Project
| # | Title | Team Members | TA | Documents | Sponsor |
|---|---|---|---|---|---|
| 42 | Low-Latency Analog Differential Equation Solver |
Jiachang Wang Tianyue Jia Yanzi Li Yishan Sheng |
Aili Wang | ||
| # Problem Ordinary Differential Equations (ODEs) are widely used to describe dynamic systems in the real world, such as mechanical vibration systems, electrical circuits, and thermal processes. These systems evolve continuously over time, and their dynamic behaviors are commonly modeled using differential equations. For example, a typical mass–spring–damper system can be described as $mx+cx+kx=F(t)$ where $x(t)$ represents the displacement of the system, $m$ is the mass, $c$ is the damping coefficient, $k$ is the spring stiffness, and $F(t)$ represents the external force applied to the system. Traditionally, such differential equations are solved numerically using digital computers. Numerical methods such as Euler’s method or Runge–Kutta methods discretize the equation and compute the solution iteratively. However, these approaches require repeated calculations and may introduce computational latency. In applications that require real-time response, such as dynamic system modeling, control system analysis, and rapid prototyping, digital methods may suffer from inefficiency or delay. In contrast, analog circuits can implement differential equations directly using continuous-time signal processing. By representing system variables as voltage signals and constructing differentiator, scaling, and summation circuits with operational amplifiers, the mathematical relationships of differential equations can be implemented directly at the circuit level. For example, the proposed system can simulate the real-time response of dynamic systems such as mechanical vibration models or RLC circuit responses. In this approach, system variables such as displacement or current can be observed directly on an oscilloscope as time-varying signals. This provides a low-latency solution for dynamic system analysis, control system education, and rapid engineering prototyping. # Solution Overview This project aims to design and implement an analog differential equation solver using operational amplifier circuits. The system directly implements the mathematical relationships of a differential equation through analog signal processing, enabling continuous-time computation of dynamic system responses. In the proposed system, variables in the differential equation are represented as voltage signals. For instance, the system state $x(t)$ is represented by a voltage signal, while its time derivative $dx/dt$ is generated using an operational amplifier differentiator circuit. At the same time, weighted amplification and summation circuits are used to construct the algebraic terms on the right-hand side of the differential equation, such as coefficient multiplications and signal combinations. The entire circuit forms a feedback structure that ensures the signals within the system satisfy the constructed differential equation. When an input signal $u(t)$ is applied as an external excitation, the circuit generates the corresponding system response in real time. The resulting waveform can then be observed on an oscilloscope. Compared with conventional digital methods, this analog approach performs computation continuously without discretization or iterative numerical algorithms. As a result, the system can achieve low-latency computation and demonstrate the feasibility of using analog electronics for real-time dynamic system modeling. # Solution Components The system consists of several subsystems, each responsible for implementing a specific function required for solving the differential equation. ## Subsystem I: Differentiation Module - **Hardware I.a – Differentiator Circuit**: This module implements an operational amplifier differentiator circuit that computes the time derivative of an input signal. The circuit uses a capacitor–resistor network together with an op-amp to produce an output voltage proportional to the derivative of the input voltage. This module provides the fundamental operation required for representing derivative terms in the differential equation. - **Hardware I.b – Summation and Scaling Circuit**: This module uses operational amplifier summing amplifiers and resistor networks to implement weighted combinations of signals. By adjusting resistor values, the circuit can scale signals to represent coefficients in the differential equation. The circuit performs operations such as coefficient multiplication and signal addition, for example implementing expressions such as $ax$ or $ax+bu(t)$. ## Subsystem II: Input Signal Module - **Hardware II.a – Signal Generation**: This subsystem provides the external input signal $u(t)$ to the system. A function generator will be used to produce different types of excitation signals, such as step signals, sinusoidal signals, or square waves. These signals simulate external inputs to the dynamic system modeled by the differential equation. ## Subsystem III: Output Observation Module - **Hardware III.a – Oscilloscope Visualization**: The output voltage of the circuit represents the solution of the differential equation, corresponding to the system response over time. This signal will be connected to an oscilloscope, allowing real-time observation of system behavior such as oscillations, damping, or steady-state responses. ## Subsystem IV: Power Supply Module - **Hardware IV.a – Dual Power Supply**: Operational amplifiers require both positive and negative supply voltages to process signals that vary around zero. Therefore, the system will use a dual DC power supply providing approximately ±12 V to power the analog circuits. # Criterion for Success The project will be considered successful if the following criteria are satisfied: - **Accurate Differentiation**: The differentiator circuit must correctly compute the time derivative of the input signal and operate stably within the expected frequency range. - **Correct Equation Implementation**: The summation and scaling circuits must correctly implement the coefficients and mathematical structure of the target differential equation. - **Real-Time System Response**: When an excitation signal is applied, the system should produce a continuous output signal representing the system response in real time. - **Consistency with Theoretical Behavior**: The waveform displayed on the oscilloscope should match the expected theoretical behavior of the modeled differential equation, such as exponential decay, oscillatory motion, or steady-state response, within reasonable tolerance. |
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