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13.9: AC Bridges

We  have already met, in Chapter 4, Section 4.11, the Wheatstone bridge, which is a DC (direct current) bridge for comparing resistances, or for "measuring" an unknown resistance if it is compared with a known resistance.  In the Wheatstone bridge (figure \(\text{IV.9}\)), balance is achieved when \(\frac{R_1}{R_2}=\frac{R_3}{R_4}\).  Likewise in a AC (alternating current) bridge, in which the power supply is an AC generator, and there are impedances (combinations of \(R\), \(L\) and \(C\) ) in each arm (figure \(\text{XIII.8}\)),

                           
FIGURE \(\text{XIII.8}\)

balance is achieved when

\[\label{13.9.1}\frac{Z_1}{Z_2}=\frac{Z_3}{Z_4}\]

or, of course, \(\frac{Z_1}{Z_3}=\frac{Z_2}{Z_4}\). This means not only that the RMS potentials on both sides of the detector must be equal, but they must be in phase, so that the potentials are the same at all times.  (I have drawn the "detector" as though it were a galvanometer, simply because that is easiest for me to draw.  In practice, it might be a pair of earphones or an oscilloscope.)  Each side of equation \ref{13.9.1} is a complex number, and two complex numbers are equal if and only if their real and imaginary parts are separately equal.  Thus equation \ref{13.9.1} really represents two equations – which are necessary in order to satisfy the two conditions that the potentials on either side of the detector are equal in magnitude and in phase.

We shall look at three examples of AC bridges.  It is not recommended that these be committed to memory.  They are described only as examples of how to do the calculation.

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