Lecture 4: Sensor interface circuits
g Review of circuit theory
n Voltage, current and resistance
n Capacitance and inductance
n Complex number representations
g Measurement of resistance
n Voltage dividers
n Wheatstone Bridge
n Temperature compensation for strain gauges
g AC bridges
n Measurement of capacitance
n Measurement of impedance
Intelligent Sensor Systems
Ricardo Gutierrez-Osuna
Wright State University
1
Voltage, current, resistance and power
g Voltage
n The voltage between two points is the energy required to move a unit of positive
charge from a lower to a higher potential. Voltage is measured in Volts (V)
g Current
n Current is the rate of electric charge through a point. The unit of measure is the
Ampere or Amp (A)
g Resistance
n Given a piece of conducting material connected to a voltage difference V, which
drives through it a current I, the resistance is defined as
R=
V
I
g As you will recall, this is known as Ohm’s Law
g An element whose resistance is constant for all values of V is called an ohmic resistor
n Series and parallel resistors…
g Power
n The power dissipated by a resistor is
V2 2
P = VI =
=I R
R
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Wright State University
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Kirchhoff’s Laws
g 1st Law (for nodes)
n The algebraic sum of the currents into any node of a circuit is zero
g Or, the sum of the currents entering equals the sum of the currents leaving
g Thus, elements in series have the same current flowing through them
I2
I1
I1=I2+I3
I3
g 2nd Law (for loops)
n The algebraic sum of voltages in a loop is zero
g Thus, elements in parallel have the same voltage across them.
VA
VB
VAB+ VBC+VCD+VDA=0
VD
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Ricardo Gutierrez-Osuna
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VC
3
Capacitors and inductors
g A capacitor is an element capable of storing charge
n The amount of charge is proportional to the voltage across the capacitor
Q = CV
g C is known as the capacitance (measured in Farads)
n Taking derivatives
dV
dQ d(CV )
=
⇒I= C
dt
dt
dt
n Therefore, a capacitor is an element whose rate of voltage change is
proportional to the current through it
g Similarly, an inductor is an element whose rate of current
change is proportional to the voltage applied across it
dI
V =L
dt
g L is called the inductance and is measured in Henrys
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Wright State University
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Frequency analysis
g Consider a capacitor driven by a sine wave voltage
I(t)
C
V(t)=V0sin(ωt)
n The current through the capacitor is
I=C
dV
d
= C (V0 sin( W )) = C V0cos( t)
dt
dt
g Therefore, the current phase-leads the voltage by 900 and the ratio of
amplitudes is
V(t)
I(t)
=
1
C
g What happens when the voltage is a DC source?
Intelligent Sensor Systems
Ricardo Gutierrez-Osuna
Wright State University
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Voltages as complex numbers
g At this point it is convenient to switch to a complex-number
representation of signals
n Recall that ejϕ=cosϕ+jsin ϕ
V0cos(ωt+ϕ)
⇓
V0ejϕ
Circuit voltage
versus time:
V(t)=V0cos(ωt+ϕ)
Multiply by ejϕ and
take real part
Complex number
representation:
V0ejϕ=a+jb
From [HH89]
g Applying this to the capacitor V(t)/I(t) relationship
dV
= C V0cos( t)
dt
⇓
1
V
V0 sin( W)
cos( W − ) e − j
=
=
=
=
C
C H− j
I C V0cos( t)
C e j0
I=C
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Ricardo Gutierrez-Osuna
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=
1
j C
6
Impedance
g Impedance (Z) is a generalization of resistance for circuits that
have capacitors and inductors
n Capacitors and inductors have reactance, while resistors have
resistance
1
-j
ZC =
j &
ZL = j /
=
&
ZR = R
g Ohm’s Law generalized
V
=Z
I
g Impedance in series and parallel
Z S = Z1 + Z 2 + L ZN
1
1
1
1
=
+
+L+
ZP Z1 Z 2
ZN
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Wright State University
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Example: High-pass filter
g High pass filter
C
n The current through cap and resistor is
I=
Vout
Vin
Vin
Vin
=
Z R+ 1
j &
R
n The output voltage is equal to the voltage differential across the resistor
Vout = RI = R
Vin
R+
1
j
&
n If we focus on amplitude and ignore phase
Vout = R
Vin
R+
1
j
&
=R
Vin
 1 
R +

