Thesis Organization

Besides, the circuit designers generally have critical concern about the accuracy of simulation models for active and passive components. As a result, an accurate RF device model suitable for various manufacturing technologies is strongly demanded.

The mentioned requirement triggers our motivation of this work to build an accurate and scalable model for on-chip spiral inductors in RF circuit applications. Besides the accuracy and scalability, a reliable de-embedding method and an efficient model parameter extraction flow are the primary goals of this work. The accurate extraction of intrinsic device characteristics is prerequisite to accurate modeling while the challenges become tougher for miniaturized devices. An efficient model parameter extraction flow can be automated through commercial extraction tool to expedite the model extraction and optimization.

1.2 Thesis Organization

The theme of this thesis is the development of an accurate and scalable on-chip inductor model applicable for RF circuit simulation and design over broadband up to 20 GHz and beyond. In Chapter 2, I will discuss the existing issues for current inductor models, e.g. π-model. Also, I will introduce briefly the application of pi-model which is used to build in passive model.

In Chapter 3 and Chapter 4, I will focus on the development of a broadband and scalable model for on-chip Inductor. Both single-end and symmetric inductors have been covered in this work. A new symmetric inductor of fully symmetric layout as well as taper metal line have been fabricated and a new de-embedding method has been

derived to realize accurate extraction of the intrinsic device parameters. A parameter extraction flow has been established through equivalent circuit analysis to enable automatic parameter extraction and optimization. The equivalent circuit, physics phenomenon that is observation from 3D EM simulation, and analysis of extracted parameters will all be explained in these chapters. According to above concepts, we will design new model to present different inductor at the high frequency characteristics. We also improve asymmetrystructures for spiral and conventional symmetry inductor between the S11 and S22. But it can decrease the quality factor (Q) and self-resonant frequency (fSR). So we will design taper inductor to increase quality factor. For the above reason, how to improve the characteristics of passive devices and achieve low cost and high competition simultaneously is worth trying.

In Chapter5, the lump-element equivalent circuit verified and analyzed by ADS circuit simulator is to simulate circuit level for different inductor modification.

Chapter7 is discussed the future work and Appendixes related to analytical formula for lump-element equivalent circuit. Our analysis and inference will be verified through ADS simulation result for equivalent circuit. And we gives the conclusions to this work and its development in the future.

Chapter 2

Review on Existing Inductor Models – Remaining Issues

2.1 Requirements for inductor models for RF circuit simulation

The rapid growth of the wireless communication market has fueled a large demand for low cost, high competitive, portable products. Traditionally, radio systems are implemented on the board level incorporating a lot of discrete components.

Recently, compared with discrete and hybrid designs, the monolithic approach offers improved reliability , lower cost and smaller size, broadband performance, and design flexibility. In conventional design, bonding wires having a relatively high Q were used to replace on-chip inductors. However, the bonding wires generally suffer worse variations in inductance value because that they cannot be as tightly controlled as the on-chip inductors implemented by integrated circuit process. Recent advancement in silicon based RF CMOS technology can provide RF passive components such as inductors with fair performance suitable for analog and RF IC design up to several giga-hertz, then it can be integrated on a chip to match market demands. Therefore, an accurate on-chip RF passive device model applicable for circuit simulation and design becomes indispensable and the mentioned requirement triggers our motivation of this work.

Extensive research work has been done to investigate inductors of various layouts and topologies such as spiral inductor, conventional symmetric inductor, and fully

symmetric inductors of single-end and differential configuration. All the mentioned inductors have been fabricated on semi-conducting Si substrate for measurement, characterization as well as model parameter extraction for circuit simulation model development. In this chapter, we will introduce existing inductor models targeted for Si based RF circuit simulation. Comparison will be done for various models in terms of accuracy and bandwidth of validity, scalability and geometry of validity as well as model parameter extraction methodologies, etc.

