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

Traffic Detector Handbook:Third Edition—Volume II

APPENDIX D. ELECTRICAL CHARACTERISTICS OF WIRE AND CABLE

This appendix describes the calculation of the electrical characteristics of twisted lead-in wire composed of Belden #14 AWG copper conductor wire. The wire size and spacing definitions are shown in Figure D-1. Here, D represents the distance between the center of the wire cores in the twisted pair and d is the diameter of the wire core.

Figure D-1 shows the circular cross sections of two number 14 AWI copper wires. The diameter of the insulation is represented graphically by the outer white circle and symbolically by the algebraic variable

Figure D-1.Wire spacing and size definitions used in wire inductance and capacitance calculations.

CAPACITANCE OF PARALLEL CONDUCTORS

The capacitance C of the twisted wire pair is given by

Equation D-1. Capital C is equal to the quotient of the following terms. The numerator is the product of the quotient of epsilon divided by epsilon subscript 0 multiplied by 10 to the negative 9 power. The numerator is the product of 36 multiplied by the inverse hyperbolic cosine of the quotient of capital D divided by lowercase d where the units of the result are in farads per meter. (D-1)

where the factor ε/ε0 is the relative dielectric constant of the wire cover material.

#14 AWG Belden 9438 is commonly used to construct inductive loops.

For Belden 9438, the relative dielectric constant of the polyethylene covering of the wire core is 2.3. Therefore,

Equation D-2. Capital c is equal to the quotient of the following terms. The numerator is equal to the product of 2.3 multiplied by 10 to the negative 9 power. The denominator is the product of 36 multiplied by the inverse hyperbolic cosine of the quotient of 139 mils over 64.1 mils. (D-2)

The capacitance can also be expressed as

Equation D-3. Capital c is equal to the quotient of the following terms. The numerator is equal to the product of 2.3 multiplied by 10 to the negative 9 power. The denominator is the product of 36 multiplied by the natural log of the quotient of 139 mils over 32.05 mils. (D-3)

or

Equation D-4. Capital c is equal to the product of 4.355 multiplied by 10 to the negative 11 power farads per meter, which in turn is equal to 43.55 picofarads per meter, which is equal to 13.3 picofarads per foot. (D-4)

Figure D-2 depicts the electromagnetic field lines that surround the dielectric material in a twisted pair of wires.

Figure D-2 shows elliptically shaped dielectric figures around the two copper wires described in Figure D-1. The larger elliptical field is centered around the tangent point between the two wires. Two more fields extend from where the two wires touch with the center being at the location where they touch.

Figure D-2. Dielectric field surrounding a pair of wires.

This geometry is simplified as shown in Figure D-3 by including the region that encompasses most of the electromagnetic field.

Figure D-3 shows that the dielectric fields of Figure D-2 can be approximated by a simple rectangle enclosing the sides of the circular outlines of the wires. This is described further in the text.

Figure D-3. Simplified geometry that includes most of the electromagnetic field surrounding a twisted pair of wires.

The following heuristic approach modifies the capacitance value given by Equation D-1 for estimating the capacitance of a twisted wire pair based on the geometry of Figure D-3. It assumes that all the electromagnetic field energy is contained in a box surrounding the two wires and their insulation. It gives a reasonable approximation to the capacitance in a closed form equation. An exact closed form solution for the capacitance is not known.

The area of the cable is given by

Equation D-5. Capital a subscript cable is equal to the sum of the quotient of the product of pi multiplied by capital L squared all divided by 4, added to the quotient of the product of pi multiplied by capital L squared all divided by 4, which in turn is equal to the quotient of the product of pi multiplied by capital L squared all divided by 2. (D-5)

where L is the diameter of the wire, including the polyethylene insulation.

The area of the box surrounding the twisted wire pair is given as

Equation D-6. Capital A subscript rectangle is equal to the product of two multiplied by capital L, multiplied by capital L, which in turn is equal to the product of 2 multiplied by capital L squared. (D-6)

The ratio of the cable area to the box area is thus:

Equation D-7. "Ratio of Areas" is equal to the quotient of the following terms. The numerator is equal to the quotient of the product of pi multiplied by capital L squared all divided by 2. The denominator is the product of 2 multiplied by capital L squared. This quotient is equal to pi divided by 4, or approximately 0.79. (D-7)

From Equation D-3, the capacitance of the twisted wire pair becomes

Equation D-8. Capital c is equal to the product of 0.79 multiplied by 13.3 picofarads per foot, which in turn is equal to 10.45 picofarads per foot. (D-8)

For 100 ft (30 m) of wire,

Equation D-9. Capital C is equal to 1045 picofarads. (D-9)

Table D-1 shows that the actual measured capacitance is in the range of 997 to 1006 pF for 100 ft (30 m) of wire. All units of picofarads per foot may be converted to picofarads per meter by multiplying by 0.305 meters per foot. Thus, the heuristic calculation produces a reasonable value. The heuristic result may be fed into finite element analysis models as a first approximation to a final "exact" answer.

