Chapter 6
Analog Electrical Devices
and Measurements
Material from Theory and Design for Mechanical Measurements; Figliola, Third Edition
Analog Devices
– Most devices measure in one form or another the
common electrical response to a physical
phenomenon. Voltage, current, and resistance-based
measurements are the most common.
„
Current Measurement
– Most analog meters are calibrated to indicate
another variable than what it is actually sensing,
• current through a resistor can be measured as voltage
• Ex: current flow through galvanometer
„
DC
– Basic analog measurement of current uses the
inductive forces on the current carrying conductor in
magnetic field. This force can be used to measure
the needle deflection on a display.
„
Force F = Ilβ for
90o (angle between
field and current)
Else F = Il k xβ
^
Figliola, 2000
„
The direction of
force is dependent
on the right-hand
rule.
1
„
Current loop will
experience a torque if
not aligned with the
magnetic field.
Torque on loop
Figliola, 2000
Tµ = NIAβ sin α
N = turns
I = current
A = cross-sectional area
defined by perimeter of
current loop
α = normal to current loop
and magnetic field
Figliola, 2000
„
A Galvanometer is
sensitive to current flow
through torque exerted
on current loop in
magnetic field.
– It’s useful to turn an
electric signal into a
visibly discernable signal.
„
The deflection of pointer
represents the current
through the loop and is
held steady by spring.
„
„
Figliola, 2000
Multi-range meter,
where shunt
resistors are
selected to reduce
current flow through
meter movement.
This system
requires current to
power movement. It
therefore puts a load
on the current.
2
• Voltage measurement = can vary from micro-volt
range on some transducers to several volts at the output
stage of the system. Power systems can be in the
kilovolt or larger range.
Figliola, 2000
„
The measurement of dc voltage by using analog
instruments may be accomplished through this
circuit, where a D’Arsonval movement is
employed in series with a register.
Analog Meters
„ The
D’Arsonval movement is
fundamentally sensitive to current flow,
but in conjunction with an appropriate
known fixed resistance can be
calibrated in terms of voltage.
„ This basic circuit is employed in the
construction of analog voltage dials, volt
ohmmeters (VOM).
Analog Meters
„
„
The measurement of ac voltage can be
accomplished through rectification of the ac
signal or through use of an electromagnet,
either in an electrodynamometer, or with a
movable iron vane.
These instruments are basically sensitive to
the rms value of a simple periodic ac current
and can be calibrated in terms of voltage;
shunt resistors can be used to establish the
appropriate scale.
3
The voltage divider circuit is constructed by using a
resistor made of a length of wire wound around an
appropriate insulating support.
„ The point labeled A represents a sliding contact,
which makes an electrical connection with the
resistor R at any point along its length.
„ The resistance between point A and point B in the
circuit is a linear function of the distance from A to
B, for an ideal resistor.
„
Figliola, 2000
Voltage Divider Circuit
„ The
output voltage sensed by the
voltmeter is given by:
Eo =
Lx
Rx
Ei =
Ei
LT
RT
if the internal resistance of the meter is
very large relative to RT so as to ignore
the current draw of the meter.
Potentiometer Circuit
„ Potentiometer
= useful to measure
voltages in the micro-volt and milli-volt
range. Equivalent to a balance scale, it
balances an unknown input voltage
against a known internal voltage.
„ A simple potentiometer circuit is derived
from the voltage divider circuit.
„ In this circuit, a galvanometer is used to
detect current flow.
4
Potentiometer Circuit
„ Any
current flow through the
galvanometer, G, would be a result of
an imbalance in the measured voltage,
Em, and the voltage imposed across
points A to B, EAB.
„ If Em is not equal to EAB, a current will
flow through the galvanometer, G.
„
Figliola, 2000
Galvanometer detects
current flow, due to
imbalance in voltage Em
and EAB. When EAB=Em,
there is a balance and
no current means no
displacement in
galvanometer. The
position of A indicates
the voltage Em.
„
„
Practical circuit with
standard cell. S cell
is balanced to W cell
using potentiometer
and calibration
resistor.
When balanced
G = 0 and A gives
value of Em.
