How to estimate the Q of spiral inductors using a planar EM simulator, Part 1 of 2

How to estimate the Q of spiral inductors using a planar EM simulator, Part 1 of 2

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Electromagnetic (EM) simulators are commonly used to model spiral inductors in various planar technologies: for example, silicon and gallium arsenide chips, as well as the module and board level. The most common figure of merit for an inductor is its quality factor, or as it is commonly called, the Q.  Unfortunately, accurately getting the Q for an inductor is a difficult task for an EM simulator.  Planar simulators have a difficult time correctly calculating the resistive loss of the metal.  They rely on impedance boundary conditions, the accuracy of which is suspect at the corners of the conductors.  A second problem is that the mesh on the surface of the conductor must be extremely fine, leading to an impractically large mesh.  3D EM simulators have similar problems.  The interior of the metal must be excessively meshed to model the resistive loss accurately.  Second, most finite element simulators, the most popular 3D simulator, iteratively refine the mesh until the problem converges.  Unfortunately, unless care is taken, the simulator can mistakenly determine it has converged; yet, the calculated value of the Q still has significant error.

The method we demonstrate here has a number of distinct advantages.  First, a planar simulator can be used with normal mesh densities, resulting in a more realistic simulation time.  The only other required simulation tool is a cross-sectional transmission line solver that can accurately give the electrical, transmission line parameters for coupled lines.  Such tools are available at very low cost, and are extremely fast to run.  Second, because of the speed of the simulation, the designer can quickly adjust design parameters to get the desired response.  Tuning and optimization studies become realistic.

The technique works by taking advantage of the fact that most spiral inductors can be viewed as straight sections of coupled lines tied together at the ends.  Of course this is somewhat technology and geometry dependent. The current in spirals mostly flows in approximately straight line sections, not in the bends and corners connecting the sections.  We first simulate the inductor using an EM planar simulator and normal mesh densities on the surface of the conductors.  Unfortunately, the Q is off because of the inaccurate resistance modeling, due to the coarse mesh.  We now apply a correction factor by simulating the cross section of the straight line sections of the inductor in two different ways.  First, we use an EM simulator to simulate parallel line geometries.  The parallel, coupled lines are an approximation to the actual coupled lines of the real, multi-turn spiral.  The conductor loss is calculated.  We then use available models finite element to calculate the loss of the same, parallel lines.  Very accurate finite element method solvers are available which mesh the cross sectional area of the lines and surrounding media.  [1, 2]   (The cross sectional solver result of resistance/length is multiplied by the length of the lines to get the total loss.)  This simulation is fast to carry out, as it is just a cross-sectional solve.  The resistance will be higher for the cross-sectional solver than for the planar solver.  The ratio is then used to correct the Q of the original inductor.

Spiral inductors are one of the most common passive elements used in modern circuit designs.  They appear in virtually every manufacturing technology used today: integrated circuits in silicon, RF and microwave integrated circuit (MMIC) technologies of various kinds, multilayer modules in ceramic or organic technologies, and, of course, printed circuit boards. They are used for a variety of reasons, most commonly for filtering, biasing, and signal shaping purposes.  The designer typically wants to use an inductor with the best possible circuit performance that meets the various other manufacturing constraints of the problem.  The most common figure-of-merit for inductors is the quality factor, usually referred to as the “Q”.

A figure-of-merit is useful because it (hopefully) gives the designer a quick indication as to how well the inductor will perform at frequencies of interest.  The higher the Q, the better is the inductor’s performance.  Q is defined in a variety of ways, all of which are consistent with each other.  The most fundamental physical definition of Q for a time harmonically varying system is [3]:

Q is dimensionless.  The angular frequency is ω.  Q is therefore the ratio of the average energy stored in the system to the dissipated power and multiplied by the angular frequency.  Is this definition useful to the designer?  If we think of a simple inductor model, the reactive power for time harmonic excitation is directly proportional to the inductance.  Therefore, higher reactive power at a given frequency indicates a “better” inductor.  The resistive power is due to the losses in the inductor.  The main resistive losses are due to the finite conductivity of the metal conductors and the dielectric losses of the surrounding materials.  The resistive behavior is particularly important for noise figure in RF circuitry.

