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AlternatingCurrent BridgesAlternatingcurrent bridges^{16} are used for measuring the effective resistance and inductance of inductors, the effective resistance and reactance of capacitors, and the input impedance of lines and networks. The condition of bridge balance usually is determined with a telephone receiver if the test frequency is audible. Above the audiblefrequency range the null detector often is an amplifier and a rectifier. At higher frequencies, a radioreceiving set makes an excellent detector.^{17,} ^{18} In the alternatingcurrent bridge of Fig. 19 R_{1} and R_{2} are resistors with negligible reactance. Impedance Z_{s} may be a standard inductor or capacitor, and Z_{x} is the unknown inductor or capacitor to be measured.
At balance, negligible current flows through the receiver, minimum tone is heard in the receiver, and the points across which the receiver is connected are at the same potential. Thus I_{1}R_{1} = I_{2}R_{2} in both magnitude and phase, and I_{1}Z_{s} = I_{2}Z_{x} in both magnitude and phase. Then,
When this equation is generalized, Because resistances and reactances produce different effects, in balancing an alternatingcurrent bridge, first one quantity, and then the other, is varied, until minimum tone of the test frequency is heard. The inphase and outofphase terms of equation 39a must be separated as follows: R_{x}R_{1} + jX_{x}R_{1} = R_{2}R_{s} + jX_{s}R_{2}, R_{x}R_{1} = R_{2}R_{s}, and X_{x}R_{1} = X_{s}R_{2}. Solving for the unknown terms gives where R_{x} is the effective resistance and X_{x} is the reactance of the unknown. When a standard inductor is used to measure an unknown inductor, equation 40 becomes When a standard capacitor is used to measure an unknown capacitor, equation 40 becomes in which the ratio R_{1}/R_{2} is opposite from equation 41. In the preceding equations the units are ohms, henrys, farads, and cycles per second. Bridge with Standard Inductor. The unknown inductor or capacitor of Fig. 20 is measured in terms of the standard inductor L_{s} and standard resistor R_{s}. The effective resistance of the standard inductor necessitates a correction. If Fig. 20 (a) is used, then R_{s} of equation 40 equals R_{s} of Fig. 20(a) plus the effective resistance of the standard inductor. If Fig. 20(b) is used, then the effective resistance of the unknown capacitor is R_{x} of equation 40 minus the effective resistance of the standard inductor. Often, in bridges, such as Figs. 20(a) and (b), R_{1} and R_{2} each equal some value, such as 1000 ohms, and are fixed. Also, a resistor, equal to the effective resistance of the standard inductor, is connected in the arm opposite the standard variable inductor. The bridge of Fig. 20(a) then becomes direct reading; at balance the setting of R_{s} gives the effective resistance of the unknown inductor, and the setting of L_{s }gives the inductance of the unknown.
The bridge of Fig. 20(6) is adjusted until the inductive reactance 2πfL_{s} equals the capacitive reactance l/(2πfC). For this condition,
As will be noted, the frequency must be known. Bridge with Standard Capacitor. For many purposes the standard capacitor of Fig. 21 has negligible effective resistance and no resistance corrections are necessary. For the circuit of Fig. 21(a), when resonance is obtained the value of the unknown inductor is
If the circuit of Fig. 21 (b) is used, then the capacitance of the unknown is given by equation 42.
Bridge Measurements of Mutual Inductance. The circuits of Figs. 20(a) and 21(a) are used for measuring mutual inductance. Suppose that the two coils of Fig. 22 are connected aiding so that the magnetic effects add; then, because of the mutual inductance between the coils, the back voltage between terminals 1 and 4 may be considered as composed of four components: (1) the back voltage caused by the selfinductance of coil A, (2) the voltage induced in coil A by the current in coil B and the mutual inductance between B and A, (3) the back voltage caused by the selfinductance of coil B, and (4) the voltage induced in coil B by the current in coil A and the mutual inductance between A and B. This total back voltage appears as an inductive effect at the terminals, and hence the equivalent selfinductance as measured by a bridge is
But if the coils are connected opposing so that the magnetic effects subtract, then the induced voltages caused by the mutual inductances subtract from those caused by the selfinductances, and
Subtracting equation 45a from equation 45 gives
The mutual inductance between two coils is, accordingly, onefourth the difference in the inductance measured with the two coils aiding and the two coils opposing.
Bridge Measurements of Incremental Inductance. The magnitudes of the incremental inductance and effective resistance of an inductor with a ferromagnetic core will vary with the magnitude of both the direct and the alternating currents through the coil (see SelfInductance). The bridge of Fig. 23 provides both direct current and alternating current. The direct current from the battery is regulated by resistance r and measured with the milliammeter. It is difficult to measure directly the magnitude of the alternatingcurrent component, although it can be done. A highimpedance vacuumtube voltmeter connected as indicated can be used to maintain constant the alternating voltage drop across the inductor. This is accomplished by varying the voltage divider R. Both the direct current and the alternating voltage must be kept constant as the bridge is balanced. If the inductor has many turns (a filter choke for instance), then a low frequency of about 100 cycles must be used to avoid the effects of the distributed capacitance. For this reason, null detector D is often a tunable vacuumtube amplifier and detector, or some similar device. It should be noted that resistor S must pass the directcurrent component. At balance,^{16}
where all values are in henrys, farads, and ohms, and ω equals 2π times the frequency.


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