New Technology for Automatic Spiral Inductor Model Generation Speeds and Enhances Circuit Design for
Ultra-Wideband Applications
by Mounir Adada, Agilent EEsof EDA

Introduction
The availability of spiral inductor models that meet the demands of the emerging wireless communication designs is a crucial element of a successful design flow. Yet, the challenges of modeling spiral inductors for narrow-band applications are increasing along with emerging Ultra-Wideband (UWB) wireless applications. The challenge is getting an accurate model fit for UWB with traditional, narrow-band modeling methodologies.

This article discusses integrated spiral inductor metrics, key physical design challenges, and current modeling approaches and limitations. It then introduces a new spiral inductor modeling methodology and application example that is well suited to UWB wireless applications.

Figure 1: An equivalent network representation of a spiral inductor on silicon,
showing the added parasitic capacitance and resistance.

Spiral Inductor Metrics
To efficiently break down the modeling tasks into more understandable and manageable parts, we will look at the key inductor metrics that designers look for. Three key metrics accompany integrated inductors- Self and mutual inductance, quality factor (Q), and self-resonance frequency (SRF).

Inductors are used in circuit design as storage bins for magnetic energy, in contrast to capacitors, which are used as storage bins for electric energy. The inductance value is composed of self and mutual inductance. The self-inductance is a measure of the magnetic field-generated by a time-varying current- external to the wire, and is more dependent on the length of the wire than on its cross-section. Mutual inductance is the measure of mutually coupled magnetic fields of adjacent wires with current flowing in the same direction. With spiral inductors, mutual inductance tends to be the dominant portion of the inductor's overall inductance value, and its value is more dependent on the spiral wire's pitch than the wire's spacing.

The next important metric concerning integrated inductors is the quality factor, or Q. Q is a measure of how good an inductor you have. The simple definition of Q is the amount of energy stored over the energy loss in one cycle. With regard to inductors, Q becomes the ratio of peak magnetic energy minus peak electric energy (an unwanted side effect observed in the spiral inductor's parasitic capacitance) over energy loss in one cycle.

Figure 1 shows a typical circuit model of an integrated spiral inductor. Note that because there are added parasitic capacitances to the model, stored electric energy becomes an unwelcome addition to the inductor, negatively affecting the inductor's quality.

Figure 2: A cross-section of a typical spiral
inductor with its associated parasitic capacitance and resistance.

The third inductor design metric is known as the self-resonance frequency (SRF). At this frequency, the inductor stops behaving like an inductor and starts to resemble a poor capacitor. This is best illustrated in Figure 2. Simply, at high enough frequency, the spiral inductor associated electric field increases to a point beyond the oxide layer(s) and capacitively couples thru the semi-conducting substrate to the low potential point, whether it is a backside ground or a package ground. Designers often choose a frequency operating range that is safe enough from the SRF (within 65 - 75% of the inductor's frequency band).

Design Challenges: High Q Isn't the Answer for Everything!
With key design metrics established, designers of spiral inductors still face tricky questions about the overall performance of spiral inductors for specific applications. Inductors can range in size, Q-factor, frequency band, and other application-specific factors. For example, high Q isn't always the most desired feature of an inductor. For certain designs, such as LC tanks, an inductor impedance value may play a more important role. In other designs, such as shunt-peak circuits, it is more important to measure the inductor parasitic capacitance. Access to a design-kit-supplied library of high-Q inductors doesn't solve all the design issues high-frequency engineers face, and the need for custom tailored inductors remains.

Figure 3: Metal and substrate losses typical of spiral inductors on silicon

Key Physical Design Issues
Now that we have a good idea of what inductor qualities we would like to have in certain classes of inductors, let us examine the physics behind integrated inductors to get a better understanding of how these qualities can be controlled. For integrated spiral inductors, there are some key physical design issues that modeling engineers need to be concerned with. These include accurate substrate modeling and analysis, spiral-to-substrate interaction, and spiral conductor physical properties.