Most circuit designers prefer to use definitions of Q based on circuit network theory.  They are, of course, consistent with the physical definition given above.  For example, for a one port network it is possible to define Q as the ratio of the real to the imaginary part of the admittance matrix:

Other definitions are possible, for example a two port network definition can be used.  

But how can you effectively calculate the Q of a spiral inductor using an electromagnetic (EM) simulator?  At first glance, the answer seems obvious.  First, draw the spiral and its associated environment.  Second, put ports on the spiral.  Third, simulate the spiral at the desired frequencies, and extract the value of Q from the resulting S parameters.  (If the S parameters are known, it is trivial to obtain Y parameters and use equation 2.)  The problem is that Q is a very sensitive measurement.  There are a number of places where EM simulation can go wrong.  In the first part of this paper we summarize the more common pitfalls that can occur.  In order to understand what can happen, we discuss both 3D simulators and planar simulators.  The physical issues in the two simulators are the same; however, the numerical implementation details are different.  The problems are grouped into three main areas: accurately calculating the conductors’ resistances; understanding the ground return issues; and making sure the ports used are properly calibrated.  In addition, depending on the technology being used there can be other problems.  For example, spiral simulations in silicon have to include the lossy silicon substrate.

In principle, 3D EM simulators are the most accurate tool for simulating the spiral.  They mesh the entire geometry using small 3D cells.  The electric field is then calculated for each cell.  If the mesh is fine enough, the relevant physics is captured, and the Q calculation will be captured.  However, the mesh must be carefully constructed.  Default settings are normally not adequate.  Normally, these simulators do not mesh the interior metal of the inductor.  Rather, they use an impedance boundary condition on the surface of the metal and only mesh the exterior regions to the metal.  This usually results in an inaccurate estimate of the Q.  The correct method is to also mesh the interior metal. 

However, this results in a very large cell count for the mesh.  The mesh must be fine compared to the spatial variation in the current for the conductor.  The current density is concentrated near the surface of the conductor, and decays exponentially as one goes into the metal.  Furthermore, the details of the current near corners in the metal lines cross section are important.  The size of the mesh can easily be increased by an order of magnitude or more if the Q is to be accurately calculated.  Therefore, an alternative procedure is preferred.

Planar EM simulators are a popular tool for circuit simulation.  In these simulators, the current is solved for on the surface of the conductors after the surface of the metal is meshed.  There is no need to mesh the entire spatial domain of the problem, which includes the dielectric regions surrounding the inductor.  The advantage is that planar simulators can look at much larger problems than 3D simulators.  For example, the layouts of entire circuits can be studied, depending on the complexity of the circuit.  Unfortunately, by restricting the solution to currents on the surface of the conductors, it is difficult to get an accurate answer for the Q of the spiral, for the same reasons that the 3D simulators cannot get by with using an impedance boundary condition on the surface of the conductors. 

In this paper, we explain a technique for calculating the Q that does not require meshing the interior of the conductors.  Therefore, planar simulators can be used.  (The method also allows 3D simulators to be used without meshing the interior of the conductors.)  The idea is to simulate the inductor using the surface impedance boundary conditions normally available in EM simulators.  Then, a second canonical problem is simulated, which consists of straight, coupled lines with the same cross sectional geometry as the original spiral.  The loss of the coupled lines can also be calculated using available cross sectional line modeling methods.  Most typically, a cross sectional EM tool is used which includes the detailed behavior of the current distribution in the lines [2].  The results from the full EM simulation and cross sectional model are compared.  The difference is used to correct the Q from the original simulation.  The technique assumes that the spiral can be approximately modeled as straight line sections, which is usually the case.  The corners of the spiral are usually a secondary effect when estimating the spiral’s performance.  In this way, the designer can get a good estimate of the Q without resorting to unrealistic mesh densities for 3D simulators, and can still use a planar simulator if desired.

