Fractional-order elements (FOEs) serve as fundamental components in fractional-order circuits, forming the cornerstone of research into fractional-order circuit systems. Unlike single-component counterparts, the multi-component method offers greater flexibility in selecting constituent elements, enabling the adjustment of order and impedance coefficients for enhanced practicality. This paper provides a comprehensive overview of prevailing construction methods to facilitate the selection of appropriate multi-component FOEs for specific application needs. These methods are classified into three categories based on the type of constituent devices: passive devices, operational amplifiers, and power electronic converters.

In the construction method based on passive devices, two approaches involving Foster RLC ladder circuits are discussed. Passive RLC networks are used to create circuits that match the impedance characteristics of the transfer function. Reducing phase error requires increasing the order and using numerous components, leading to complex structures and cumbersome calculations. When standard off-the-shelf components cannot be used, replacing analytical parameters with standard ones increases the phase angle deviation. Adjusting the FOE order or impedance coefficients requires replacing all circuit components. This method is most suitable for fixed FOE order and impedance coefficients, which are effective in medium to low-frequency scenarios.

Next, the paper introduces the construction method based on operational amplifiers. The method of constructing FOEs based on generalized impedance converter (GIC) circuits can realize FOEs with orders varying from -2 to 2. However, the limited open-loop gain and gain instability of operational amplifiers often result in significant deviations of the obtained FOE performance from its ideal characteristics. Therefore, exploring how to use other active devices, such as operational transconductance amplifiers (OTA) and current feedback operational amplifiers (CFOA), to construct FOEs is an exploration direction. GIC circuits offer great integration and functionality, making them suitable for applications where precise impedance and phase characteristics are crucial.

The construction method based on power electronic converters is also discussed in detail. Multi-component FOEs based on power electronic converters have wide applications because their power level depends on the inverter, and their order and impedance can be adjusted by changing the control parameters. However, different structures of filters affect the operating performance of fractional-order elements. Therefore, exploring the application of different filter structures on multi-component FOEs and optimizing the parameters of the filters become the direction of development for power electronic converter-based FOEs. Power electronic converters provide the advantage of handling higher power levels and dynamic adaptability. The ability to digitally control the fractional order and impedance in real-time makes these elements highly versatile.

Lastly, this paper proposes a three-phase fractional-order electrical spring (TPFES). TPFES controls the order of the equivalent fractional-order capacitance of each phase and the pseudo-capacitance. The effect of stabilizing the load voltage is realized, the power factor is improved, and the power is balanced. The application in grid power compensation demonstrates that multi-component FOEs can effectively enhance the performance of practical circuit systems. This work provides a reference for future applications of fractional order components in electrical engineering.