EM Tracker HFluxPerI Derivation - Revision history

Traneus at 01:09, 19 November 2017

2017-11-19T01:09:28Z

Traneus at 01:05, 19 November 2017

2017-11-19T01:05:47Z

Traneus at 00:48, 19 November 2017

2017-11-19T00:48:45Z

Traneus at 00:47, 19 November 2017

2017-11-19T00:47:45Z

Traneus: Expanded discussion to coil trios

2017-11-19T00:28:57Z

Expanded discussion to coil trios

Traneus at 00:11, 19 November 2017

2017-11-19T00:11:03Z

Traneus: Bolded equations for better readability, and corrected some errors.

2017-11-19T00:05:05Z

Bolded equations for better readability, and corrected some errors.

Traneus at 23:56, 18 November 2017

2017-11-18T23:56:01Z

Traneus at 00:29, 21 December 2015

2015-12-21T00:29:23Z

Traneus at 00:18, 21 December 2015

2015-12-21T00:18:53Z

@@ Line 107: / Line 107: @@
 Aeff_tmtr_mat is known from the transmitter coil characterization process.
-Calculating the sum of the squares of the nine elements in HFluxPerI, permits calculating Rmag independently of the rotation matrices. This needs more detail here, see the Raab and raab etal papers.
+Calculating the sum of the squares of the nine elements in HFluxPerI, permits calculating Rmag independently of the rotation matrices. This needs more detail here, see the Raab and Raab Blood Steiner Jones papers.
 The left side is thus known. The tracker position-and-orientation algorithm does the equivalent of calculating the two rotation matrices from the known left side and the known Ro. Note that if Ro were the identity matrix, the two rotation matrices could not be calculated separately.

@@ Line 77: / Line 77: @@
 <br>
 (+0, +0, -1)
 </tt></b>
@@ Line 82: / Line 92: @@
 <b>HFluxPerI_mat = (1/(4 pi Rmag^3)) Aeff_rcvr_mat Rcvr_rotation_mat Ro Tmtr_rotation_mat Aeff_tmtr_mat</b>

← Older revision		Revision as of 00:48, 19 November 2017
Line 81:		Line 81:
	Putting it all together, we get:		Putting it all together, we get:

−	<b>HFluxPerI_mat = (1/(4 pi Rmag^3)) Aeff_rcvr_mat Rcvr_rotation_mat Ro ~~Tmtr_rotation_mat_Aeff_tmtr_mat~~</b>	+	<b>HFluxPerI_mat = (1/(4 pi Rmag^3)) Aeff_rcvr_mat Rcvr_rotation_mat Ro Tmtr_rotation_mat Aeff_tmtr_mat</b>

@@ Line 67: / Line 67: @@
 <b>Tmtr_rotation_mat Aeff_tmtr_mat</b>
-The parenthetical expression containing Runit_vect becomes a 3x3 matrix Ro. Assume we are working in a coordinate system where Runit_vect = (1,0,0). Then
+Assume we are working in a coordinate system where Runit_vect = (1,0,0). This leaves a free choice of rotations about Runit_vec, called "position roll" in Raab's 1981 paper and in the Raab, Blood, Steiner, Jones 1979 paper. Position roll drops out when converting the two rotation matrices to receiver position and orientation.
 Ro is:
 <b><tt>
@@ Line 78: / Line 79: @@
 </tt></b>
-Next...
+Putting it all together, we get:

@@ Line 31: / Line 31: @@
 Dividing by the transmitter current Itmtr, gives scalar HFluxPerI (measured in meters):
-<b>HfluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect</b>
+<b>HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect</b>
 HFluxPerI is a purely geometrical property of the coils and their relationship in space.
@@ Line 57: / Line 57: @@
 So far, we have considered one transmitter coil and one receiver coil, deriving the receiver HFluxPerI scalar:
-<b>HfluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect</b>
+<b>HFluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect</b>

← Older revision		Revision as of 00:11, 19 November 2017
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	The minus sign comes from the electromagnetics of induced voltages.		The minus sign comes from the electromagnetics of induced voltages.
		+
		+	So far, we have considered one transmitter coil and one receiver coil, deriving the receiver HFluxPerI scalar:
		+
		+	<b>HfluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect</b>