INDUCTANCE OF PARALLEL CONDUCTORS

The internal inductance Li is calculated using Equation A-23 of Appendix A with µr =1 and µ0 =4 p x 10-7 H/m.

µH/m can be converted to to µH/ft by dividing by 0.3048.

The inductance L of the twisted wire pair is given by

Equation D-10. Capital L is equal to the sum of the product of 0.4 multiplied by the natural logarithm of the quotient of 2 multiplied by capital D divided by lowercase d, added to the product of 2 multiplied by capital L subscript lowercase i measured in units of microhenrys per meter. (D-10)

The internal inductance Li of copper at 1 kHz is 0.05 µH/m. Thus,

Equation D-11. Capital L is equal to the sum of the product of 0.4 multiplied by the natural logarithm of the quotient of 2 multiplied by capital D divided by lowercase d, added to 0.1 measured in units of microhenrys per meter. (D-11)

For Belden 9438 wire,

Equation D-12. Capital L is equal to the sum of the product of 0.4 multiplied by the natural logarithm of the quotient of the product of 2 multiplied by 139 mils divided by 64.1 mils, added to 0.1 measured in units of microhenrys per meter. (D-12)

or

Equation D-13. Capital L is equal to 0.69 microhenrys per meter or 0.21 microhenrys per foot. (D-13)

For 100 ft (30 m) of wire,

Equation D-14. L is equal to 0.21 microhenrys. (D-14)
 
Table D-1. Measured lead-in cable electrical characteristics of #14 AWG twisted pair.
Frequency (kHz)Open circuit measurement of capacitance (pF)Open circuit measurement of conductance (µmhos) Short circuit measurement of inductance (µH)Short circuit measurement of resistance (Ω)
0.1 997-0.000522.870.57
1.0 999-0.008222.950.57
5.0 1000-0.05622.970.58
10.0 1001-0.1422.910.60
15.0 1002-0.2422.830.63
20.0 1002-0.3622.720.67
25.0 1002-0.5122.600.72
30.0 1002-0.6722.470.77
35.2941 1003-0.8522.330.82
40.0 1003-1.0222.210.87
45.4545 1003-1.2122.090.92
50.0 1003-1.4021.990.97
54.5454 1003-1.5821.901.01
60.0 1003-1.7721.81.06
66.6666 1003-2.0321.691.12
71.4286 1003-2.2621.621.16
75.0 1004-2.4321.571.19
80.0 1004-2.6721.511.24
85.714 1004-2.9521.441.28
96.0 1004-3.4621.331.37
100.0 1005-3.7921.291.4
120.0 1005-4.7421.121.56
125.0 1006-5.0221.091.59
150.0 1006-6.5220.931.78
Type: Twisted pair of Belden 9438 wire (not shielded)
Gauge: #14 AWG
Twists per foot: 5.5 (18.1 twists per meter)
Pair length: 100 ft (30 m)
Wire location: Laboratory floor, Turner-Fairbank Highway Research Center
Measuring Instrument: HP 4284A
Note: Balun unavailable; instrument unbalanced during measurements
 
 
Table D-2. Measured lead-in cable electrical characteristics: Shielded cable lead-in.
Frequency (kHz)Belden 8718 cable (#12 AWG)Belden 8720 cable
Inductance (µH)Resistance (Ω) Inductance (µH)Resistance (Ω)
0.1 19.780.3521.000.59
1 19.960.3521.140.59
5 19.940.3721.160.60
10 19.780.4321.070.64
15 19.540.5120.940.71
20 19.260.6220.760.80
25 18.960.7420.560.91
30 18.650.8720.331.04
35.2941 18.331.0220.081.19
40 18.051.1619.841.33
45.4545 17.741.3319.571.50
50 17.491.4719.341.64
54.5454 17.241.6219.111.79
60 16.961.7918.841.97
66.6666 16.632.0018.512.20
71.4286 16.392.1618.282.36
75 16.232.2718.112.48
80 16.002.4317.882.65
85.714 15.752.6017.782.89
96 15.322.9117.183.18
100 15.163.0317.013.30
120 14.433.5816.253.90
125 14.273.71116.074.04
150 13.524.2915.274.67
Type: Shielded cable
Gauge: #12 or #14 AWG
Length: 100 ft (30 m)
Wire location: Laboratory floor, Turner-Fairbank Highway Research Center
Measuring Instrument: HP 4284A
Note: Balun unavailable; instrument unbalanced during measurements
 
