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Publication Number: FHWAHRT06139
Date: October 2006 
This appendix describes the derivation and application of formulas to calculate the vehicle detection sensitivity of singleturn and multiturn rectangular loops of coaxial wire. The effect on detection sensitivity of reinforcing steel mesh in the pavement and leadin cable inductance is accounted for in the models. The loop is installed in the roadway pavement and connected to a roadside electronics unit by leadin 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, singleturn, 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.
The inductiveloop 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 3turn, 6 x 6ft (1.83 x 1.83m) loop of #14 AWG wire embedded in the pavement, a leadin cable to roadside, and roadside electronics unit. The vehicledetection sensitivity is the normalized inductance change at the terminals of the electronics unit when a vehicle is sensed by the loop. Highgroundclearance 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 vehicledetection sensitivity of an inductiveloop detection system is maximized when the spacing between loop turns is correctly selected. The increased sensitivity should improve detection of highgroundclearance trucks.
This appendix is based on a previous paper that developed a detectionsensitivity formula for a singleturn 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 selfinductance of the singleturn loop and the selfinductance of the shortedturn 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 leadin cable. A formula for the selfinductance 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 selfinductance to decrease, which results in increased detection sensitivity until the decrease in mutual coupling to the shorted turn becomes significant.
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.
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 E1 illustrates the geometry of the vehicle and reinforcing steel shorted turn model for a twoturn 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 E1. Twoturn loop in roadway.
Figure E2 shows the air core transformer model of a oneturn 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 E2. Singleturn inductiveloop circuit model.
The circuit equations corresponding to Figure E2 are:
Z is impedance in ohms. I is current in amperes.Equations E2 and E3 are set equal to 0 since there is no external driving voltage in the two secondary windings in Figure E2 that contain L_{22} and L_{33}. 
(E1) 
(E2) 
(E3) 
where


All mutual impedances are symmetrical (i.e., Z_{12} = Z_{21}, Z_{13} = Z_{31}, Z23 = Z_{32}).
Experimental measurements show that the quality factor ωL_{11} /R_{11} of the loop driving point impedance Z_{11} is ≥ 10.
Assuming small circuit loss to simplify calculations gives
(E9) 
(E10) 
(E11) 
The loop drivingpoint impedance Z_{1} from Equations E1, E2, and E3 is
(E12) 
If the reinforcing mesh is spaced a great distance from the inductive loop,
Then
(E13) 
In this case, the loop drivingpoint impedance Z_{1} depends only on the effect of the vehicle.
The sensitivity S_{L} of an inductive loop is defined as
NV represents the value of the variable when no vehicle is present.V represents the value of the variable when a vehicle is present. 
(E14) 
The change in loop drivingpoint impedance ΔZ_{1} is given by
(E15) 
where ΔZ_{1}^{NV} is the loop drivingpoint impedance with no vehicle present and ΔZ_{1}^{V} is the loop drivingpoint impedance when sensing a vehicle.
The loop drivingpoint impedance Z_{1} can also be expressed as
(E16) 
Then, if no vehicle is present,
(E17) 
(E18) 
(E19) 
The loop sensitivity S_{L} is given by
(E20) 
if
(E21) 
Applying Equations E9, E10, and E11 to Equation E20 yields the loop sensitivity S_{L} in percent as
(E22) 
where  
S_{L}  = inductive loop sensitivity (percent)  
M_{12}  = mutual inductance between inductive loop and loopsimulating vehicle (H)  
M_{13}  = mutual inductance between inductive loop and loopsimulating reinforcing mesh (H)  
M_{23}  = mutual inductance between loopsimulating vehicle and loopsimulating reinforcing mesh (H)  
L_{11}  = selfinductance of inductive loop (H)  
L_{22}  = selfinductance of loopsimulating vehicle (H)  
L_{33}  = selfinductance of loopsimulating reinforcing steel mesh (H). 
When reinforcing steel mesh is not present,
(E23) 
Figure E3 shows the air core transformer model of a twoturn inductive loop, vehicle undercarriage, and reinforcing steel mesh. Mutual inductance terms that model the effects of the multipleturn loop have been added to the model.
Figure E3. Twoturn inductiveloop circuit model.
From Figure E3, the circuit equations for the twoturn loop are given by
(E24) 
(E25) 
(E26) 
where  
Z_{12}^{11} = mutual impedance between primary turn 1 of the transformer and secondary shorted turn 2, which models the vehicle,  
Z_{12}^{22} = mutual impedance between primary turn 2 of the transformer and secondary shorted turn 2, which models the vehicle,  
Z_{11}^{12} = mutual impedance between primary turn 1 and primary turn 2. 
Then
(E27) 
(E28) 
(E29) 
(E30) 
(E31) 
(E32) 
(E33) 
(E34) 
(E35) 
As compared to the oneturn loop model, the selfimpedance Z_{11} is replaced by the total twoturn loop impedance. Each mutual impedance is replaced by the sum of the mutual impedances from each turn.
Circuit equations for a multiturn loop are given by
(E36) 
(E37) 
(E38) 
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
(E39) 
(E40) 
(E41) 
(E42) 
(E43) 
Equations E39–E43 are generalized for the multiturn case by letting
(E44) 
(E45) 
(E46) 
where L^{*}_{N} is the total selfinductance of the loop.
Detection sensitivity is expressed as
(E47) 
For a twoturn (N = 2) loop,
(E48) 
The formula for multiturn loopsensitivity 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 loopsystem inductance caused by vehicle passage or presence is given by
(E49) 
Equation E47 can be applied to a multiturn rectangular loop with N coaxial, equalspaced, identical turns. The lowfrequency inductance L_{N} for such a loop is given by^{(3)}
(E50) 
or
(E51) 
where L_{11} is the selfinductance 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 twoturn loop,
(E52) 
The total selfinductance L*_{N} of the loop is equal to the sum of the low–frequency inductance L_{N} and the highfrequency, skineffect inductance L_{H} and is expressed as
(E53) 
where
(E54) 
l_{1} and l_{2} are turn width (m) and turn length (m), respectively, and the internal inductance L^{i} is 0.036 µH/m (0.111 µH/ft) at 47 kHz.
The series inductance L_{C} of the transmission line connecting the loop in the roadway to the electronics unit reduces the inductiveloop system sensitivity S_{L} at the electronics unit to
(E55) 
Equations E47, E51, E53, and E55 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 lowfrequency inductance of a coaxial, multiturn, rectangular loop of round wire. This lowfrequency inductance subroutine calls a subroutine, which calculates the mutual inductance between two coaxial, singleturn, rectangular loops. The singleturn, mutual inductance subroutine calls a subroutine, which calculates the mutual inductance between two parallel current elements.
The only known measured loopdetector sensitivity data^{(4)} versus number of loop conductor turns are presented in Table E1. These data were obtained by driving a compact automobile over a 6 x 6ft (1.83 x 1.83m) 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.
Since the average undercarriage height of the compact automobile used was unknown, the automobile was modeled with a 6 x 6ft (1.83 x 1.83m) shorted turn coaxially located 0.71 ft (216 mm) over the loop of #14 AWG wire. A loopwire spacing of 150 mils (38 mm) and leadin cable inductance of 0.22 µH/ft (0.72 µH/m) were assumed.
Table E1 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 inductiveloop design parameters on loop sensitivity as described below.
Loop turns  Measured loop detector sensitivity** (%)  Calculated loop detector sensitivity (%)  Difference (%) 