5
Analog Resistance Measurements
– Checking continuity in electric circuit
– Measure resistance
– Measure change in resistance in order of
10-6 Ω over range 10-5 to 1015 Ω
– Compensate for change in ambient
condition
„ Working
principle is to measure change
in resistance relative to a measured
standard.
Ohmmeter Circuit
„
„
Impose a known
voltage across an
unknown resistance
and measure
resulting current
flow using
galvanometer.
Shunt resistors and
D’Arsonval
mechanism is used
to measure wide
range of
resistances.
Ohmmeter Circuit
„
„
Shunt resistor protects
meter from excess
current
Maximum current is
limited by heat
dissipation in shunt
resistor
– Power = I2R
„
Make sure the watt
ratings of resistors are
ok for the current
applied
Figliola, 2000
6
„
„
„
Bridge circuits – used to
measure capacitance,
inductance, and most often
resistance
Wheatstone bridge –
accurately measures
resistance and detects
small changes in resistance
Changes in R1 reflect
change in system
– When Ig =0, the bridge is
balanced and there is no
voltage drop from B to C
– Then (node current)
Figliola, 2000
• I1R1-I3R3=0
• I2R2-I4R4=0
– Also when balanced I1=I2, I3=I4
– Solving: R2/R1 = R4/R3
Application
1.
If R1 varies, adjust R2 to maintain
balanced system – called nulling
(zeroing).
2.
If the galvanometer is replaced with a
voltmeter, the voltage drop is an
indication of the change in the
resistance R1.
Null Method
„
„
„
If R1 varies, R2 is adjusted to directly
compensate for change in R1. R2 is a
calibrated variable resistance which can be
controlled manually or automatically. In this
case we do not care about the value of “Ei”only if there is a current flow through the
galvanometer.
There is error introduced by the
galvanometer, which does not have exactly
zero current flow.
Note: This can be used to
Uncertainty in R1:
select the right battery and
– ur1/R1 = Ig(R1+Rg)/Ei
galvanometer. Caution must
be taken to dissipate heat
properly. (Watts = I12R1)
7
Deflection Method
„
„
In an unbalanced condition, the magnitude of
the current or voltage drop for the meter or
galvanometer portion of a bridge circuit is a
direct indication of the change in resistance of
one or more of the arms of the bridge.
Consider first the case in which the voltage
drop from node B to node C in the basic
bridge is measured by a meter having an
infinite internal impedance, so that there is no
current flow through the meter, as shown on
the next slide.
Deflection Method
„
„
„
Since Im=0 and balanced I1 = I2, I3 = I4, then
E0 = I1R1 - I3R3 through substitution of prior
equations
E0 = Ei(R1/(R1 + R2) – R3 / (R3 + R4))
If there is a change in R1, R1' = R1 + δR
R1'
R3
E0 + δE0 = Ei [(
)
R1'+R2 R3 + R4
=
„
Ei[(R1' R4 - R3R2)
(R1' + R2)(R3 + R4)
In many cases, R1 = R2 = R3 = R4
– δE0 / Ei = (δR / R) / 4 + 2(δR / R)
8
„
If voltage meter is replaced by a low impedance
current measuring device and bridge operated
un-balanced, we have a current sensitive bridge.
–
–
Then, Ei = I1R1 + I2R2
I2 = I1 – Ig
•
Then, Ei = I1(R1+R2) - IgR2
Consider voltage drop through loop R1, Rg, R3
1. I1R1 + IgRg – I3R3 = 0
For Rg, R4, R2
2. IgRg + I4R4 – I2R2 = 0
Substituting I2 = I1 – Ig and I4 = I3 + Ig from node current
3. IgRg + (I3 + Ig)R4 – (I1 - Ig)R2 = 0
Solving 3 equations simultaneously for Ig
Ig=
[Ei(R3R2-R1R4)]
[R3(R1+R2)(Rg+R2+R4)+R1R2R4-R3R22+RgR4(R1+R2)]
„ Then,
change in R1 in terms of bridge
deflection voltage E0
E0
R2
[ +
]
δR R 3 Ei (R 2 + R 4)
−1
=
R2
R 1 R 1 [1 − E0 −
]
Ei (R 2 + R 4)
„ If
all resistors equal to R and R1 changes
δR, the current Ig = Ei (δR / R) / (4(R+Rg))
– output voltage E0=IgRg
– E0=Ei (δR / R) / (4(1 + R / Rg))
9
Bridge Impedance
„ The
bridge impedance can affect output
of constant voltage source having an
internal resistance Rs.
„ RB= bridge resistance
– RB=R1R3/(R1+R3) + R2R4/(R2+R4)
„ If
RS is voltage source internal
resistance, ES is voltage
– Ei = ESRB/RS+RB
„ RB=R1R3 /
„
(R1 + R3) + R2R4 / (R2 + R4)
Ei =
EsRb
Rs + Rb
„ Similarly,
bridge impedance can affect
voltage indicated by voltage measuring
device, if Rg is internal impedance of
VOM.
• EM = E0RB' / Rg + RB'
• RB' = R1⏐⏐R2 + R3⏐⏐R4
• RB' = (R1R2 / (R1 + R2 ))+ (R3R4 / (R3 + R4 ))
Deflection Method
„ This
can be written as:
Eo =
R 3R 4
Em R 1R 2
+
+ Rg)
(
Rg R 1 + R 2 R 3 + R 4
„ The
difference between the measured
voltage, Em, and the actual voltage, Eo,
is called a loading error, in this case
caused by the bridge impedance load.
10
Loading Errors and Impedance
Matching
„
„
„
„
„
„
„
Ideally an instrument should not in itself affect the variable
being measured (non-invasive)
Loading – when measurement system alters variable being
measured
Loading Error – difference between unaltered value and
observed/recorded value of variable
Process Loading Error – when the insertion of a sensor into
the process alters in some way the physical variable
Interstage Loading Error – if output from one stage is
affected by subsequent stages.
Goal – minimize loading errors
Ex: Insert a thermometer at room temperature into a
specimen at some higher temperature. Heat removed to
achieve equilibrium is loading.
„
Ex: galvanometer in wheatstone bridge
removes current from circuit to activate meter
movement.
„
Note: Null balance methods generally minimize
magnitude of loading errors to negligible levels,
while deflection method can add significant
loading if care is not taken.
„ Loading
Error in Voltage Divider
• Total: R1 + R2 = RT
R1||Rm
RL = R1Rm / (R1 + Rm)
• Req = R2 + R1Rm / (R1+Rm)
• I = Ei / Req = E1/ (R2 + R1Rm/(R1 +Rm))
Figliola, 2000
11
„
Output Voltage
– E0 = Ei – IR2 or
Eo
1
=
Ei 1 + (R 2 / R 1)(R 1 / Rm + 1)
„
As Rm goes to infinity
„
Then, using
„
Error: eI = Ei [(E0 / Ei)' – E0 / Ei]
Then
• E0 / Ei = R1 / (R1 + R2) ; E0 / Ei = R1 / R1 as R2 Æ 0
– (E0 / Ei)' = R1 / (R1 + R2) = R1 / RT
„
– eI = Ei (R1 – RT + (RT – R1) [(R1 / Rm) + 1]
RT + [(RT2 / R1) – RT] [(R1 / Rm) + 1]
The loading error goes to zero as Rm Æ
∞
Interstage Loading Errors
Figliola, 2000
• Consider the common situation in which the output voltage signal from
one measurement system provides the input to the following device.
• The open circuit potential, E1, is present at the output terminal of
device 1 with output impedance, Z1.
• However, the output signal from device 1 provides the input to a
second device, which, at its input terminals, has an input impedance,
Zm.
Interstage Loading Errors
„
Using Thevenin’s equivalent of Device 1
– Em = I * Zth = E1(1 / (1 + z1/zm))
„
„
„
„
E1 = Em + Em(z1/zm)
Em = E1 – Em(z1/zm)
E1 potential at output of Device 1 is input to
Device 2
Device 1 has output voltage E1 and output
impedance z1
Device 2 has input impedance z2
The original potential has been charged by
interstage connection causing loading error,
eI = E1 – Em
• eI = E1 (1 – (1/(1 + z1/zm)))
eI Æ 0 as zm >> z1
12
Analog Signal Conditioning
„
Amplifiers scale the magnitude of analog
input signal
Eo( t ) = h{Ei( t )}
„
A linear scaling amplifier
h{Ei( t )} = GEi( t )
G = gain
It is a constant that may be positive or negative
Analog Signal Conditioning
„
„
Amplifiers have a finite frequency
response and limited input voltage
range.
