This work proposes general formulas for designing two different topologies of fractional-order relaxation oscillators. One topology contains an Operational Amplifier and the other one relies on an Operational Trans-Resistance Amplifier. The design procedure hinges on the general fractional-order natural and step responses of RC, which is proved in this work depending on Mittag Leffler function. The proposed topologies can be controlled to generate symmetrical and non-symmetrical square wave signals. They also benefit from the employment of fractional-order capacitors (FOCs), which makes it possible to obtain higher frequencies using simple components. Furthermore, these topologies are verified through numerical solutions, circuit simulations, and experimental implementations. This encourages the authors to build circuit emulators for the FOC, which is achieved by applying the Foster-I synthesizing technique to the Matsuda’s approximation of \(s^\alpha\).

这项工作提出了用于设计分数阶张弛振荡器的两种不同拓扑的通用公式。一种拓扑包含一个运算放大器，另一种拓扑依赖于一个运算跨阻放大器。设计过程取决于RC的一般分数阶自然响应和阶跃响应，这在这项工作中得到了证明，具体取决于Mittag Leffler函数。可以控制提出的拓扑以生成对称和不对称的方波信号。它们还受益于分数阶电容器（FOC）的使用，这使得可以使用简单的组件获得更高的频率。此外，这些拓扑已通过数值解，电路仿真和实验实现得到了验证。这鼓励作者为FOC构建电路仿真器，这是通过将Foster-I合成技术应用于Matsuda的\（s ^ \ alpha \）近似来实现的。