On the Use of a Relaxation Oscillator as a Characterized Voltage-to-Frequency Converter

This paper considers the circuit synthesis of a particular type of voltage-to-frequency converter with a predetermined, but arbitrary, voltage-to-frequency characteristic. The basis of this converter is a relaxation oscillator consisting of a linear energy storage network, a combined signal and bias voltage source which provides power to the oscillator, and a single four-layer diode or similar relaxation switch. The paper first attempts to show what forms of characterization such a circuit might reasonably be expected to give in a voltage-to-frequency conversion. A more rigorous argument follows where it is shown that the voltage waveform out of such a network is a linear function of both the signal voltage and the diode current, the current being treated as an impulse discharge from some effective capacity. The sum equation, thus derived, can be solved using a remarkable theorem about the Möbius function of number theory to give the exact form of the desired impulse response of the linear network. As an example of the application and practical limitations of the approach, a network for a fractional root converter is derived and discussed in terms of a completed square-root converter.