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Traffic Detector Handbook:Third Edition—Volume II

APPENDIX E. VEHICLE DETECTION SENSITIVITY FORMULAS FOR RECTANGULAR LOOPS

ABSTRACT

This appendix describes the derivation and application of formulas to calculate the vehicle detection sensitivity of single-turn and multiturn rectangular loops of coaxial wire. The effect on detection sensitivity of reinforcing steel mesh in the pavement and lead-in cable inductance is accounted for in the models. The loop is installed in the roadway pavement and connected to a roadside electronics unit by lead-in cable, which is modeled as a transmission line. When a vehicle passes over the loop, eddy currents induced in the vehicle undercarriage cause a decrease in loop inductance, which is sensed by the electronics unit.

The multiturn rectangular loop is modeled as a number of series connected, single-turn, rectangular loops that exhibit mutual coupling between each turn of wire. The multiturn loop is represented by the primary of an air core transformer, which contains two secondary windings composed of shorted turns that simulate the vehicle undercarriage and reinforcing steel mesh. The derived equations are used to calculate tables of vehicle detection sensitivity, expressed as vehicle undercarriage detection height, as a function of number of loop turns, loop size, and loop spacing.

INTRODUCTION

The inductive-loop detector (ILD) system is used nationwide to detect stopped or moving vehicles for traffic surveillance and control systems. A typical ILD system consists of a 3-turn, 6- x 6-ft (1.83- x 1.83-m) loop of #14 AWG wire embedded in the pavement, a lead-in cable to roadside, and roadside electronics unit. The vehicle-detection sensitivity is the normalized inductance change at the terminals of the electronics unit when a vehicle is sensed by the loop. High-ground-clearance vehicles such as trucks cause a small change in loop inductance, which sometimes results in nondetection of part of the vehicle. Increasing the sensitivity of the electronics unit to sense such small inductance changes usually increases the response time of the electronics unit, which causes errors in ILD applications for measuring vehicle speed.

The vehicle-detection sensitivity of an inductive-loop detection system is maximized when the spacing between loop turns is correctly selected. The increased sensitivity should improve detection of high-ground-clearance trucks.

This appendix is based on a previous paper that developed a detectionsensitivity formula for a single-turn rectangular loop.⁽¹⁾ It was shown that, with no reinforcing steel, the sensitivity is proportional to the mutual inductance squared, divided by the product of the self-inductance of the single-turn loop and the self-inductance of the shorted-turn loop simulating the vehicle. If the loop turns of a multiturn, rectangular loop are tightly coupled, the sensitivity is independent of the number of loop turns, provided the loop inductance is approximately ten times larger than the inductance of the lead-in cable. A formula for the self-inductance of a multiturn rectangular loop, which accounts for leakage flux, was developed in a second paper.⁽²⁾ As the loop turns become more widely spaced, the increased leakage flux causes the loop self-inductance to decrease, which results in increased detection sensitivity until the decrease in mutual coupling to the shorted turn becomes significant.

VEHICLE-DETECTION SENSITIVITY THEORY

VEHICLE MODEL

The vehicle is modeled as a flat, perfectly conducting plate at a height above the loop approximately equal to the average undercarriage height of the vehicle. The width and length of the plate are equal to the width and length of the vehicle. The continuous, perfectly conducting plate can be simulated by a wire grid, provided the mesh elements are sufficiently small.

In order to simplify the calculations of loop system sensitivity, the wire grid simulating the vehicle is set equal to the size of the inductive loop in the roadway. When the wire grid is coaxially located over the loop, the interior currents induced in the grid by the loop cancel, leaving a current flowing around the perimeter of the grid, provided the grid width and length are equal to those of the loop. Thus, the grid can be replaced with a wire loop or shorted turn equal to the grid perimeter. This case represents a vehicle centered over the loop (i.e., maximum percentage inductance change of the loop). Magnetic effects on the loop due to the permeable material of the vehicle are assumed to be negligible.

REINFORCING STEEL MODEL

The mesh elements of the reinforcing steel in the pavement are considered sufficiently small and the size of the mesh sufficiently large compared to the loop so that the steel mesh can be replaced with a perfectly conducting plane of infinite extent. Based on image theory, the infinitely conducting plane is replaced with an image loop at twice the distance from the loop to the reinforcing steel. Figure E-1 illustrates the geometry of the vehicle and reinforcing steel shorted turn model for a two-turn rectangular inductive loop. The direction of induced currents determines the sign of the mutual inductance terms used in the circuit model. A plus sign indicates current flow in the same direction and a minus sign current flow in opposite directions.

Figure E-1. Two-turn loop in roadway.

