Sensor placement

ION MagNav Workshop, PLANS 2023, Monterey, CA

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Distribution A: Authorized for public release. Distribution is unlimited. Case No. 2023-0427.

How-to

There’s a docker container with a jupyter notebook that contains this exercise:

docker run  -p 8888:8888 anielsen/magnav-plans-workshop:latest

Then select work/sensor_placement.ipynb

Cylindrical Dipole magnet

Cylindrical Dipole magnetized along \(z\)

Near-field magnetic dipole

You may or may not remember from electromagnetics that the magnetic field from a dipole decays as the inverse cube of the distance from the dipole center \[ B(r) \propto \frac{1}{(r/b)^3} \] See dipole (Griffiths 1999)

This fact is essential for understanding

1. How to position magnetometers in a vehicle
2. Essential differences from free-space propagation (e.g. radio waves)

Code

from matplotlib import pylab as plt
import numpy as np

plt.ion();

Sensor placement

Signal decay with distance

• The power-density of radio-frequency wave propation falls as \(1/r^2\).
  • reflects that energy spreads out over surface of a sphere area \(A=4\pi r^2\)
• The magnetic field falls as \(1/r^3\).
  • No, magnets aren’t lazy.

Differences between \(1/r^3\) and \(1/r^2\)

Code

nx = 1_000
x_max = 100
x_min = 1.0
x_span = x_max - x_min
dx = x_span / nx
x = np.arange(nx)*dx + x_min

y_r2 = 1/x**2
y_r3 = 1/x**3

Code

fig,ax = plt.subplots()
ax.plot(x,y_r2,label=r"$1/x^2$")
ax.plot(x,y_r3,label=r"$1/x^3$")
ax.legend()
ax.set_xlabel("Distance from source")
ax.set_ylabel("Amplitude")
ax.grid();

Figure 1: \(\frac{1}{r^3}\) compared to \(\frac{1}{r^2}\)

Code

fig,ax = plt.subplots()
ax.semilogy(x,y_r2,label=r"$1/x^2$")
ax.semilogy(x,y_r3,label=r"$1/x^3$")
ax.set_ylim([1e-6,1])
ax.legend()
ax.set_xlabel("Distance from source")
ax.set_ylabel("Amplitude")
ax.grid();

Figure 2: \(\frac{1}{r^3}\) compared to \(\frac{1}{r^2}\)

Code

fig,ax = plt.subplots()
ax.loglog(x,y_r2,label=r"$1/x^2$")
ax.loglog(x,y_r3,label=r"$1/x^3$")
ax.set_ylim([1e-6,1])
ax.legend()
ax.set_xlabel("Distance from source")
ax.set_ylabel("Amplitude")
ax.grid();

Figure 3: \(\frac{1}{r^3}\) compared to \(\frac{1}{r^2}\)

For fun add \(e^{-x}\)

Code

y_exp = np.exp(-x)

Code

fig,ax = plt.subplots()
ax.loglog(x,y_r2,label=r"$1/x^2$")
ax.loglog(x,y_r3,label=r"$1/x^3$")
ax.loglog(x,y_exp,label=r"$e^{-x}$")
ax.set_ylim([1e-6,1])
ax.legend()
ax.set_xlabel("Distance from source")
ax.set_ylabel("Amplitude")
ax.grid();

Figure 4: \(\frac{1}{r^3}\) compared to \(\frac{1}{r^2}\) and \(\exp{(-x)}\)

References

Griffiths, David J. 1999. Introduction to Electrodynamics. 3rd ed. Prentice Hall.