Q is a Very Sensitive Measurement
Why is Q such a sensitive measurement to make?  In this section we will look at a number of reasons beginning with the focus of this paper, getting the conductor loss correct.  To illustrate the problem, we look at a spiral inductor in a microwave integrated circuit (MMIC) process.  Similar problems exist in other technologies: integrated inductors in package modules and silicon chip technologies.  Figure 1 shows a MMIC spiral on gallium arsenide (GaAs).  The spiral is typical of MMIC technology.  It is placed on top of the GaAs substrate, which is 100 um thick.  Gold lines are used.  The total thickness of the line is approximately 3 um.  The lines are 10 um wide with 6 um gaps.  The right side of the figure shows the details of the mesh used by the planar simulator.  The size of the mesh is approximately 9500 unknowns.

Figure 1: A MMIC Spiral and Details of the Mesh

Notice that the sides of the metal are meshed.  It is important that the simulator include the non-zero thickness of the conducting lines, as the spacing between turns of the spiral is the same order as the thickness of the lines.  This is typical of spiral inductors in all technologies: module, MMIC, and silicon.  The mesh density shown is typical of spirals; there is more than one cell across the line width, and a few cells up the vertical sides. 

Figure 2 shows the two port S parameters for the spiral from 0.1 to 10 GHz.  Notice that the simulated and experimental data are very close to the human eye.  Typically, this sort of agreement is considered acceptable to the designer using the spiral, well within normal process variations of the process technology being used.

Figure 2: The Numerical and Experimental Data are in Good Agreement for Normal Design Criteria

Figure 3 shows the Q for the spiral using the definition in equation 1 for the Q.  (The Q can also be measured using the second port of inductor.  The results are similar to the Q calculated from port 1.)  

Figure 3: Q values Calculated Using the Numerical and Experimental Data

The results noticeably differ.  The numerical simulation calculation differs from the numerical data by 27 percent at 5 GHz.  This is typical of Q calculations using numerical simulators.  The S parameter data apparently agree well; yet the derived Q’s from the data are noticeably different.  

The problem is that Q is a ratio of the reactance to the resistance.  The resistance is small at low frequencies, getting larger as the frequency increases.  The numerical calculation for the resistance can have a small absolute error, but a large relative error.  For example, if the actual resistance is 0.1 Ohms at a given frequency, and the numerical calculation is 0.2 Ohms, the absolute error is 0.1 Ohms, which is a reasonably small number for circuit design.  The relative error is 100 percent, leading to a Q in error of 50 percent.  Figure 4 shows the resistance for our example, where the resistance is defined as the real part of the impedance matrix looking into port 1 when port 2 is shorted.  There is a 36 percent difference in the resistances at 5 GHz.

Figure 4: The Resistance of the Experimental and Numerical Data Differ by 36 Percent  

For completeness, we also show the inductance for the measured and experimental data.  Figure 5 shows the inductance, which is defined to be the reactance looking into port 1 when port 2 is shorted, divided by the angular frequency. 

Figure 5: The Inductance Measured at Port 1

The results differ by 19 percent at 5 GHz.  Typically, numerical inductance calculations agree with experimental data better than those for resistance.  Notice that the error in inductance goes in the direction of helping the Q values agree as the inductance values are smaller for the numerical data, making the Q smaller, and the resistance values are low, making the Q bigger.  There are a variety of reasons for the inductance error.  First, inductors have internal and external inductance.  Inductance is defined as a ratio of magnetic flux through a surface to the current that produces the magnetic field.  The contribution from magnetic fields inside the conductors gives internal inductance; the magnetic fields outside the conductors give the external inductance.  The relative importance of the two terms depends on the manufacturing technology and the geometry.  An EM simulator should be able to calculate the external inductance accurately as the measurement depends on the total  current flowing in the conductors, which is not a very sensitive parameter with mesh density.  The internal inductance is more difficult to calculate accurately, as the detailed current distribution in the conductors needs to be known.  