CIRCUIT MODEL FOR ONE-TURN LOOP

Figure E-2 shows the air core transformer model of a one-turn inductive loop, vehicle undercarriage, and reinforcing steel mesh. The inductive loop is modeled as the primary winding of the transformer, while the vehicle and reinforcing steel are modeled as shorted turn secondary windings. The secondary winding simulating the vehicle is movable to simulate changes in vehicle undercarriage height H_V. The secondary winding modeling the shorted turn reinforcing steel mesh is located at a distance of twice the steel mesh to inductive loop spacing H_S.

Figure E-2. Single-turn inductive-loop circuit model.

Circuit Equations for One-Turn Loop

The circuit equations corresponding to Figure E-2 are:

(E-1)

(E-2)

(E-3)

where

(E-4) (E-5) (E-6) (E-7) (E-8)

R is resistance in ohms. j represents the square root of -1 in the imaginary term. ω is frequency in radians. L is inductance in henries.

All mutual impedances are symmetrical (i.e., Z₁₂ = Z₂₁, Z₁₃ = Z₃₁, Z23 = Z₃₂).

Experimental measurements show that the quality factor ωL₁₁ /R₁₁ of the loop driving point impedance Z₁₁ is ≥ 10.

Assuming small circuit loss to simplify calculations gives

(E-9)

(E-10)

(E-11)

Inductive-Loop Driving-Point Impedance

The loop driving-point impedance Z₁ from Equations E-1, E-2, and E-3 is

(E-12)

If the reinforcing mesh is spaced a great distance from the inductive loop,

Then

(E-13)

In this case, the loop driving-point impedance Z₁ depends only on the effect of the vehicle.

Inductive-Loop Sensitivity

The sensitivity S_L of an inductive loop is defined as

(E-14)

The change in loop driving-point impedance ΔZ₁ is given by

(E-15)

where ΔZ₁^NV is the loop driving-point impedance with no vehicle present and ΔZ₁^V is the loop driving-point impedance when sensing a vehicle.

The loop driving-point impedance Z₁ can also be expressed as

(E-16)

Then, if no vehicle is present,

(E-17)

(E-18)

(E-19)

The loop sensitivity S_L is given by

(E-20)

if

(E-21)

Applying Equations E-9, E-10, and E-11 to Equation E-20 yields the loop sensitivity S_L in percent as

(E-22)

When reinforcing steel mesh is not present,

(E-23)

CIRCUIT MODEL FOR TWO-TURN LOOP

Figure E-3 shows the air core transformer model of a two-turn inductive loop, vehicle undercarriage, and reinforcing steel mesh. Mutual inductance terms that model the effects of the multiple-turn loop have been added to the model.

Figure E-3. Two-turn inductive-loop circuit model.

Circuit Equations for Two-Turn Loop

From Figure E-3, the circuit equations for the two-turn loop are given by

(E-24)

(E-25)

(E-26)

Then

(E-27)

(E-28)

(E-29)

(E-30)

(E-31)

(E-32)

(E-33)

(E-34)

(E-35)

As compared to the one-turn loop model, the self-impedance Z₁₁ is replaced by the total two-turn loop impedance. Each mutual impedance is replaced by the sum of the mutual impedances from each turn.

CIRCUIT EQUATIONS FOR MULTITURN LOOP

Circuit equations for a multiturn loop are given by

(E-36)

(E-37)

(E-38)

where δ_ij is the Kronecker Delta function (i.e., δ_ij = 1 for i = j and δ_ij = 0 for i not equal to j). Then

(E-39)

(E-40)

(E-41)

(E-42)

(E-43)

Equations E-39–E-43 are generalized for the multiturn case by letting

(E-44)

(E-45)

(E-46)

where L^*_N is the total self-inductance of the loop.

VEHICLE-DETECTION SENSITIVITY FOR A MULTITURN LOOP

Detection sensitivity is expressed as

(E-47)

For a two-turn (N = 2) loop,

(E-48)

The formula for multiturn loop-sensitivity can be applied to any loop geometry, provided the vehicle and reinforcing steel mesh can be modeled as shorted turns.

When no reinforcing steel mesh is present, the sensitivity of the electronics unit to a change in loop-system inductance caused by vehicle passage or presence is given by

(E-49)

Equation E-47 can be applied to a multiturn rectangular loop with N coaxial, equal-spaced, identical turns. The low-frequency inductance L_N for such a loop is given by⁽³⁾

(E-50)

or

(E-51)

where L₁₁ is the self-inductance of a single turn and M_1,i+1 is the mutual inductance between turn one and the i + 1 turn. This formula assumes that the smallest dimension of a loop turn length or width is much greater than the largest spacing between loop turns.

