9.2 On-chip inductor modelling


9.2 On-chip inductor modelling

Inductors are an essential component of RF circuit design and are used in impedance matching networks, resonant networks, etc. Accurate compact modelling of on-chip inductors is very important because their parasitics can significantly degrade the performance of high-frequency circuits. Moreover, analytical representations of the parasitic components are needed in order to consider them as part of the parasitic-aware design and optimisation processes. The two types of inductors used in RF integrated circuit design are on-chip spirals and bond-wire inductors.

9.2.1 Spiral inductors

Monolithic spiral inductors play a key role in increasing the integration level of RF chips. However, their performance is far inferior to discrete off-chip inductors because of their large series and substrate loss resistances. Figure 9.1 shows the cross-sectional and top views of a square spiral inductor. Two types of parameters are needed to model the spiral inductor, namely, process-and design-controlled parameters. Process-controlled parameters (Figure 9.1a) include oxide thickness, metal thickness, substrate resistivity, etc., while design-controlled parameters (Figure 9.1b) include metal width, number of turns, centre spacing, metal line spacing, etc. The relationships between the parasitic component values and the design and process parameters are complex and difficult to model accurately. For example, to reduce the series and substrate resistances of the spiral inductor and increase its quality factor (Q), the metal width can be increased. However, this increases the oxide capacitance between the spiral and the substrate, thus reducing the self-resonance frequency (Fr).

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Figure 9.1: (a) Cross-sectional and (b) top views of a parasitic-laden monolithic square spiral inductor

There are several ways to find the parasitic component values associated with an inductor. One method involves the development of approximate analytic equations [1], and another uses electromagnetic (EM) simulations. A purely empirical method involves first fabricating and characterising all inductors, in the chosen CMOS technology, before they are used in subsequent circuit designs. The first method is easy to use and moderately accurate to frequencies of about 2 GHz. However, at higher frequencies, the compact model component values are insufficiently accurate due to various high-frequency effects such as the skin effect, etc. The EM simulation method can be more accurate for higher frequencies depending on knowledge of the CMOS process details, the complexity and type of the compact model, and the capabilities of the simulator. However, EM simulations require very long simulation times and are often not sufficiently accurate. The full-blown empirical characterisation method provides good models for all frequencies, but it is expensive and time consuming to fabricate inductors prior to the design and optimisation processes.

An alternative approach for modelling spiral inductors combines aspects of EM simulation and full-blown characterisation. We first design, fabricate and characterise three spirals with identical geometric features spanning a range of values. (We call this the ‘three bears’ approach because the smallest (baby), a mid-size (mother), and the largest (father) inductor values provide sufficient data.) After fabrication and characterisation, the results are used to calibrate an EM simulator to fit the measured data. It can now accurately predict parasitics for interpolated inductance values.

Figure 9.2 shows a typical measured frequency response of an on-chip spiral inductor. In 0.35 μm CMOS, a metal-3 spiral with 6.25 turns (9 nH, metal width = 15 μm, centre spacing = 101.4 μm, and metal spacing = 1.2 μm) exhibits a peak Q of 4 at 3.2 GHz and a self-resonant frequency of 3.2 GHz. Parasitic-aware optimisation requires that the parasitic values be expressed versus inductance (Figure 9.3). The model equations are obtained using the three bears modelling approach, EM simulations for interpolation, and the polyfit command in MATLAB.

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Figure 9.2: Typical measured frequency response of an on-chip square spiral inductor

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Figure 9.3: Parasitic values (see Figure 9.1) versus inductance for on-chip square spiral inductors. The dark circles represent values extracted from measurements using the ‘three bears’ modelling approach. The ‘x’ values are obtained using a calibrated EM simulator

9.2.2 Bond wire inductor

Another possible on-chip inductor is the bond wire, which has several advantages over the spiral inductor. Since the bond wire is often gold, rather than aluminium, and its radius is large (about 30 μm) compared to the dimensions of the spiral, its series resistance is much smaller, and therefore, its quality factor is also much higher. Figure 9.4 shows a bond wire inductor with its important parasitics. Note that the bonding pads contribute Cox and Rsub which are constant with respect to the inductance value (i.e. the length of the bond wire).

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Figure 9.4: A typical bond wire inductor with parasitics

As a rule of thumb, each 1 mm length of the bond wire contributes 1 nH of inductance. A first-order formula commonly used to calculate the bond wire inductance value is [2]

where L is the inductance in nH, l is the length of the bond-wire in mm, and r is its radius in mm. Parasitic component values for a typical bond pad are Cox = 200 aF and Rsub = 20 Ω.

The series resistance is

where ρ is the resistivity of gold, 2.35 μohm-cm, l is the length of the wire in cm, and A is its cross-sectional area in cm2. For use at high frequencies, the skin effect that increases the effective resistivity at high frequencies must be considered. The associated skin depth δ is

Since the permeability (μ) of gold is 1.26 μF/m, the skin depth is 2.57 μmat 900 MHz, for example. Assuming a typical radius for the bond wire inductor of 30 μm, δ is clearly much smaller than r; hence, the approximate skin area of conduction is

Using (9.3) and (9.4), the series resistance is 0.05 ohm per mm.

Traditionally, the manufacturability and repeatability of bond wire inductors have been concerns. However, it has recently been shown that machine-bonded wires of the type used in manufacturing exhibit less than 5 per cent inductance variations and less than 6 per cent Q variations [3].




Wireless Communication Circuits and Systems
Wireless Communications Circuits and Systems (IEE Circuits, Devices and Systems Series 16)
ISBN: 0852964439
EAN: 2147483647
Year: 2004
Pages: 100
Authors: Yichuang Sun

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