Substrate modeling and analysis is an important step in successful inductor modeling. When it comes to Si-based designs, the importance of accurately modeling the substrate effects is further enhanced. This is mainly due to the semi-conducting, or semi-insulating, nature of Si. Of course, the substrate effects depend on the doping level of the Si substrate. This presents a great challenge for spiral inductor designers, where controlling energy coupling from the spiral inductor to the substrate is an important and challenging task. Note that the substrate coupling effect is a function of frequency, and as shown in Figure 3 , this phenomenon dominates spiral inductor losses, affecting inductor Q around the 5 GHz region and above. A simple illustration, as shown in Figure 4, can explain this physical phenomenon. The electric field generated due to the inductor's magnetic flux is confined within the oxide layers at low frequencies. However, at higher frequencies, the electric field becomes large enough that it capacitively couples through the oxide layer(s) to the substrate. And, at even higher frequency, the electric field breaks thru the substrate capacitor and shorts to the low potential point, whether it is a backside ground or a package ground. This is the self-resonance frequency (SRF) of the spiral inductor described earlier.

Figure 4: A cross-sectional view of a spiral inductor on a silicon substrate with SiO2 layers inbetween.

Another key physical effect associated with spiral inductors are the so-called substrate currents. Substrate currents are mainly composed of two parts: displacement currents from spiral traces to the substrate thru the oxide capacitance, and eddy currents in the substrate (Figure 5). Displacement currents are a product of the time varying electric field thru the oxide capacitance, and increase with higher frequencies as described earlier. Eddy currents, often associated with transformer applications, are a product of the spiral inductor time-varying magnetic field penetrating the conductive substrate. The induced currents in the conductive substrate flow in opposite direction to the current flow of spiral inductor, producing a negative effect on the performance of the integrated inductor. Eddy currents are hard to predict and quantify. However, if they become acute (depending on the topology of the spiral inductor and doping characteristics of the substrate), patterned ground shields may be applied, as shown in Figure 6. Patterned grounds solve two interrelated issues: first, they provide adequate isolation between the spiral inductor metal and the conductive substrate; and second, the breaks in the metal prevent image currents from flowing in close proximity and in opposite direction to the current flowing thru the inductor metal tracks. Having a solid metal patch between the inductor and the conductive substrate causes image currents to flow in the opposite direction as the inductor current, hence producing a counterproductive electromagnetic (EM) field.

Figure 5: Substrate currents are composed of displacement currents via the SiO2
capacitance and eddy currents due to time-varying fields penetrating Si substrate.

Metal Current Distribution and Skin Effects
At low frequencies, current flow distribution inside a wire tends to be evenly distributed. However, at high frequencies, current flow distribution becomes non-uniform and affected by eddy currents. As described earlier, eddy currents are a product of time-varying magnetic fields and adhere to Faraday's law. The effects of eddy currents can be observed in proximity effects, as discussed earlier regarding substrate currents, and as skin effects. Skin effects are essentially a measure of field penetration into nearby metal inducing eddy currents inside the metal, which in turn produce fields running in opposite direction to the impinging fields- and affecting the current distribution inside the metal conductor. Skin effects are measured by the so-called skin depth and are a function of frequency.

Figure 6: Patterned ground shields often provide a good solution when substrate
currents become too severe.

To better illustrate skin effects on a typical spiral inductor conductor, we will examine the current distribution of a given cross-sectional metal strip as a function of frequency, as shown in Figure 7.

Figure 7: Current distribution and resistive losses inside a metal conductor at
different frequencies.

Depending on the width (w), thickness (t), and position of the given metal trace, skin effects may manifest themselves as edge effects (in other words, when t < ds and w > ds), or single- or double-sided skin effects (in other words, when t = ds and w > ds) depending on whether the metal trace is in microstrip or stripline configuration. At low frequencies, with the current uniformly distributed through the metal cross-sectional area, DC resistance determines the loss factor. However, at higher frequencies, skin effects cause the current flow area to decrease and hence, the metal trace resistivity losses to increase. As skin depth is a function of frequency, these resistive losses are further increased with even higher frequencies, and definitely between 3 and 10 GHz.

