Using least squares data fitting on a spiral inductor, long process with little knowledge on how to even tackle a problem like this

Question

Approximate inductance formula. A planar spiral inductor, implemented in CMOS, for use in RF circuits.

The inductor is characterized by four key parameters:

• 𝑛, the number of turns (which is a multiple of 1/4, but that needn’t concern us)

• 𝑤, the width of the wire

• 𝑑, the inner diameter

• 𝐷, the outer diameter

The inductance 𝐿 of such an inductor is a complicated function of the parameters 𝑛, 𝑤, 𝑑, and

𝐷. The inductance 𝐿 can be found by solving Maxwell’s equations, which takes considerable

computer time, or by fabricating the inductor and measuring the inductance. In this problem

you will develop a simple approximate inductance model of the form

𝐿̂ = 𝛼n^ß₁w^ß₂d^ß₃D^ß₄,

The inductance where 𝛼, 𝛽₁, 𝛽₂, 𝛽₃, 𝛽4 ∈ ℝ are constants that characterize the approximate

model. (since 𝐿 is positive, we have 𝛼 > 0, but the constants 𝛽₂, ... , 𝛽₄ can be negative.) This

simple approximate model, if accurate enough, can be used for design of planar spiral

inductors.

The accompanied data file inductor_data.m contains data for 50 inductors. (The data is

real, not that it would affect how you solve the problem ...) For inductor i, we give parameters

𝑛𝑖, 𝑤𝑖 , 𝑑𝑖 , and 𝐷𝑖 (all in μm), and also, the inductance 𝐿𝑖 (in nH) obtained from measurements.

(The data are organized as vectors of length 50. Thus, for example, 𝑤12 gives the wire width of

inductor 12.) Your task, i.e., the problem, is to find α, 𝛽1, ... , 𝛽4 so that

𝐿̂ 𝑖 = 𝛼𝑛_i^𝛽₁ 𝑤_i^𝛽₂ 𝑑_i^𝛽₃ 𝐷_i^𝛽₄ = 𝐿_i, for 𝑖 = 1, ... ,50 .

Your solution must include a clear description of how you found your parameters, as well as

their actual numerical values. Note that we have not specified the criterion that you use to

judge the approximate model (i.e., the fit between 𝐿̂ 𝑖 and 𝐿𝑖); we leave that to your

engineering judgment. But be sure to tell us what criterion you use. We define the percentage

error between 𝐿̂ _i and 𝐿𝑖 as

𝑒_i = 100 *|𝐿̂ _i − 𝐿_i|/𝐿_i

Find the average percentage error for your model, i.e., (𝑒₁ + ··· + 𝑒₅₀)/50. (We are only

asking you to find the average percentage error for your model; we do not require that your

model minimize the average percentage error.) Hint: you might find it easier to work with

𝑙𝑜𝑔 𝐿.

All I know is that I can use the least squares data fitting and I think I can solve it via MATLAB but besides that I am unsure how one is suppose to explain the steps leading up to before solving it with MATLAB. The question confuses me on how one can solve for 5 unknowns and only one equation.

Answer 1

The way in which you have posed the question is ill-defined as there are a lot of other factors that need to be considered. You mention that this spiral inductor is going to be fabricated in CMOS ---- you have not specified the environment in which this inductor is to be fabricated. What is the die size? What is the dielectric constant of the material? What frequency is this inductor going to be operating?

If you are looking to solve this problem in its most general case, the only way to do it is as you say ---- use a 3D field solver or build it and measure it directly. For pedagogical purposes, neither of these approaches is convenient.

If you want to solve this problem in a way that is tractable and could possibly be put into a closed form solution, you could calculate the DC inductance of a spiral inductor in free space. This is a multi-step process:

1. Using the Biot-Savart Law, you can integrate the cross-product of each differential current element with the unit vector pointing to a field point in question to obtain the magnetic field intensity H at that point.
2. The inductance is going to be approximately equal to the total flux passing through a plane surface that is nearly coincident with the surface of the planar spiral divided by the current flowing in the spiral.

The Biot-Savart Law is given by:

dB = (μ/4π) ∫ (I dl × r') / | r'|³

where r' is the displacement vector between a differential current element on the spiral inductor and the field point.