Spiral Inductor Design

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Reference

Simple Accurate Expressions for Planar Spiral Inductances, S. Mohan et al., JSSC Oct 1999, pp. 1419

Simple Spiral Inductor Behaviour

A very simple approximation to the inductance of a spiral inductor is $L = \mu_o n^2 r = 1.2\times 10^{-6}n^2 r$ , where $n$ is the number of turns and $r$ is the radius.

Spiral Inductor model

Reference: Lee sec. 4.5

Layout of a spiral inductor.

An equivalent model of a spiral inductor.

We need to be able to generate an equivalent circuit model of the spiral inductor. We will work with a fairly simple yet accurate inductor model that includes inductance $L$ , ohmic resistance $R s$ , capacitance to substrate $C O X$ , substrate effects $R s u b$ and $C s u b$ , and shunt capacitance $C P$ due to the capacitance between the main spiral and the crossunder (shown in red).

We will look at how to estimate these parameters
Spiral inductors generally have higher $Q$ if implemented as a circular spiral, but many processes restrict layouts to a discrete set of angles - 'Manhattan geometries'
We will focus on square spiral inductors for simple analysis, but using octagonal inductors is preferable for implementation

Inductance

Inductance can be expressed quite accurately (usually within 5%) by

$L \approx \frac{37.5 \mu_0 n^2 a^2}{22r - 14a}$

where $n$ is the number of turns, $a$ is the distance from the center to the mean of the winding radii

This models hollow spiral inductors, which have higher Q. When the trace runs to the center of the inductors, the magnetic field lines start cancelling for the sections of the inductor that are very close together, degrading the inductance. Removing the windings at the center of the inductor improves the $Q$ because the inner windings contribute to series resistance, but not as significantly to inductance.
Another equation for inductance is

$L = 1.17 \mu_o \frac{n^2 (D_{out} + D_{in})}{1+2.75 \psi}$

where $n$ is the number of turns, $D o u t$ is the diameter of the spiral, $D i n$ is the width of the inductor center, and $\psi = \frac{D_{out} - D_{in}}{D_{out} + D_{in}}$

A tool that can be used to compute inductance,and compare the calculated inductance to ASITIC results, is available in the File:Spiral inductor calculator.xls.

Ohmic Resistance

Equivalent circuit model for a spiral inductor.

If skin effect is not significant, the series resistance is calculated based on the cross sectional area of the inductor

$R_s \approx \frac{l}{w t \sigma}$ where $l$ is the total length of the inductor winding, $w$ is the width of the winding, $t$ is the thickness of the metal layer, and $σ$ is the conductance of the metal

When the skin depth $δ$ becomes less than the thickness of the metal, it can be used instead of the metal thickness $t$ . A more accurate result can be achieved if the true nature of the exponential decay of the electric field is used, yielding

$R_s \approx \frac{l}{w \sigma \delta(1-e^{-t/\delta})}$

Capacitances

The simplest way of estimating the capacitance between the conductor and the substrate is using a parallel plate capacitance:

$C_{ox} = wl \frac{\epsilon_{ox}}{t_{ox}}$ where $ε o x$ is the permittivity of the oxide under the spiral inductor, and $t o x$ is the distance between the spiral inductor and the substrate

The shunt capacitance $C p$ can be approximated by

$C_p = n w^2 \frac{\epsilon_{ox}}{t_{ox}'}$ where $t o x'$ is the distance between the main spiral layer and the crossunder layer

Substrate

Substrate modeling is the most complex, as it includes current flowing into the substrate due to the displacement current flowing through $C o x$ and the image current flowing in in the substrate
Image currents: recall that current flowing in a spiral loop induces a magnetic field with direction given by the right-hand-rule, and magnitude

$B = \frac{n \mu I}{2 r}$

where $r$ is the radius of the loop (Ampere's law)

A changing magnetic field $\mathbf{B}$ induces an electric field 'opposite' to the direction of the right-hand rule (negative sign): Faraday's Law

$\oint \mathbf{E} \cdot \mathbf{dl} = -\int \int \frac{\partial \mathbf{B}}{\partial t} \mathbf{dA}$

Eddy currents induced in the substrate.

