Inductive Sensors

Electromagnetic Induction

Electromagnetic induction: We know that a current flowing in a wire would produce a magnetic field. To increase the measurable effect of the field we can spin the wire into a coil.

For one turn of the coil, the magnetic flux is:

\[\Phi=\frac{ni}{\mathcal{R}}\]

• where $\mathcal{R}$ is the reluctance, a property of the core material. Magnetic reluctance is analogous to electrical resistance in an electrical circuit.

\[\mathcal{R}=\frac{\mathcal{l}}{\mu_0 \mu A}\]

• $\mathcal{l}$ - total length of flux path
  • A path which is followed by magnetic lines of force and in which the magnetic flux density is significant.
• $\mu_0$ - permeability of vacuum
• $\mu$ - permeability of core material
• A - cross sectional area of the coil

\[\Psi=n\Phi = \frac{n^2 i}{\mathcal{R}}\]

• where $\Phi$ is magnetic flux for one turn of the coil, unit Tesla. And $\Psi$ is the total flux, unit Weber.

Definition: The inductance L of a coil is the total flux per unit current:

\[L = \frac{\Psi}{i}=\frac{n^2}{\mathcal{R}}\]

• The relationship between inductance and current is: change in current causes a measurable voltage.

\[V_o = L \cdot \frac{di}{dt}\]

Further, change in the magnetic field can induce a change in current and vice versa.

\[\frac{d\Psi}{dt} \propto \frac{di}{dt} \rightarrow V\propto \frac{d \Psi}{dt}\]

Mutual Inductance

• An alternating current flow in one coil can induce a current flow in another coil.

\[V_2 = - M \frac{di_1}{dt}\]

• where M is coefficient of mutual inductance.
  • In general, it depends on the reluctance(distance and material between coils) with which a magnetic flux can flow between inductors.
• $i_1$ would induce another current flow $i_2$ in the RHS circuit

Usage

The Linear Variable Differential Transformer (LVDT)

Linear variable differential transformer is a type of electrical transformer used for measuring linear displacement (position).

• A video explanation

The red bar is the core moving in x direction, and Vs is the source power. According to effect of mutual inductance, we know $V_1 = -M_1\frac{di_s}{dt}$ and $V_2 = -M_2\frac{di_s}{dt}$. Red bar’s change $\Delta x$ in position will cause a change in $M_1$ and $M_2$ due to reluctance changes. Thus,

\[V_0 = V_1 - V_2 = (-M_1\frac{di_s}{dt}) - (-M_2\frac{di_s}{dt}) = (M_2 - M_1)\frac{di_s}{dt}\]

Why we need 2 secondary coils here? - It gives the linearity, ($M_2-M_1$) can show it’s linear. With one coil, the change in M is non-linear.

How do we measure?

• At $x=0$, $V_0=0$, $M_1=M_2$
• Connect in open-wiring configuration
• As core moves, it will change $M_1, M_2$, this will change the induced voltages $V_1, V_2$, and the amplitude of $V_0$ is linear with $\Delta x$.

Electromagnetic Flow Meter

• Electromagnetic flow meters are comprised of a transmitter and sensor that together measure water flow.

• If a conductor of length l(m) is moving perpendicular to a magnetic field $\Phi$ at a velocity e (m/s), the voltage $V_o$ is induced:

\[V_0=\Phi l e\]

• One might more familiar with the form $V = Blv$ but they are the same.
• With Faraday’s law, we can build flow meter from it.

In the case of a flow meter, the conductor is the liquid. We know that

\[Q = A \cdot e \qquad A = \frac{\pi D^2}{4}\]

• e is the velocity of the flow m/s
• A is the cross-sectional area of the pipe
• Q is the flow of the liquid $m^3/s$
• D is the diameter

\[V_0=\Phi l e=\Phi D \frac{4Q}{\pi D^2}=\frac{4\Phi Q}{\pi D}\]

• Measuring the voltage can help us to measure the flow, although it is very difficult to build flow meter.

Inductive-loop traffic detectors (RLC circuit)

Vehicle detection loops can detect vehicles passing or arriving at a certain point. When no car is present, the inductance of the loop is a steady, known value. When a car passes over the loop, the inductance changes. We can simplify it to a circuit like in the figure below, which is a common tool for measuring the change of an unknown inductor.

Definition: Impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.

• Impedance for a pure resistor is its resistance
• Let the capacitance of a capacitor is C and the alternating current passing through the capacitor circuit have the angular frequency $\omega$. Then the impedance experiences by the current passing through the capacitor is $z = \frac{1}{j\omega C}$
• If L is the inductance of an inductor operating by an alternating voltage of angular frequency $\omega$, then the impedance offered by the pure inductor to the alternating current is, $z=j\omega L$.
• Source

To obtain the value $L_3$, a variable capacitor $C_2$ is digitally adjusted until the bridge circuit is balanced ($V_0 = 0$). Then, it can be shown that $z_1 \cdot z_4 = z_3 \cdot z_2 $. Thus,

$R_1\cdot R_4 = (j\omega L_3)(\frac{1}{j\omega C_2})$

$ L_3 = R_1\cdot R_4 \cdot C_2 $

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This is my notes from the class I am taking “CEE 575 Infrastructure Sensing” offered by Prof. Branko Kerkez at University of Michigan.