&


2
2
= Vin
RC
(
RC ) + 1
2
g Asymptotic behavior…
V
1
1
g Corner frequency CORNER =
⇒ 20log10 out = 20log10
= -3.010 dB
RC
Vin
1+ 1
Intelligent Sensor Systems
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Wright State University
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Measurement circuits
g Resistance measurements
n Voltage divider (half-bridge)
n Wheatstone bridge
g A.C. bridges
n Measurement of capacitance
n Measurement of impedance
Intelligent Sensor Systems
Ricardo Gutierrez-Osuna
Wright State University
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Voltage divider
g Assumptions
VCC
n Interested in measuring the fractional change in
resistance x of the sensor: RS=R0(1+x)
RL= R0k
g R0 is the sensor resistance in the absence of a stimuli
Vout
n Load resistor expressed as RL=R0k for convenience
RS= R0(1+x)
g The output voltage of the circuit is
RS
=
R S + RL
R 0 (1+ x )
1+ x
= VCC
= VCC
R0 (1+ x ) + R 0k
1+ x + k
g Questions
n What if we reverse RS and RL?
n How can we recover RS from Vout?
k=0.1
0.9
0.8
0.7
Vout/ VCC
Vout = VCC
k=1
0.6
0.5
0.4
0.3
0.2
k=10
0.1
1
2
3
4
5
6
7
8
9
10
x
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Wright State University
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Voltage divider
g What is the sensitivity of this
circuit?
dVout
1+ x 
d 
=
 VCC
=
1+ x + k 
dx
dx 
(1+ x + k ) − (1+ x ) =
= VCC
(1+ x + k )2
k
= VCC
(1+ x + k )2
0.2
S=
0.15
k=1
S
0.1
0.05
k=10
k=0.1
1
2
3
4
5
6
7
8
9
10
x
g For which RL do we achieve
maximum sensitivity?
2

dS
d 
k
1+ x + k ) − k2(1+ x + k )
(
 VCC
=0⇒
=0⇒
= 0 ⇒ k = 1+ x
2 
2
dk
dk 
(1+ x + k )
(1+ x + k ) 
n This is, the sensitivity is maximum
when RL=RS
Intelligent Sensor Systems
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Wright State University
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Wheatstone bridge
g A circuit that consists of two dividers
n A reference voltage divider (left)
n A sensor voltage divider
R1
g Wheatstone bridge operating modes VCC
n Null mode
g R4 adjusted until the balance condition is
met:
Vout = 0 ⇔ R 3 = R 4
R2
Vout
R4
R3= R0(1+x)
R2
R1
n Advantage: measurement is independent
of fluctuations in VCC
n Deflection mode
g The unbalanced voltage Vout is used as
the output of the circuit
Vout
 R3
R4 


= VCC 
−
 R2 + R 3 R 3 + R 4 
n Advantage: speed
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Wright State University
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Wheatstone bridge
g Assumptions
R
R
k= 1 = 2
R 4 R0
0.35
k=1
0.3
Vout/ VCC
n Want to measure sensor fractional
resistance changes RS=R0(1+x)
n Bridge is operating near the
balance condition:
0.4
k=10
0.25
0.2
0.15
0.1
k=0.1
0.05
1
2
3
4
g The output voltage becomes
5
x
6
7
8
9
10

 R0 (1+ x )
R4
 =
Vout = VCC 
−
 R0k + R0 (1+ x ) R 4k + R 4 
 (1+ x )
kx
1 
 = VCC
−
= VCC 
(1+ k )(1+ k + x )
 k + (1+ x ) k + 1
Intelligent Sensor Systems
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Wright State University
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Wheatstone bridge
g What is the sensitivity of the Wheatstone bridge?