Boundary conditions for static and time-varying magnetic fields state that a normal $\mathbf{B}$ field is continuous across a boundary, so the same field exists in the substrate
By Ohm's law, $\mathbf{J} = \sigma \mathbf{E}$ , so the electric field in the susbstrate induces a current in the same direction as the electric field
Hence, the changing magnetic field due to the current in the spiral induces a current to flow in the opposite direction
These are eddy currents
Eddy current flow tends to decrease the effective inductance (i.e. the substrate currents work against the spiral current), but they also add loss to the substrate loss
This loss is modeled by $R s u b$

Substrate resistance is given by

$R_{sub} \approx \frac{2}{w l G_o}$

where $G o$ is a fitting parameter typically around $10 - 7$ S/ $μ m 2$ , depending on the distance to the substrate and the physical properties of the substrate

The capacitance $C s u b$ models displacement current flow in the substrate and effects of the eddy currents

$C_{sub} \approx \frac{w l C_o}{2}$

where $C o$ is a fitting parameter typically between $10 - 3$ and $10 - 2$ fF/ $μ m 2$

Improved Models

A more complex but more accurate model will use more elements. One such scalable model is Scalable Compact Circuit Model and Synthesis for RF CMOS Spiral Inductors, W. Gao, IEEE MTT, March 2006, p. 1055

Extracting a spiral inductor model from measurements

Extracting a spiral inductor model from measurements.

Often a lumped element model is constructed by measuring many spiral inductors, and developing a model from the measured results
A common extraction method is to measure S-parameters (vector network analyzer), then convert to Y-parameters (since substrate elements are in shunt with inductance). Conversion charts available in Pozar and Rogers.
Recall: to measure $Y_{11} = \frac{I_1}{V_1}$ when $V 2 = 0$ we short circuit port two of the inductor model, which short circuits the $C o x$ and substrate elements at port 2
$Y 11 = Y A - Y 12$ , where $Y A$ is the admittance of $C o x / 2$ and the substrate elements at port 1. Similarly, $Y 22 = Y B - Y 12$ , where $Y B$ is the admittance of the elements at port 2
$Y 12 = Y 21 = - j ω C p + 1 / (R s + j ω L)$
Problem: too many unknowns. The real and imaginary parts of the admittance allow us to solve for two unknowns with each equation, but that leaves two unknowns to be solved
Solution? Assume that $C p$ and $C o x$ are fixed for all frequencies of interest, and use multiple frequency points as additional equations
For more information, see A BROADBAND PARAMETER EXTRACTION TECHNIQUE FOR THE EQUIVALENT CIRCUIT OF PLANAR INDUCTORS, I. Shih, MTT-S 1992

Inductor Q

At frequencies not close to the self resonant frequency, $Q$ can be estimated as $Q = \frac{|\Im(Z)|}{|\Re(Z)|}$ where $Z$ is the impedance of the inductor
A more accurate approach is to use

$Q = \frac{\omega_o}{2} \left| \frac{d\phi}{d\omega}\right|_{\omega=\omega_o}$

where $ω o$ is the resonant frequency

For frequencies not at the self-resonant frequency of the spiral inductor, this can be calculated by adding an ideal shunt capacitor with an admittance equal to the imaginary part of the $Y 11$ of the spiral inductor at each frequency of interest. Then $Y'(ω) = j ω C + Y 11$ , and we can solve for the Q using

$\left . \frac{d\phi}{d\omega} \right |_{\omega=\omega_o} = \frac{2 Q}{\omega_o} = \frac{\angle Y'(\omega_o + \delta \omega) - \angle Y'(\omega - \delta \omega)}{2 \delta \omega}$

To improve $Q$ , we want to minimize resistive elements
For more background, see Niknejad:1998.

Effect of inductor Q on Circuit Performance

High Q results in:

Inductor with low phase noise
Low noise figure LNA
Low loss matching networks
Narrowband filters with reduced passband insertion loss
Lower power consumption
Improved receiver sensitivity

Inductor Self-Resonance (sec. 5.13)

Since the spiral inductor has both inductive and capacitive behaviour, the parasitic capacitances will resonate with the inductance at a certain frequency, roughly given by $\omega_o = 1/\sqrt{L C_{tot}}$
As the inductor size increases the capacitance $C o x$ does as well, and so the self-resonant frequency decreases
At frequencies above the self-resonant frequency the inductor will look capacitive (why?)

Caption

Generally, the closer the spiral inductor is to the substrate, the larger the total parasitic capacitance and the lower the self resonant frequency
How will self-resonant frequency scale with the inductance? E.g. what will the relative values of self-resonant frequency be for a 0.1 nH inductor compared to a 10 nH inductor?