1  3.25  3.72  14.47 
2  4.75  4.75  0 
3*  5.20  5.20  0 
4  5.50  5.47  0.55 
5  5.60  5.68  1.43 
6  5.65  5.83  3.19 
8  5.75  6.07  5.57 
10  6.05  6.25  3.31 
*  The height of 0.71 ft (0.22 m) was selected so that the difference equaled zero. 
**  Data measured at f = 50 kHz. 
Vehicledetection height slowly increases when the turn spacing exceeds 1 inch (25 mm). 
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 toptobottom turn spacing is used. A loop slot depth of 13/8 inches (35 mm) is recommended in the Traffic Detector Handbook for a twoturn loop. A nominal sensitivity of 0.098 percent was assumed for the detection threshold.
A leadin cable length of zero was used so that the effect of loop turns on sensitivity would be independent of leadin 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 loopdetector system sensitivity computer program are presented in Table E2 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 × 6ft (1.83 × 1.83m) by 2000mils (50.8mm) loop is 6 ft^{3} (0.171 m^{3}), as illustrated by the results in Table E2.
Vehicledetection height slowly increases when the turn spacing exceeds 1 inch (25 mm). 
Since Table E2 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 threeturn loop was calculated as shown in Table E3. The vehicle undercarriage detection height slowly increases when the turn spacing exceeds 1 inch (25 mm). The volume of the 6 × 6ft (1.83 × 1.83m) by 300mils (7.6mm) loop is 0.90 ft^{3} (0.25 mm^{3}) with a detection height of 4.7 ft (1.43 m). Increasing the loop volume by a factor of 10 results in a 0.3ft (9.1cm) increase in vehicle detection height.
Loop turns  Loop turn spacing (mils)  Loop turn spacing (mm)  Vehicle detection height for 5 × 5ft (1.52 × 1.52m) loop (ft)  Vehicle detection height for 5 × 5ft (1.52 × 1.52m) loop (m)  Loop inductance for 6 × 6ft (1.83 × 1.83m) loop (µh)  Vehicle detection height for 6 × 6ft (1.83 × 1.83m) loop (ft)  Vehicle detection height for 6 × 6ft (1.83 × 1.83m) loop (m) 