For a two-turn loop,

(E-52)

The total self-inductance L*_N of the loop is equal to the sum of the low–frequency inductance L_N and the high-frequency, skin-effect inductance L_H and is expressed as

(E-53)

where

(E-54)

l₁ and l₂ are turn width (m) and turn length (m), respectively, and the internal inductance Lⁱ is 0.036 µH/m (0.111 µH/ft) at 47 kHz.

INDUCTIVE-LOOP DETECTOR SENSITIVITY

The series inductance L_C of the transmission line connecting the loop in the roadway to the electronics unit reduces the inductive-loop system sensitivity S_L at the electronics unit to

(E-55)

LOOP-DETECTOR SENSITIVITY MEASURED DATA AND CALCULATED RESULTS

LOOP-DETECTOR SENSITIVITY COMPUTER PROGRAM

Equations E-47, E-51, E-53, and E-55 were programmed in BASIC and run on an IBM computer. The main program calls a subroutine, which calculates the sum of the mutual inductance from each loop turn to the image loop simulating the vehicle and image loop simulating the reinforcing steel. The main program also calls a subroutine, which calculates the external, or low-frequency inductance of a coaxial, multiturn, rectangular loop of round wire. This low-frequency inductance subroutine calls a subroutine, which calculates the mutual inductance between two coaxial, single-turn, rectangular loops. The single-turn, mutual inductance subroutine calls a subroutine, which calculates the mutual inductance between two parallel current elements.

MEASURED LOOP-DETECTOR SENSITIVITY DATA

The only known measured loop-detector sensitivity data⁽⁴⁾ versus number of loop conductor turns are presented in Table E-1. These data were obtained by driving a compact automobile over a 6- x 6-ft (1.83- x 1.83-m) loop with 10 ft (3.05 m) of #14 underground feeder (UF) cable connected between the loop and the laboratory equipment. Loop sensitivity as a function of the number of loop turns was calculated from the change in loop inductance recorded with and without the vehicle. The sensitivity magnitude increased approximately 5 percent when the operating frequency changed from 15 to 100 kHz for four or more loop turns.

COMPARISON BETWEEN MEASURED AND CALCULATED SENSITIVITY DATA

Since the average undercarriage height of the compact automobile used was unknown, the automobile was modeled with a 6- x 6-ft (1.83- x 1.83-m) shorted turn coaxially located 0.71 ft (216 mm) over the loop of #14 AWG wire. A loop-wire spacing of 150 mils (38 mm) and lead-in cable inductance of 0.22 µH/ft (0.72 µH/m) were assumed.

Table E-1 shows that the agreement between measured and calculated loopdetector sensitivity is good. Therefore, values calculated from the computer program model can be used to infer the effects of inductive-loop design parameters on loop sensitivity as described below.

EFFECT OF LOOP TURNS ON SENSITIVITY

A survey by Dorsey⁽⁵⁾ found that 15 of 21 States cut loop slots greater than 2 inches (50.8 mm) in the pavement. The spacing between the center of the conductor of the top loop turn and the center of the conductor of the bottom loop turn was assumed to be 2 inches (50.8 mm) for analysis. In practice, a much closer top-to-bottom turn spacing is used. A loop slot depth of 1-3/8 inches (35 mm) is recommended in the Traffic Detector Handbook for a two-turn loop. A nominal sensitivity of 0.098 percent was assumed for the detection threshold.

A lead-in cable length of zero was used so that the effect of loop turns on sensitivity would be independent of lead-in cable length.

The effect of the reinforcing steel mesh was removed by spacing the mesh 1000 ft (304.8 m) from the loop. Results from the loop-detector system sensitivity computer program are presented in Table E-2 for 1 to 8 loop turns. The vehicle undercarriage detection height is approximately proportional to the volume enclosed by the loop conductors and is approximately independent of the number of loop turns for a given volume. For example, the volume of the 6- × 6-ft (1.83- × 1.83-m) by 2000-mils (50.8-mm) loop is 6 ft³ (0.171 m³), as illustrated by the results in Table E-2.

EFFECT OF LOOP VOLUME ON SENSITIVITY

Since Table E-2 demonstrates that the undercarriage detection height is approximately independent of the number of loop turns for three or more turns, the effect on loop sensitivity and hence detection height of increasing the spacing between turns in a three-turn loop was calculated as shown in Table E-3. The vehicle undercarriage detection height slowly increases when the turn spacing exceeds 1 inch (25 mm). The volume of the 6- × 6-ft (1.83- × 1.83-m) by 300-mils (7.6-mm) loop is 0.90 ft³ (0.25 mm³) with a detection height of 4.7 ft (1.43 m). Increasing the loop volume by a factor of 10 results in a 0.3-ft (9.1-cm) increase in vehicle detection height.