Simulating Spiral Inductors

Caption

Full 3-D electromagnetics solvers can be used to simulate spiral inductors, e.g. FDTD simulators like Fidelity or EMPIRE, FEM simulators like HFSS, or custom code - however, significant time and computer memory needed
Planar simulation software more suited - e.g. method of moments simulators like IE3D or Momentum
Faster (but slightly less accurate) solution: dedicated spiral inductor software that incorporates the model we discussed, and calculates Q. The most common one is ASITIC - written by Ali Niknejad. It is very fast, and good for initial design

ASITIC - Planar Spiral Inductors and Transformers

Screenshot of ASITIC, used for spiral inductor simulations

ASITIC is a simple program commonly used for spiral inductor design. It is installed on the UNIX system, though there are bugs that cause it to crash when eddy currents are simulated. The Linux version, installed on europa.ee.queensu.ca, works fine. You can also install it on your own Linux machine by downloading from the ASITIC website.

On Linux systems, ASITIC may be started using asitic_linux -t cmosp18_sub_contact.tek, where the cmosp18_sub_contact.tek file specifies the metal and dielectric properties for the CMOSp18 process. It is available from the /home/barnard/microwave_common directory. Note that the default tek file for the CMOSP18 technology, which is available in /CMC/kits/cmosp18/asitic, does not have a substrate contact layer defined (which is essentially a p+ doped area), which prevents the definition of a ground in the substrate.

A script plotsp is installed at /home/centauri/microwave_common/bin which may be run from the command line to plot S-parameter data after simulation using the 2portx command if the file is set to be in real and imaginary format (using "set polar=false"). If you create a file called .asitic in the directory in which you start ASITIC, and copy the following lines to it, the settings will be used by default:

set polar=false
set eddy=on
set ofile=sparam.s2p

You can also set ofile to be any filename you'd like. The setting set eddy=on will turn eddy currents on by default. It makes the simulations more accurate, but also more time-consuming.

To set your start and stop frequencies to, e.g., 1 to 30 GHz, type

set freq1=1 set freq2=30

Example ASITIC simulation

Let's create a symmetric square inductor, around 2.6 nH. Bold denotes ASITIC commands.

symsq name=a len=200 w=15 s=2 n=4 metal=metal6 exit=metal5 xorg=200 yorg=200

wire name=gnd len=200 wid=20 xorg=180 yorg=180 metal=msub

Then to create a PI model at 10 GHz we could do

pix a 10 gnd

This yields

maxL = 750.00, maxT =  1.17, maxW =  1.17 (lambda = 15000.00, delta =  1.46)
Performing Analysis at 10 GHz
Generating capacitance matrix (101x101)...
Generating inductance matrix (793x793).....
Pi Model at f=10 GHz:  Q = 2.218 , 2.289 , 4.438
L = 2.627 nH R = 12.95
Cs1 = 46.24 fF Rs1 = 244.5
Cs2 = 45 fF Rs2 = 219      Est. Resonance = 14.44 GHz

To get a more accurate picture of the frequency response of the spiral we could simulate S-parameters from 1 to 20 GHz in steps of 2 GHz:

set freq1=1

set freq2=20

set fstep=2

set ofile=sparam.dat

2portx a gnd

which will spit out a bunch of lines that look like:

maxL = 7500.00, maxT =  3.70, maxW =  3.70 (lambda = 150000.00, delta =  4.62)

and write the S-parameters to the file sparam.dat. This can then be printed using asitic-plot, or possibly plotsp-new on some machines (a general S-parameter plotting utility that uses GNUplot).

Some other sample commands that may be useful:

#
#Useful ASITIC analysis commands (after an inductor has been created):
#
#pix inductor_name frequency gnd
#
# will calculate the equivalent pi model of inductor_name at frequency (in GHz), 
# using gnd as the ground plane (the ground is optional)
#
#2portx inductor_name gnd
#
# will calculate 2 port parameters for inductor_name. The name of the output file 
# is determined by "set ofile=filename", and S-parameter is selected by 
# "set style=s-parameters". 
#
# The command "plotsp sparam.s2p" may be run from the Linux command line to 
# plot the data after simulation data if the file is set to be in real and 
# imaginary format (using "set polar=false"). Use the command 
# "set ofile=filename.s2p" to tell asitic where to save the output.
#
# To create a roughly 4nH inductor:
sq name=a len=175 w=10 s=.5 n=5 xorg=200 yorg=200 metal=metal6 exit=metal5
# To create a ground plane underneath it:
wire name=gnd len=200 wid=20 xorg=180 yorg=180 metal=msub
# ground halo
sq name=halo len=200 wid=200 w=20 n=1 xorg=180 yorg=180 s=10 metal=msub

