7 Variable conversion elements

 Outputs from measurement sensors that take the form of voltage signals can be measured using the voltage indicating and test instruments discussed in the last chapter. However, in many cases, the sensor output does not take the form of an electrical voltage. Examples of these other forms of sensor output include translational displacements and changes in various electrical parameters such as resistance, inductance, capacitance and current. In some cases, the output may alternatively take the form of variations in the phase or frequency of an a.c. signal.

For sensor outputs that are initially in some non-voltage form, conversion to a measurement signal that is in a more convenient form can be achieved by various types of variable conversion element in the measurement system. Bridge circuits are a particularly important type of variable conversion element, and these will be covered in some detail. Following this, the various alternative techniques for transducing the outputs of a measurement sensor will be covered.

 

7.1 Bridge circuits

Bridge circuits are used very commonly as a variable conversion element in measurement systems and produce an output in the form of a voltage level that changes as the measured physical quantity changes. They provide an accurate method of measuring resistance, inductance and capacitance values, and enable the detection of very small changes in these quantities about a nominal value. They are of immense importance in measurement system technology because so many transducers measuring physical quantities have an output that is expressed as a change in resistance, inductance or capacitance. The displacement-measuring strain gauge, which has a varying resistance output, is but one example of this class of transducers. Normally, excitation of the bridge is by a d.c. voltage for resistance measurement and by an a.c. voltage for inductance or capacitance measurement. Both null and deflection types of bridge exist, and, in a like manner to instruments in general, null types are mainly employed for calibration purposes and deflection types are used within closed-loop automatic control schemes.

7.1.1 Null-type, d.c. bridge (Wheatstone bridge)

A null-type bridge with d.c. excitation, commonly known as a Wheatstone bridge, has the form shown in Figure 7.1. The four arms of the bridge consist of the unknown resistance R_u, two equal value resistors R₂ and R₃ and a variable resistor R_v (usually a decade resistance box). A d.c. voltage V_i is applied across the points AC and the resistance R_v is varied until the voltage measured across points BD is zero. This null point is usually measured with a high sensitivity galvanometer.

To analyse the Whetstone bridge, define the current flowing in each arm to be I₁ ...I₄ as shown in Figure 7.1. Normally, if a high impedance voltage-measuring instrument is used, the current I_m drawn by the measuring instrument will be very small and can be approximated to zero. If this assumption is made, then, for I_m = 0:

                                                              I₁ = I₃ and I₂ = I₄

Looking at path ADC, we have a voltage V_i applied across a resistance R_u + R₃ and by Ohm’s law:

                                                    I₁ = V_i/(R_u + R₃)

Similarly for path ABC:

                                                    I₂ = V_i/(R_v + R₂)

Now we can calculate the voltage drop across AD and AB:

          V_AD = I₁R_v = V_iR_u/(R_u + R₃) ;    V_AB = I₂R_v = V_iR_v/(R_v + R₂)

By the principle of superposition,

                          V₀ = V_BD = V_BA + V_AD = - V_AB + V_AD

Thus:

                         V₀ = - V_iR_v/(R_v + R₂) + V_iR_u/(R_u + R₃)                 (7.1)

At the null point V₀ = 0, so:

                                 R_u/(R_u + R₃) = R_v/(R_v + R₂)

Inverting both sides:

(R_u + R₃)/R_u = (R_v + R₂)/Rv  i.e. R₃/R_u = R₂/R_v or R_u = R₃R_v/R₂     (7.2)

Thus, if R₂ = R₃, then R_u = R_v. As R_v is an accurately known value because it is derived from a variable decade resistance box, this means that R_u is also accurately known.