ME483
Wheatstone Half-Bridge Calibration
Wheatstone Half-Bridge for Strain Measurement
Background
The strain lab uses two strain gauges mounted on the top and bottom surface of a beam in bending to
measure the surface strain due to bending. These two strain gauges are wired in a half-bridge
configuration to measure the strain on the beam. This document describes the theoretical background
behind this measurement system.
Wheatstone Bridge
A Wheatstone bridge consists of 4 resistors connected in a
configuration as shown on the right. Across two opposite
connection points (C and D) a voltage Ei is applied (also
called the excitation voltage).
i0 ≈ 0
i2
R2
Eo
-
R4
i2
i2
B
-
+
Ei
0 , therefore
Ei
R2
+
C
D
Using Kirchhoff’s 2nd law, which states that the sum of the
voltage in a closed loop has to be zero, we can write for the
loop, which goes from the excitation voltage through
resistors 2 and 4:
R4 )i2
i1
R3
R1
We can make the following assumption:
i0 ≈ 0 (high impedance voltmeter)
Ei ( R2
A
i1
Figure 1: Wheat-stone bridge circuit
R4
For the loop going through resistors 1 and 3 we get
Ei
( R1
R3 )i1
0
; therefore
i1
Ei
R1
R3
From these equations the unknown currents have been determined. However, we are interested in the
voltage difference between points C and D ( the output voltage of the bridge, E0), which we want to relate
to resistance changes in the bridge. This can be found from a 3rd voltage loop across resistors 3 and 4 and
E0.
E0
R4 i2
R3 i1
0
Solving for the output voltage and substituting the earlier results for the currents gives an equation for the
bridge output voltage as a function of the resistors and the input excitation voltage Ei.
E0
R3
R1
R4
R3
R2
R4
Ei or
E0
Ei
R3
R1
R4
R3
R2
R4
This equation can be used to determine, how a resistance change in the bridge changes the output voltage.
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ME483
Wheatstone Half-Bridge Calibration
A “balanced bridge” has zero output voltage. Balancing is achieved by choosing resistors that make the
expression
R3
R1
R4
R3
This is equivalent to
R2
R4
1
R1
R3
go to zero.
1
1
R2
R4
0
or
1
R1
R3
R2
R4
R1
R2
or
R3
R4
The equations show that for balance the ratio of resistances across neighboring arms have to be equal.
Wheatstone Half-Bridge with two strain gauges
The Wheatstone bridge is now modified so that resistor 2 and 4 are strain gauges (choosing 1 and 3
would be an equivalent choice, the only requirement is that the resistors are in series with the
excitation voltage). This is called the half-bridge configuration. Another half-bridge configuration
would be possible by choosing resistors 1 and 2 as strain gauges. This would lead to a different set of
equations and is not considered here.
A strain gauge produces a resistance change proportional to strain. The proportionality factor GF is called
the gauge factor and depends on the strain gauge material.
R
R
GF
L
L
GF
We can therefore write for the resistance change in the top
and bottom strain gauge
R2
R4
RG 0
RG 0
R
R
RG (1 GF )
RG (1 GF )
where RG0 is the nominal (unstrained) gauge resistance.
Assume that Rm is the nominal value of R1 and R3.
Substitution of these results into the equation for the
bridge output voltage yields
E0
Ei
Rm
Rm
Rm
1 1 GF
2
2
RG 0 (1 GF )
RG 0 (1 GF ) RG 0 (1 GF )
GF
2
thus
E0
Ei
F
RG=RG0+ R
GF
2
This equation is linear in the strain and only depends on
the gauge factor independent of the resistor values of the
strain gauges or the opposing resistors.
RG=RG0 - R
Figure 2: Strain gauge application on cantilever
beam
Rm
A
Rm
C
D
Eo
-
RG
RG
B
-
+
Ei
Figure 3: Half Bridge configuration with two
strain gauges with resistance RG
Note that since typical strains are small and excitation voltages have to be kept low to manage current
flow and power dissipation in the circuit, very small voltages have to be measured reliably.
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ME483
Wheatstone Half-Bridge Calibration
Effect of bridge imbalance
In practical applications the bridge is not perfectly balanced at zero strain. This can be due to several
reasons such as slightly different bridge resistors. The
resulting effect will be a voltage offset, which has to be
subtracted from the measured voltage. The subsequent
A
analysis shows the equations, when resistors and strain
RmRm+
gauges are different from the nominal values. Assume
C
that each resistor can be expressed as having a nominal
D
value plus a small deviation from nominal as follows
+
-
R1
Rm
R2
RG 0
R3
Rm
R4
RG 0
Rm (1
RG+
)
Eo
B
RG (1 GF
Rm (1
RG-
)
)
-
+
RG (1 GF
Ei
)
Figure 4: Half Bridge configuration with unequal
where and are the percentage deviations from the
resistors leading to bridge imbalance at zero
nominal values of the bridge resistors and the strain
strain
gauge resistance. Here it is assumed for positive and
, R1 is smaller than nominal, R2 is larger than nominal
and the respective opposed resistors have an equivalent offset in the opposite direction. The effect of
strain is also included in the strain gauge resistors.
Substitution into the bridge equation yields
E0
Ei
Rm (1 )
Rm (1 ) Rm (1
)
1
1 GF
2
2
or separated by terms
E0
Ei
RG 0 (1 GF
)
RG 0 (1 GF
) RG 0 (1 GF
)
GF
2
GF
2 
2

Strain
Bridge
Im balance
The imbalance contribution adds a constant shift to the bridge output voltage determined by the initial
imbalance of the resistors and the strain gauges. Note that the sign of can be changed by switching the
strain gauge connections. This can be utilized to minimize the imbalance.
Numerical examples
A typical value for gauge factors used in the lab is around 2; strains of the order of 10 microstrains or
less need to be resolved with a maximum strain around 1000 microstrain ( ).
E0
Ei
GF
2
2 *10 *10
2
6
10
5
With an excitation voltage of 2.5V, the output voltage change to detect 10 microstrains is only
E0
10 5 Ei
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10 5 * 2500 mV
0.025 mV
25 V
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ME483
Wheatstone Half-Bridge Calibration
The change in output due to a change in strain of 10
is thus only 25 V. At full range of
1000 the voltage change is 2500 V or 2.5mV. This implies that the A/D conversion has to be
quite accurate to capture the small voltage changes.
If a bridge imbalance is introduced with the strain gauges being 1% different ( =±0.01) in
resistance at the“unstrained condition” and the opposing resistors are 0.5% different in value
( =0.005), then the total bridge imbalance is either 1.5% or -0.5% depending on how the strain
gauges are connected into the bridge. The contribution from the imbalance would therefore be
E0
Ei
2
E0
Ei
2
0.0075
.0025
750 *10
250 *10
5
5
With an excitation voltage of 2.5V, the output voltage of the unstrained circuit due to imbalance would be
either
E0
E0
750 *10 5 Ei
250 *10 5 Ei
750 *10 5 * 2500 mV
250 *10 5 * 2500 mV
18.8 mV
6.25 mV
The numerical example shows the challenges of measuring strain due to the high resolution and
signal quality required to detect strain and thus voltage changes in the signal and the fact that an
unbalanced bridge can produce a rather large voltage offset, which has to be “nulled out”.
Labview implements “nulling” through a calibration step. During the calibration step the offset
voltage is measured and subtracted from all subsequent measurements.
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