Fractional-Order Circuit Models and Signal Processing

Fractional-order circuit models extend classical integer-order descriptions by employing elements whose impedance scales with a non-integer exponent of frequency. These elements—such as fractance devices embodying fractional-order capacitance and inductance—enable more accurate representation of real-world behaviours exhibiting memory and non-locality. In signal processing, fractional-order filters and operators provide enhanced control over frequency responses, phase characteristics and dynamic performance. They allow for adjustable passband and stopband slopes, compact implementations of complex filter types and improved transient responses. Recent advances in approximation techniques, component realisations and modelling methods have made practical implementation of fractional circuits increasingly feasible, leading to applications in power converters, high-frequency communications and adaptive filtering. The convergence of fractional calculus with modern circuit synthesis offers a unified framework for designing robust, tunable and miniaturised systems that reconcile theoretical flexibility with engineering constraints.

Research from Nature Portfolio

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Research from all publishers

Recent developments have focused on both the synthesis of high-order fractional filters and the physical realisation of fractional elements. A 2023 study demonstrated the design of higher-order fractional filters with fully controllable frequency characteristics, using minimum-phase state-space models to fit desired magnitude responses and implementing these via operational transconductance amplifiers and field-programmable analogue arrays for electronic adjustability of type, order and corner frequencies. In 2021, a novel coaxial structure was introduced for the direct realisation of half-order inductors, exploiting the transverse electromagnetic mode in a conductive filling material to yield a stable 0.5-order inductive response over the 18 MHz–1 GHz range with minimal phase deviation. Another 2021 contribution applied classical circuit theory to analogue modelling of fractional-order elements, presenting canonical RC Cauer- and Foster-form networks that approximate arbitrary phase responses. This work included synthesis procedures for one-port and two-port fractional integrators and differentiators, accompanied by circuit diagrams, simulations and practical implementations that validate the proposed approach.

Fractional-Order Circuit Models and Signal Processing publication trend

The graph below shows the total number of articles in fractional-order circuit models and signal processing across all publications each year (not limited to Nature Index journals).

Technical terms

Fractional calculus: A generalisation of differentiation and integration to non-integer orders, allowing operators that capture memory and hereditary properties.

Fractional-order element (FOE): An electrical component whose impedance follows Z ∝ (jω)^α with α non-integer, encompassing fractional-order capacitors and inductors.

Fractance: A portmanteau of ‘fractional’ and ‘reactance’, used to describe FOEs exhibiting frequency responses intermediate between capacitors and inductors.

Transfer function: A mathematical representation of the relationship between input and output in the frequency domain, extended here to fractional orders.

Constant Phase Element (CPE): A simplified model of an FOE characterised by a constant phase angle over a wide frequency band.

References

1. Design of Higher-Order Fractional Filters With Fully Controllable Frequency Characteristics. IEEE Access (2023).
2. Fractional-Order Inductor: Design, Simulation, and Implementation. IEEE Access (2021).
3. Analog Modeling of Fractional-Order Elements: A Classical Circuit Theory Approach. IEEE Access (2021).