2.2 Analysis and comparison of existing models

Monolithic inductors have drawn increasing interest for applications in radio frequency integrated circuit (RF ICs), such as low noise amplifier (LNA), voltage controlled oscillator (VCO), Mixer , input and output match network. It is believed that SoC approach can provide benefit of lower cost, higher integration, and better system performance. However, some inherent limitations originated from the low resistivity substrate of bulk Si should be overcomed through effort in process technology and layout or new configurations in circuit operation, e.g. differenentially driven instead of single end operation. To facilitate the RF circuit simulation accuracy and prediction capability, the physical limitation coming from substrate loss, conductor loss, and the mutual interaction should be carefully considered and implemented in the circuit level models. The physical mechanisms, which are well recognized for on Si chip inductors include eddy currents on spiral metal coils and semiconducting substrate due to instantaneous electromagnetic field coupling, crossover capacitance between the spiral coils and under-pass, coupling capacitance between monolithic inductor and substrate, substrate capacitance and substrate ohmic loss, etc. In the following, the

discussion on mentioned model features will be provided.

2.2.1 Accuracy and bandwidth of validity

The lack of accurate model for on-chip inductors presents one of the most challenging problems for silicon-based RF IC design. In conventional IC technologies, inductors are not considered as standard components like transistors, resistors, or capacitors, whose equivalent circuit models are usually included in the Spice model for circuit simulation. However, this situation is rapidly changing as the demand for RF IC’s continues to grow. Various approaches for modeling inductors on silicon have been reported in past decade. Most of these models are based on numerical techniques, curve fitting or empirical formulae and therefore are relatively inaccurate

for higher frequencies. For monolithic inductor design and optimization, a compact physical model is required. The difficulty of physical modeling stems from the

complexity of high frequency phenomena such as the eddy currents in the coil conductor and semiconducting substrate as well as the substrate loss in the silicon.

The key to accurate physical modeling is firstly to identify all the parasitic and loss effects and then to implement a physics based model for simulating the identified parasitic and loss effects. Since an inductor is intended for storing magnetic energy, the inevitable resistance and capacitance in a real inductor are counter-productive and thus are considered parasitic effects. The parasitic resistances dissipate energy through ohmic loss while the parasitic capacitances store electric energy. A traditional equivalent circuit model of an inductor generally called π-model is shown in Fig. 2.1

(a)

(b)

Figure 2.1 (a) Top (die photo);Middle, 3-D view (b)the lumped physical model of a spiral inductor on silicon

The inductance and resistance of the spiral and underpass is represented by the series inductance, Ls, and the series resistance, Rs, respectively. The overlap between the spiral and the underpass allows direct capacitive coupling between the two terminals of the inductor. The feed-through path is modeled by the parallel capacitance, Cp. The oxide capacitance between the spiral and the silicon substrate is modeled by Cox. The silicon substrate capacitance and resistance are modeled by Csi

and Rsi. There are several sources of loss in a monolithic inductor. One relatively obvious loss comes from the series winding resistance. This is because the interconnect metal used in most CMOS processes. The DC resistance of the inductor is easily calculated as the product of this sheet resistance and the number of squares in the metal strip. However, at high frequencies the resistance of the strip increases due to skin effect, proximity effect and current crowding. The substrate loss will increase with frequency due to the dissipative currents that flow in the silicon substrate. In fact, there are two different physical mechanisms that cause the induction of these currents and opposition flux.

Although physical considerations are included in such a structure, the original π-model lacks the following import feature:

1. Strong frequency dependence of series inductance and résistance as a result of the current crowding in the crowding

2. Frequency-independent circuit structure that is compatible with transient analysis and broadband design

3. It is difficulty to match high frequency behaviors, especially for thick metal case where metal-line-coupling capacitance is not negligible and substrate loss.

