Closed-Form Field Solution for Finite-Length Harmonic Linear Current Source
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Solution Overview
Problem
Current methods for geophysical electromagnetic exploration fail to provide a closed-form exact solution for the field generated by a finite-length harmonic linear current source, especially in the near zone, due to the inability to express the exact solution in a closed form using integral expressions, which limits their applicability and accuracy.
Innovation Solution
A method is developed to obtain a closed-form exact solution for the field generated by a finite-length harmonic linear current source using a cosinoidal expression for the uniform current distribution, allowing for elementary quadrature of the integral formula and expressing the field in a form suitable for all zones, including the near zone, by processing the current with a cosine function and solving the integral to obtain the vector magnetic potential and magnetic field intensity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If electric dipole approximation is adopted (point source assumption), then closed form solution can be obtained, but accuracy deteriorates in near zone
Solution Approach 1:
The patent changes the current distribution parameter from uniform (dipole approximation) to cosinoidal distribution, which allows the integral to be solved in closed form while accurately representing the physical current distribution. The cosinoidal current distribution I(z') = I0 cos(πz'/l) matches the actual current behavior in finite-length antennas, resolving the contradiction between mathematical tractability and physical accuracy.
2Device complexity
If uniform current distribution is assumed, then integral expression can be simplified, but closed form quadrature cannot be realized for finite-length source
Solution Approach 1:
The patent transforms the current distribution function from uniform to cosinoidal, which introduces a specific functional form that makes the integral solvable in closed form. The cosinoidal distribution I(z') = I0 cos(πz'/l) is chosen because it naturally satisfies the boundary conditions (current is maximum at center and zero at endpoints) while allowing analytical integration, thus resolving the contradiction between simplification and exactness.
3Manufacturing precision
If sinusoidal distribution with terminal current set to 0 is used, then closed evaluation can be carried out, but expression is inconsistent with actual current distribution
Solution Approach 1:
Instead of using the conventional sinusoidal distribution that forces zero current at terminals, the patent inverts the approach by using cosinoidal distribution with maximum current at the center and natural zero at endpoints. This reversal of the standard assumption (from sine to cosine) provides both closed-form solvability and physical accuracy, as the cosinoidal form I(z') = I0 cos(πz'/l) correctly represents the current standing wave pattern in a resonant antenna.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
The solution provides a more accurate representation of the harmonic current distribution along a line, enhancing the applicability across different frequency changes and conductivity variations, and intuitively reveals the change rules of the electromagnetic field, aiding in theoretical studies on antennas and radio wave propagation.
Implementation Method 1
listing integral formula for vector magnetic potential containing a source point position vector of a finite-length harmonic linear current source
Data Source
AI summary
A method of solving closed form exact solution for the field generated by a finite-length harmonic linear current source in a whole space. The vector magnetic potential formula of the finite-length harmonic linear current source containing a source point position vector is listed, the uniform current is subjected to cosine processing, and the current in the vector magnetic potential formula of the harmonic linear current source is expressed by a cosine function. The vector magnetic potential formula can be subjected to quadrature by an elementary function to obtain the closed form exact solution for the field generated by the finite-length harmonic linear current source in whole space. The cosine expression of the linear current source can better reflect the fundamental attributes of the electric dipole and harmonic current of linear current source in the conductive whole space. The obtained closed form exact solution is applicable in the all zone.