4.3 Analytical modeling of square coil
4.3.1 Modeling of mutual inductance
The circuit topology of the two coupled coils EC and OC resembles an inductively coupled trans- former, which is represented by an equivalent circuit as shown in Figure 4.4.
M
EC OC
Source I
VO
Figure 4.4: Equivalent circuit model of an inductive coil.
Consider EC and OC placed near to each other as shown in figure. When EC is connected to the power supply, current (I) flows in it, which produces a magnetic flux (λ1) and a part of the flux (λ12) links with the secondary side coil OC. Then, the MI of the coil is represented as M and is given by (4.1).
M = λ12
I (4.1)
The flux linked with the OC due to current in the EC can be calculated analytically by considering the flux distribution of each individual coil turns of the OC. The method works by approximating the area of OC with small regions, encompassing the entire square and thereby considering the complete spiral square coil. The flux through each small region of the OC is taken into account to calculate the flux linked to OC due to EC. To carry out this process, a sequence of program routine have been used. The total flux linked in the OC is obtained by the sum of the flux linked in each small grid for all the turns of the OC. Assuming ϕn is the flux linked with the nth region of the area enclosed by single turn of the OC, then λ12 can be estimated by (4.2). The limit of the summation depends on
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the number of small regions formed in a single turn of OC.
λ12=X
ϕn (4.2)
The flux linked to each of these small regions of OC due to EC is calculated with the following assumptions.
• The insulated coil conductors are placed such that there is no space between conductors of any two loops and they are touching each other.
• The magnetic field in the small region of OC caused due to EC (Figure 4.5(b)) is assumed to be constant and its value is calculated at the center of that small region.
Having made the above assumptions, ϕn for each small region is calculated using (4.3) and (4.4), which depends on the magnetic field at the center (−B→c) of the small region, nth area (An) and normal vector of thenth area (Acn).
ϕn =−B→c·(An·Abn) (4.3)
An= ∆xn·∆yn (4.4)
Here, ∆xnand ∆ynare the length and width of the small divided region as shown in Figure 4.5(b).
The magnetic field −B→c at the center of the small region is caused due to EC.−B→c is calculated for P turns of EC and is given by (4.5). The individual coil turns of EC is modeled by four straight current carrying conductors. Let −→
B is the magnetic field due to one current carrying loop of EC as shown in Figure 4.5(a) and is given by (4.6).
−→ Bc =
XP
m=1
−
→B (4.5)
−
→B = X4
n=1
−→Bn (4.6)
In the above equation, nrepresents the four sides of the single current carrying loop and−B→1,−B→2,
−→
B3 and −B→4 are the magnetic fields of the sides of the square coil AB, BC, CD and DA. −→Bn can be calculated from the Bio-Savart law for magnetic field. The basic magnetic field equation (−→Bn) at any point in space due to a straight current carrying conductor is given by (4.7).
−→Bn=
Z µ0I−→ ds×Rb
4πR2 (4.7)
4.3 Analytical modeling of square coil
The vector R, in the above equation is the unit vector in the direction of position vector of theb observation point, originating from the differential element of current carrying conductor (−→
ds). The direction of −→
ds is in the direction of current in the conductor. The integration in (4.7) is performed over the length of the conductor. Similarly, the magnetic field at a point in space and flux linkage calculations can be done for spiral square coils with multiple turns as shown in Figure 4.5(c), where L and W are the length and width of the coil.
C
−
→ B4
−→ B3
D
A −→ B
B1
−
→ B2
(a)
An
∆Xn
∆Yn D
A
C
B flux linkage (ϕn)
(b)
W
L
(c)
Figure 4.5: Square current carrying coil (a) single turn (b) single turn segmented (c) multiple turn.
4.3.2 Numerical evaluation
The MI for the mutually coupled coils is calculated numerically using the equations discussed in the previous section. The procedure for numerical calculation is explained in the following steps.
• The total number of turns for EC (P) and OC (Q) are determined.
• 3D co-ordinates of a single turn of EC and OC are determined.
• Diameter of conductor and distance between EC and OC (depending on the type of variation) is determined.
• The selected turn of the coil is divided into multiple small areas from A1...An as shown in Figure 4.5(b).
• Calculate the total flux linked to each small region of OC using (4.8) by calculating the magnetic field at the center .
ϕ1= X4
n=1
µ0
4π
Z I−→ ds×Rb1
R12 · A1·Ab1
(4.8)
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• The flux linked in a single turn (ϕm) of OC is obtained by summing all the fluxes using (4.9).
ϕm = XP
k=1
X(ϕ1+ϕ2...+ϕn) (4.9)
• This procedure is repeated for all turns of EC and OC andλ12 is calculated using (4.10).
λ12= XQ
m=1
ϕm (4.10)
• Therefore from these equations MI is calculated using (4.1).
The detailed description of this evaluation is given in the flowchart shown in Figure 4.6. This procedure has been used for all variations of EC and OC with their corresponding new coordinates and vertices. Thus, the method adapted in this work has used only Biot-Savart law for a straight current carrying conductor; which is the basic equation for calculating the magnetic field. It does not require any double or triple integration functions for its computation. This gives an interesting compactness for the calculation of MI by avoiding complicated mathematical equations. An example for particular case of variations is given in Appendix (see Appendix A.5).