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Publication Number: FHWA-HRT-06-139
Date: October 2006

Traffic Detector Handbook:Third Edition—Volume II

APPENDIX B.CURRENT SHEET FORMULA FOR CALCULATION OF LOOP INDUCTANCE

A formula for calculating the inductance produced by a rectangular current sheet is given by(1)

Equation B-1. Capital L is equal to the product of 0.004 multiplied by pi multiplied by Capital N squared multiplied by the quotient of quantity capital A multiplied by capital A subscript 1 divided by capital B, multiplied by capital F prime measured in mu multiplied by capital H. (B-1)

where

Equation B-2. Capital F prime is equal to the summation of the product of Beta subscript 1 multiplied by gamma, added to the product of Beta subscript 1 prime multiplied by gamma multiplied by the natural log of one divided by gamma, added to the product of Beta subscript 2 multiplied by gamma squared, added to the product of Beta subscript 3 multiplied by gamma cubed minus the product of Beta subscript 5 multiplied by gamma to the fifth power added to a continuation of the expansion. (B-2)
Equation B-3. Gamma is equal to the quotient of capital B over capital A, which is equal to the quotient of the length of current sheet divided by the longer side of rectangle. (B-3)
N = number of turns and
α1 = length of shorter side of loop.

Inductive loops are modeled as very short solenoids.

The term, “length of current sheet,” is defined as the axial length of a coil or solenoid. For an inductive loop, the length of the current sheet is equivalent to the height of the wires in the slot, as described in the sample calculation that follows. Since the length of the current sheet is very small compared to the longer side of the rectangle, γ is also very small. The factor F’ adjusts for the fact that inductive loops perform as very short solenoids (i.e., the loop area is much greater than the turn spacing).

The values of β are obtained from Table B-1.  The factor K that appears in the table is given by

Equation B-4. Capital K is equal to the quotient of capital A subscript 1 divided by capital A. (B-4)
Table B-1. Values of β coefficients for short rectangular solenoids.(2)
Kβ1β1β2β3β5β7
1.00 0.46220.63660.2122−0.00460.0046−0.0382
0.95 0.45740.65340.2234−0.00460.0053 
0.90 0.45120.67200.2358−0.00460.0064−0.0525
0.85 0.44480.69280.2496−0.00420.0080 
0.80 0.43640.71620.2653−0.00310.0103−0.0831
0.75 0.42600.74270.2829−0.00100.0141 
0.70 0.41320.77300.30320.00260.0198−0.1564
0.65 0.39710.80800.32650.00850.0291 
0.60 0.37670.84880.35370.01790.0432−0.3372
0.55 0.35000.89700.38580.03310.0711 
0.50 0.31510.95490.42440.05780.1183−0.7855
0.40 0.18361.11410.53050.16970.3898−2.403
0.30 0.03141.33590.70740.54332.0517−7.85
0.20 −0.64091.90991.06102.323014.50715.51
0.10 −3.23093.50142.122022.5480497.3614280.00

SAMPLE INDUCTANCE CALCULATION

An example from reference 1 is used to demonstrate the calculation of the self inductance for a short, three-turn, 6- by 6-foot (1.8- by 1.8-m) loop, using the current sheet model described above.

The following parameters are used:
a1  = 6 ft = 182.88 cm
a   = 6 ft = 182.88 cm
N   = 3 turns
P  = turn spacing = 150 mils = 0.381 cm
b  = N P = (3)(0.381) = 1.143 cm
γ  = (b/a) = 1.143/182.88 = 0.00625 cm
K  = (a1 / a) = 1.

From Table B-1,

 β1 = 0.4622
 β1’ = 0.6366
 β2 = 0.2122
 β3= -0.0046
 β5= 0.0046
 β7= -0.0382.

Solving Equation B-2 for F’ yields

Equation B-5(B-5)
Equation B-6(B-6)

where

Equation B-7(B-7)

Using Equation B-1, the inductance of the loop is found as

Equation B-8(B-8)

or

Equation B-9(B-9)

REFERENCES

  1. Grover, F.W. Inductance Calculations. Dover Publications, Inc. 1962.
  2. Y. Niwa, “A Study of Coils Wound on Rectangular Frames with Special Reference to the Calculation of Inductance,” Research of the Electrotechnical Laboratory, No. 141, Tokyo, Japan, 1924.

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