30Sep/100

## The Hall effect

Since i have found a position with Carnegie Mellon, i now have the time to reflect on my job search and more to the point, the interviews.  While most of the time was spent discussing the experience listed on my resume, the question i remember hearing the most was if i could explain what the Hall effect is.  Now, i have designed circuits leveraging Hall effect sensors in the past - in motor controllers as angular position sensors, in magnetic switches, and in scroll wheels as linear position sensors, but i never took the time to dig in and understand how the sensors worked.  For all i cared it could have been voodoo, black magic, or a combination of the two - one way or another the sensor is able to tell me where it is in relation to a nearby magnetic field.  That only gets you so far, so i guess it's time to demystify these little ICs.

What is the Hall effect:
The Hall effect, named after the scientist who first detected it Edwin Hall, is the opposing force a flow of charge carriers (current) induces to cancel out a Lorentz force acting upon it.  This opposing  forces is equal in magnitude and opposite in sign to the Lorentz force in order for the steady state net force on the carries to be zero, satisfying Newtons 3rd law.  This answer requires a basic understanding of the fundamental laws of classical electromagnetic fields - Maxwell's equations, and  the Lorentz force where Maxwell's equations describe how moving charge carriers induce EM fields and the Lorentz force is how EM fields effect moving charge carriers.

How to leverage the Hall effect:
One generally uses Hall effect sensors to sense the position of a magnet, or the magnitude of a current.  There are 3 main types of position sensors - binary, linear and radial.  For each, it is assumed that there is a magnetic field (generally generated by a magnet) that is a part of the component you want to sense the position of, in a location where it will pass over the sensor's active area.

• For a binary sensor, it will switch logic states when the active area is in proximity of a north or south pole (axially magnetized).
• For a linear sensor, it will give you an analog or multiple bit digital output corresponding to the distance the active area is away from a north or south pole.
• For a radial sensor, it will give you an analog or multiple bit digital output corresponding to the angle at which the north/south pole is to the active area.  Radial sensors require a diametrically magnetized magnet persistently centered on the sensors active area.
• For current sensing, the Hall sensor's active area is placed perpendicularly to the currents induced magnetic field and will output an analog or multiple bit digital signal corresponding to the current's magnitude.

How do Hall effect sensors work:
Internal to the IC, there is a rectangular plate that a small current travels through in the y-direction, with voltage sensing circuitry connected across the plate in the x-direction.  While there is no external magnetic field and therefore no Lorentz force, the current travels linearly through the plate and no voltage difference can be sensed at the sides of the plate.  However, in the presence of a magnetic field perpendicular to the current flow (through the large surface of the plate in the z-direction), an electric field is formed called the Hall field which when integrated across the length of the plate gives you the a voltage differential called the Hall voltage.

The Hall voltage is dependent on on the sensor's plate current, plate dimensions, and sensed Lorentz's force.  We first need to extrapolate the charge carrier's velocity which we can find using the plate current and plate dimensions.

$\begin{matrix} \vec{I}&=&\vec{J}\cdot\left(\ell\cdot w\right)\\\vec{I}&=&N\cdot q\cdot\vec{u}\cdot\left(\ell\cdot w\right)\\\vec{u}&=&\frac{\vec{I}}{N\cdot q\cdot\left(\ell\cdot w\right)}\end{matrix}$

Where $\vec{I}$ is the plate current vector, $\left(\ell,w\right)$ is the length in the x-direction and width in the z-direction respectively of the plate, $\vec{J}$ is the current density vector, N is the charge carrier density, q is the carrier charge which is $1.602\cdot10^{-19}\left[C\right]$ for an electron, and $\vec{u}$ is the charge carrier's velocity vector.  We note that the charge carriers are flowing in only the y-direction therefore the current vector, current density vector, and velocity vector are all pointing only in the y-direction.

We then use the Lorentz force equation to find the Hall field remembering that the Hall field is equal in magnitude and opposite in direction to the field created by the Lorentz force.

$\begin{matrix}\vec{F_L}&=&&&q\cdot\left(\vec{E}+\vec{u}\times\vec{B}\right)\\\frac{\vec{F_L}}{q}&=&\vec{E_L}&=&\vec{E}+\vec{u}\times\vec{B}\\\vec{E_H}&=&-\vec{E_L}&=&-\left(\vec{E}+\vec{u}\times\vec{B}\right)\end{matrix}$

Where $\vec{F_L}$ is the Lorentz force vector, $\vec{E}$ is the external electric field intensity vector, $\vec{u}$ is the hall sensor's charge carrier velocity vector from above, $\vec{B}$ is the external magnetic flux density vector, $\vec{E_L}$ is the electric field vector produced by the Lorentz force, and $\vec{E_H}$ is the induced Hall field vector.  We note that because we are sensing the hall voltage across the length of the plate in the x-direction, the voltage will only be induced by a Lorentz magnetic force in the z-direction and skewed by Lorentz electric force in the x-direction, therefore we only consider the magnitude of the external electric and magnetic fields in those directions.  This simplifies our Hall field to:

$\begin{matrix}\vec{E_H}&=&-\left(E_0\,\hat{x}+u_0\,\hat{y}\times B_0\,\hat{z}\right)\\E_H\,\hat{x}&=&-E_0\,\hat{x}-u_0\!\cdot\!B_0\,\hat{x}\end{matrix}$

To find the induced Hall voltage $V_H$, we must integrate the Hall field across the entire length $\ell$ of the plate:

$\displaystyle V_H=\int_0^{\ell}E_H\,dx=-E_0\cdot\ell-u_0\cdot B_0\cdot\ell$

In most cases, the sensor's intended use is to detect and measure the magnetic field; in electrically noisy environments, the external electric field adds noise to the measurement.  This noise can be mitigated by increasing the strength of the magnetic field, or increasing the velocity of the charge carriers across the Hall plate.  Going on the assumption that $u_0\!\cdot\!B_0\!\gg\!E_0$ the Hall voltage simplifies to:

$V_H=-u_0\cdot B_0\cdot\ell$

Plugging back in the equation for $\vec{u}$ we get:

$\begin{matrix} V_H&=&-\frac{I_0}{N\cdot q\cdot\left(\ell\cdot w\right)}\cdot B_0\cdot\ell\\&=&-\frac{I_0\cdot B_0}{N\cdot q\cdot w}\end{matrix}$

Another term known as the Hall coefficient which plugs the current density equation in for $\vec{u}$ into the simplified Hall field equation resulting in the ratio:

$\begin{matrix} E_H\,\hat{x}&=&-E_0\,\hat{x}-\frac{1}{N\cdot q}\cdot J_0\,\hat{y}\cdot B_0\,\hat{z}\\R_H&=&\frac{\left(E_H+E_0\right)\,\hat{x}}{J_0\,\hat{y}\,\cdot\, B_0\,\hat{z}}=-\frac{1}{N\cdot q}\end{matrix}$

Conclusion:
The Hall effect is how moving charge carriers react to a Lorentz force, by producing a voltage differential known as a Hall voltage.  This Lorentz force can be from a magnet, or from the induced magnetic field produced by a flow of current.  Hall effect sensors can be used as non-obstructive current sensors, as well as wireless position sensors.