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The theory of the frequency-discriminator circuit which is used in radio receivers equipped with automatic frequency control is worked out. Formulas for computing gain, frequency-control sensitivity, adjacent-channel attentuation, pull-in range, etc., are developed. These formulas apply to any value of primary or secondary inductance and to any value of coil quality factor. The loading represented by the plate impedance of the driver tube and by the diodes is also taken into account. After slight modification, the resulting formulas can be used for various different circuit arrangements. Nonlinear audio-frequency, distortion is found to exist in the case where the channels for signal and automatic frequency control are common; these distortions, however, are small. It is found that the loading effect of the diodes is responsible for the poor selectivity resulting for most automatic-frequency-control transformers. A circuit in which the automatic frequency control and the signal channel are separated is preferable if better selectivity is desired.