## The Measurement Of Mutual Inductance

• MUTUAL INDUCTANCE is one of the most important properties of electrical circuits. By means of it the world's power is transmuted from generator voltage to transmission line voltage and back to the voltage suitable for our motors and electric lights. Yet the measurement of mutual inductance is of little importance as compared to measurements of self-inductance, capacitance, and resistance.

The major reason for this slighting of mutual inductance lies in the interesting fact that iron-core transformers are nearly perfect. The mutual inductance between the primary and secondary windings has so nearly its maximum value that no ordinary measurement of it could distinguish the minute difference. Such differences are of course of great importance and are measured by some characteristic of the transformer itself, the voltage regulation of a power transformer or its leakage reactance, and the frequency characteristic of an audio transformer.

The general theory of coupled circuits involves at one extreme the closely coupled iron-core transformers just mentioned and at the other extreme the loosely coupled tuned circuits of radio-frequency amplifiers, of which the neutrodyne receiver of Haz-eltine is an historical example. Such loosely coupled circuits are now being used as band-pass filters at the intermediate frequency of heterodyne receivers.

Mutual inductance is unique in that it can exist only in the presence of self-inductance. An important measure of mutual inductance is its ratio to the geometric mean of the two self-inductances it connects, called the coefficient of coupling, k.

which may have any value from zero to unity. In loosely-coupled tuned circuits the resonance curve has a single peak for all values of coupling coefficient

ALSO IN THIS ISSUE: New Wave Filters A Handy Voltage Divider A Wide-Range R-F Choke

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less than a so-called critical coupling.1 A larger coupling coefficient results in a flattened top with the steepness of the sides maintained. The measurement of this small mutual inductance is of considerable importance in the adjustment of i-f filters.

Mutual inductance may be measured by using as a standard of comparison a mutual inductance, a self-inductance, or a capacitance. When the standard mutual inductance is continuously variable and of such a range that it can be made equal to the unknown mutual inductance, the Felici mutual-inductance balance is the simplest method. As

shown in Figure 1, the primaries of both mutual inductances, unknown and standard, are connected in series to a power source. Their secondaries are also connected in series to a suitable detector, head telephones, or oilier a-c operated meter, in such direction that their induced voltages oppose. The standard is then varied until a balance is obtained, when

The error in this measurement is essentially that of the standard, provided that the capacitive reactances between the coils of the mutual inductances are large compared with their mutual reactances and that the impedance of the detector is small.

Type 107 Variable Inductors are calibrated in such a way that the mutual inductance between their rotor and stator windings can be easily obtained from their dials calibrated in self-inductance. For a 1 % full-scale accuracy of this dial calibration, the error in the mutual inductance will range from 2.5% to 10% over the portion of the scale normally used.

When the primary and secondary windings of a mutual inductance are connected in series, the self-inductance L of the pair is

L = ii + Z-2 =b 2M (3) The mutual inductance AT may be calculated from the two self-inductances La and L,0 obtained with the two coils aiding and opposing (using the + and — signs before 2M).

For a coefficient of coupling nearly unity where L„ is very small compared to La, the error in the determination of M is that of La itself. For smaller coupling coefficients this error increases, as always happens when the difference of two nearly equal numbers enters in any

Figure 2. Type 107 Variable Inductor. Mutual inductance at any setting is one-half the difference of the scale reading and the value of self-inductance at zero mutual inductance as entered on the nameplale

1 fcc = y/Dx D3 or coMc — VLi Lx where Di «nd are the dissipation factors (reciprocal of storage factor Q) of primary and secondary.

Figure 2. Type 107 Variable Inductor. Mutual inductance at any setting is one-half the difference of the scale reading and the value of self-inductance at zero mutual inductance as entered on the nameplale formula. For example, when K = 0.1 and L\ = L2, the error will be increased fivefold. The measurement of self-in-ductance may be made on a Type 650-A Impedance Bridge with an error of 2 ¿til or 2% and on a Type 667-A Inductance Bridge with an error of 0.1 /uh or 0.2%- The errors which occur in inductance measurements were discussed in considerable detail in the General Radio Experimenter for March, 1934.1 The Type 293-A Universal Bridge may also be used with resultant errors which lie between those of the other bridges mentioned. In bridges having a decade ratio arm, from the setting of which the self-inductance is calculated, such as the Type 667-A and Type 293-A Bridges, the increase in the error as the two separate bridge balances approach one another is minimized if one ratio arm is kept fixed and the change in balance taken up by minimum changes in the other. The error is then that of the change in resistance of the decade ratio arm.

Mutual inductance may be compared with a self-inductance on the Campbell mutual-inductance bridge shown in Figure 5.

The self-inductance Lp of the winding

1R. F. Field, "The Measurement of a Small Inductance," Genera] Radio Experimenter, Vol. VIII, No. 10, March, 1934.

The self-inductance Lp of the winding

1R. F. Field, "The Measurement of a Small Inductance," Genera] Radio Experimenter, Vol. VIII, No. 10, March, 1934.

Figure 3. Type 650-A Impedance Bridge

Figure 5. The Campbell Mutual-Inductance

Bridge

### Figure 5. The Campbell Mutual-Inductance

Bridge connected into the bridge arm may be measured by removing the other winding from the detector circuit. Denoting these bridge readings by primes, B'

Such measurements are easily made on the Type 667-A Inductance Bridge by connecting one winding of the mutual inductance to the unknown terminals and the other winding in series with the detector. The errors are slightly

Figure 4. Type 667-A Inductance Bridge co

Figure 3. Type 650-A Impedance Bridge

Figure 4. Type 667-A Inductance Bridge

less than for the previous case because for a given mutual inductance the dif-

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