OP-Amp is most common
1. Very high input impedance zi>107Ω
2. Low output impedance
zo<100Ω
3. High internal gain
A = 105
Figliola, 2000
„ Since zi >> 0, I ≈ 0
„ Signal in-phase for
„ Signal
non-inverting input
180o out of phase for inverting
input
„ Requires dual polarity dc excitation
voltage ranging from ±5 V to ± 15V
13
„
High internal open-loop gain, A, (no external loads)
„
Gain “A” is flat at low frequency, but drops off quickly
at high frequency.
By using External Resistances at input and at
feedback, the circuit is stabilized and gain is fixed.
Resistors R1 and R2 are used to form a feedback loop
and controls overall amplifier circuit gain.
„
„
Eo( t ) = A[ Ei 2( t ) − Ei1( t )]
Figliola, 2000
For Non-Inverting Circuit
„
„
„
„
„
„
Eo - io(R1 + R2) = 0
Ei - io(R2) = 0
Eo = io(R1 + R2)
Ei = io(R2)
Ei / R2 = Eo / (R1 + R2)
G = Eo / Ei = (R1 + R2) / R2
Figliola, 2000
Inverting Amp Circuit
„
„
„
„
„
„
„
Ei - ii(R1 + R2) + Eo = 0
Ei - iiR1 = 0
ii = Ei / R1
Ei – (Ei / R1) (R1+R2) +
Eo = 0
Ei –Ei – Ei (R2 / R1) +
Eo = 0
Eo = Ei (R2 / R1)
G = Eo / Ei = (R2 / R1)
Figliola, 2000
14
„
Differential Amplifier Circuit (OP-Amp)
• Use both inputs to compare input voltage
“voltage comparator”
• Eo(t) = [Ei1(t) – Ei2(t)] (R2 / R1)
Figliola, 2000
Figliola, 2000
„
Special – Analog Voltage Comparator
„
E0 = G(Ei1 – Ei2) for |Ei1 – Ei2| < ET
E0 = + E bias for Ei1 – Ei2 > ET
E0 = - E bias for Ei1 – Ei2< -ET
Often Ei2 is equal to some reference voltage, which
allows the comparator to detect if the input is greater
or less than reference used in A/D converters.
The threshold Et is fixed by bias voltage
– Output proportional to difference in input voltage
„
„
„
„
– Output saturates at Ei values = ET
15
Sample and Hold Circuit
„
„
„
The sample and hold circuit (SHC) is used to
take a narrow-band measurement of a timechanging signal and to hold that measured
value until reset.
It is widely used in data-acquisition systems
using A/D converters.
The circuit tracks the signal until it is triggered
to sample the signal and hold it.
Charge Amplifier
„A
charge amplifier is used to convert a
high-impedance charge, q, into an
output voltage, Eo.
„ The circuit consists of a high gain,
inverting voltage operational amplifier.
„ These circuits are commonly used with
transducers that utilize piezoelectric
crystals.
16
Current Loop: 4-20 mA
„
„
„
„
A problem with voltage signals below ~100
mV is that they are quite vulnerable to noise
along the transmission lines.
One means of transmitting low-level voltage
signals over long-distances is by signal
boosting.
A common alternative method is a 4-20 mA
current loop (read as 4 to 20).
The low-level voltage is converted into a
standard current loop signal of between 4 and
20 mA, the lower constant current value for
the minimum voltage and the higher value for
the maximum voltage in the range.
Analog Signal Conditioning
Figliola, 2000
• Filters are used to remove undesirable frequency information.
• M(f) is magnitude ratio of dynamic system response.
• Fc = cut off frequency
• Low-pass filter permits frequencies below cut off frequency and blocks frequencies
above it.
