Spiritual – Temporal
1 Nephi 15:32
32 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 2:41
41 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.
ECEN 301
Discussion #6 – Things Practical
1
Lecture 6 – The Wheatstone Bridge
An Applications of Things Electrical
ECEN 301
Discussion #6 – Things Practical
2
Wheatstone Bridge
A circuit used to find:
• An unknown resistance Rx
• The unknown voltage vab
• Another unknown value (such as a force)
c
c
R1
vs
+
_
R3
va
a
vb
R2
b
vs
+
_
Rx
d
ECEN 301
R1
R3
a
va
vb
b
Rx
R2
d
Discussion #6 – Things Practical
3
Wheatstone Bridge
•
•
The circuit consists of the parallel combination of 3
subcircuits with the same voltage:
a) The voltage source
b) 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
•
R2
vad  vs
R1  R2
c
vx = vbd
R1
vs
+
_
R3
va
a
vb
R2
b
Rx
d
ECEN 301
Discussion #6 – Things Practical
vbd  vs
Rx
R3  Rx
4
Wheatstone Bridge
• KVL around the bottom loop:
vab  vad  vbd
 R2
Rx 

 vs 

 R1  R2 R3  Rx 
c
R1
vs
+
_
R3
va
a
vb
R2
b
Rx
d
ECEN 301
Discussion #6 – Things Practical
5
Wheatstone Bridge
• Example1: when is vab = 0?
c
R1
vs
+
_
R3
va
a
vb
R2
b
Rx
d
ECEN 301
Discussion #6 – Things Practical
6
Wheatstone Bridge
• Example1: when is vab = 0?
 R2
Rx 

vab  vs 

R

R
R

R
2
3
x 
 1
c
R1
vs
+
_
R3
va
a
vb
R2
b
Rx
d
 R2
Rx 
  0
vs 

R

R
R

R
2
 1
3
x 
Rx
R2

R1  R2 R3  Rx
R2  R3  Rx   Rx  R1  R2 
R2 R3  R2 Rx  Rx R1  R2 Rx
R2 R3  Rx R1
R3 Rx

R1 R2
R2 Rx

R1 R3
ECEN 301
Discussion #6 – Things Practical
7
Wheatstone Bridge
• Find Rx:
 R2
Rx 

vab  vs 

 R1  R2 R3  Rx 
c
R1
vs
+
_
R3
va
a
vb
R2
ECEN 301
b
Rx
d

 R2
vab 


R3 

R  R2 vs 
Rx   1

vab 
R2
1 


 R1  R2 vs 
Discussion #6 – Things Practical
8
Wheatstone Bridge
•
Example2: find Rx
•
R1 = R2 = R3 = 1kΩ, vs = 12V, vab = 12mV
c
R1
vs
+
_
R3
va
a
vb
R2
b
Rx
 R2
v 
R3 
 ab 
R1  R2 vs 

Rx 

v 
R2
1 
 ab 
 R1  R2 vs 
d
ECEN 301
Discussion #6 – Things Practical
9
Wheatstone Bridge
•
Example2: find Rx
•
R1 = R2 = R3 = 1kΩ, vs = 12V, vab = 12mV
vs
+
_
R3
va
a
vb
R2
b
Rx
d
ECEN 301
 103
12  10 3 

10 
3 
2

10
12




103
12  10 3 
1 

3 
12 
 2 10
3
c
R1
 R2
v 
R3 
 ab 
R  R2 vs 
Rx   1

v 
R2
1 
 ab 
 R1  R2 vs 
Discussion #6 – Things Practical
1

103   10 3 
2


 1
3 
1   10 
 2

499

0.501
 996
10
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
L
• NB: cylindrical resistance: R 
A
L – length of cylindrical resistor
σ – conductivity of the resistor
A – resistor cross-sectional area
• Compression/elongation will change resistance
Compression
ECEN 301
lower resistance
Stretch
Discussion #6 – Things Practical
higher resistance
11
Resistance Strain Gauges
• Gauge factor (GF): the relationship between change in
resistance and change in length
• value of about 2 is common
R / R0
GF 
L / L
R0 – the zero strain resistance
• 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
ECEN 301
L

