Recent models for on-chip spiral inductors have been extensively examined and compared by transfer function analysis. Through the calculation of the transfer functions for the models, including T-, 1-π, and 2-π models, the pros and cons of equivalent circuit topology of each model are evaluated. It is found that the number of poles provided by a certain topology is a constant, while complex poles are responsible for the broadband fitting capacity of the model. The 2-π model has the most poles and the best broadband fitting capacity, while 1-π and T-models are better solutions considering both accuracy and efficiency. A novel broadband model combining the advantages of the physics-based circuit model and behavioral macro-model is proposed for accurately characterizing RF behaviors of spiral inductors. A number of inductors with various geometries have been fabricated to verify the model. Excellent agreements are obtained between the measured data and calculation from the proposed model up to 40 GHz. This modeling method is also applicable to other passive components such as transmission lines and transformers.

Transfer Function Analysis and Broadband Scalable Model for On-Chip Spiral Inductors

Passive devices and spiral inductors in particular suffer severely from losses in the silicon substrate underneath. This major component in limiting the quality factor of spiral inductors can be mitigated by using higher substrate resistivity or by locally increasing the resistivity below the inductor. These approaches have been modelled mostly ad hoc with phenomenological π models guiding the design. In this paper we use HFSS 3D finite element simulations to simulate the response of spiral inductors above high resistivity substrates of various resistivities achievable with Si. We relate these simulations to a simple one-port model of the Q-factor of spiral inductors to show in physical terms that the effective substrate thickness of around 30 μm of the spiral inductor is equal to half the electric field penetration depth. The value of the penetration depth depends on the lay-out of the spiral inductor, but, critically, not on the substrate resistivity, and hence the one-port model maintains the proportionality between resistance parameter and substrate resistivity.

Analytical and numerical model of spiral inductors on high resistivity silicon substrates

A lumped-element model for ON-chip interconnects is proposed in this paper. The effect of current return path of interconnects is analyzed. An efficient analytical parameter extraction method for the model is proposed. The inductor $L_{g}$ was adopted in the shunt branch of this model to characterize the induced time-varying magnetic field. Thus, the bandwidth of the model is extended over the resonance frequency. A group of interconnects with different structures were fabricated in a 0.18- $\mu \text{m}$ CMOS process to investigate the effects of current return path. The good agreement of $S$ -parameters between the model and measurements suggests that the proposed model can accurately predict the performance of interconnect in a wide frequency band up to 80 GHz, even over the resonance frequency.

An Equivalent Circuit Model With Current Return Path Effects for ON-Chip Interconnect up to 80 GHz

Because of the compact layout of RFICs and mm-wave ICs, coupling effects among neighboring inductors may seriously degrade circuit performance. This paper analyzes the coupling effects among multiple on-chip spiral inductors and develops a fully scalable compact lumped element model for multiple coupled inductors by using four-port S-parameters. Each single inductor model is directly extracted from four-port S-parameters based on a one-port extraction algorithm. Mutual inductance and coupling capacitance among inductors is extracted and added to the single inductor models. Compared with the measurement, the proposed model can accurately predict the physical behavior of multiple coupled inductors from dc to self-resonant frequencies, as well as EM simulation results. The test structures were fabricated in a commercial 0.18 μm RFCMOS process.

Characterization and Modeling of Multiple Coupled Inductors Based on On-Chip Four-Port Measurement

In this brief, closed-form analytical expressions are obtained to accurately calculate the oxide and substrate (both lateral/vertical) capacitances of the $\pi $ -equivalent circuit model for on-chip tapered spiral inductors. A lateral RC substrate network using the above-mentioned expressions is shown to significantly improve the model accuracy, especially with lower substrate resistivities. Furthermore, improvement in Qmax with reverse excitation in tapered spirals is also accurately predicted by the proposed model. The accuracy of the proposed model is validated till 15 GHz using several inductor geometries across process parameters suitable for the design of RF circuits. Excellent agreement is observed between the model, electromagnetic simulations, and measurements.

A Substrate Model for On-Chip Tapered Spiral Inductors With Forward and Reverse Excitations

A systematic model-parameter-extraction technique is presented for accurately modeling on-chip spiral inductors in radio frequency integrated circuits (RFICs). The model is a pi-circuit with an additional parallel RC network connecting both vertical branches to account for substrate coupling. The extraction starts with extracting the series inductance and resistance at low frequencies. Then, the oxide capacitance is evaluated in an intermediate frequency range. Afterward, the substrate effects including the substrate resistance and capacitance, as well as coupling, are extracted at higher frequencies. All the lumped circuit element values are analytically determined by the network analysis from the measured network parameters (S- or Y-parameters). The proposed approach thus can provide better circuital interpretations of the inductor behaviors for facilitating the design of RFIC inductors. Square and circular CMOS spiral and octagonal BiCMOS7 spiral inductors are investigated to test this technique. Highly accurate frequency responses by the extracted parameters are obtained over a wide frequency band without any optimization. This reveals the validation and capability of the proposed parameter-extraction method.

https://ir.nctu.edu.tw:443/bitstream/11536/8173/1/000260899000050.pdf

Tzung-Chi Wu

Papers

Accurate Systematic Model-Parameter Extraction for On-Chip Spiral Inductors