# Mathematical model based on current-carrying wire tracking intelligent vehicle

The current-carrying conductor is laid on the ground, and a periodic alternating current of a certain frequency and amplitude is used as a trace for guiding the traveling of the smart car. Use a large inductor as a sensor, with a certain circuit to obtain the corresponding voltage. This is a relatively new way of tracking, appearing in the 5th "Freescale" Cup National College Student Smart Car Competition. At present, there are two main methods for solving trace information. One is to check the table after rectification and filtering, so that the voltage corresponds to the distance; the other is to use a row of dense inductors as the sensor, and the inductance near the trace is high voltage, away from the trace. The display voltage is low. Since there is no accurate mathematical model to solve the trace information before the competition, this article will try to make efforts in this regard.

1. The mathematical model of sensor arrangement and track information calculation assumes that the current carrying wire is infinitely straight and the current is . It can be assumed that the inductor is an ideal inductor, that is, the space size is negligible, and the inductor core operates in the linear region  .

The magnetic induction in a certain direction of space is By Biot-Savar's Law ,among them Determined by the spatial location, . May wish to call For the space function, the different placement of the inductors corresponds to different . If the trace is an infinitely long straight wire, then the same inductor placement mode, The expression does not change.

Assume The open-circuit induced electromotive force for the inductor is obtained by Faraday's law of electromagnetic induction  : ,among them Determined by the parameters of the inductor, it is a constant.

will Performing a Laplace transform ,among them ,

will Performing a Laplace transform ,among them .

The circuit between the inductor and the A/D conversion is designed as a linear time-invariant system, and its transfer function is set. ,then

Measured before the car started â€¢â€¢â€¢â€¢â€¢â€¢(1)

Measured after the car started â€¢â€¢â€¢â€¢â€¢â€¢(2)

Equation (2) divided by (1) It is found that the transfer function and the current, such as the circuit, which are independent of the space function, are eliminated. For different inductance arrangement methods, as long as the space function is obtained first, then the corresponding trace information solution expression can be obtained by using this method. This is exactly what this article is going to do. For convenience, this method is referred to as "Fa Yi."

This method has several advantages: First, the expression has nothing to do with the circuit transfer function, that is, it is independent of the circuit, which makes the circuit design theoretically extremely simple, only needs to satisfy the linear time constant; second, the expression and the track The current is irrelevant, which makes the model derived by this method theoretically highly adaptable; the third is that the model derived by this method is simple and efficient to use.

1.1 Double one inductance ranging As shown in Figure 1, the two inductors A and B are placed horizontally, and the axes are parallel to the x-axis. Height from the ground . Remember . The method of  will be used below to avoid opening the root number.

The variable of the A inductor is indicated by the subscript A, and the variable of the B inductor is indicated by the subscript B. In order to facilitate the use of Act One, it may be .

Obviously , easy ,

Then ,among them Assume Smaller, can be ignored. then â€¢â€¢â€¢â€¢â€¢â€¢(3)

Same principle ,among them â€¢â€¢â€¢â€¢â€¢â€¢(4) â€¢â€¢â€¢â€¢â€¢â€¢(5)

Lianli (3), (4), (5), solved. note It is positive or negative.

The experimental data is shown in Table 1. All units are standard points, the voltage is the peak-to-peak value read by the oscilloscope, and the error is the relative error of not taking the absolute value. Conclusion: Except for the second group, the errors of other groups are less than 10%, most of them are less than 5%, and have no obvious relationship with the track. Explain that the mathematical model is correct, and it also proves the correctness of the law 1.1.

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