Analog Circuit for Solving Linear Ordinary Differential Equations

This paper designs an analog circuit for solving linear ordinary differential equations using two LM324AD operational amplifiers and fundamental circuit components within the Multisim environment. Accompanying this, a signal generation circuit is constructed to input sinusoidal and triangular waves, with the design's accuracy further validated through Simulink simulations. The signal generation section employs a Wien bridge oscillator to produce a sinusoidal wave with a frequency of w = 600 rad/s and amplitude exceeding 2V. Subsequently, an amplification circuit amplifies the waveform. For generating a triangular wave of the same frequency, considering the LM324's relatively low slew rate, stability and precision are maintained by first producing a square wave from an oscillation circuit with adjustable duty cycle, which is then converted into a triangular wave via an RC integration circuit. To ensure the waveform's suitability for subsequent solution processes, bias voltage adjustments are made for fine-tuning. The analog circuit for equation-solving comprises two integrators, a phase shifter, and a differential amplifier, collectively enabling solutions for linear ODEs. The integrators handle the differential aspects of the equations, while the differential amplifier implements the proportional coefficients. The phase shifter is pivotal in maintaining circuit oscillation stability, adhering to the Barkhausen stability criterion. Integration of these modules necessitates closed-loop signal stability due to the sensitivity of linear ODEs to signal variations; thus, meticulous tuning of the differential circuit parameters ensures stable output solutions. Finally, the design framework is replicated in Simulink, with the computed results compared against simulation outputs to confirm the design's correctness.