Chapter 13. Wheatstone Bridge and Sensors
Recommended Article : [Circuit Theory] Table of Contents
3. Applications of Wheatstone Bridge Circuit
1. Wheatstone Bridge Circuit
⑴ (Conceptual Distinction) Ballast circuit
Figure 1. Ballast circuit
① Mathematical Analysis
② Drawbacks
○ Inaccurate zero position possible
○ Temperature compensation not possible
○ Poor sensitivity & accuracy
⑵ Wheatstone Bridge Circuit : Proposed for improving the Ballast circuit
Figure 2. Overview of Wheatstone Bridge
Let’s assume that a resistance R is connected between terminals A and B.
Let V be the voltage of the battery, VA be the voltage at node A, and VB be the voltage at node B.
Then, we can obtain the following KCL equation.
Solving the above equations, we get the following results. (Where 1/R is defined as G.)
If terminals A-B are open (G = 0), the voltages at each node are as follows.
This is consistent with the expected voltage division across each series circuit (R1-R3 loop, R2-R4 loop).
In this case, the currents are as follows.
If terminals A-B are shorted (G = ∞), the voltages at each node are as follows.
This is the voltage exactly as expected in a series circuit of two parallel circuits.
Since the method to calculate currents in each case is the same, we will omit it.
When R takes an arbitrary value, the current flowing from node A to node B is as follows.
Note that the denominator is always positive.
Let’s assume an ammeter with an internal resistance R is connected between terminals A-B.
Let’s say if the absolute value of current I exceeds a certain value, the ammeter needle points to full scale.
Therefore, the condition where the ammeter needle does not move (balanced Wheatstone bridge) is as follows.
This conclusion also holds for AC bridge circuits.
⑶ Arrangement of Wheatstone Bridge : Explained with a focus on strain gauges
① Quarter bridge, half bridge (one in tension, one in compression), full bridge (two in tension, two in compression)
② Possion arrangement : Half bridge, full bridge, etc., are available
③ Dummy method : Arranging strain sensors that are not subjected to strain to eliminate factors like temperature
④ Gauge method : Increasing sensitivity by allowing factors like temperature to affect all sensors and eliminating those factors
⑷ Shunt calibration : Connected in parallel with Rc
Figure 3. Shunt calibration
2. Sensor Physics
⑴ Piezoelectric Effect
① Gauge factor : The resistance value of a metal coil increases with length expansion in a strain gauge
② Sauerbrey equation
Figure 4. Piezoelectric Effect
○ f0 : Resonant frequency (Hz)
○ Δf : Frequency change (Hz)
○ Δm : Loaded mass (g)
○ A : Piezoelectrically active area (electrode area) of crystal (cm²)
○ ρq : Density of quartz (2.648 g/cm³)
○ μq : Shear modulus of AT-cut quartz crystal (2.947 × 10¹¹ g/cm³·s²)
○ vq : Transverse wave velocity in quartz (m/s)
⑵ Pyroelectric Effect
⑶ Hall Effect
⑷ Photoconductive Effect
⑸ Magnetoelastic Effect : Phenomenon where the elasticity of a material changes due to a magnetic field
Figure 5. Magnetoelastic Effect
① E : Modulus of elasticity
② σ : Poisson’s ratio
③ P : Mass density of the sensor material
④ L : Longitudinal dimension of the sensor
⑤ f : Initial resonance frequency
⑥ M : Initial mass
⑹ Magnetoresistance Effect
⑺ Magneto-optic Effect
3. Applications of Wheatstone Bridge Circuit
⑴ Smoke Detector
Figure 6. Smoke Detector
A smoke detector is a sensor that detects fires to activate sprinklers.
Every public building should have one on every room ceiling!
Examining the detailed structure of this device, we find the following.
Figure 7. Detailed Structure of Smoke Detector
Two variable resistors that change their resistance value according to the light intensity are built-in.
The light from the light source is applied to the variable resistors through two mirrors (reflectors).
(In other words, the light from the light source spreads radiantly, but only the light incident at a specific angle can reach each resistor.)
If a fire occurs and smoke is generated, the amount of light reflected on the bottom mirror decreases.
This is because the light is scattered by smoke particles.
Thus, the difference in resistance values between the reference cell and the smoke detector increases significantly, triggering a fire alarm.
The circuit representation of this setup is as follows.
Figure 8. Detailed Circuit Diagram of Wheatstone Bridge
Let’s assume the voltage of the DC power supply is V.
If we let Rbalance = R, then Vbalance is as follows.