According to above theory and original π-model, we modify π-model for on-chip spiral

inductors over again to fit measurement data. Moreover, we add two new element Rp

and Lsub to improve above third item, as shown figure 2.2. A parallel Rp is to simulate current crowding in coil’s RLC network and series Lsub1,2 are placed under the Cox1,2

to be represent eddy effect in the substrate RLC network. In order to verify the accuracy of the modify π-model, spiral inductors with various geometrical configurations were fabricated using 0.13 μm eight-metal CMOS technology. To assess the model validity, we compare difference with model and measurement.

Figure 2.2 Modify π-model for on-chip spiral inductors.

Figure 2.3 and 2.4 show the measured and modeled S-parameters, mag(S21) and phase(S21) for a varying coil number of turns. As can be seen from these figures, the S-parameters of model match the measured data worst, especially a lager turn (N=3.5, 4.5, 5.5) at the high frequency. Figure 2.5 (a) ~ (d) reveal the exact match of Mag(S11)

for smaller coils (N= 2.5, 3.5) over full frequency range up to 20GHz, but the other figure 2.6 shows enormous error of phase(S11). Due to above match condition, modify π-model may be not suit to simulate measured S-parameters for spiral inductors.

Besides, we also make comparison with performance parameters for spiral inductor, i.e., L(ω), Re(Zin(ω)), and Q(ω). From figure 2.7 ~ 2.9 illustrates, we find that the modify π-model provides very good match with the measurement for L(ω), Re(Zin(ω)), and Q(ω) before self-resonance frequency. According to above comparison, modify π-model may be not simulate all parameters of spiral inductors and maybe can simulate certain specific parameters, especially L(ω), Re(Zin(ω)), and Q(ω). Hence, in the following chapter, we will change equivalent circuit structure over again. We use 3D EM simulation by Ansoft HFSS to simulate on-chip inductor and discover truly conforms to the physics significance parameter to establish new equivalent circuit.

0 2 4 6 8 10 12 14 16 18 20

Figure 2.3 Comparison of S21 (magnitude) between π-model simulation and measurement for spiral inductors. Coil numbers (a) N=1.5, (b) N=2.5, (c) N=3.5, (d)

N=4.5

Figure 2.4 Comparison of S21 (phase) between π-model simulation and measurement for spiral inductors. Coil numbers (a) N=1.5, (b) N=2.5, (c) N=3.5, (d) N=4.5

Figure 2.5 Comparison of S11 (magnitude) between π-model simulation and measurement for spiral inductors. Coil numbers (a) N=2.5, (b) N=3.5, (c) N=4.5, (d)

N=5.5

Figure 2.6 Comparison of S11 (phase) between π-model simulation and measurement for spiral inductors. Coil numbers (a) N=2.5, (b) N=3.5, (c) N=4.5, (d) N=5.5

Figure 2.7 Comparison of L(ω) between π-model simulation and measurement for

spiral inductors. Coil numbers (a) N=2.5, (b) N=3.5, (c) N=4.5, (d) N=5.5

800 π_model_2.5 meal_2.5

Re

(

Zin

)

(Ohm)

frequence, f (GHz)

0 2 4 6 8 10 12 14 16 18 20 0

frequence, f (GHz)

0 2 4 6 8 10 12 14 16 18 20

1000 π_model_4.5 mea_4.5

Re

(

Zin

)

(Ohm)

frequence, f (GHz)

0 2 4 6 8 10 12 14 16 18 20 0

frequence, f (GHz)

Figure 2.8 Comparison of Re(Zin(ω)) between π-model simulation and measurement for spiral inductors. Coil numbers (a) N=2.5, (b) N=3.5, (c) N=4.5, (d) N=5.5

Figure 2.9 Comparison of Q(ω) between π-model simulation and measurement for

spiral inductors. Coil numbers (a) N=2.5, (b) N=2.5, (c) N=3.5, (d) N=4.5

2.2.2 Scalability and geometries of validity

There are various geometries available for a monolithic inductor to be implemented, e.g. rectangular, hexagonal, octagonal, and circular as shown in figure 2.10.

(d) (c)

(a) (b)

Figure 2.10 Spiral inductor geometries.

