High inductance fractal inductors for wireless applications

Turkish Journal of Electrical Engineering & Computer Sciences http://journals.tubitak.gov.tr/elektrik/ Turk J Elec Eng & Comp Sci (2017) 25: 3868 – 3880 c TÜBİTAK ⃝ doi:10.3906/elk-1607-190 Research Article High inductance fractal inductors for wireless applications Akhendra Kumar PADAVALA∗, Bheema Rao NISTALA Department of Electronics and Communication Engineering, National Institute of Technology, Warangal, India Received: 17.07.2016 • Accepted/Published Online: 18.05.2017 • Final Version: 05.10.2017 Abstract: This paper presents fractal-based inductors for industrial, scientific, and medical applications in a frequency range of 3–500 MHz. The proposed inductors are designed based on the Hilbert space-filling curve and omega-shaped space-filling curve. The fractal inductors are designed and simulated by using a full wave high frequency structural simulator. The Hilbert curve-based fractal loop inductor and omega curve-based fractal loop inductor achieve improvements in the inductance value of 21% to 31% and 11% to 30.88%, respectively, over reported standard inductors. The printed inductors are constructed on 3.2 mm RT/Duroid 5770 substrate and measured with a network analyzer (E8363B). It was found that the experimental results are almost in good agreement with the simulation results. It was also observed that the proposed fractal inductors have poor radiating power, indicating no significant electromagnetic radiation. Key words: High frequency structural simulator, inductance value, printed circuit board, quality factor, self resonant frequency 1. Introduction Passive components play a vital role in the overall system performance of industrial, scientific, and medical wireless communication systems. The inductor is a critical and extensively used component among all of the passive components in many circuit applications such as power amplifiers [1], matching networks [2] and DCDC converters [3]. High inductance values (L), high quality factor (Q), and maximum achievable self resonant frequency are the three important aspects of inductor design. Obtaining larger values of inductance usually implies longer conductive segments, leading to a larger on-chip area. Inductors designed by using fractal geometry could potentially solve this problem. Fractal inductors were first reported in [4–6] but suffer from anticurrent pathways. An intuitive study of mathematically defined fractal space-filling inductors was carried out in [7–9]. The inductor designs were more competitive at lower fractal iterations but, at higher fractal iterations, the designs yield lower value of inductance compared to serpentine structures along with anticurrent pathways. The fractal loop inductors reported in [10] had higher inductance values at higher iterations. However, these designs were restricted to lower order frequencies and single layer fabrication process. In the current study, anticurrent pathways were reduced by adopting a loop structure and a multilayer fabrication process. This multilayer fabrication of the component can further increase the L. This paper is organized as follows. Section 2 presents the constructional details of fractal inductors. Section 3 presents the simulation and experimental results, and Section 4 presents the conclusion. ∗ Correspondence: 3868 PADAVALA and NISTALA/Turk J Elec Eng & Comp Sci 2. Proposed fractal inductors In general, fractal curves are characterized by two parameters, namely the iteration order (IO) and the indentation factor (IF). The proposed Hilbert fractal inductor is designed by using first order IO with indentation depth (ID) as the IF. An omega-based fractal inductor is designed by using an omega curve with the IR as ’3’ and the indentation angle (IA) as the IF. The proposed fractal inductors are fabricated using a RT/Duroid 5770 with a thickness of 1.6 mm, a dielectric constant of 2.33, and a loss tangent of 0.0009. The proposed fractal inductor structure is shown in Figure 1. The corresponding equivalent circuit is shown in Figure 2. In the figure, ‘R ’ indicates losses associated with a metal wire that is very limited for high conductivity metals. Proximity and eddy current losses associated with ‘R ’ are also less significant at the operating frequency of 3–500 MHz. ‘L ’ is the actual L, ‘Cint ’ is the intertwining capacitance between the metal lines, and ‘Csub ’ is the capacitance between the metal lines and the substrate. The effective inductance is a combination of ‘L ’ (inductance value) and the two parasitic capacitance values forming a frequency dependent inductor. At lower frequencies, the effect of parasitic capacitances are lower and its value increases with frequency. The variation in L with frequency is especially required for the optimal performance of circuits. Figure 1. Proposed fractal inductor structure. Figure 2. Equivalent fractal inductor structures on PCB. 2.1. Construction of the Hilbert curve-based fractal inductor The proposed Hilbert curve-based fractal inductor is shown in Figure 3. It was designed by using the Hilbert curve with the IO as ‘1’ and the ID as the IF. The space-filling property of the Hilbert curve has been exploited to increase the metal run of the conductor, which leads to an increase in the L. 2.2. Construction of the omega curve-based fractal loop inductor The proposed omega curve-based fractal loop inductor is shown in Figure 4. It was designed by using an omega curve with the IO as ’3’ and the IA as the IF. The L of the omega curve-based fractal loop inductor increases with the order. 2.3. Mathematical extraction of inductance For the proposed operating range of 3–500 MHz, inductance can be extracted using analytical expressions, as reported in [11]. According to Green’s function, the total inductance of the proposed inductor is the sum of the self inductance of each segment and mutual inductance between the segments [12], given by Eq. (1). LT = Lself + Lm (1) 3869 PADAVALA and NISTALA/Turk J Elec Eng & Comp Sci Figure 3. Hilbert curve-based fractal loop inductor. Figure 4. Omega curve-based fractal loop inductor. 2.3.1. Self inductance calculations The self inductance of each segment shown in Figure 5 is calculated by axial filament approximation given by Neumann’s inductance formula [13], given by Eq. (2): 1 Lself = 2 w ∫w ∫w Mf dx1 dx2 x2 =0 x1=0 3870 (2) PADAVALA and NISTALA/Turk J Elec Eng & Comp Sci Figure 5.Trace of rectangular segment. ′ Here ′ Mf is the mutual inductance between the two assumed filaments, which are part of the segment, separated by a distance ‘ d’ and given by Eq. (4): d2 = (x1 − x2 ) Mf = µ0 [f (z)] 4π Here f (z) = z ln(z + 2 (3) l, −l (Z) 0, 0 (4) √ √ z 2 + d2 ) − ( z 2 +d2 ) (5) After integrating Eq. (2), the self inductance is obtained as µ0 1 Lself = [ lw2 ln 4π w2 ( l w + ... √( ) l 2 w ) ( 2 + 1 + l w ln w l + √( ) w 2 l ) +1 ) ) 3/ 1 (3 1 (2 l + w3 − l + w2 2 3 3 + ... (6) Similarly, the self inductance of (...truncated)