Page 485 - Instrumentation Reference Book 3E

468 Electrical measurements

balance then involves balancing the bridge with by tan tJ and D by tan 6, where 5 is the loss angle.
the detector across BC by adjusting R6 and then Generally, the quality of an inductance is measured
balancing the bridge with the detector across AB by its Q factor and the quality of a capacitor by its
by adjusting &. The procedure is then repeated D value or loss angle.
until a balance is achieved under both conditions.
The first balance condition ensures that there is 20.7.3 Four-arm a.c. bridge measurements
no potential drop across Rshz and thus no current
flows through it. If the resistive elements of the Wheatstone bridge
are replaced by impedances and the d.c. source
and detector are replaced by their a.c. equiva-
20.7.2 A.c. equivalent circuits of resistors, lents, as shown in Figure 20.59, then if 21 is the
capacitors and inductors unknown impedance the balance condition is
Resistors, capacitors, and inductors do not exist given by
as pure components. They are in general made up
of combinations of all three impedance elements.
For example, a resistor may have both capacitive
and inductive parasitic elements. Figure 20.57 or
shows the complete equivalent circuits for phys-
ical realizations of the three components together
with simplified equivalent circuits which are com-
monly used. Further details of these equivalent iz1 = iz2 + iz3 ~ iz4
circuits can be found in Oliver and Cage (1971). There are therefore a very large number of pos-
At any one frequency any physical component sible bridge configurations. The most useful can
can be represented by its complex impedance be classified according to the following scheme
Z = R ijX or its admittance Y = G &jB. Since due to Ferguson. Since the unknown impedance
I' = 1/Z and Z = 1/Y then has only two parameters RI and XI, it is there-
fore sufficient to adjust only two of the six avail-
able parameters on the right-hand side of the
balance equation. If the adjustment for each
and
parameter of the unknown impedance is to be
independent then the variables should be
adjusted in the same branch. Adjusting the para-
These two representations of the component meters R2, X2, is the same as adjusting para-
correspond to series and parallel equivalent cir- meters R3, X3, and thus, four-arm bridges can
cuits. If at a given frequency the impedance is be classified into one of two types. either ratio
Z = R +jX then the equivalent circuit at that bridges or product bridges.
frequency in terms of ideal components is a resis- In the ratio bridge the adjustable elements in
tor in either series or parallel with an inductor, as either Z2 or Z3 are adjacent to the unknown
shown in Figure 20.58(a). This figure also gives impedance and the ratio, either ZslZ4 or 22/24,
the conversion formulae between the two repre- must be either real or imaginary but not complex
sentations. For components whose impedance at if the two elements in the balance condition are to
any given frequency is given by Z = R - jX the be independent. In product bridges the balance is
equivalent circuits are series or parallel combin- achieved by adjusting the elements in Z,, which is
ations of a resistor and a capacitor, as in Figure opposite the unknown. For the adjustments to be
20.58( b). independent requires Zz . Z3 to be real or imagin-
The quality factor, (2: is a measure of the abil- ary but not complex.
ity of a reactive element to act as a pure storage Figure 20.60 gives examples of a range of com-
element. It is defined as monly used four-arm bridges for the measure-
ment of C and L. For further details concerning
27r x maximum stored energy in the cycle the application of such bridges the reader should
Energy dissipated per cycle consult Hague and Foord (1971).
The dissipation factor, D, is given by
20.7.3.1 Stra.v impedances in a. c. bridges
1
D=- Associated with the branches, source, and detect-
Q or of an a.c. bridge there are distributed capacit-
The Q and D factors for the series and parallel ances to ground. The use of shields around these
inductive and capacitive circuits are given in Figure elements enables the stray capacitances to be
20.58. From this figure it can be seen that Q is given defined in terms of their location, magnitude, and