Sunday, May 26, 2024

Analysis and Loss Mechanism of a Spiral Inductor Via the Green’s Function Model

Parent Category: 2015 HFE

By Pasquale Dottorato

A spiral inductor was analyzed through appropriate Green’s function derived from solution of Maxwell’s equations. The loss mechanism at the substrate was investigated, which included consideration of dissipative coupling, Eddy currents and edge proximity effects. A spiral inductor in 0.35 µm CMOS was simulated, built, and measured. S-parameter, inductance, series resistance and Q-factor were calculated and measured.  They were subsequently compared with several models provided in technical literature. Finally, a modified π circuit model was extracted.


A spiral was built by using one of the metal layers embedded in silicon oxide and placed at some distance from the substrate. When a time varying voltage is applied between ends of the spiral, electrical and magnetic fields are generated. Magnetic Field b(t) time variant current flowing along a conductor induces a magnetic field and two effects are produced:

  • Self and mutual inductance coupling among the metal tracks of the inductor.
  • Eddy currents are induced in the substrate and metal tracks.

The Electric Field, e(t), generated as a consequence of the existing voltage difference between the two ends of the spiral, produces ohmic losses in the spiral due to the resistivity of the metal tracks. Furthermore, the electric field passing through the oxide between strips is generated as a result of the existing voltage difference between adjacent strips and the inner connection. It produces the capacitive coupling within the coil. The electric field passing through the oxide and substrate is generated as a result of the existing voltage difference between spiral and substrate. This produces two effects:

  • Capacitive coupling between the spiral circuit and substrate;
  • Ohmic losses due to displacement currents induced through the capacitive coupling between the spiral and the substrate.

The Theory

Maxwell’s equation for curl are [4], [6]:

1503 HFE spiral eq01   


with ε, µ complex permittivity electric and magnetic respectively equal: 

ε=ε’-jε” and µ=µ’-jµ”

Managing and invoking a Lorentz’s gauge the electromagnetic field is [3], [5], [14]:

1503 HFE spiral eq02    

(2) (3)   


1503 HFE spiral eq03


1503 HFE spiral eq02c

Φ(r,t), A(r,t), ρ(r’) and J(r’) are defined in [4]; while, GE(r,r’) and GM(r,r’) are the Green’s function with k run through the conductors which make up device; V’ is the volume of all conductors.

From (1), (2) and (3) we obtain for i-th segment:

1503 HFE spiral eq04


In general, the current density is non-uniform across the cross-section of each segment. The non- uniformity is due to current constriction and proximity effect. If, however, we discretize [13], [14] and [28] the cross-section into k sections of constant current density, then (4) will take another form: 

1503 HFE spiral eq05


where Iim, is current inside the sub-segment m of segment i,lm is vector in the direction of the segment, and wi,m is a function that has a volume of 1/ai,m  inside the cross-sectional area of sub segment m of segment i. Therefore:

1503 HFE spiral eq06


1503 HFE spiral eq07


and Sk is kept the volume Vk

The equation (6) can be simplified [10], [20] if we make the assumption that the potential drops across parallel sub-segments are equals. We will simplify the above equation to:

1503 HFE spiral eq08


Or in matrix notation:

1503 HFE spiral eq09


The matrix [M] has has diagonal entries representing the internal impedance of each segment and the off-diagonal terms represent the external magnetic coupling from segment to segment. The Green’s function is [2], [3], [5]:

1503 HFE spiral eq10


Where r, r’ identifies the position; therefore the equation (5), (6) and (7) leads to the following definition for  Mik about magnetic coupling [6]:

1503 HFE spiral eq11


Vi and Vj denotes the conductive region (segments), while Ii, Ij are the total currents through their segments. The (10) is handled differently depending whether self-inductances (i=j) a mutual inductances (i≠j) are calculated. This relates the inductances only geometry; however, this result in closed form expression for elementary shapes of geometries. The vector dot product quantifies into the well-known observation that the mutual inductance between two elements is a maximum when the current flow is in the same direction. When the current flow is in opposite direction, the mutual inductance is a minimum and negative; while, if the current elements are in orthogonal directions, then, the mutual inductance is zero.  Likewise for electric field we obtain [11]:

1503 HFE spiral eq12



1503 HFE spiral eq13


 [C] is the capacitance matrix, it’s obtained from inverse potential matrix [P] [20] that define as:

1503 HFE spiral eq14


In (13) we can identify the real part as due to the finite conductivity of the substrate, in fact apply the charge conservation equation at each segment we obtain:

1503 HFE spiral eq15

(14) (15)

The inversion of the matrix [P] is neither simple nor quickly solved; however, the use of techniques such as Cholesky decomposition or DCT (Discrete Cosine Transform) methods help the development of efficient algorithms making calculations [14]. 

Spiral Inductor

For evaluation (8) and (12) we introduce the GMD AMD and AMSD concept, yielding the following general approximate expression [4] for two parallel conductor (i and j) straight with rectangular cross-section :

1503 HFE spiral eq16


Equations (16) are expressed in nH. There are four cases possible for calculating the mutual inductance; 

1503 HFE spiral fg01

Figure 1 • Example of calculation of M for two planar conductors.

1503 HFE spiral eq17


These equations  apply from (a) to (c) case. The index i and j are conductors; p and q are the index of the length for the difference in the length of two conductors and its angle tilt. A matrix representation allows one to visualize the terms better. Since reciprocity holds (Mij=Mji), we obtain a symmetric matrix implying that we only need to compute half of (n2-n) mutual inductance terms:

1503 HFE spiral eq18


Where Mij elements of the matrix is the mutual inductance between I and j conductor if i ≠ j, while Lij, is self-inductance of the ith conductor if i = j. In order, the rectangular coil of n turn has N=4n segments, 2[n] of which are orthogonal to the other 2[n] segment of the 2[n] parallel segment, n have current flow in the opposite direction to the other n. Using symmetry we can write the total inductance as

1503 HFE spiral eq19


Where M+ij denotes the positive mutual inductance terms and M-ij denotes the negative mutual inductance terms. As well from trace of matrix (17) the coefficient of self-induction of the coil.

Loss Mechanism

All integrated passive devices suffer from substrate effects, in fact the spiral inductor fabricated on substrate experienced significant losses due to the magnetically and electrically induced Eddy current [1], [9], [20] and [22]. The objective of the parasitic extraction is to compute the capacitance matrix C for a multi conductor geometry [11]. Eddy current densities are higher at the edges of the conductor body where they are developed  and the skin effect is formed. Analysis shows that the magnetic field and thus inductance of one conducting loop is dramatically affected by eddy currents distributed within a nearby conducting ground plane.

1503 HFE spiral fg02

Figure 2 • Loss mechanism for on-chip spiral inductor.

The use of the Green’s function greatly simplifies the problem by implicitly taking into account the boundary conditions, making it unnecessary to discretize the boundaries. The Green function has been previously computed in analytical form from [20]. Replacing this in (6), (11) and solving for the source leads to the finding in [14]. Follow immediately from here: 

1503 HFE spiral eq20


The capacitance connected between contact-i and contact-j is then given by:  

1503 HFE spiral eq21


where φmi is the potential on the m-th subdivision of contact-I; while the conductance between contact-i and contact-j is given by:

1503 HFE spiral eq22


Let contact-i have N subdivisions, and contact-j have Mh subdivisions, furthermore the element of [C] representing interaction between the m-th division of ‘i’ and the n-th division of ‘j’ be given by . 

With equation (19), (20) and (21) we account for all loss mechanisms. From this we can extract an π circuit model as shown in figure 2. Here ohmic loss is a factor limiting the inductor performance. For spiral inductors operating at high frequencies, the series resistance is frequency dependent; substrate parasitics result from the electrical coupling between the metal track and substrate, as the metal track of a spiral inductor can be considered as a microstrip on substrate with waves passing through it. Three elements, COX, GSUB, and CSUB, are used to model the substrate. The frequency-dependent permittivity εeff (f) is needed to model the frequency-dependent capacitance. 