17
Filters Types
Passive filters are circuits made up of resistors, capacitors,
and inductors.
„ Active filters incorporate operational amplifiers.
„ The sharp cut off of an ideal filter can not be realized.
„ Roll off designated in decibels per decade
„ Phase shift between input and output
„ Filter design is based on its cut off frequency, which is the
frequency where the signal power is reduced to ½, which is
equivalent to m(w) = 0.707
„ Decibels; db = 20 log m(w) = 20 log (0.707) = -3 db
„
Figliola, 2000
Butterworth Low Pass Filter
„
A simple passive low-pass Butterworth filter
can be constructed by using the resistor and
capacitor (RC) circuit.
Figliola, 2000
Butterworth Low Pass Filter
„
„
„
„
„
o
RC E 0 + E0 = Ei
o
RC E 0 + E0 = KA sin wt
-t/τ
E0(t) = ce + B(w)sin [wt+ø(w)]; τ=RC w = 2πf
Phase shift: ø(w) = -tan -wτ
Magnitude ratio: m(w) = B/KA
1
=
1/2
[1 + (wτ)2]
„
„
Design: fc occurs at m(w) = 0.707 or –3dB
τ = RC = 1/2πfc ; so fc = 2πRC
18
The roll off slope can be improved by cascading low
pass filter stages
K-Stage Filter:
Figliola, 2000
„
M(f) = 1/ [1 + (f / fc)2k]1/2
„
Φ (f ) =
K
∑
φ i( f )
i=1
„
Alternation at any frequency
„
Values and Expressions for sizing Li and Ci are given
in Table 1 and Eq 6.60
– dB = 10 log [1 + (f / fc)2k]
Figliola, 2000
Roll off is a function of the K number of stages.
Note: steeper slope of attenuation for K = 4 & 5
„
1st order – single stage
low pass active
Butterworth filter
„
High Pass
– fc = 1/ (2πR2C2)
– fc = 1/ (2πR1C1)
Magnitude Ratio
R2
(f/fc)
KM(f) = ( ) (
2 1/2 )
R1 (1 + (f/fc) )
„
Figliola, 2000
19
Grounding and Shielding
„
„
Type of connecting wire can have a significant
impact on noise levels, especially for low-level
signals (<100 mV).
Rule of Thumb:
1. Keep wires as short as possible
2. Keep signal wires away from noise sources
a) Separate conduit from high-voltage AC, and especially
variable speed PWM drives
b) Avoid motors when possible
c) Run perpendicular to AC lines
3. Use individual wire shielded cables
a) 2 conductor shielded cables
b) Twisted pair with individual shields
c) Drain wires
„
Ground – a conductor connected to a probe
that is driven far into the earth provides a
return path to earth.
„
The ground at the feeder box may not be at
the earth ground potential due to the voltages
that have been induced into the down stream
ground conductor.
– The difference between the two ground point
voltages is called common-mode voltage.
„
Ground loops exist when a signal circuit is
connected to ground at two or more points
having different ground potential. This
potential induces a current in circuit that can
bias the signal or create unwanted
frequencies (usually 60 hz).
„ Proper
Connection
„ Shields
– Long wires act like antennas to pick up wet
noise. A shield is used to protect the signal
conductors from RF (radio frequency) and
EMI noise. The shield is either metal foil or
wire braid with a drain wire that is
connected on one end (only) to ground.
• If the shield is connected on both ends, a
ground loop is created.
20
Noise Sources
„
„
„
„
„
„
„
Motors
Transformers
Solenoids
Relays
Motor drives
Switch gears
Motor contactors
Noise Solutions
„
„
„
„
„
„
„
Filters
Shields
DC chokes
Diodes
Avoidance
Separate conduits
Perpendicular
crossing
Connecting Wires
„
Twisted pairs tend to cancel induced voltage,
common for analog signals
„
Conductor size should be selected to
minimize voltage drop in conductor.
„
Coaxial cable (single internal conductor with
return path through shield) is common for
high frequency applications, and can be sent
over long distances. A triaxial cable is better
(2 internal wires) and gives excellent noise
suppression. But it is very expensive!
21