1  NA  NA  3.8  1.16  10.42  4.5  1.37 
2  2,000  50.8  4.1  1.25  29.15  4.8  1.46 
3  1,000  25.4  4.2  1.28  60.15  5.0  1.52 
4  660  16.8  4.3  1.31  103.47  5.0  1.52 
5  500  12.7  4.3  1.31  158.72  5.0  1.52 
6  400  10.2  4.3  1.31  226.48  5.0  1.52 
7  330  8.4  4.3  1.31  307.23  5.0  1.52 
8  286  7.3  4.3  1.31  399.10  5.0  1.52 
Conductor type:  #14 AWG Cable 
Leadin cable length:  0 ft 
Mesh spacing:  1,000 ft (305 m) 
Detection sensitivity threshold:  0.098 percent 
Loop turn spacing (mils)  Vehicle detection height for 6 × 6ft loop (ft)  Loop turn spacing (mm)  Vehicle detection height for 1.83 × 1.83m loop (m) 

150  4.7  3.8  1.43 
300  4.8  7.6  1.46 
450  4.8  11.4  1.46 
600  4.9  15.2  1.49 
750  4.9  19.1  1.49 
900  4.9  22.9  1.49 
1,050  5.0  26.7  1.52 
1,200  5.0  30.5  1.52 
1,350  5.0  34.3  1.52 
1,500  5.0  38.1  1.52 
Conductor type:  #14 AWG cable 
Leadin cable length:  0 
Mesh spacing:  1,000 ft (305m) 
Detection sensitivity threshold:  0.098 percent 
Leadin cable inductance reduces loop sensitivity and hence vehicle detection height. 
Leadin 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 leadin cable inductance. Tables E4 and E5 illustrate this effect.
Reinforcing steel mesh reduces loop sensitivity and hence vehicle detection height. 
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 E4 and E5.
Combined effects of leadin cable inductance and steel mesh accentuate the reduction of loop sensitivity and hence vehicle detection height. 
The combined effects of leadin cable inductance and steel mesh accentuate the reduction of loop sensitivity and hence vehicle undercarriage detection height. Tables E4 and E5 show that the number of loop turns should be five or more for detection of 4fthigh (1.22mhigh) groundclearance trucks when leadin cable and steel mesh are present.
Loop turns  Loop turn spacing (mils)  Vehicle detection height with leadin cable (ft)  Vehicle detection height with mesh (ft)  Vehicle detection height with leadin cable and mesh (ft) 

1  NA  2.3  3.9  1.9 
2  2,000  3.5  4.3  3.0 
3  1,000  4.1  4.4  3.6 
4  660  4.4  4.4  3.9 
5  500  4.6  4.5  4.1 
6  400  4.7  4.5  4.2 
7  330  4.8  4.5  4.3 
8  286  4.9  4.5  4.3 
Conductor type:  #14 AWG Cable 
Leadin cable length:  250 ft 
Mesh spacing:  3,000 mils 
Detection sensitivity threshold:  0.098 percent 
Loop turns  Loop turn spacing (mm)  Vehicle detection height with leadin cable (m)  Vehicle detection height with mesh (m)  Vehicle detection height with leadin cable and mesh (m) 

1  NA  0.70  1.19  0.58 
2  50.8  0.94  1.31  0.91 
3  25.4  1.25  1.34  1.10 
4  16.8  1.34  1.34  1.19 
5  12.7  1.40  1.37  1.25 
6  10.2  1.43  1.37  1.28 
7  8.4  1.46  1.37  1.31 
8  7.3  1.49  1.37  1.31 
Conductor type:  #14 AWG Cable 
Leadin cable length:  76.2 m 
Mesh spacing:  72.6 mm 
Detection sensitivity threshold:  0.098 percent 
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 × 6ft (1.83 × 1.83m), 3turn 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 leadin cable. For example, the selfinductance of a 6 × 6ft (1.83 × 1.83m), 3turn 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 leadin cable is 6 µH and the maximum leadin cable length is 27 ft (8.23 m), then the leadin cable inductance is 0.22 µH/ft (0.72 µH/m). The selfinductance of a 6 × 6ft (1.83 × 1.83m), 6turn loop of #14 AWG wire with a turn spacing of 0.4 inches (10.2 mm) is 226 µH, which allows a leadin cable length of 104 ft (31.7 m) when constrained by the requirement to limit the leadin 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 3turn loop with a loopturn spacing of 1 inch (25.4 mm) can be used with 250 ft (76.2 m) of leadin cable, as shown in Table E4. 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 3turn loop.
A 5turn 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 0.5inch (12.7mm) spacing is to use loop wire centered in a flexible, waterblocked, 0.5inch (12.7mm) height density polyethylene tube. Leadin cables longer than 250 ft (76.2 m) require increasing the sensitivity of the electronics unit to a value greater than 0.098 percent.
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FHWAHRT06139 
Keywords: Traffic detectors, sensors, detector installation, detector maintenance, signalized intersections, intersection safety, intersection treatments, infrared sensor, inductive loop, magnetometer, video image processor, microwave radar sensor, laser radar sensor, acoustic sensor, ultrasonic sensor, magnetic sensor Updated: 03/08/2016