Let’s now consider the following facts.
a. Photoconductive cells have a linearly changing resistance with light intensity.
b. RReference = R
As a result, the output voltage Vbalance is proportional to the change in resistance value.
The reason for setting Rbalance as a variable resistor is to compensate for manufacturing errors.
⑵ Silicon Pressure Sensor
The pressure sensor utilizes the piezoresistive effect where resistance value changes due to external force.
The shape of the pressure sensor is shown below.
Figure 9. Shape of Pressure Sensor
For convenience, let’s name the resistances in a clockwise direction from the 9 o’clock position as R1, R2, R3, R4.
The right diagram is a top view of the left diagram.
In the right diagram, the length of the thick part is about 0.5 mm, and the vertical length of the thin diaphragm is about 1-2 nm.
The horizontal length of the thin diaphragm is in millimeters.
Silicon is used for the pressure sensor because it etches easily in KOH due to its good crystal structure.
The vertical stress (longitudinal stress) of R1 and R3 is the same as the trans
verse stress of R2 and R4.
The transverse stress of R1 and R3 is the same as the longitudinal stress of R2 and R4.
Figure 10. Material and Stress Representation
If one resistance receives a vertical stress of σl, it also receives a transverse stress of σt = -νσl.
Here, ν is the Poisson’s ratio.
Figure 11. Tensile Stress and Poisson’s Ratio of Material[Note:9]
Note that strain (ε) is a strain ratio, calculated by dividing the change in length (δ) by the initial length.
Figure 12. Meaning of Strain
Figure 13. Strain-Stress Curve
The resistance value of the strain gauge has properties similar to length.
In other words, just like strain is proportional to stress (σ = Eε, E : Young’s modulus), ΔR / R is also proportional to stress.
Similarly, the transverse stress and longitudinal stress of R1 and R3 independently affect ΔR / R.
Therefore, we can express the changes in R1 and R3 due to external stress as follows.
Similarly, we can express the changes in R2 and R4 due to external stress as follows.
One product has the values of the following constants.
Thus,
(Note : There may be errors in calculations.)
As a result, we can express this as follows.
Let’s represent this sensor in a circuit.
Figure 14. Wheatstone Bridge Circuit for the Given Scenario
In this case, due to the small values of α1 and α2 (0.02 or smaller), and the fact that there is at most a 10% difference,
Since external pressure and stress are proportional, this sensor outputs a value proportional to pressure.
⑶ Strain Gauge Bridge
A transducer that measures Strain induced by force is designed as follows.
Figure 15. Structure of Strain Gauge Bridge[Note:12]
Blue or yellow boxes represent piezoelectric elements (proportional to stress) where resistance values change according to Strain.
When force is applied vertically and horizontally, the device deforms as follows.
Figure 16. Operating Principle of Strain Gauge Bridge
The area where the blue box is placed becomes concave, exerting compressive force on the resistance marked by the blue box.
Similarly, the area where the yellow box is lifted becomes convex, exerting tensile force on the resistance marked by the yellow box.
As a result, the resistance value of the blue-boxed resistance decreases, and the resistance value of the yellow-boxed resistance increases.
Let’s represent this setup in a circuit.
Figure 17. Wheatstone Bridge Circuit for the Given Problem Scenario[Note:13]
This circuit has already been discussed extensively, so let’s consider the following circuit.
Figure 18. Application of Wheatstone Bridge Circuit[Note:14]
By examining the bridge circuit, we can find vi.
Therefore, due to the dependent voltage source, the voltage between the wire ends with the 50 Ω resistor and the voltmeter is ΔR / R × 50b mV.
However, since the internal resistance of the DMM is around 10-11 MΩ, an approximation can be made as follows.
As a result, a circuit that amplifies the value due to external Stress has been completed.
⑷ Murray’s Loop Method
One of the methods for electrical fault detection, using the principles of Wheatstone Bridge to detect faults (1-wire short circuit) on a circuit.
This method requires a healthy auxiliary wire 1-wire.
Figure 19. Murray’s Loop Method Diagram[Note:15]
If no current flows in the ammeter, it is in equilibrium state.
However, L : Total length of the line (m), x : Distance from measurement point to fault point (m)
The Murray’s Loop Method is a measurement method for 1-wire short circuit and short circuit between wires.
The measurement method for 3-wire short circuit and short circuit between wires is the pulse measurement method (pulse radar).
The measurement method for single-wire fault is the capacitive bridge method.
Input: 2016.01.06 22:05
Modified: 2020.09.04 00:08