EFFECT OF LEAD-IN CABLE INDUCTANCE ON SENSITIVITY

Lead-in cable inductance reduces loop sensitivity to vehicles and hence vehicle undercarriage detection height. Additional loop turns increase the loop inductance, which reduces the detrimental effect of the lead-in cable inductance. Tables E-4 and E-5 illustrate this effect.

EFFECT OF MESH ON SENSITIVITY

Reinforcing steel mesh also reduces loop sensitivity to vehicles and hence vehicle undercarriage detection height. Adding loop turns has little effect on negating the reduced sensitivity caused by the mesh, as shown in Tables E-4 and E-5.

EFFECT OF LEAD-IN CABLE AND MESH ON SENSITIVITY

The combined effects of lead-in cable inductance and steel mesh accentuate the reduction of loop sensitivity and hence vehicle undercarriage detection height. Tables E-4 and E-5 show that the number of loop turns should be five or more for detection of 4-ft-high (1.22-m-high) ground-clearance trucks when lead-in cable and steel mesh are present.

RESULTS AND CONCLUSIONS

A formula for calculating the vehicle detection sensitivity for a multiturn loop was presented. Results from this formula indicate that the vehicle detection sensitivity for a 6- × 6-ft (1.83- × 1.83-m), 3-turn loop of #14 AWG wire increases as the spacing between turns increases. Although a maximum detection height of 5.05 ft (1.54 m) occurs with a turn spacing of 2 inches (50.8 mm), a detection height of 4.99 ft (1.52 m) occurs at the more practical turn spacing of 1 inch (2.54 cm). A very small increase in detection height to 5.06 ft (1.54 cm) was noted when #12 AWG wire was used instead of #14 AWG wire.

When the loop turns are spaced to use the maximum practical installation space in the pavement, the undercarriage detection height is approximately independent of the number of turns, provided that the loop inductance is approximately ten times larger than the inductance of the lead-in cable. For example, the self-inductance of a 6- × 6-ft (1.83- × 1.83-m), 3-turn loop of #14 AWG wire with a turn spacing of 1 inch (25.4 mm) is 60.15 µH. If the maximum inductance of the lead-in cable is 6 µH and the maximum lead-in cable length is 27 ft (8.23 m), then the lead-in cable inductance is 0.22 µH/ft (0.72 µH/m). The self-inductance of a 6- × 6-ft (1.83- × 1.83-m), 6-turn loop of #14 AWG wire with a turn spacing of 0.4 inches (10.2 mm) is 226 µH, which allows a lead-in cable length of 104 ft (31.7 m) when constrained by the requirement to limit the lead-in cable inductance to no more than 23 µH (i.e., 1/10 the value of the loop inductance).

If a vehicle undercarriage detection height of 4.1 ft (1.25 m) is acceptable, then the 3-turn loop with a loop-turn spacing of 1 inch (25.4 mm) can be used with 250 ft (76.2 m) of lead-in cable, as shown in Table E-4. The presence of reinforcing steel 3 inches (76.2 mm) from the top loop turn reduces the vehicle detection height to 3.6 ft (1.10 m) for the 3-turn loop.

A 5-turn loop spaced 0.5 inches (12.7 mm) between turns with a detector sensitivity of 0.098 percent has adequate vehicle detection height sensitivity for most applications. One possible method to obtain the 0.5-inch (12.7-mm) spacing is to use loop wire centered in a flexible, water-blocked, 0.5-inch (12.7-mm) height density polyethylene tube. Lead-in cables longer than 250 ft (76.2 m) require increasing the sensitivity of the electronics unit to a value greater than 0.098 percent.

REFERENCES

1. Mills, M.K. "Inductive Loop Detector Analysis," 31st IEEE VTG Conference Record. April 1981.
2. Mills, M.K. "Self Inductance Formulas for Multi-Turn Rectangular Loops used with Vehicle Detectors," 33rd IEEE VTG Conference Record. May 1983. pp 65–73.
3. Mills, M.K. op. cit., p. 70.
4. Link, Advanced Product Division. Link Inductance Loop Analysis Project, Final Report. August 1968. p. 21.
5. Dorsey, W.F. Status Report on Vehicle Detectors, Report No. FHWARD-77-137. U.S. Department of Transportation, Federal Highway Administration, Washington, DC. November 1976.
6. DeLaski, A.B., and P.S. Parsonson. Traffic Detector Handbook, Report No. FHWA-IP-81-1. U.S. Department of Transportation, Federal Highway Administration, Washington, DC. 1985. p. 113.

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