spiritual – temporal

ECEN 301 Discussion #6 – Things Practical 1

Spiritual – Temporal1 Nephi 15:3232 And it came to pass that I said unto them that it was a

representation of things both temporal and spiritual; for the day should come that they must be judged of their works, yea, even the works which were done by the temporal body in their days of probation.

Mosiah 2:4141 And moreover, I would desire that ye should consider on the

blessed and happy state of those that keep the commandments of God. For behold, they are blessed in all things, both temporal and spiritual; and if they hold out faithful to the end they are received into heaven, that thereby they may dwell with God in a state of never-ending happiness. O remember, remember that these things are true; for the Lord God hath spoken it.


Lecture 6 – The Wheatstone Bridge

An Applications of Things Electrical


Wheatstone BridgeA circuit used to find:• An unknown resistance Rx

• The unknown voltage vab • Another unknown value (such as a force)

vs+_

R2

R1 R3

Rx

bava vb

c

d

c R1

vs+_

R3

R2 Rx

d

a bva vb


Wheatstone Bridge• The circuit consists of the parallel combination of 3

subcircuits with the same voltage:a) The voltage sourceb) Series combination of R1 and R2

c) Series combination of R3 and Rx• Voltage divider between:

a) R1 and R2• v2 = vad

b) R3 and Rx• vx = vbd 21

2

RRRvv sad

vs+_

R2

R1 R3

Rx

bava vb

c

d x

xsbd RRRvv

3


Wheatstone Bridge• KVL around the bottom loop:

x

xs

bdadab

RRR

RRRv

vvv

321

2

vs+_

R2

R1 R3

Rx

bava vb

c

d


Wheatstone Bridge• Example1: when is vab = 0?

vs+_

R2

R1 R3

Rx

bava vb

c

d


Wheatstone Bridge• Example1: when is vab = 0?

31

2

21

3

132

21232

2132

321

2

321

2

321

2

0

RR

RR

RR

RR

RRRRRRRRRRRRRRRRRR

RRR

RRRRR

RRR

Rv

RRR

RRRvv

x

x

x

xxx

xx

x

x

x

xs

x

xsab

vs+_

R2

R1 R3

Rx

bava vb

c

d


Wheatstone Bridge• Find Rx:

s

ab

s

ab

x

x

xsab

vv

RRR

vv

RRRR

R

RRR

RRRvv

21

2

21

23

321

2

1

vs

+_

R2

R1 R3

Rx

bava vb

c

d


Wheatstone Bridge• Example2: find Rx

• R1 = R2 = R3 = 1kΩ, vs = 12V, vab = 12mV

s

ab

s

ab

x

vv

RRR

vv

RRRR

R

21

2

21

23

1vs

+_

R2

R1 R3

Rx

bava vb

c

d


Wheatstone Bridge• Example2: find Rx

• R1 = R2 = R3 = 1kΩ, vs = 12V, vab = 12mV

996501.0

499

10211

102110

121012

102101

121012

1021010

1

3

33

3

3

3

3

3

33

21

2

21

23

s

ab

s

ab

x

vv

RRR

vv

RRRR

R

vs+_

R2

R1 R3

Rx

bava vb

c

d


Resistance Strain Gauges• Strain gauge: device bonded to the surface of an object and

whose resistance varies as a function of surface strain• Used to perform measurements of:

• Strain• Stress,• Force• Torque• Pressure

• NB: cylindrical resistance:

• Compression/elongation will change resistance

ALR

Compression lower resistance Stretch higher resistance

L – length of cylindrical resistorσ – conductivity of the resistorA – resistor cross-sectional area


Resistance Strain Gauges• Gauge factor (GF): the relationship between change in

resistance and change in length • value of about 2 is common

LL

• Strain (ε): the fractional change in length of an object• Max strain that can be measured is about 0.4 – 0.5 percent

• i.e. ε = 0.004 to 0.005• For a 120Ω resistor: +/– 1.2 Ω

LLRRGF

// 0

R0 – the zero strain resistance


Resistance Strain Gauges• Change in resistance due to strain:

GFRR 0

RGCircuit symbol for a strain gauge


Wheatstone Bridge• Wheatstone bridge commonly used to measure force

using strain gauge resistors• Example: force applied to a cantilever beam

• Two strain gauges (R1 and R4) on top• Two strain gauges (R2 and R3) on bottom• Under no strain: R1 = R2 = R3 = R4 = R0

R1

vs+_

R3

R2 R4

d

a bva vb

ia ib

c

R4

R1

R2 & R3

FL

w

h


Wheatstone Bridge• Under the strain of force F we have:

R4

R1

R2 & R3

FL

w

hvs

+_

R0 + ∆R

bava vb

c

d

R0 + ∆R

R0 - ∆R

R0 - ∆R

ia ib

tension

tension

compression

compression

RRRR 041

RRRR 032


Wheatstone Bridge• From elementary statics it can be shown that:

R4

R1

R2 & R3

FL

w

hvs

+_

R0 + ∆R

bava vb

c

d

R0 + ∆R

R0 - ∆R

R0 - ∆R

ia ib

YwhLF

2

6 Y is the beam’s modulus of elasticity


Wheatstone Bridge• Using Ohm’s Law:

vs+_

R0 + ∆R

bava vb

c

d

R0 + ∆R

R0 - ∆R

R0 - ∆R

ia ib

43 RRvi s

b

21 RRvi s

a

R1

vs+_

R3

R2 R4

d

a bva vb

ia ib

c


Wheatstone Bridge• Using Ohm’s Law:

vs+_

R0 + ∆R

bava vb

c

d

R0 + ∆R

R0 - ∆R

R0 - ∆R

ia ib

GFvRRv

RRRRRRv

RRRRRRv

RRRv

RRRv

RiRivvvvvv

s

s

ss

ss

ab

adbdabbaout

0

00

0

00

0

21

2

43

4

24


Wheatstone Bridge• Find vo in terms of force F:

kF

FYwhGFLv

YwhLFGFv

GFvv

s

s

so

2

2

6

6

k is a calibration constant

vs+_

R0 + ∆R

bava vb

c

d

R0 + ∆R

R0 - ∆R

R0 - ∆R

ia ib


Wheatstone Bridge• Example3: using the Wheatstone bridge as a strain measurement tool find the

maximum vo• the bridge measures forces ranging from 0 to 500 N• L = 0.3m, w = 0.05m, h = 0.01m, GF = 2, Y = 69x109N/m2, vs = 12V

NmVFNVF

F

FYwhGFLv

kFv

s

o

/125.0/1025.1

1069)01.0)(05.0()3.0)(2)(12(6

6

4

92

2

R4

R1

R2 & R3

FL

w

h


Wheatstone Bridge• Example3: using the Wheatstone bridge as a strain measurement tool find

the maximum vo• the bridge measures forces ranging from 0 to 500 N• L = 0.3m, w = 0.05m, h = 0.01m, GF = 2, Y = 69x109N/m2, vs = 12V

mV

NmVFvo

5.62)500(1025.1

/125.0: voltageimumMax

4

R4

R1

R2 & R3

FL

w

h


Practical Sources


Ideal Sources• Ideal current source

• 3A source

012345678

0 1 2 3 4 5 6 7 8

v

i

• Ideal voltage source• 6V source

012345678

0 1 2 3 4 5 6 7 8

v

i

Provides a prescribed voltage across its terminals irrespective of the current flowing through it.

Provides a prescribed current irrespective of the voltage across it.


