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where <math>N</math> is the number of coil turns and <math>\Phi_{i}</math> is the magnetic flux intersecting the sensor due to the <math>i^{th}</math> transmitter coil PCB. <math>\frac{d\Phi_i}{dt}</math> depends on the position, orientation, core material and frequency of the emitter coil w.r.t. the sensor. To derive the voltage across on the sensor in terms of system variables we first look at the magnetic field intensity <math>\mathbf{H_i}</math> at a particular point in space due to a single PCB coil. This intensity is calculated from the PCB coil model described in section 3a and can be written as
<mathdisplay="block">\mathbf{H_i} = H_0\sin(\omega_i t)\mathbf{n_H} </math>
where <math>H_0</math> is the magnitude of the intensity, <math>\omega_i</math> is the frequency of the time varying field and </math>n_H</math> is a unit vector pointing in the direction of the field intensity. By placing a sensor at this point in space, the magnetic flux intersecting the sensor can be written as
<mathdisplay="block">\Phi_i = \mu_0\mu_r A H_0\sin(\omega_i t)\mathbf{n_H . n_A}</math>
where <math>\mu_0</math> and <math>\mu_r</math> are absolute and relative permeabilities of free space and ferrite core respectively, <math>A</math> is the cross sectional area of the sensor and <math>n_A</math> is a unit vector pointing along the axis of the sensor. Multiplying by the number of turns <math>N</math> and taking the time derivative results in
<mathdisplay="block">v_i(t) = \frac{d\Phi_i}{dt}= - \mu_0\mu_r A N H_0 \omega_i\cos(\omega_i t)\mathbf{n_H . n_A}</math>
The negative sign (<math>-</math>) is added to satisfy Lenz's law. This equation highlights some important aspects of the system:
Anser EMT utilises 8 transmitter PCB coils in total. Therefore the induced sensor voltage on the sensor is a superposition of voltages due to each of the 8 coils:
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The sensor that was used to test the Anser EMT was the NDI model 610099 5-DOF sensor. The sensor measures 0.5mm in diameter with a rigid tip of 8mm. The sensor has been electrically characterised with a DC resistance of ~<math>70\Omega$ </math> and series inductance of <math>2\text{mH}</math>. A parasitic capacitance of <math>900\text{pF}</math> is also present. A photograph of the sensor shown below:
