Spiral Inductors

Introduction:

In contrast with digital circuits which use mainly active devices, on-chip passive components are necessary and imperative adjuncts to most RF electronics. These components, which include inductors, capacitors, varactors, and resistors, have been known as performance as well as cost limiting elements of radio frequency (RF) integrated circuits. While all of these components can be realized using MOS technology, their specific designs necessitate special consideration due to the requirement of high-quality factor Q at relatively high frequencies. Inductors in particular are critical components in oscillators and other tuned circuits. For low-frequency applications, passive devices can be connected externally, but as the frequency increases, the characteristics of the passive devices would be overwhelmed by parasitic effect. For instance, a voltage-controlled oscillator (VCO) of 10 MHz needs a tank inductance on the order of several µH, whereas at 10 GHz the inductance is around 1 nH. It’s impossible to access such a small inductance externally, since the inductance associated with the package pin and bond wire can exceed 1 nH. As a result, on-chip passive components are commonly used in RF applications.

This blog will focus on the on-chip inductors. Basically, there are three types of on-chip inductors. The most widely used type is the planar spiral inductor, and a square shaped spiral inductor. Although a circular shaped inductor may be more efficient and yield better performance, the shape of inductor is often limited to the availability of fabrication processes. Most processes restrict all spiral angles to be 90°, and a rectangular/square pattern (hereafter called square pattern) is a nature choice, but a polygon spiral inductor can serve as a compromise between the square and circular shaped inductors. Structural parameters such as the outer dimension, number of turns, the distance between the centres of lines (or pitch), and substrate property are all important factors in determining the performance of on-chip inductors.

Concepts and Modelling of Spiral Inductors:

Traditionally, spiral inductors are made in square shape due to its ease of design and support from drawing tools. From the performance point of view, however, the most optimum pattern is a circular spiral because it suffers less resistive and capacitive losses. But the circular inductor is not widely used because only a few commercial layout tools support such a pattern. Hexagonal and octagonal structures are good alternatives, as they resemble closely to the circular structure and are easier to construct and supported by most computer-aided design tools. It has been reported that the series resistance of the octagonal and circular shaped inductors is 10% smaller than that of a square shaped spiral inductor with the same inductance value.

Circuit Equivalent:

This blog will discuss the concepts and formulas for series inductance (LS), resistances (RS and RSi), capacitances (CS, CSi, and COX), and quality factor and substrate loss.

Series Inductance:

In 1946, Grover derived formulas for the inductance of various inductor structures. Greenhouse later applied the formulas to calculate the inductance of a square shaped inductor. He divided the inductor into straight-line segments, and calculated the inductance by summing the self-inductance of the individual segment and mutual inductance between any two parallel segments. The model has the form of

LS = L0 + (M+)–(M−)

where LS is the total series inductance, L0 is the sum of the self inductance of all the straight segments, M+ is the sum of the positive mutual inductances and M- is the sum of the negative mutual inductances. Self inductance L’0 of a particular segment can be expressed as

L’0 is the inductance in nH, l is the length of a segment in cm, w is the width of a segment in cm, and t is the metal thickness in cm. The mutual inductance between any two parallel wires can be calculated using

M = 2lQ'

where M is the mutual inductance in nH and Q’ is the mutual inductance parameter

GMD denotes the geometrical mean distance between the two wires. When two parallel wires are of the same width, GMD is reduced to

d is the pitch of the two wires. Note that the mutual inductance between two segments that are perpendicular to each other is neglected. As the number of segments increases, the calculation complexity is increased notably because it is proportional to (number of segments) ^2. Another drawback of the model is its limitation to only square shaped inductors. The above model could be simplified using an averaged distance for all segments rather than considering the segments individually. Based on this approach, the self and mutual inductances are calculated directly as

where µ0 is the permeability of vacuum, lT is the total inductor length, n is the number of turns, and d’ is the averaged distance of all segments.

Resistance:

Series resistance RS (see Fig. 4(c)) arises from the metal resistivity in the inductor and is closely related to the quality factor. As such, the series resistance is a key issue for inductor modelling. When the inductor operates at high frequencies, the metal line suffers from the skin and proximity effects, and RS becomes a function of frequency [19]. As a first-order approximation, the current density decays exponentially away from the metal-SiO2 interface.

Where ρ is the resistivity of the wire, and t(eff) is given by

t is the physical thickness of the wire, and δ is the skin depth which is a function of the frequency:

where µ is the permeability in H/m and f is the frequency in Hz. The most severe drawback of a frequency-dependent component, such as RS, in a model is that it cannot be directly implemented in a time domain simulator, such as Cadence Spectre. Researchers have proposed to use frequency-independent components to model frequency dependent resistance. Ooi et al replaced RS with a network of 2 R’s and 1 L, where R and L are frequency-independent components, in the inductor equivalent circuit. The total equivalent resistance R(total) of the box is

where R0 is the steady-state series resistance, ω is the radian frequency, P is the turn pitch, t is the inductor thickness, w is the inductor width, σ is the conductivity, N is the total number of turns, and M is the turn number where the field falls to zero. The substrate resistance is given by

where l is total length of all line segments, G(sub) is the conductance per unit area of the substrate.

Capacitance:

There are basically three types of capacitances in an on-chip inductor: the series capacitance CS between metal lines, the oxide capacitance COX associated with the oxide layer, and the coupling capacitance CSi associated with the Si substrate.

where n is the number of overlaps, w is the spiral line width, Csub is the capacitance of the substrate, tox is the oxide thickness underneath the metal, and toxM1-M2 is the oxide thickness between the spiral. An improved method [26], which evaluates the voltage and energy stored in each turn, leads to the equivalent capacitances of Cp and Csub, as shown in Fig. 8. Compared to the model in Fig. 4(c), Cp and Csub in this model are equivalent to CS and the combination of Cox and CSi, respectively,

where Cms represents the capacitance per unit area between the mth metal layer and the substrate, Cmm represents the capacitance per unit length between adjacent metal tracks, Ak is the track area of k th turn and lk is the length of kth turn. The model also implies that CS is a function of the index difference of each adjacent segment pair. This means that the larger the index difference between the two adjacent lines, the higher the capacitance.

Quality Factor and Substrate Loss:

The quality factor Q is an extremely important figure of merit for the inductor at high frequencies. For an inductor, only the energy stored in the magnetic field is of interest, and the quality factor is defined. Basically, it describes how good an inductor can work as an energy-storage element. In the ideal case, inductance is pure energy-storage element (Q approaches infinity), while in reality parasitic resistance and capacitance reduce Q. This is because the parasitic resistance consumes stored energy, and the parasitic capacitance reduces inductivity (the inductor can even become capacitive at high frequencies). Self-resonant frequency fSR marks the point where the inductor turns to capacitive and, obviously, the larger the parasitic capacitance, the lower the fSR.

If the inductor has one terminal grounded, as in typical applications, then the equivalent circuit of the inductor can be reduced to that shown in Fig. 9. From such a model, the quality factor Q of the inductor can be derived

where ω is the radian frequency, LS is the series inductance, RS is the series resistance, RP is the coupling resistance, and CP is the coupling capacitance.