7 Variable conversion elements

 

7.1.3 Error analysis

In the application of bridge circuits, the contribution of component-value tolerances to total measurement system accuracy limits must be clearly understood. The analysis below applies to a null-type (Wheatstone) bridge, but similar principles can be applied for a deflection-type bridge. The maximum measurement error is determined by first finding the value of R_u in equation (7.2) with each parameter in the equation set at that limit of its tolerance which produces the maximum value of R_u. Similarly, the minimum possible value of R_u is calculated, and the required error band is then the span between these maximum and minimum values.

Example 7.3

In the Wheatstone bridge circuit of Figure 7.1, R_v is a decade resistance box with a specified inaccuracy ±0.2% and R₂ = R₃ = 500 Ω ± 0.1%. If the value of R_v at the null position is 520.4 Ω, determine the error band for R_u expressed as a percentage of its nominal value.

The cumulative effect of errors in individual bridge circuit components is clearly seen. Although the maximum error in any one component is ±0.2%, the possible error in the measured value of R_u is ±0.4%. Such a magnitude of error is often not acceptable, and special measures are taken to overcome the introduction of error by component-value tolerances. One such practical measure is the introduction of apex balancing. This is one of many methods of bridge balancing that all produce a similar result.

Apex balancing

One form of apex balancing consists of placing an additional variable resistor R₅ at the junction C between the resistances R₂ and R₃, and applying the excitation voltage V_i to the wiper of this variable resistance, as shown in Figure 7.6.

For calibration purposes, R_u and R_v are replaced by two equal resistances whose values are accurately known, and R₅ is varied until the output voltage V₀ is zero. At this point, if the portions of resistance on either side of the wiper on R₅ are R₆ and R₇ (such that R₅ = R₆ + R₇), we can write:

                                    R₃ + R₆ = R₂ + R₇

We have thus eliminated any source of error due to the tolerance in the value of R₂ and R₃, and the error in the measured value of R_u depends only on the accuracy of one component, the decade resistance box R_v.

Example 7.4

A potentiometer R5 is put into the apex of the bridge shown in Figure 7.6 to balance the circuit. The bridge components have the following values:

Determine the required value of the resistances R₆ and R₇ of the parts of the potentiometer track either side of the slider in order to balance the bridge and compensate for the unequal values of R₂ and R₃.

Solution

For balance, R₂ + R₇ = R₃ + R₆; hence, 515 + R₇ = 480 + R₆

Also, because R₆ and R₇ are the two parts of the potentiometer track R₅ whose resistance is 100 Ω:

R₆ + R₇ = 100; thus 515 + R₇ = 480 +(100 - R₇); i.e. 2R₇ = 580 - 515 = 65 Thus, R₇ = 32.5; hence, R₆ = 100 - 32.5 = 67.5 Ω.

 

7.1.4 A.c. bridges

Bridges with a.c. excitation are used to measure unknown impedances. As for d.c. bridges, both null and deflection types exist, with null types being generally reserved for calibration duties.

Null-type impedance bridge

A typical null-type impedance bridge is shown in Figure 7.7. The null point can be conveniently detected by monitoring the output with a pair of headphones connected via an operational amplifier across the points BD. This is a much cheaper method of null detection than the application of an expensive galvanometer that is required for a d.c. Wheatstone bridge.

Referring to Figure 7.7, at the null point,

If Z_u is capacitive, i.e. Z_u = 1/jωC_u, then Z_v must consist of a variable capacitance box, which is readily available. If Z_u is inductive, then Z_u = R_u + jωL_u.

Notice that the expression for Z_u as an inductive impedance has a resistive term in it because it is impossible to realize a pure inductor. An inductor coil always has a resistive component, though this is made as small as possible by designing the coil to have a high Q factor (Q factor is the ratio inductance/resistance). Therefore, Z_v must consist of a variable-resistance box and a variable-inductance box. However, the latter are not readily available because it is difficult and hence expensive to manufacture a set of fixed value inductors to make up a variable-inductance box. For this reason, an alternative kind of null-type bridge circuit, known as the Maxwell bridge, is commonly used to measure unknown inductances.

Thus, the Maxwell bridge can be used to measure the Q value of a coil directly using this relationship.

Example 7.5

In the Maxwell bridge shown in Figure 7.8, let the fixed-value bridge components have the following values: R₃ = 5 Ω; C = 1 mF. Calculate the value of the unknown impedance (L_u, R_u) if R₁ = 159 Ω and R₂ = 10 Ω at balance.

Deflection-type a.c. bridge

A common deflection type of a.c. bridge circuit is shown in Figure 7.9. For capacitance measurement:

                          Z_u = 1/jωC_u; Z1 = 1/jωC₁

For inductance measurement (making the simplification that the resistive component of the inductor is small and approximates to zero):

                                      Z_u = jωL_u; Z₁ = jωL₁ 

Analysis of the circuit to find the relationship between V0 and Z_u is greatly simplified if one assumes that I_m is negligible. This is valid provided that the instrument measuring V₀ has a high impedance. For I_m = 0, currents in the two branches of the bridge, as defined in Figure 7.9, are given by:

This latter relationship (7.15) is in practice only approximate since inductive impedances are never pure inductances as assumed but always contain a finite resistance (i.e. Z_u = jωL_u + R). However, the approximation is valid in many circumstances.