SaM - MEMORIZE (Magnetism)

List of things to memorize:

SaM - Magnetism

• SaM - Magnetic Permeability • Magnetic Susceptibility • Magnetic Field
• SaM - Types of Magnetic Materials (Diamagnetic, Paramagnetic and Ferromagnetic Material)
• SaM - Geometrical Magneto Resistance
  • SaM - Drift Current • Current Density Equation
• SaM - Types of Magneto Resistances
• SaM - Anisotropic Magneto Resistance (AMR) • Easy Axis • Barber Pole • Honeywell
• SaM - Inductive Sensors

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SaM - Magnetic Permeability • Magnetic Susceptibility • Magnetic Field

$\begin{array}{lc}\text{In a vacum}& \vec{B}={\mu }_0\vec{H}\\ \text{In a material}& \vec{B}=\mu \vec{H}\end{array}$Where:$\mu ={\mu }_0{\mu }_r$And: ${\mu }_r=1+{\mathcal{X}}_M$

• Terminology:
  • $\mu$ : magnetic permeability $\left(\mu ={\mu }_0{\mu }_r\right)$
    • ${\mu }_0$ : magnetic permeability in the void $\left({\mu }_0=4\pi \cdot 10^{-7}\frac{\text{T}\cdot \text{m}}{\text{A}}\right)$
    • ${\mu }_r$ : relative magnetic permeability
  • ${\mathcal{X}}_M$ : magnetic susceptibility $\left({\mu }_r=1+{\mathcal{X}}_M\right)$
  • $B$ : magnetic field density $\left(B=\mu H\right)$, and $\left[B\right]=T\left(\text{Tesla}\right)$
  • $H$ : magnetic field intensity $\left[H\right]=\frac{\text{A}}{\text{m}}$
  • $M$ : magnetization $\left(M={\mu }_0\mathcal{X}H\right)$

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SaM - Types of Magnetic Materials (Diamagnetic, Paramagnetic and Ferromagnetic Material)

==Both diamagnetic and paramagnetic materials have ${\mu }_r\simeq 1$==, more specifically: - Diamagnetic material: ${\mu }_r<1$ & ${\mathcal{X}}_M\simeq -10^{-4}$ - Paramagnetic material: ${\mu }_r>1$ & ${\mathcal{X}}_M\in \left[+10^{-3}÷+10^{-5}\right]$ - While ferromagnetic material shows “hysteresis”, so ${\mu }_r$ is not constant:

• Terminology:
  • ${\mu }_r$ : relative magnetic permeability
  • ${\mathcal{X}}_M$ : magnetic susceptibility $\left({\mu }_r=1+{\mathcal{X}}_M\right)$
  • $H$ : magnetic field intensity $\left[H\right]=\frac{\text{A}}{\text{m}}$
  • $M$ : magnetization $\left(M={\mu }_0\mathcal{X}H\right)$

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SaM - Geometrical Magneto Resistance

• Current density equation:$\vec{J}=nq\vec{v}=\sigma \vec{E}$Can be rewritten as:$\vec{v}=\mu \vec{E}$Where:$\mu =\frac{\sigma }{nq}$
• Geometrical magneto resistance formula:$\vec{v}=\mu \vec{E}+\mu (\vec{v}\times \vec{B})$
• Terminology:
  • $\vec{J}$ : current density vector, $\left[J\right]=\frac{\text{A}}{{\text{m}}^2}$
  • $n$ : density of carriers.
  • $q$ : charge of an electron $\left(-1.6\cdot 10^{-19}\text{C}\right)$ or of a hole $\left(+1.6\cdot 10^{-19}\text{C}\right)$
  • $\vec{v}$ : velocity of carriers.
  • $\sigma$ : conductivity of the material.
  • $\vec{E}$ : electric field vector.
  • $\mu$ : mobility of carriers.
  • $\vec{B}$ : magnetic field density.

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SaM - Drift Current • Current Density Equation

Transport Current Equation in one dimension:$J_n={\sigma }_{n}E$Velocity of an electron: $\vec{v}=\frac{q\tau }{m_e^{\ast }}\vec{E}$Relation between the avereage time between collisions ($\tau$) and the average free path ($\lambda$):$\tau =\frac{\lambda }{V_{TH}}$Where $v_{TH}$ is called the “mean thermal velocity”, and it is defined as:$V_{TH}=4\sqrt{\frac{kT}{2\pi m_e^{\ast }}}$The transport current, with a symmetrical distribution of carriers:${\vec{J}}_n=nq{\mu }_{n}E$

• Terminology:
  • $m_e^{\ast }$ : effective mass

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SaM - Types of Magneto Resistances

Given a certain magnetic field intensity $(H)$ a magneto-resistive material will change its resisitivity.