L
Discussion #6 – Things Practical
12
Resistance Strain Gauges
• Change in resistance due to strain:
R  R0GF
RG
ECEN 301
Circuit symbol for a strain gauge
Discussion #6 – Things Practical
13
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
c
F
L
ia
R1
h
R1
vs
R4
w
+
_
a
va
ib
R3
vb
b
R4
R2
R2 & R3
d
ECEN 301
Discussion #6 – Things Practical
14
Wheatstone Bridge
• Under the strain of force F we have:
R1  R4  R0  R
tension
R2  R3  R0  R
c
compression
R0 - ∆R
R0 + ∆R
F
L
vs
R4
w
ECEN 301
a
vb
b
ia ib
h
R1
R2 & R3
+
_
va
R0 - ∆R
compression
Discussion #6 – Things Practical
R0 + ∆R
d
tension
15
Wheatstone Bridge
• From elementary statics it can be shown that:
6 LF
 2
wh Y
Y is the beam’s modulus of elasticity
c
R0 - ∆R
R0 + ∆R
F
L
vs
+
_
a
R4
w
b
R0 - ∆R
R2 & R3
ECEN 301
vb
ia ib
h
R1
va
R0 + ∆R
d
Discussion #6 – Things Practical
16
Wheatstone Bridge
• Using Ohm’s Law:
vs
ia 
R1  R2
vs
ib 
R3  R4
c
c
ia
R1
vs
+
_
a
va
ib
R3
vb
b
vs
+
_
a
va
vb
b
ia ib
R4
R2
R0 - ∆R
R0 + ∆R
R0 - ∆R
R0 + ∆R
d
d
ECEN 301
Discussion #6 – Things Practical
17
Wheatstone Bridge
• Using Ohm’s Law:
vout  vba  vb  va  vbd  vad
c
 ib R4  ia R2
R0 - ∆R
R0 + ∆R
vs
+
_
a
va
vb
b
i a ib
R0 - ∆R
R0 + ∆R
d
ECEN 301

vs R4
vR
 s 2
R3  R4 R1  R2
 vs
R0  R
R0  R
 vs
R0  R  R0  R
R0  R  R0  R
R
 vs
R0
 vs GF
Discussion #6 – Things Practical
18
Wheatstone Bridge
• Find vo in terms of force F:
vo  vs GF
c
R0 - ∆R
R0 + ∆R
vs
+
_
a
va
vb
b
i a ib
R0 - ∆R
R0 + ∆R
d
ECEN 301
6 LF
 vs GF 2
wh Y
 6vs GFL 

F
2
 wh Y 
 kF
k is a calibration constant
Discussion #6 – Things Practical
19
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
vo  kF
F
L
h
R1
R4
w
R2 & R3
 6vs GFL 