Electromagnetic (EM) simulation can help to verify the layout geometry effect on inductors and the results suggest that circular spiral can provide the best performance

in terms of higher quality factor and smaller chip area. The mechanism responsible for the improved performance realized by circular spiral comes from the reduced current crowding effect. The circular inductor as shown in figure 2.2 (d) can place the largest amount of conductor in the smallest possible area, reducing the series resistance and parasitic capacitance of the spiral inductors. However, one major drawback of the circular structure is its layout complexity. It is because that its metal line consists of many cells rotated with different angles. In general, specific coding is required to generate this structure by layout tools.

In fact, a good model is developed to accurately simulate the broadband characteristics of on-Si-chip for different geometries of the inductive passive components, up to 20GHz. Besides the broadband feature, scalability is justified by good match with a liner function of geometries of the inductive passive components for all model parameters employed in the RLC network. The satisfactory scalability manifest themselves physical parameters rather than curve fitting.

2.2.3 Model parameter extraction flow and automation

Which a new model or a conventional model has been developed to accurately simulate the broadband characteristics, its all the unknown R, L,C parameters haven’t been determined initial value. So we must establish a parameter extraction flow through equivalent circuit analysis to determine initial guess value and to enable automatic parameter extraction and optimization. All the unknown R,L,C parameters are extracted from analytical equations derived from different equivalent circuit analysis. We can use Z-matrix and /or Y-matrix to extract all parameters. Above extraction and optimization principle, we use some principle to define a set of

analytical equation from measurement and to generate all unknown parameters at the equivalent circuits. Due to the necessary approximation, the extracted R,L,C parameters in the first run of low are generally not the exactly correct solution but just serve as the initial guess or further optimization through best fitting to the measured S-parameters, L(ω), Re(ω), and Q(ω).

2.3 Model enhancement strategies

The lack of an accurate and scalable model for on-chip inductors becomes one of the most challenging problems for Si-based RF IC design. The existing models suffer two major drawbacks in terms of accuracy for limited bandwidth and poor scalability.

Many reference publications reported improvement on the commonly adopted π-model by modification on the equivalent circuit schematics. However, limited band width to few gigahertz remains an issue for most of the modified π-models. A two π-model was proposed to improve the accuracy of R(ω) and L(ω) beyond self-resonance frequency. Unfortunately, this two π-model suffers a singular point above resonance. Besides, the complicated circuit topology with double element number will lead to difficulty in parameter extraction and greater time consumption in circuit simulation. Recent work using modified T-model demonstrated promising improvement in broadband accuracy and suggested the advantage of T-model over π-model. However, the scalability of model’s major concern was not presented. To solve the mentioned issues, a new T-model was proposed and developed in this work.

This T-model is proposed to realize two primary features, i.e., broadband accuracy and scalability. The T-model is composed of two RLC networks to account for spiral coils, lossy substrate, and their mutual interaction. Four physical elements, Rs Ls Rp

and Cp are incorporated to describe the spiral coils above Si substrate and other elements. All the physical elements are constants independent of frequencies and can

be expressed by a close form circuit analysis on the proposed T-model. Parameter extraction and optimization can be conducted with an initial guess extracted by approximation valid for specified frequency range.

All the model parameters manifest themselves with predictable scalability w.r.t. coil numbers and physical nature. A parameter extraction flow has been established to enable automatic parameter extraction and optimization that is easy to be adopted by existing circuit simulators like Agilent ADS or parameter extractor such as Agilent IC-Cap. The model accuracy over broadband is validated by good agreement with the measured S-parameters, L(W), Re(Zin(W)), and Q(W) up to 20GHz that this scalable inductor model can effectively improve RF circuit simulation accuracy in broad bandwidth and facilitate the design optimization using on-chip inductors.

2.4 Fundamental of quality factor for an inductor

For an ideal inductor free from energy loss due to parasitic resistance and substrate coupling effect, the magnetic energy stored can be given by (2.1),

1 ²

L 2

E = L i _L

(2.1) Where

i _L

is the instantaneous current through the inductor.