1503 HFE spiral fg03

Figure 3 • Improved inductor model with horizontally coupled Resistance and Capacitance RSUB and CSUB.

Modified π Circuit  Model

Some studies have been conducted to improve the accuracy of the simple lumped models. Though Cao et al. [28] proposed a double π model to account for the frequency-dependant resistance and inductance, here, we refer to Gil and Shin [31] modified the simple π model by adding the horizontally coupled substrate resistor and capacitor. The equivalent circuit and results of Q factor are given in Figure 3 and Figure 7.

1503 HFE spiral fg04

Figure 4 • (a) Topology (b) cross section of the on-chip square shaped inductor.

Model Verification and Result

A symmetrical spiral inductor fabricated with the 0.35µm CMOS technology was first simulated with Sonnet simulator [33] and measured to verify the model developed. The inductor was built on a 9.59µm oxide and 500µm silicon substrate. The inductor has 6 turns with metal width of 15μm, metal thickness of 2μm, and metal line spacing of 8μm. Two-port parameters were measured, and the inductance, resistance and quality factor of the inductor were extracted. In addition to the present model, two existing inductor models Yue and Wong [28], Mohan et al. [32] intended for the asymmetrical inductors were considered. The topology of the inductors and its layers are shown in Figure 4. 

1503 HFE spiral fg05

Figure 5 • Comparisons of the present model, existing models, and measurements of (a) Quality Factor (b) Inductance and (c) Series Resistance for built symmetrical inductor.

The inductors were measured using an Agilent network analyzer and properly de-embedded (see Figure 8). We carry out the simulated and measured data from Figure 5 to Figure 7 and in Table 1.

1503 HFE spiral fg06

Figure 6 • Comparisons of the present model, existing models, and measurements of (a) Magnitude and (b) Phase of S21 for built symmetrical inductor.

The present model demonstrates a better accuracy over the existing models for a wide range of operating frequencies. Thus, our results suggested that it is erroneous and impractical to use the inductor model developed intended for asymmetrical inductors for predicting the characteristics of symmetrical inductors.

1503 HFE spiral fg07

Figure 7 • Q Factors Measured and Simulated with and without RSUB and CSUB.

1503 HFE spiral fg08

Figure 8 • Block Diagram of the VNA De-embedding DUT from Fixture Setup Measurement.

1503 HFE spiral tb01

Table 1 • Extraction Modified π Circuit Model.


A numerical compact and accurate model for spiral inductors has been developed through the Green’s function. The Sonnet 2,5D EM simulator tool was used to numerical model. Unlike the existing inductor models which were developed intended only for asymmetrical inductors, the present model is shown capable of predicting accurately both the symmetrical and asymmetrical inductors. The concept of the effective line width was introduced to account for the effect of non-uniform current distribution in the metal lines, and overlap parasitics and geometry factors have also been included. Comparisons among the present model, existing models, and measured data were presented to illustrate the usefulness of this work.

About the Author:

Dr. Dottorato was born in November 23, 1969. He is a Sr. RF & Microwave Designer Consultant and member of IEEE. He received the “Laurea” degree in electrical engineering “cum laude” and Ph.D. degree from University of Naples. After an interesting experience at IRECE and electronics and telecommunication department at University “Federico II” of Naples, he worked in the design of microwave equipment for Oerlikon Contraves, an electronics defense company in Rome. Since 2005 he has been working in the R&D department of an electronics company in Bologna. Interests include numerical modelling of inverse and scattering electromagnetic problems, the design of microwave devices with operating frequencies up to 110 GHz such as: antennas, conformal antennas for EW applications, LNAs, HPA pHmet and GaN technology processes, MMIC/RFIC in CMOS GaAs and InP technology processes.


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