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Silicon-based radio-frequency integrated circuits are becoming more and more competitive in wide-band frequency range. An essential component of these ICs is on-chip (integrated) transformer. It is widely used in mobile communications, microwave integrated circuits, low-noise amplifiers, active mixers, and baluns. This paper deals with the design, simulation, and analysis of novel fractal configurations of the primary and secondary coils of the integrated transformers. Integrated stacked transformers, which use fractal curves (Hilbert, Peano, and von Koch) to form the primary and secondary windings, are presented. In this way, the occupied area on the chip is lower and a number of lithographic processes are decreased. The performances of the proposed integrated transformers are investigated with electromagnetic simulations up to 20 GHz. The influence of the order of fractal curves and the width of conductive lines on the inductance and quality factor is also described.

Constant growth of wireless applications brought to an intensive need for mobile communications and mobile communication devices. Due to a growing need for wireless communication devices, radio frequency and wireless market is continuing its development. The integrated transformer is an essential component in many RF and
microwave integrated circuits [

The unique property of fractal curves is that, after an infinite number of iterations, their length becomes infinite although the entire curve fits into the finite area. This space-filling property can be exploited for the miniaturization of the integrated transformers. Due to the technology limitations such as a minimal line width and spacing achievable by the fabrication process and because of its degree of complexity, the ideal fractal cannot be built. Our research is limited to prefractals with a low degree of iteration (or low order). In this work, we present novel layouts of the primary and secondary windings in the shape of fractal curves and
demonstrate a comprehensive analysis of the shape and order fractal curves
influence on the inductance and quality factor of the stacked transformers for RFICs applications. The simulation has been generated using the Microwave Office software package [

Fractals are a whole new set of geometrical objects featuring two main common properties: self-similarity and fractional dimension. There are many mathematical
structures that are fractals; for example, Sierpinski’s gasket, Peano curve, von Koch’s snowflake, the Mandelbrot set, the Hilbert curve, and so forth [

Hilbert curves are built through an iterative procedure that generates almost self-similar structures. In addition, Hilbert curves are space-filling curves, meaning that in the limit the fractal curve fills the whole space. The capability to pack conductive lines in a small space following a Hilbert curve is very appealing for manufacturing windings of on-chip transformers. The first three steps in the construction of the Hilbert curve are shown in Figure

Hilbert curve (iteration 1, iteration 2, and iteration 3).

The original Peano curve is a base-motiffractal that uses a line segment for the base and the motif depicted in Figure

Peano curve (iteration 1, iteration 2, and iteration 3).

Koch curve is a fractal curve characterized by such properties as a curve that is infinitely long, contained within a finite region, and not differentiable at any point (they just have corners). A geometric construction scheme for the Koch curve is shown in Figure

A geometrical construction of standard Koch curves.

Implementations and design of monolithic transformers consist of different trade-offs, which need to be considered in the geometry of the transformer layout. The inductance is determined by the primary or secondary windings (coils) lateral dimensions. Parasitic capacitances and resistances are determined by both lateral and
vertical dimensions. Conventional configurations include interleaved and stacked transformer, with the spiral geometry of coils. These configurations offer varying trade-offs among self-inductance, mutual coupling coefficient,

We have designed novel configurations of on-chip stacked transformers, where the
primary and secondary windings have shapes of different fractal curves. Figure

A stacked fractal transformer: (a) schematic symbol, (b) 3D model, and (c) the cross-section.

A stacked transformer depicted in Figure

The losses in the conductive segments are taken into account through two parameters, which are presented in Figure

The electrical parameters of conductive material (Al) including high frequency loss coefficient.

The layout topology of the windings of the integrated transformers strongly depends on the application of the transformer. In this paper, the stacked configuration of monolithic transformer is analyzed. Stacked transformer, or vertical coupling structure, represents a multiple conductor layer structure. This configuration has the
advantage of area efficiency and higher mutual coupling between the windings
due to placing the primary coil on top of the secondary. Stacked transformers
mainly have high coupling factor (

In this subsection, we investigate behavior of transformer parameters with variations of the Hilbert curves iteration and the width of conductive segments of the primary and secondary parts of the transformer.

In the first example, the primary and secondary windings are made in the form of Hilbert curve of the third order

A monolithic stacked transformer realized with two 3rd-order Hilbert curve, (a)

The inductance and quality factor as a function of frequency for stacked transformer, (a) 3rd-order Hilbert curve,

From Figure

In the next simulation the order of fractal curve is increased. The primary and secondary winding are realized using 4th-order of Hilbert curve

A top view of fractal transformers realized with two Hilbert curves

The simulation results for the inductance and quality factor as a function of frequency are depicted in Figure

Frequency characteristics of the inductance and quality factor of integrated stacked transformer realized with 4th order of Hilbert curve (a)

The proposed Hilbert transformer with

To conclude this subsection, it is important to point out that a good

The performance comparison of different transformer realizations.

Ref. | Configuration | |||
---|---|---|---|---|

[ | Differential square spiral transformer | 0.8 | 5.2 peak @ 5 GHz | 10 |

[ | Interleaved 3 turn square spiral transformer | 5.32 | 5.77 peak @ 2.95 GHz | 6.2 |

[ | Interleaved square spiral transformer | 8.5 | NA | 4.9 |

[ | Differential square spiral transformer | 5.5 | 10 peak @ 1.7 GHz | NA |

This work | Stacked transformer, Hilbert | 0.6 | 5.95 peak @ 8.4 GHz | 18.8 |

This improving performance is evident regarding increasing

In this subsection, the Koch fractal curves of the third and forth order are used for realization of the primary and secondary windings of the stacked transformers. The geometrical and technological parameters are the same as in the earlier simulations. In the first realization, the primary and secondary coils are designed of three serially connected 3rd-order Koch curves with the width of conductive (aluminum) lines

A Koch fractal transformer, (a) 3D view (b) the top view.

In Figure

Dependence of

Although the primary and the secondary are designed to have the same shape and the width of conductive lines, obtained values of their inductance and corresponding values of quality factor are slightly different due to the existence of the back-side metallization. The current induced in the metallization layer lowers the inductance values in both transformer structures, since it flows in the opposite direction than the current in windings. The inductance of the secondary winding is more affected because it
is placed closer to the metallization. It can be observed that with an increase of frequency,

The Peano fractal curves of the second order are also used for realization of the primary and secondary coils of integrated stacked transformer. The layout of the transformer is shown in Figure

The Peano

The simulation results for the inductance and quality factor are presented in Figures

Comparing proposed integrated transformers it can be concluded that the configuration with 3rd-order Hilbert curve has a maximal
value of the

The integrated transformer characteristics and performances greatly depend on geometrical and process parameters. In this paper, the novel fractal stacked transformers were analyzed using full-wave EM simulations and compared in terms of the inductance and quality factor. Simulation results show that using fractal layouts for the primary and secondary windings, similar or better performances can be achieved in comparison with earlier published results for monolithic transformers with square spiral geometry. The presented results mean that transformer configurations with fractal curves can be very useful for RF-IC designers to design high-performance RF and microwave integrated circuits.