Figure 8a: An octagonal spiral inductor referenced in [2] for UWB applications.

The Need for Spiral Inductor Design Models
Next, let's examine current modeling approaches and associated issues. In the early 1990s, when integrated inductors started to gain popularity, spiral inductor models were often generated manually, using measured data as the source of the model. Early models were built using a discrete model library setup, where a number of perturbed spiral inductors were fabricated and the measured data tabulated in lookup tables. This provided the end user with a model database that offered a limited number of spirals' topologies and an even more limited parameter sets. This approach, even though it offered very good accuracy at the selected points, greatly limited design options, and if the process was changed, the entire effort of manually building the model needed to be performed all over again.

Figure 8b: Close correlation between measured and modeled by ADS Momentum
of the spiral's inductance value.

Later, PI networks (see Figure 1) gained more popularity among the modeling community. They allowed modeling engineers to build equivalent networks based on measured data. The early PI network representation of integrated spiral inductors was based on seven elements, and later evolved to include nine or more elements. Early on, the accuracy checks behind these equivalent networks were measured data, and later included a mix of both measured and modeled data. PI networks offer a good way to harness the performance of a given spiral inductor in a compact model that can simulate fast within a given EDA tool. However, they present some real challenges when it comes to spiral inductor modeling for UWB applications, because PI networks are narrow-band models, and extending them beyond their traditional use model is very challenging. Other related modeling issues include the need to fabricate a large number of varying spiral inductor topologies (with a limited type and number of parameters), and the need for specific modeling expertise. These issues drive the cost of spiral inductor model generation up significantly.

Figure 8c: Close correlation of the spiral's Q factor as measured
and modeled by ADS Momentum.

To overcome some of these challenges, many modeling engineers are beginning to use EM tools to build accurate spiral inductor models over wide frequency bands. This approach provides many benefits, such as minimal EM modeling knowledge, good accuracy over the defined frequency and parameter space, more freedom to try different design variations, and the ability to verify the model performance within its intended physical environment. A good characterization of the process parameters (for example, substrate and metallization) is needed, however. Figure 8 shows a reference spiral inductor model for UWB application modeled using Agilent EEsof Momentum EM simulator. Very good agreement was reached between modeled and measured data.

Taking Spiral Inductor Modeling to the Next Level
Although the EM approach to spiral inductors modeling has many benefits, it also has some limitations, mainly because the EM model is often a snapshot of a given set of parameter and process specifications. In other words, if the user requires a different mix of component parameters or a slightly modified layer stack, a new model is needed. This may be manageable when working on a single inductor at a time, but when working with bigger circuits, this can be painful.

Figure 9: An overview of the inner workings behind the Advanced Model
Composer modeling technology.

Over the years, many have come out with clever techniques to make this iterative process more efficient with well-designed database management and interpolation techniques. However, these attempts, as worthy as they may be, still do not deliver what circuit designers of emerging wireless applications need- a parameterized, broadband, EM-accurate model that simulates at very fast speeds that are comparable to those of standard analytical models, without compromising accuracy or speed.

Fortunately, Advanced Model Composer (AMC), a new technological innovation from Agilent EEsof EDA, allows spiral inductor modeling engineers to develop, with minimal effort, libraries of custom, parameterized spiral inductor components that exhibit EM accuracy and run at ultrafast speeds that are comparable to the speed of analytical models.

Figure 10: Advanced Model Composer user interface allows modeling engineers to
set up their modeling jobs using a graphical interface.

Using this technology, spiral inductor modeling engineers can now generate, with minimal effort, custom-tailored spiral inductor component libraries based on the frequency bands of interest and the process properties specific to current and emerging technologies. These spiral inductor model libraries exhibit EM accuracy over the entire frequency and parameter space, and run at fast simulation speeds while maintaining a compact footprint, and with no associated model database to maintain. They can also be readily shared with other users of Agilent EEsof Advanced Design System (ADS), to allow them to achieve the same design accuracy as they make their contributions to the design process.

