominal 350 Ω value is +1%. Calculate the offset error in the bridge output due

to the lead resistance.

Errors Produced by Wiring Resistancein Remote Resistive Bridge Sensor

Offset error due to lead resistance

The resistance of each copper cable is 10.5Ω for 100ft, and the total lead resistance in series with the 350Ω strain gage is therefore 21Ω at 25°C. The

bridge output voltage is simply the difference between the output of two voltage dividers, each driven from a +10V source.

When the load on the gauge is zero, the bridge would have an output offset voltage of 145.63mV for a nominal strain gage resistance of 350Ω.14563.0

700

350

21700

2135010

32

2

41

1

RR

R

RR

RVV Bo


If a 21Ω compensating resistance RCOMP

is used as shown to compensate for the lead resistance, Calculate the full scale bridge output

at 25°C and the percentage zero error (offset error) and the sensitivity error (gain error) due to a +10°C temperature rise in the cable.


Full scale bridge output at 25°C

The full scale bridge output voltage takes place when the gauge resistance is of 350 Ω + (350 Ω × 1%) = 353.5 Ω

The full scale output of the bridge at 25°C is 23.45 mV.

02345.021700

21350

215.703

215.35310

32

2

41

1

RR

R

RR

RVV Bo

Example: Errors Produced by Wiring Resistancein Remote Resistive Bridge Sensor

Zero error (zero offset) due to 10°C

With +10°C temperature rise, ΔRLEAD

= 10 × (0.385/100) × 10.5 = 0.404 Ω, RLEAD

= 10.5 Ω + 0.404 Ω = 10.904 Ω.

Total lead resistance in series with the gauge = 2 × 10.904 = 21.808

With the compensating resistance, an offset error of 5.434 mV is present at 10°C temperature rise which is (5.343/23.45) = 23.2% of full scale output.005434.0

21700

21350

808.21700

808.2135010

32

2

41

1

RR

R

RR

RVV Bo


Gain error due to 10°C temperature rise

The full scale bridge output at 35°C is calculated based on a gauge resistance of 350Ω + (350Ω × 1%) = 353.5Ω. The lead wire resistance, however, is (2

×10.904Ω = 21.808Ω

The full scale output due to 10°C temp rise is 28.83 mV giving a deflection from zero load of (28.83 - 5.434 = 23.396 mV) with an error of (23.396 –

23.45 = -0.054 mV) or (-0.054/23.45) = -0.23% of full scale output

02883.021700

21350

808.215.703

808.215.35310

32

2

41

1

RR

R

RR

RVV Bo


The effects of wiring resistance on the bridge output can be minimized by the 3-wire connection. The sense lead measures the voltage output

of a divider: the top half is the bridge resistor plus the lead resistance, and the bottom half is strain gage resistance plus the lead resistance.


The nominal sense voltage is independent of the lead resistance. When the strain gage resistance increases to fullscale (353.5W), the bridge

output increases to +24.15mV.


Gain error due to 10°C temperature rise

Increasing the temperature to +35ºC increases the lead resistance by +0.404W in each half of the divider. The fullscale bridge output voltage decreases to

+24.13mV because of the small loss in sensitivity, but there is no offset error. The gain error due to the temperature increase of +10ºC is therefore only –

0.02mV, or –0.08% of fullscale.

Documents

Instrumentation and Measurements Dr. Mohammad Kilani Analog Electrical Devices and Voltage Dividing Circuits