Planar solvers use an impedance boundary condition placed on the conductors’ boundaries.  This boundary condition is based on approximations that break down near corners of the conductors.  Depending on the geometry, the accuracy of the surface impedance approach should be suspect.  Alternatively, 3D simulators can use either impedance boundary conditions on the surface of the conductors, or mesh up the metal interiors volume.  However, this method can lead to unrealistic computation times because of the large mesh required for the conductors.  Therefore, it appears the calculation of the internal inductance of the spiral has the same problems as in calculating the conductors’ resistance.  Fortunately, the internal inductance is normally a small part of the total inductance, with the external inductance dominating.  After all, if this weren’t the case, we wouldn’t have a very good inductor.

We are therefore faced with the prospect of either meshing up the interior of the conductors with a very fine mesh using a 3D EM simulator, or trying to use a surface boundary condition in a planar simulator and risk an inaccurate answer.  This paper shows how we can correct the surface impedance approach such that we get a much improved estimation of the Q and maintain reasonable mesh sizes and computational times.  

Other Problems with Calculating Q for Spirals
For completeness, we mention a couple of other common problems with estimating Q using EM simulators.  While not being the main focus of this paper, these other issues should always be addressed when looking at Q.  We start by discussing ground return.  Inductance is only uniquely defined if the complete current path is known, including the ground return [4].  Otherwise, it is not possible to give an unambiguous answer to the magnetic flux through the current loop, as there is no known completed current loop.  (It is possible to define a partial inductance where the complete current loop is not known.  When the various partial inductances are added together, the resulting answer should be the correct, total inductance.  The topic of partial inductance is beyond the scope of this paper.)

EM simulators use ports to excite the circuit, and measure the resulting S parameters.  All ports have a notion of a local ground.  The easiest way to determine the local ground is to think of the current.  Current flows out of the port down the line.  Where is it coming from?  It is coming from the ground of the port.  Figures 6 and 7 illustrate the idea with some common ports used in 3D and planar simulators.

Figure 6: A Wave Port in a 3D Simulator and an Internal Port Used in 3D or Planar Simulators

A wave port is shown on the left side of the figure.  Current goes down the line out of the port.  The current return is on the outside of the port, where typically some type of metal ground plane is used.  The port on the right is an internal, circuit type port as used in both planar and 3D simulators.  The line is cut and the port inserted.  It can be thought of as a two point probe with an internal impedance (usually 50 Ohms).  The red probe tip is attached to one side.  The current flows out of the port onto the line.  The other side is the local ground for the port.

Figure 7 shows two types of grounding schemes for ports placed at the end of lines in planar simulators.  This is the most common type of port used in planar simulators.  The left side of the figure shows an explicit grounding scheme.  A metal strap is attached to the ground plane.  It is obvious that the return current is coming up the grounding strap.   The right side shows an implicit grounding scheme.  There is no grounding strap.  The return current actually comes from infinity in this situation.  The advantage of the explicit ground is that the designer knows well where the current is coming from, at the expense  of higher port parasitics because of the grounding strap.  The implicit port has lower parasitics, but care must be taken to not get confused with the ground return. 

Figure 7:  Edge Ports in Planar Simulators with Implicit and Explicit Grounding Schemes

A common problem that occurs when simulating spiral inductors is that the ground reference for the simulation does not agree with the actual physical situation.  For example, an on-chip spiral is measured using a wafer probe.  The probe has a signal contact and one or more ground contacts.  How is the current return path for the spiral defined on the chip?  Typical situations include a ground ring drawn around the spiral on the same metal layer as the spiral itself, or by using vias to go down to lower levels of metallization in the stackup and returning on some form of ground nets or ground plane.  The designer must make sure that the EM simulation mimics the actual ground return as closely as possible.  

