Loading Effects
The output of a sensor device may deviate from the correct
value due to loading effect. We can categorize two types
of loading effect:
• Inter element loading
A given element in the system may modify the
characteristics of the previous element.
• Process loading
The introduction of the sensing element into the process or
system being measured causes the value of the measured
variable to change.
Sensors and Interfacing
Loading effect
1
Electrical loading (two-port networks)
A. A sensor device is represented by Thevenin equivalent circuit
VL = iZ L = E th
ZL
,
Z th + Z L
loading of the Thevenin equivalent circuit
ZL≧Zth, VL→ETH
maximum voltage transfer from the network to the load.
Eth: Voltage source, open circuit voltage of the network across the
output terminal.
All voltage sources reduced to zero and replaced by their internal
impedance.
Zth: The impedance looking back into the terminal.
Sensors and Interfacing
Loading effect
2
Example: Temperature measurement system
6
4
2
10
10
×
, VL = 1000VIN
VIN = 40 ×10 −6 T
6
2 × 10 + 20
75 + 10 4
2 × 106
104
TM = 25VL = 40 × 10 × 25 × 1000T ×
×
6
2 × 10 + 20 75 + 104
= 0.99257T, loading error = 0.0075T
−6
Sensors and Interfacing
Loading effect
3
Use of buffer amplifier to reduce loading effects
• PH transducer (glass electrode): use of Buffer Amplifier
Sensitivity: Eth = 59 PH(mV) or 1 PH →59mV Sensitivity=59mV/PH
Zth = 109Ω
Indicator : Zth = RL =104Ω
1
PH/mV
59
4
10
1
−5
PH M = 59 PH ( 4
)
×
≈
10
PH
9
59
10 + 10
Scale sensitivity :
Sensors and Interfacing
Loading effect
4
Buffer Amplifier
→ Buffer Amplifier
PH M
1012
10 4
1
= 59 × 12
×
×
= 0.998003
9
4
10 + 10 10 + 10 59
Loading error: -0.002 PH
Sensors and Interfacing
Loading effect
5
Loading Effect of Potentiometer
• The fraction displacement: x = d/dT
• total resistance: RP Ω
Open circuit voltage
E th R p x
across the output
• E th = ?
=
⇒ E th = Vs x
Vs
Rp
thermals AB
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Loading effect
6
Loading Effect of Potentiometer
• R th =?
Vs = 0
⎧
⎨
⎩internal impedance = 0
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Loading effect
7
Loading Effect of Potentiometer
1
1
1
=
+
R th R p x R p (1 − x)
R th =
R p (1 - x)R p x
R p (x + 1 − x)
= R p x(1 - x)
RL
RL
1
• VL = E th
= Vs x
= Vs x
Rp
R L + R th
R p x(1 − x) + R L
x(1 − x) + 1
RL
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Loading effect
8
Loading Effect of Potentiometer
• Loading effect
the relationship between VL and x is non-linear,
the amount of non-linearity depending on the
ratio RP/PL
N(x) = E th − VL = Vs x{1 −
Sensors and Interfacing
1
}
Rp
( )x(1 − x) + 1
RL
Loading effect
9
Loading Effect of Potentiometer
Sensors and Interfacing
Loading effect
10
Loading Effect of Potentiometer
™Design
if
Rp
RL
≤1
2
x (1 - x)(
Rp
)
RL
N(x) = Vs {
}
Rp
1 + ( )x(1 − x)
RL
N(x) ≈ Vs (
Rp
RL
N(x) has a maximum value of
)(x 2 − x 3 )
Rp
4
ˆ = Vs ( )
N
27 , R L
when x = 2/3
as a percentage of full-scale deflection
Rp o
Rp
400
ˆ =
≈ 15
percent
N
o
RL
27 R L
Sensors and Interfacing
Loading effect
11
Loading Effect of Potentiometer
N̂ = 2 ﹪
dT = 10 cm
RL = 10 K Ω
RP
20
→ 15
≤ 2 ⇒ R P ≤ × 103 Ω
RL
15
1K potentiometer (=RP)
dVL
≈ Vs → the greater VS, the higher sensitivity
dx
But considering the power dissipation
⇒ VS ≤ 0.1× 103 ≤ 10V
2
VS
≤ 0.1W
RP
→ Sensitivity = 1.0 Vcm-1
Sensors and Interfacing
Loading effect
12
Norton equivalent circuit
•
Norton equivalent circuit
ZN: the impedance looking back into the output terminals with all
voltage source reduced to zero and replaced by their internal
impedance.