Practical Sources• Actual voltage sources have limitations

• There is a limit to the number of total electrons any battery can motivate through a circuit

• How to measure limitations?• Total number of electrons? (huge number)• Use coulombs? (also too huge)• amp-hour – unit invented for this purpose

• 1 amp-hour = 1 amp for 1 hour = 2 amps for ½ hour

= 1/3 amp for 3 hours

• Batteries have ratings indicating their current limitations• Car battery – 12V, 70 amp-hours (A-h) @ 3.5 A (for 20 hours)• D – cell (1.5V) carbon-zinc battery – 4.6 amp-hours @ 100mA (for 46 hours)• 9 – volt carbon-zinc battery – 400 mA-hours @ 8mA (for 50 hours)

1 amp = 1 coulombs/second1 amp-hour = 3600 coulombs



• As the load resistance (RL) decreases, the voltage source (vs) is required to provide increasing amounts of current (i) in order to maintain vs

+_vs v

+

_

i

Load (RL)

L

s

Rvi

As RL 0, vs has to provide an infinite amount of current!



A series resistance rs poses a limit to the maximum current the voltage source can provide

Ls

ss Rr

vi

+_vs

+

vL

–

is

rs

Practical voltage source

Load (RL)




+_vs

+

vL

–

is

rs


Load (RL)Ls

Ls

LsL

RrRv

Riv

NB: vs = vL



+_vs

+

vL

–

is

rs

s

sS

s

ssR

Ls

ss

rvi

rvi

Rrvi

L

max

0lim




Practical Sources• Actual current sources have limitations

• As the load resistance (RL) increases, the current source (is) is required to provide increasing amounts of current (v) in order to maintain is

is

+

vs

–

Load (RL)

Ls Riv

As RL ∞, is has to provide an infinite amount of voltage!



A series resistance rs poses a limit to the maximum voltage the current source can provide

)||( LssS Rriv

is

+

vs

–

rs

Practical current source

Load (RL)



ssS

sssR

LssS

riv

rivRriv

L

max

lim)||(

A series resistance rs poses a limit to the maximum voltage the current source can provide

is

+

vs

–

rs

Practical current source


Practical Sources

012345678

0 1 2 3 4 5 6 7 8

v

i

012345678

0 1 2 3 4 5 6 7 8

v

i

• Actual current and voltage sources have limitations

Ideal 3A current source Ideal 6V votage source


Practical Sources

012345678

0 1 2 3 4 5 6 7 8

v

i

012345678

0 1 2 3 4 5 6 7 8

v

i

• Actual current and voltage sources have limitations

Practical 3A current source Practical 6V voltage source


Measuring Devices


Ohmmeter• Ohmmeter: measures the resistance of a circuit

elementNB: the resistance of an element can only be measured when the element is disconnected from all other circuit elements

Ω Ω ROhmmeter symbol

Ohmmeter connection setup for circuit element resistance measurement


Ammeter• Ammeter: a device that can measure the

current flowing though a circuit element when connected in series with that circuit elementNB: 1. the ammeter must be connected in series with the circuit element 2. the ammeter should not restrict the flow of current (i.e. cause a voltage drop) – an ideal ammeter has zero resistance

A Ammeter symbol

iR2vs

+_

R1 R1

iR2

vs+_

A


Voltmeter• Voltmeter: a device that can measure the

voltage across a circuit element when connected in parallel with that circuit element

NB: 1. the voltmeter must be connected in parallel with the circuit element 2. the voltmeter should not draw any current away from the element – an ideal voltmeter has infinite resistance

V Voltmeter symbol vs

i

+R2

–

+_

R1 R1

i

+R2

–vs

+_ V+v2

–


Wattmeter• Wattmeter: a device that can measure the

power dissipated by a circuit element

NB: 1. the wattmeter must be connected in parallel with the circuit element, but also in series with the circuit. – a wattmeter is simply the combination of a voltmeter and an ammeter

W Wattmeter symbol vs

i

+R2

–

+_

R1 R1

i

+R2

–vs

+_

W


Practical Voltmeters and Ammeters• In reality, voltmeters and ammeters have internal

resistances• Practical ammeters will always add some resistance• Practical voltmeters will always draw some current

V

IdealVoltmeter

Vrm

PracticalVoltmeter

A

IdealAmmeter

PracticalAmmeter

A

rm

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