A related problem with ports is the issue of deembedding.  Deembedding a port in an EM simulator resolves two issues.  First, the parasitics of the port are removed.  For example, in the left picture in Figure 7 the ground strap has parasitic inductance and capacitance.  Deembedding removes the effects of these parasitics from the S parameters.  Second, deembedding sets the reference plane for the port.  The reference plane is where the incident wave coming from the port has zero degrees phase.  All ports have a reference plane; usually it is at the port itself unless a deembedding distance is set up.  Deembedding works by simulating canonical structures that have the same port configuration as the original port.  The most common problem for spirals is that the reference plane used for the spiral in the circuit does not agree with the reference plane location in the EM simulation [5]. 

Calculation of Conductor Loss and Meshing
The Q of a spiral inductor depends critically on the conductor loss.  An accurate calculation of the conductor loss requires that the current distribution in the spiral be calculated accurately.  Unfortunately, the distribution needs to be more accurately determined than for normal EM simulations.  Normally, conductor loss is a secondary effect in order to obtain accurate S parameters.  Therefore, it is adequate to include the loss of the metal as an impedance boundary condition on the surface of the conductor.  A relatively coarse mesh can be used on the surface of the conductor.  For example, the mesh shown on the right side of Figure 1 is normally quite adequate for S parameter calculations.  A good rule of thumb is that five or so cells across a line is a fine enough cell size.  Occasionally, tightly coupled lines in filter structures require a cell size of one seventh of line.  Any finer mesh is a waste of computational resources and can lead to numerical problems.  (This can occur because the matrix is getting larger and the condition number of the matrix is getting bigger with the smaller cells.)

When we look at calculating Q, the normal simulation assumptions for conductor loss are a problem.  First, surface impedance approximations make two important assumptions:  the conductor cross- sectional dimensions (height and width) are large compared to the skin depth of the metal, and that there are no nearby conductors influencing the currents.  Both of these assumptions are violated in spiral conductor topologies.  Second, the normal meshing density rule of five cells across a line is not adequate for the accuracy needed for the Q calculation.  The surface impedance problem can be removed by using a 3D EM simulator and meshing inside the conductors, but at a heavy price.  The size of the problem increases by an order of magnitude.  In part two of this article, we will look at each of these two problems in more detail.

[1] GFMCLIN model in Microwave Office, version 2010, AWR Inc.
[2] David Meeker home page: Follow this link for information: FEMM Manual:
[3] The Design of CMOS Radio-Frequency Integrated Circuits, Second Edition, Thomas H. Lee, pp 88 – 92, 2004, Cambridge University Press.
[4] Fields and Waves in Communication Electronics, Ramo, Whinnery and Van Duzer, Third Edition, pp. 81 -84, 1994, Wiley.
[5] “A potentially significant on-wafer high-frequency measurement calibration error”, James C. Rautio and Robert Groves, IEE Microwave Magazine, December, 2005, pp. 94 – 100.
[6] Classical Electrodynamics, David Jason, Wiley, 1975.

About the Authors
Dr. John Dunn is a senior applications engineer at AWR.  His areas of expertise include electromagnetic modeling and simulation for high speed circuit applications.  Prior to joining AWR, he was head of the interconnect modeling group at Tektronix, Beaverton, Oregon,  and a professor of electrical engineering at the University of Colorado, Boulder.

Dr. Vladimir V. Veremey was born in Kramatorsk, Ukraine. He earned his M.Sc. and Ph.D. degrees from the Kharkov State University, Kharkov, Ukraine. In 1978, he joined the Institute of Radiophysics and Electronics (IRE) of the Ukrainian National Academy of Sciences, and in 1996 he became an associate professor at the Gebze Institute of Technology in Turkey. From 1997 to 2000 he was a visiting scholar fellow at Pennsylvania State University. In 2000 he joined Xpedion Design Systems and from 2006 to 2009 he was with Agilent Technologies, where he worked on development of nonlinear circuit simulators for RFIC. In 2009 he joined AWR Corporation. Dr. Veremey’s active research interests currently include the theoretical and numerical modeling of electromagnetic problems, numerical-analytical methods in electromagnetic-field theory, the scattering of waves from resonance obstacles, and antenna design. He has published over 100 papers in refereed journal and symposium proceedings.

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