# 0.4 nH inductor
sq name=a len=100 w=10 s=0.5 n=1.5 metal=metal6 exit=metal5 xorg=75 yorg=75

# 1.725 nH inductor
sq name=a len=150 w=10 s=1 n=2.5 metal=metal6 exit=metal5 xorg=50 yorg=50

# 1.5 nH inductor
sq name=a len=150 w=10 s=1 n=2.75 metal=metal6 exit=metal5 xorg=50 yorg=50

#2 nH inductor
sq name=a len=150 w=10 s=3.5 n=2.5 metal=metal6 exit=metal5 xorg=50 yorg=50

#3 nH inductor (self resonates at 25 GHz, good for bias network)
sq name=a len=170 w=10 s=1.5 n=3.25 metal=metal6 exit=metal5 xorg=50 yorg=50

#0.32 nH single turn, Q=5.2
sq name=a len=110 w=2 s=2 n=1 metal=metal6 exit=metal5 xorg=50 yorg=50

# 0.48 nH, Q=5
symsq name=a len=80 w=4 s=3 n=2 metal=metal6 exit=metal5 xorg=50 yorg=50
# 0.2 nH, Q=4.4
symsq name=a len=50 w=4 s=3 n=2 metal=metal6 exit=metal5 xorg=50 yorg=50
# 0.38 nH, Q=4.8
symsq name=a len=70 w=4 s=3 n=2 metal=metal6 exit=metal5 xorg=50 yorg=50
# 0.222 nH, Q=5.7
symsq name=a len=70 w=8 s=3 n=2 metal=metal6 exit=metal5 xorg=50 yorg=50
# 0.3 nH, Q=5.7
symsq name=a len=80 w=8 s=3 n=2 metal=metal6 exit=metal5 xorg=50 yorg=50

There is a quick overview of the use of the program on the CAD tool tips page.

Momentum simulation of inductors

Screenshot of ADS Momentum being used to simulate a spiral inductor.

The screen above shows ADS Momentum being used to simulate a planar inductor (in this case a horseshoe inductor for a differential VCO). The inductor is simulated with ports, indicated as arrows. Since creation and simulation generally take longer with Momentum, it is often used as the final step in simulating an inductor, or more complex metallic structure. ASITIC can usually be used more quickly to create an inductor.

Instructions for using Momentum can be accessed by selecting Help in the main ADS window, or by pointing a browser on the departmental UNIX system to file:/ece/vlsi/eesof_ads/doc/mom/wwhelp.htm.

Patterned ground shields

Current in spiral inductor induces eddy currents in substrate, which causes losses and reduces inductance
Can place metal ground shield under inductor, but the the eddy currents induced in it will still reduce inductance
Need to provide a shield to prevent fields from inducing current in substrate, but which will not allow eddy current to flow

Patterned ground shield under a spiral inductor

Patterned ground shields can reduce eddy currents by disrupting current flow.

Solution: patterned ground shields that prevent eddy currents from flowing. They can be formed from the polysilicon layer (Yue:1998) and $n +$ diffusion regions (Chen:2001). Using a higher metal layer is possible, but increases the capacitance to ground (and how will this affect the self-resonant frequency?)
Result: larger inductance, and reduced capacitance to the substrate $C s u b$

Q of Patterned ground shields

Measured Q of spiral inductors with patterned ground shields, from Chen:2001

Chen 2001 compared Q and inductance of spiral inductors using polysilicon, metal 1, or $n +$ diffusion layers
Notice: patterned ground plane improves Q over certain range of freqencies
Inductance not affected by ground planes at low frequencies (why?), but is increased over certain range of frequencies
$n +$ diffusion produces highest Q and inductance in this paper, but relative advantage of three layers depends on the geometry and material properties
When operating at frequencies close to the self resonant frequency, patterned ground planes are not helpful

Symmetric spiral inductors

It is important that spiral inductors be symmetric in certain applications, for example when being used in the drains of a differential pair in an LNA or oscillator. In this case both outputs of the inductor are on the outside; neither one ends up in the middle of the inductor. The crossover is done by moving both arms closer to the center at each winding, maintaining symmetry in the spiral.

The symsq command in Asitic is useful to draw these inductors.

0.88 nH symmetric spiral inductor

Off-chip spiral inductors

Some work is being done to improve the Q of spiral inductors by fabricating them off of the silicon substrate, and coupling them to the active devices using novel methods. For example, see

An Ultralow-Loss and Broadband Micromachined RF Inductor for RFIC Input-Matching Applications, T. Wang, IEEE Trans. Elec. Dev., March 2006, pg. 568-570.