F
2
 wh Y 


6(12)( 2)(0.3)
F
 
2
9 
 (0.05)(0.01) 69 10 
 1.25F 10 4 V / N
 0.125F mV / N
ECEN 301
Discussion #6 – Things Practical
20
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
F
Max imum voltage :
L
h
R1
R4
w
R2 & R3
ECEN 301
vo  0.125 F mV / N
 1.25 10  4 (500)
 62.5 mV
Discussion #6 – Things Practical
21
Practical Sources
ECEN 301
Discussion #6 – Things Practical
22
Ideal Sources
• Ideal current source
• Ideal voltage source
• 3A source
• 6V source
i
8
7
6
5
4
3
2
1
0
v
0
1
2
3
4
5
6
7
8
Provides a prescribed voltage across its
terminals irrespective of the current
flowing through it.
ECEN 301
i
8
7
6
5
4
3
2
1
0
v
0
1
2
3
4
5
6
7
8
Provides a prescribed current
irrespective of the voltage across it.
Discussion #6 – Things Practical
23
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
1 amp = 1 coulombs/second
1 amp-hour = 3600 coulombs
• 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)
ECEN 301
Discussion #6 – Things Practical
24
Practical Sources
• Actual voltage sources have limitations
• 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
i
RL
i
+
vs
+
_
v
Load (RL)
_
ECEN 301
As RL 0, vs has to
provide an infinite amount
of current!
Discussion #6 – Things Practical
25
Practical Sources
• Actual voltage sources have limitations
A series resistance rs poses a limit to the
maximum current the voltage source can provide
is
rs
vs
+
_
vs
is 
rs  RL
+
vL
Load (RL)
–
Practical voltage source
ECEN 301
Discussion #6 – Things Practical
26
Practical Sources
• Actual voltage sources have limitations
A series resistance rs poses a limit to the
maximum current the voltage source can provide
is
rs
vs
+
_
v L  i s RL
 vs
+
vL
RL
rs  RL
Load (RL)
–
NB: vs = vL
Practical voltage source
ECEN 301
Discussion #6 – Things Practical
27
Practical Sources
• Actual voltage sources have limitations
A series resistance rs poses a limit to the
maximum current the voltage source can provide
is
rs
vs
+
_
+
vL
vs
is 
rs  RL
vs
lim is 
RL  0
rs
–
iS max
Practical voltage source
ECEN 301
vs

rs
Discussion #6 – Things Practical
28
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
vs  i  RL
+
is
vs
Load (RL)
As RL ∞, is has to
provide an infinite amount
of voltage!
–
ECEN 301
Discussion #6 – Things Practical
29
Practical Sources
• Actual current sources have limitations
A series resistance rs poses a limit to the
maximum voltage the current source can provide
vS  is (rs || RL )
+
is
rs
vs
Load (RL)
–
Practical current source
ECEN 301
Discussion #6 – Things Practical
30
Practical Sources
• Actual current sources have limitations
A series resistance rs poses a limit to the
maximum voltage the current source can provide
vS  is (rs || RL )
+
is
rs
vs
–
lim vs  is rs
RL  
vS max  is rs
Practical current source
ECEN 301
Discussion #6 – Things Practical
31
Practical Sources
• Actual current and voltage sources have
limitations
Ideal 3A current source
Ideal 6V votage source
i
8
7
6
5
4
3
2
1
0
v
0
ECEN 301
1
2
3
4
5
6
7
8
i
8
7
6
5
4
3
2
1
0
v
0
1
Discussion #6 – Things Practical
2
3
4
5
6
7
8
32
Practical Sources
• Actual current and voltage sources have
limitations
Practical 3A current source
i
8
7
6
5
4
3
2
1
0
v
0
ECEN 301
1
2
3
4
5
6
7
8
Practical 6V voltage source
i
8
7
6
5
4
3
2
1
0
v
0
1
2
Discussion #6 – Things Practical
3
4
5
6
7
8
33
Measuring Devices
ECEN 301
Discussion #6 – Things Practical
34
Ohmmeter
• Ohmmeter: measures the resistance of a circuit
element
NB: the resistance of an element can only be measured when the element is
disconnected from all other circuit elements
Ω
ECEN 301
Ohmmeter symbol
Ω
R
Discussion #6 – Things Practical
Ohmmeter connection
setup for circuit element
resistance measurement
35
Ammeter
• Ammeter: a device that can measure the
current flowing though a circuit element when
connected in series with that circuit element
NB: 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
R1
A
Ammeter
symbol
R1
A
vs
+
_
R2
i
vs
+
_
i
R2
ECEN 301
Discussion #6 – Things Practical
36
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
R1
R1
V
Voltmeter
symbol
ECEN 301
vs
+
_
i
+
R2
–
vs
+
_
Discussion #6 – Things Practical
i
+
R2
–
+
v2 V
–
37
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
R1
W
Wattmeter
symbol
ECEN 301
vs
+
_
i
R1
+
R2
–
vs
+
_
Discussion #6 – Things Practical
i
W
+
R2
–
38
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
A
V
rm
V
A
rm
Ideal
Voltmeter
ECEN 301
Practical
Voltmeter
Ideal
Ammeter
Discussion #6 – Things Practical
Practical
Ammeter
39