@@ Line 23: / Line 23: @@
 Place a dipole receiver coil (with effective-area vector Aeff_rcvr_vect measured in square meters) at the observation point Rvect. Scalar HFlux (measured in amperes times meters) through the receiver coil is defined as:
-HFlux(Rvect) = Aeff_rcvr_vect .dotproduct. Hvect(Rvect)
+<b>HFlux(Rvect) = Aeff_rcvr_vect .dotproduct. Hvect(Rvect)</b>
 Substituting for Hvect(Rvect) gives:
-Hflux(Rvect) = Aeff_rcvr_vect .dotproduct. (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect
+<b>Hflux(Rvect) = Aeff_rcvr_vect .dotproduct. (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect</b>
 Dividing by the transmitter current Itmtr, gives scalar HFluxPerI (measured in meters):
-HfluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect
+<b>HfluxPerI(Rvect) = (1 / (4 pi Rmag^3)) Aeff_rcvr_vect .dotproduct. ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect</b>
 HFluxPerI is a purely geometrical property of the coils and their relationship in space.
@@ Line 41: / Line 41: @@
 The induced voltage Vrcvr (measured in volts) across the receiver coil is:
-Vrcvr = -d/dt(Uo  HFluxPerI  Itmtr)
+<b>Vrcvr(t) = -d/dt(Uo  HFluxPerI  Itmtr(t))</b>
 Uo = pi 4e-07 volts*seconds/(amperes*meters) is the magnetic permeability of free space, usually called mu-nought.
-If Itmtr is sinusoidal at frequency F, and the receiver is moving slowly or not at all with respect to the transmitter, we have:
+If Itmtr(t) is sinusoidal at frequency F, and the receiver is moving slowly or not at all with respect to the transmitter, we have:
-Itmtr = Itmtr_peak sin(2  pi  F)
+<b>Itmtr(t) = Itmtr_peak sin(2 pi F t)</b>
-Vrcvr = -Uo  HFluxPerI  Itmtr_peak  2  pi  F  cos(2 pi F)
+<b>Vrcvr(t) = Vrcvr_peak cos(2 pi F t)</b>

@@ Line 11: / Line 11: @@
 We can write Rvect as the product of its scalar magnitude, Rmag (measured in meters), and a unit vector Runit_vect (unitless) in the direction of Rvect:
-Rvect = Rmag Runit_vect
+<b>Rvect = Rmag Runit_vect</b>
 The vector magnetic field Hvect (measured in amperes per meter) at the observation point is then:
-Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) (3 Runit_vect (Runit_vect .dotproduct. Aeff_tmtr_vect) - Aeff_tmtr_vect)
+<b>Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) (3 Runit_vect (Runit_vect .dotproduct. Aeff_tmtr_vect) - Aeff_tmtr_vect)</b>
 This can be written more compactly (being a little free with the notation) as:
-Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect
+<b>Hvect(Rvect) = (Itmtr / (4 pi Rmag^3)) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect</b>
 Place a dipole receiver coil (with effective-area vector Aeff_rcvr_vect measured in square meters) at the observation point Rvect. Scalar HFlux (measured in amperes times meters) through the receiver coil is defined as:

← Older revision		Revision as of 00:29, 21 December 2015
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	Uo = pi 4e-07 voltsseconds/(amperesmeters) is the magnetic permeability of free space, usually called mu-nought.		Uo = pi 4e-07 voltsseconds/(amperesmeters) is the magnetic permeability of free space, usually called mu-nought.
		+
		+	If Itmtr is sinusoidal at frequency F, and the receiver is moving slowly or not at all with respect to the transmitter, we have:
		+
		+	Itmtr = Itmtr_peak sin(2 pi F)
		+
		+	Vrcvr = -Uo HFluxPerI Itmtr_peak 2 pi F cos(2 pi F)

@@ Line 35: / Line 35: @@
 HFluxPerI is a purely geometrical property of the coils and their relationship in space.
-If we replace Rvect with -Rvect, Rmag is unchanged and Runit_vect is replaced by -Runit_vect. HFluxPerI is unchanged. This is the hemisphere ambiguity.
+If we replace Rvect with -Rvect, Rmag is unchanged, Runit_vect is replaced by -Runit_vect, and HFluxPerI is unchanged. This is the hemisphere ambiguity.
 If we swap Aeff_rcvr_vect and Aeff_tmtr_vect, HFluxPerI is unchanged. This is electromagnetic reciprocity.