iN: the current which flows when the terminals are short circuited.
Sensors and Interfacing
Loading effect
13
Norton equivalent ckt
VL = i N Z ⎫
ZN ZL
⎪
1
1
1 ⎬ VL = i N
=
+
ZN + ZL
⎪
Z ZN ZL ⎭
ZL << ZN, VL→iNZL
maximum current through the load.
Sensors and Interfacing
Loading effect
14
Differential Pressure Transmitter
Output: 4 ~ 20 mA current
Input: differential pressure 0~2 × 104 Pa N/m2
Sensors and Interfacing
Loading effect
15
Differential Pressure Transmitter
VL = i N
R N (R C + R R )
R N + RC + RR
total load = RC+RR
VR
RR
=
VL R C + R R
VR = i N R R
RN
= 0.995 i N R R
R N + RC + RR
™ the recorded voltage derivates from the desired range of 1 to 5 volts by
0.05 ﹪
Sensors and Interfacing
Loading effect
16
Piezoelectric Force Measurement
dt
1
1
= C N s + C Cs +
Z
RL
RL
Z=
(C N + CC )R Ls + 1
Influence the
ΔVL (s)
RL
=
Δi N (s) (C N + CC )R Ls + 1 dynamic
characteristics
Sensors and Interfacing
Loading effect
17
Process Loading
Process F = kPx+FS
FS = ksx
FS
) + Fs
kS
k p + kS
= Fs (
)
kS
F = kp (
kS
FS =
F
k p + kS
We want : ks>> kP in
order to minimize
loading error.
Sensors and Interfacing
Loading effect
18
Mechanical impedance
d 2x
dx
F =m
+ λ + kx
dt
dt
•
ΔF (s) = (ms + λ + k )Δx(s)
s
Mechanical impedance Z m =
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ΔF
•
Δx
( s ) = ms + λ + k
Loading effect
s
19
Equivalent Ckt
i
L
V
1
ΔV ( s ) = ( Ls + R + )Δi ( s )
Cs
R
Electrical impedance
m
⎧
⎫
m
→
L
⎪⎪
⎪⎪
⎨λ → R ⎬
⎪k → 1 ⎪
⎪⎩
C ⎪⎭
x&
F
di
1
V = L + Ri + ∫ idt
dt
C
1
k
λ
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ZE =
ΔV
1
( s ) = Ls + R +
Δi
Cs
Equivalent Ckt
Loading effect
20
General Transducer
•
i
x
V
F
•
i
x
F
ZM
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E th
Z th
i
•
x
V
F
ZM
Loading effect
i N ZN
V
21
Process Loading
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Loading effect
22
Process Loading
•
••
process F - k x − λ x − F = m x
p
p
•
S
••
p
F −k x −λ x = m x
S
m
•
S
S
s
dx
+ λ x + k ∫ x dt = F − F
dt
•
p
•
p
p
S
•
dx
m
+ λ x + k ∫ x dt =F
dt
k
(mps + λ + )Δ x = ΔF - ΔF
S
k
(mss + λ + )Δ x = ΔF
S
k
Process impedance Z MP (s) = m ps + λ P + P
S
k
Sensor impedance Z MS (s) = mSs + λ S + S
S
Z MS
ΔFS (s) =
ΔF(s)
Z MS + Z MP
•
s
•
S
S
S
•
p
P
S
•
S
S
Sensors and Interfacing
S
Loading effect
23