From (2.1), the peak magnetic energy stored in an inductor in sinusoidal steady state is given by,

2 2

inductor 2

1

2 2

L

peak L

E L I V

ω L

= =

(2.2)

Where

I _L

and

V _L

correspond to the peak current through and the peak voltage across the inductor.

The quality factor (Q) of an inductor is a measure of the performance of the elements defined for a sinusoidal excitation and given by,

energy stored energy stored 2 energy loss per cycle average power loss

Q = π = ω

(2.3)

The above definition is quite general which causes some confusion. However, in the case of an inductor, energy stored refers to the net peak magnetic energy.

To illustrate the determination of Q, consider an ideal inductor in series with a resistor in Figure 2.11. This models an inductor with resistance in the winding.

Figure 2.11 Inductor with a series resistance

Since the current in both elements is equal, we use the equation for the peak magnetic energy in terms of current given in (2.2) to write,

2

2

peak magnetic energy stored 2 energy loss per cycle

1 2 2

1 2 2

=

= =

=

s s

s s s s

Q

I L

L I R R

where π

π ω

τ

τ π ω

(2.4)

Where τ is the period of the sinusoidal excitation

Note that the quality factor of an inductor with a lossy winding increases with frequency. Also note that as the resistance in the inductor decreases, the quality of the inductor increases and in the limit Q becomes infinite since there is no loss. Using the above procedure, the quality factor of another pure lossy inductor can be determined.

We repeat the detail in the following.

Figure 2.12 Inductor with a parallel resistance.

Since the voltage in both elements is equal, we use the equation for the peak magnetic energy in terms of voltage given in (2.2) to write,

2

2 2

2

peak magnetic energy stored 2 energy loss per cycle

2 2

2

=

=

=

p p

p p p

p

Q

V L V

R R

L π

π ω ω τ

ω

(2.5)

Where τ is the period of the sinusoidal excitation

The definition of quality factor is general in the sense that it does not specify what stores or dissipates the energy. The subtle distinction between an inductor and an LC tank Q lies in the intended form of energy storage. For example, only the magnetic energy stored is of interest and any electric energy stored because of some inevitable parasitic capacitance in a real inductor is counterproductive. Therefore, the Q of an inductor is proportional to the net magnetic energy stored and is given by,

peak magnetic energy stored 2 energy loss per cycle

peak magnetic energy stored-peak electric energy 2

energy loss per cycle

inductor

Q π

π

=

=

(2.6)

An inductor is said to be self-resonant when the peak magnetic and electric energies are equal. Therefore, Q of an inductor vanishes to zero at the self-resonant frequency. At frequencies above the self-resonant, no net magnetic energy is available from an inductor to any external circuit. In contrast, for an LC tank, the Q is defined at the resonant frequency

ω _o

, and the energy stored term in the wxpression for Q given by (2.3) is the sum of the average magnetic and electric energy. Since at resonance the average magnetic and electric energies are equal, so we have,

average magnetic energy + average electric energy 2 energy loss per cycle

peak magnetic energy peak electric energy

2 2

energy loss per cycle energy loss per cycle

o

The average magnetic or electric energy at resonance for sinusoidal excitation is

2 2

1 1

4 L I ^L = 4 C V ^c

which are half the peak magnetic energy given by (2.2) Lets look at the parallel RLC circuit of figure 2.5 to clarify its inductor and tank Q.

Figure 2.13 Parallel RLC circuit.

The quality factor of the inductor is calculated as follows,

2

peak magnetic energy - peak electric energy 2 energy loss per cycle

1 1

where the resonant frequency

⁰ 1

p p

ω = L C

_.

Here

^p

p

R

ω L

accounts for the magnetic energy stored and ohmic loss of the parallel

ω L

accounts for the magnetic energy stored and ohmic loss of the parallel

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