• An OMR (Ordany Mangeto Resitive) material can change its reisitivity up to a $5\%$.
• There are also other types of mangeto resistances, these are all due to “quantistic effects”:
  • GMR (Giant Magneto Resitance)
  • CMR (Colossal Magneto Resitance)
  • TMR (Tunnel effect Magneto Resitance)

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SaM - Anisotropic Magneto Resistance (AMR) • Easy Axis • Barber Pole • Honeywell

• Materials that present AMR, like: Ni, Fe and Permalloy (Fe +  Ni).
• AMR Formula:$\rho (\theta )={\rho }_{\parallel }-\Delta \rho \cdot {\sin }^2\theta$
• Easy axis $(x)$, an axis taken in the same direction of ${\vec{M}}_r$
• If we have that ${\vec{H}}_{\text{ext}}\ll H_S$ than we can approximate:$\sin \theta \approx \frac{H_y}{H_S}$
• AMR Formula using the approximation:$\rho \simeq {\rho }_{\parallel }-\Delta \rho {\left(\frac{H_y}{H_S}\right)}^2$
• Total resisitance value:$R\simeq R_0\left(1-\frac{\Delta R}{R_0}{\left(\frac{H_y}{H_S}\right)}^2\right)$Where:$\begin{array}{l}\Delta R=\Delta \rho \cdot \frac{l}{S}\\ \Delta \rho ={\rho }_{\parallel }-{\rho }_{\perp }\\ R_0={\rho }_{\parallel }\cdot \frac{l}{S}\end{array}$
• Graph of $\rho (\theta )$ given $\theta$:
  
• At $\theta \approx 45°$ the sensor behaves linearly.
• Rotate the current $45°$ with a barber pole:
  
• Simplified resistance formula for a barber pole:$R(H)\approx R_0\mp \Delta R\frac{H_y}{H_S}$
• Honeywell or resistive bridge:
  
• Formula of the Honeywell:$V_{\text{TH}}={\alpha }_r\cdot V$We can also calculate the relative sensitivity $({\alpha }_r)$ as: $\frac{V_{\text{TH}}}{V}=\frac{\Delta R}{R_0}\frac{H_y}{H_S}$Remember that:$\Delta R=\left({\rho }_{\parallel }-{\rho }_{\perp }\right)\cdot \frac{l}{S}$
• Magnetic angle sensors formula:$\tan \left(2\theta \right)=\frac{V_{\text{THA}}}{V_{\text{THB}}}$Where:
  • $A$ or $V_{\text{THA}}$, refers to a honeywell bridge sensing along the $x$ axis.
  • $B$ or $V_{\text{THB}}$ refers to the honeywell bridge sensing along the $y$ axis.
• Terminology
  • $\Delta \rho ={\rho }_{\parallel }-{\rho }_{\perp }$
    • ${\rho }_{\parallel }$ is the resisitivity when $\theta =0°$.
    • ${\rho }_{\perp }$ is the resisitivity when $\theta =90°$.
  • $\theta$ is the angle formed by $\vec{J}$ and $\vec{M}$, where:
    • $\vec{J}$ : current density vector, $\left[J\right]=\frac{\text{A}}{{\text{m}}^2}$
    • $\vec{M}$ is the total magnetization vector, accounting for the external magnetic filed ${\vec{H}}_{\text{ext}}$, and for the magnetization of the material ${\vec{M}}_r$, $\left({\vec{M}}_r={\mu }_0\mathcal{X}{\vec{H}}_{\text{ext}}\right)$
    • ${\vec{H}}_{ext}=<H_x,H_y>$
  • $H_S$ is the saturation field, it is a property of the material.
  • $\rho$ : resisitvity.
  • $V_{\text{TH}}=V_{+}-V_{-}$ is the output.
  • ${\alpha }_r$ is the relative sensitivity, and for this structures it is usually $10^{-3}$.
  • $V$ is the input voltage.
  • $\vec{B}={\mu }_0{\vec{H}}_{\text{ext}}$ is the external magnetic field density.

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See Also

• SaM - MEMORIZE (Inductive Sensors)

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