Table D-3. Measured inductive-loop electrical characteristics: Belden 8718 shielded lead-in.
Frequency (kHz)Inductance (µH)Resistance (Ω) Quality factor (Q)
0.1 94.800.530.1
1 94.520.531
5 94.080.565
10 93.270.639
15 93.370.7312
20 92.980.8614
25 92.591.0115
30 92.211.1615
35.2941 91.811.3415
40 91.471.4915
45.4545 91.091.6816
50 90.791.8416
54.5454 90.502.0016
60 90.172.1816
66.6666 89.792.4116
71.4286 89.532.5716
75 89.342.6916
80 89.092.8516
85.714 88.823.0316
96 88.363.3316
100 88.203.4416
120 87.473.9317
125 87.314.0317
150 86.654.4418
Loop size: 6 x 6 ft (1.8 x 1.8 m)
Number of turns: 3 (closely wound)
Gauge: #14 AWG
        Loop location: 3 ft (0.9 m) above electronics laboratory floor, Turner-Fairbank
        Highway Research Center
        Lead-in cable: Belden 8718 (# 12 AWG)
Lead-In cable length: 100 ft (30 m)
Measuring Instrument: HP 4284A
Note: Balun unavailable; instrument unbalanced during measurements
 
Table D-4. Measured inductive-loop electrical characteristics: Belden 8720 shielded lead-in.
Frequency (kHz)Inductance (µH)Resistance (Ω) Quality factor (Q)
0.1 96.800.770.1
1 95.780.770.8
5 95.370.794
10 95.100.857
15 94.850.9410
20 94.591.0511
25 94.291.1813
30 94.001.3313
35.2941 93.671.4914
40 93.381.6514
45.4545 93.041.8414
50 92.762.0015
54.5454 92.482.1715
60 92.162.3715
66.6666 91.782.6015
71.4286 91.512.7715
75 91.202.9015
80 91.063.0715
85.714 90.773.2615
96 90.283.5915
100 90.103.7015
120 89.304.2416
125 89.124.3516
150 88.354.8017
Loop size: 6 x 6 ft (1.8 x 1.8 m)
Number of turns: 3 (closely wound)
Gauge: #14 AWG
Loop location: 3 feet (0.9 m) above electronics laboratory floor, Turner-Fairbank
          Highway Research Center
Lead-in cable: Belden 8720 (# 14 AWG)
Lead-in cable length: 100 ft (30 m)
Measuring instrument: HP 4284A
Note: Balun unavailable; instrument unbalanced during measurements
 
Table D-5. Measured inductive-loop electrical characteristics: Belden 9438 twisted pair lead-in.
Frequency (kHz)Inductance (µh)Resistance (Ω)Quality factor (Q)
0.1 98.190.750.1
1 97.830.750.8
5 97.430.77 4
10 97.200.818
15 97.040.8611
20 96.880.9213
25 96.740.9915
30 96.611.0617
35.2941 96.501.1419
40 96.431.2120
45.4545 96.371.3021
50 96.341.36 22
54.5454 96.341.4323
60 96.361.5124
66.6666 96.431.6025
71.4286 96.501.6726
75 96.561.7227
80 96.681.7827
85.714 96.831.8628
96 97.182.0029
100 97.342.0530
120 98.342.3132
125 98.632.3833
150 100.422.735
Loop size: 6 x 6 ft (1.8 x 1.8 m)
Number of turns: 3 (closely wound)
Gauge: #14 AWG
Loop location: 3 ft (0.9 m) above electronics laboratory floor, Turner-Fairbank
         Highway Research Center
Lead-in cable: Belden 9438 twisted-pair (5.5 twists/ft) (15/m)
Lead-in cable length: 100 ft (30 m)
Measuring instrument: HP 4284A
Note: Balun unavailable; instrument unbalanced during measurements
 
Table D-6. Measured inductive-loop electrical characteristics: Loop without lead-in cable.
Frequency (kHz)Inductance (µH)Resistance (Ω)Quality factor (Q)
0.1 75.020.180.3
1 74.330.193
5 74.350.2012
10 74.150.2122
15 74.040.2330
20 73.950.2537
25 73.860.2842
30 73.790.3046
35.2941 73.710.3350
40 73.660.3652
45.4545 73.610.3955
50 73.570.4157
54.5454 73.530.4359
60 73.480.4661
66.6666 73.440.4963
71.4286 73.410.5164
75 73.390.5365
80 73.360.5567
85.714 73.340.5868
96 73.300.6271
100 73.290.6472
120 73.250.7376
125 73.250.7577
150 73.250.8482
Loop size: 6 x 6 ft (1.8 x 1.8 m)
Number of turns: 3 (closely wound)
Gauge: #14 AWG
Loop location: 3 ft (0.9 m) above electronics laboratory floor, Turner-Fairbank
         Highway Research Center
Lead-in cable: None
Lead-in Length: 0 ft (0 m)
Measuring instrument: HP 4284A
Note: Balun unavailable; instrument unbalanced during measurements

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