# Design Techniques for Low-Voltage RF/mm-Wave Circuits in Nanometer CMOS Technologies
## 2 Integrated Inductive Components: Loss Phenomena and Main Design Guidelines  
Inductive components are very important in RF/mm-wave integrated circuits (ICs). Indeed, they are exploited to implement irreplaceable functionality, such as simultaneous impedance/noise matching to the 50-ohm input source in LNAs, impedance matching and tuned resonant loads in amplifiers, LC tank in voltage-controlled oscillators, and integrated single-ended-to-differential conversion [3; 4; 5; 6; 7]. The \(Q\)-factor maximization at a given value of inductance, \(L\), is the most common design issue. Other important figures of merit are the inductor \(\omega QL\) product, the transformer characteristic resistance (\(TCR\)) and the insertion loss (\(IL\)) [8; 9]. A scanning electron microscope (SEM) cross-section of a silicon-integrated spiral inductor is shown in Figure 1a. Geometrical parameters (i.e., layout parameters) can be properly tuned within the technology constraints, namely, the coil shape (i.e., circular, polygonal or squared), the number of turns, \(n\), the inner diameter, \(d_{\text{IN}}\), the metal width, \(w\), and the metal spacing, \(s\). The process parameters, such as the metal BEOL, the thickness and permittivity of the insulation layers, and the substrate conductivity, cannot be modified. Generally, the design of inductive devices is aimed at minimizing energy loss. The main energy dissipation phenomena of silicon-integrated inductive components are shown in Figure 1b. They occur in the metal layers that form the coil (series losses), as well as in the conductive layers below (parallel losses). The former is mainly due to the current crowding on the internal sides of the coil, due to skin and proximity effects. Current crowding rises with an increase in the operating frequency, thus increasing the equivalent resistance of the inductor. Thick conductive metals and multi-layer structures are common arrangements to lower the series losses, but their effectiveness degrades at RF/mm-wave frequencies. According to [PERSON]'s law, current crowding is stronger in the inner turns. Therefore, the adoption of spirals with a low fill ratio, \(\rho\), is a common rule [10]. Figure 1b also depicts two different mechanisms taking place in the substrate layers, i.e., the (vertical) displacement currents and the (horizontal) magnetically induced currents. Both electrically and magnetically induced currents increase at RF/mm-wave frequencies and dominate the inductor losses. Substrate shielding can be implemented to reduce the effect of parallel losses for RF operation [11; 12]. Figure 1: (**a**) SEM cross-section of a silicon-integrated spiral inductor; (**b**) simplified representation of main loss phenomena in silicon-integrated inductive components. Reprinted with permission from Ref [9]. Copyright 2020, IEEE. The design of an inductive component starts with its geometrical parameters, which set the low-frequency inductance, \(L_{\text{DC}}\). The inductance maximum value reduces with an increase in the operating frequency, due to self-resonance. Actual inductors have a typical inductance range from a few hundred picohenrys to tens of picohenrys, moving from tens of gigahertz to 100 GHz and beyond. Since inductance is mainly dependent on \(n\), the inner diameter, \(d_{\text{IN}}\), is adjusted to trade off the silicon area with current crowding phenomena, by using a low fill ratio, \(\rho\), while the metal width, \(w\), is typically tuned to set the peak quality factor, \(Q_{\text{MAX}}\), at the operating frequency. The metal spacing, \(s\), must be set at the minimum value to maximize the coil auto-inductance. For the purpose of completeness, Table 1 summarizes the relationship between the electrical performance and the layout/process parameters of an integrated inductor. A special role in RF/mm-wave ICs is performed by transformers [13]. Figure 2 shows the main structures of integrated transformers. The traditional arrangements are the interleaved and stacked configurations.