dVout
d 
kx
 =

= VCC
dx
dx  (1+ k )(1+ k + x ) 
k (1+ k )(1+ k + x ) − kx (1+ k )
= VCC
=
2
2
(1+ k ) (1+ k + x )
k
= VCC
(1+ k + x )2
S=
n The sensitivity of the Wheatstone bridge is the same as that of a
voltage divider
g You can think of the Wheatstone bridge as a DC offset removal circuit
g So what are the advantages, if any, of the Wheatstone bridge?
Intelligent Sensor Systems
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Wright State University
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Voltage divider vs. Wheatstone for small x
g The figures below show the output of both circuits for small fractional
resistance changes
n The voltage divider has a large DC offset compared to the voltage swing, which
makes the curves look “flat” (zero sensitivity)
g Imagine measuring the height of a person standing on top of a tall building by running
a large tape measure from the street
n The sensitivity of both circuits is the same!
g However, the Wheatstone bridge sensitivity can be boosted with a gain stage
n
Assuming that our DAQ hardware dynamic range is 0-5VDC, 0<x<0.01 and k=1, estimate the
maximum gain that could be applied to each circuit
1
2.5
k=0.1
0.6
-3
2
Wheatstone
Divider
0.8
x 10
k=1
0.4
0.2
k=1
1.5
1
k=10
0.5
k=0.1
k=10
0
0
0
0.002
0.004
0.006
x
Intelligent Sensor Systems
Ricardo Gutierrez-Osuna
Wright State University
0.008
0.01
0
0.002
0.004
0.006
0.008
0.01
x
15
Compensation in a Wheatstone bridge
g Strain gauges are quite sensitive to temperature
n A Wheatstone bridge and a dummy strain gauge may be used to
compensate for this effect
g The “active” gauge RA is subject to temperature (x) and strain (y) stimuli
g The dummy gauge RD, placed near the “active”gauge, is only subject to
temperature
n The gauges are arranged according to the figures below
n The effect of (1+y) on the right divider cancels out
R0
VCC
RD= R0(1+y)
Vout
R0
RA= R0(1+x)(1+y)
From [Ram96]
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Wright State University
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AC bridges
g The structure of the Wheatstone bridge can be used to measure
capacitive and inductive sensors
n Resistance replaced by generalized impedance
n DC bridge excitation replaced by an AC source
g The balance condition becomes
Z1 Z 2
=
Z4 Z3
Z1
VAC
n which yields two equalities, for real and
imaginary components
Z2
Vout
Z4
Z3
R1R 3 − X1X 3 = R 2R 4 − X 2 X 4
R1X 3 + X1R 3 = R 2 X 4 + X 2R 4
g There is a large number of AC bridge arrangements
n These are named after their respective developer
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Wright State University
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AC bridges
g Capacitance measurement
n Schering bridge
n Wien bridge
g Inductance measurement
n Hay bridge
n Owen bridge
C1
C1
R2
R1
VAC
R3
R1
VAC
Rx
C4
R4
Cx
Schering
Rx
Lx
Hay
R1
R2
C1
VAC
R3
VAC
Rx
R4
C4
Wien
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R4
Cx
Rx
Lx
Owen
18
References
[HH89]
[Ram96]
[Maz87]
[Die72]
[PAW91]
[Gar94]
[Fdn97]
P. Horowitz and W. Hill, 1989, The Art of Electronics, 2nd
Ed., Cambridge University Press, Cambridge, UK
D. C. Ramsay, 1996, Principles of Engineering
Instrumentation, Arnold, London, UK
F. F. Mazda, 1987, Electronic instruments and
measurement techniques, Cambridge Univ. Pr., New York
A. J. Diefenderfer, 1972, Principles of electronic
instrumentation, W. B. Sanuders Co., Philadelphia, PA.
R. Pallas-Areny and J. G. Webster, 1991, Sensors and
Signal Conditioning, Wiley, New York
J. W. Gardner, 1994, Microsensors. Principles and
Applications, Wiley, New York.
J. Fraden, 1997, Handbook of Modern Sensors. Physics,
Designs and Applications, AIP, Woodbury, NY
Intelligent Sensor Systems
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Wright State University
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