Advanced Model Composer Technology
The technology behind Advanced Model Composer, known as MAPS (Multidimensional Adaptive Parameter Sampling), selects a minimum number of EM simulations, and builds a global analytical fitting model for the scattering parameters of general planar structures as a function of the geometrical parameters and of the frequency, with a predefined accuracy.3 Data points are selected efficiently and model complexity is automatically adapted. The algorithm consists of an adaptive modeling loop and an adaptive sample selection loop (see Figure 9). The entire process is fully automated and does not require user intervention.

Application Example: UWB Spiral Inductor
To illustrate the power of this new technological innovation, we will build an octagonal spiral inductor, similar to the inductor in reference 2. Because the referenced design used ADS Momentum- the same EM engine behind AMC- the accuracy level should be similar.

Figure 11: An ADS schematic test bench setup to verify the AMC model
accuracy to direct EM simulation.

We begin by placing a layout instance of the spiral inductor macro (downloaded from the Agilent EEsof EDA Knowledge Center web site). Next, we assign ports, define a number of parameters (for example, width, spacing, and number of turns), and specify a frequency range, as shown in Figure 10. The process information is automatically read if we are working within a project that is based on the given process technology, or process information is entered via the substrate definition interface.

Next, we start the model generation process with a mouse click. Actual model generation time varies, depending on factors such as component topology, number and range of parameters, and frequency band, as well as the user's hardware configuration.

After the spiral inductor model is generated, we set up a simple test bench (Figure 11) to verify the new model accuracy to direct EM simulation. Figure 12 shows close correlation between our new AMC model and the same component analyzed with the ADS Momentum full EM simulation.

Figure 12: Close correlation between the AMC-built model and direct EM
simulation using ADS Momentum.

Spiral inductor models created with AMC simulate fast enough to allow real-time parameter tuning. Models can span the full UWB frequency band, while simulating fast enough to allow tuning and optimization of the entire circuit.

Advanced Model Composer contains an additional capability known as real-time component parameter extraction. For example, spiral inductor modeling engineers often focus on controlling the physical parameters of a given spiral inductor, such as number of turns, width, and metal spacing. However, circuit designers often are more interested in the inductance and Q values of a given inductor. With post-processing, the model designer can control which component parameter(s) are used as user input and which are extracted, in real time, so that the circuit designer gets the necessary inductor and Q values. Figure 13 shows the UWB spiral inductor with the three user defined parameters (N, W and S), and the inductance (L) as the parameter that gets extracted in real-time.

Figure 13: Advanced Model Composer post-processing, allowing modeling engineers to
provide real-time parameter extraction such as the spiral inductor's inductance value.

Summary
Spiral inductors constitute critical building blocks for emerging wireless designs. Many of the modeling techniques that worked well in the past may not be adequate to address emerging UWB wireless standards. Ultra-wide frequency bands and the requirements for accurate, yet flexible, models are raising the bar for delivery of accurate spiral inductor models that simulate fast and give circuit designers the flexibility they are used to with standard analytical models. The new modeling technology described in this article allows wireless design and modeling engineers to improve the accuracy and range of existing components or to create new, previously unavailable design models.

References
1 Source: Dr. M P Wilson, "Modeling of integrated VCO resonators using Momentum", Tality UK.
2 Youri Tretiakov, IBM Microelectronics, et al, "Improved Modeling Accuracy of Thick Metal Passive SiGe/BiCMOS Components for UWB using ADS Momentum," Microwave Product Digest, October 2004.
3 Mounir Adada and Tom Dhaene, "Advanced Model Composer: Empowering Microwave Designers with Speed and Accuracy," Microwave Product Digest, April 2004.

Mounir Adada is a Product Manager with Agilent EEsof EDA in Westlake Village, California.

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