Fractal immittance, expressed by an admittance sa (0<|a|<1), is simulated by the analog circuits composed of finite numbers of conventional elements, resistance R, capacitance C and inductance L, based on the distributed-relaxation-time models. The correlation between the number of R-C or R-L pairs and the optimum pole interval to give the widest bandwidth is estimated for each a-value by the numerical calculation for each circuit against a given criterion with respect to the phase angle. It is found that the bandwidth of 5 decades with a phase-angle error of 1 can be composed for |a|=0.1-0.9 using eighteen pairs or less of the elements.

The transient behavior in the fractal admittance acting as a non-integer-rank differential/integral operator, Y(s) ∝ sa with -1a1 and a0, is examined from the point of view of memory effects by employing the distributed-relaxation-time model. The internal state of the diode is found to be represented by the current spectrum i(λ, t) with respect to the carrier relaxation rate λ, leading to a general formulation of the long-time-tail memory behavior characteristic of the operator. One-to-one corrsepondence is found among the input voltage in the past ν(-t), the short-circuit current isc(t) and the initial current spectrum i(λ, 0) within the framework of the Laplace-type integral transformation and its inverse, assuring that each response retains in principle the entire information on the corresponding input, such as the functional form, the magnitude, the onset time, and so forth. The current and voltage responses are exemplified for various single-pulse voltage inputs. The responses to the pulse-train inputs corresponding to different ASCII codes are found to be properly discriminated between one another, showing the potentials of the present memory effects.

In this work we propose a recursive electric circuit which has a fractal structure composed of resistances and capacitors. It is first confirmed in practice that such a device, which is referred as fractance, possesses a novel characteristics between the constituent elements, or the resistance and the capacitance. That is, the absolute value of the impedance shows such a characteristics as |Z|ω-1/2, whereas the phase angle does such a constant phase property as θ=-π/4 independent of the frequency. As an application of the presently proposed fractance device, we constructed a fractional integral circuit as well as a fractional differential circuit, and confirmed that the observed output wave forms agree well with those derived from a computer simulation.