The fractional-order Laplace operator $s^{\pm \alpha }=\text{◂◽˙▸}{(j\omega )}^{\pm \alpha }$ has been used in recent years in many applications including the synthesis of fractional-order capacitors and inductors as well as in fractional-order filter design. However, the synthesis and application of fractional-order impedances with a complex-order proportional to $\text{◂◽˙▸}{s}^{(\alpha \pm j\beta )}$ have not received similar attention. In this work, we report on the possibility of synthesizing capacitive impedances with complex orders and explain the meaning of such impedances. Applications in realizing a complex order integrator and a complex order high-pass filter are demonstrated supported by circuit simulation results.

中文翻译：

---

复阶电容阻抗的可能实现及其应用

分数阶拉普拉斯算子${秒}^{\pm \alpha }=\text{◂◽˙▸}{(j\omega )}^{\pm \alpha }$近年来已用于许多应用，包括分数阶电容器和电感器的合成以及分数阶滤波器设计。然而，复阶分数阶阻抗的合成和应用与$\text{◂◽˙▸}{秒}^{(\alpha \pm j\beta )}$没有受到类似的关注。在这项工作中，我们报告了合成具有复杂顺序的电容阻抗的可能性，并解释了此类阻抗的含义。电路仿真结果支持在实现复阶积分器和复阶高通滤波器中的应用。