Physicists Pin Down the Hidden Voltage Source Behind Geomagnetically Induced Currents

Geomagnetically induced currents, the quasi-direct currents that creep through long, Earth-grounded conductors such as high-voltage power transmission networks, oil and gas pipelines, telecommunication cables and railway systems, have long been calculated with a standard circuit method whose underlying voltage source was never fully examined from first principles. A new theoretical study published in the journal Earth, Planets and Space now reinterprets where that driving voltage actually resides, generalizes the widely used Lehtinen–Pirjola formulation to include electrostatic ground potentials, and provides a quantitative justification for a practice that GIC modelers have adopted almost without question: using the electric field measured at Earth’s surface as a stand-in for the field at the altitude of an overhead transmission line.

The study, authored by Yusuke Ebihara, addresses four fundamental questions that have lingered around the standard calculation method. The Lehtinen–Pirjola formulation, introduced in 1985, assigns an open-circuit electromotive force to each segment of a transmission line and has been applied extensively to real power grids across the world. Alternative approaches, including the Nodal Admittance Matrix method, the Modified Lehtinen–Pirjola method and the more recent Graph-GIC method, have progressively refined the network equations and improved computational efficiency. Yet, as the paper notes, the physical origin of the driving source has rarely been examined from first principles. The new work tackles that gap by asking what the voltage source physically is, how electrostatic potentials generated by inhomogeneous ground resistivity should enter the formulation, what a high-impedance voltmeter connected between earthing electrodes actually measures, and whether the surface electric field is a valid proxy for the field at transmission-line altitude.

The answer to the first question comes in two complementary forms. The induced electric field that drives GICs can be explained through both the integral and the differential forms of Faraday’s law, but the differential form provides the clearer physical picture. In that view, the electric field responsible for GICs results from the superposition of a downward-propagating electromagnetic disturbance of external origin and the field reflected from the conducting Earth. The primary source, the paper concludes, is the non-conservative induced electric field distributed along the overhead transmission line itself, together with electrostatic potential differences that arise from inhomogeneous ground resistivity. These two contributions have different physical origins and must be distinguished in any first-principles interpretation.

To reach that conclusion, the study first dissects four candidate circuit representations of the driving mechanism, adapted from a 2017 analysis by Boteler and Pirjola. The first treats GICs as currents driven by a quasi-static potential difference between earthing electrodes, an approach that is intuitive for circuit engineers because ordinary DC networks are solved in terms of nodal potentials. But under uniform ground conditions the relevant electric field is purely induced and non-conservative, meaning no scalar potential exists whose gradient represents it, so the potential difference invoked in this view is not a well-defined physical quantity. The second representation invokes Faraday’s law around a loop composed of the transmission line and a return path along Earth’s surface. Because typical transmission lines sit only tens of meters high, the area enclosed by this loop is so small that the time-varying magnetic flux threading it produces an electromotive force far too small to account for observed GIC magnitudes. The third view places the induced electric field at the ground surface as the sole driving source, which is convenient in practice but mislocates the driving source, since the current actually flows along the overhead line while the driving field is assigned to the ground below it. The fourth view, the one adopted by the Lehtinen–Pirjola formulation, treats each transmission-line segment as a distributed voltage source along the conductor, an interpretation the new study identifies as the fundamental one.

The thin line-surface loop, however, is not simply wrong; it is consistent with Faraday’s law but measures the wrong quantity. The paper reevaluates the loop interpretation quantitatively using a rectangular loop consisting of a transmission line of length 200 kilometers at a height of 100 meters, with two vertical legs connecting the line to the surface. For a wave period of 60 seconds over ground with a resistivity of 10 to the third ohm-meters, the loop integral amounts to only about 2.3 millivolts, roughly one thousandth of the 2.0-volt line integral along the transmission line itself. The loop integral represents only the small residual between the induced field at line altitude and that at the ground surface, not the driving voltage. Boteler and Pirjola had reached a similar conclusion for extreme geomagnetic disturbances, obtaining a thin-loop EMF of 0.25 volts for a 200-kilometer line at 30 meters altitude under a 2000-nanotesla magnetic perturbation with a five-minute period. Applying the new formulation under the same conditions, the study finds that the line integral along the transmission line reaches 1.63 kilovolts, sufficient to drive GICs, even though the thin-loop EMF remains tiny. The voltage that actually drives the currents is the line integral of the induced field along the transmission line, precisely the source term used in the Lehtinen–Pirjola formulation.

The second major contribution is a generalization of the Lehtinen–Pirjola formulation itself. Inhomogeneities in Earth’s resistivity structure cause localized charge accumulation, generating a secondary electrostatic field, the same mechanism responsible for galvanic distortion in the magnetotelluric method. The new derivation introduces externally imposed electrostatic potentials on the ground side of the earthing line. Mathematically, the generalization replaces the conventional induced source-current vector with a corrected vector that subtracts the product of the network admittance matrix and the ground electrostatic potential vector, while the network response matrix remains unchanged. The paper demonstrates the method on a four-node network previously considered by Boteler and Pirjola, with transmission-line resistances of 5 ohms and earthing resistances of 0.5 ohms, and reproduces the earthing and transmission-line currents exactly. A key practical result follows: when a measured or modeled surface electric field already includes both the induced and electrostatic components, its line integral can be used directly as the effective open-circuit voltage in the formulation, and in that case the ground-side electrostatic correction must not be added separately.

The third question concerns what a voltmeter actually reads. In geophysical practice, the electric field is commonly measured with a high-impedance voltmeter connected between two earthing electrodes, yet the physical quantity being measured had not been clearly established. The new analysis shows theoretically that such a voltmeter measures the electrostatic ground potential difference between the electrodes as well as the line integral of the induced electric field along the lead-wire path. Because the input impedance of the voltmeter is high, the current through the leads is approximately zero, so the electrostatic field along each lead nearly cancels the induced field there. The measurement is therefore path-dependent when an induced field is present, and the result generalizes an earlier 1982 analysis by including the electrostatic ground-potential difference arising from inhomogeneous ground resistivity. The analysis assumes an idealized voltmeter with infinite input impedance and neglects instrumental effects such as electrode contact resistance, lead-wire coupling and the response of the recording system.

The fourth question, the validity of the surface-field proxy, is answered with an analytical evaluation for a plane wave normally incident on a laterally uniform ground. The difference between the induced electric field at typical transmission-line altitudes and that at the ground surface is governed by the ratio of the line altitude to the skin depth, the depth over which electromagnetic fields decay within a conducting medium. For a normally incident plane wave with periods longer than 1 second over a uniform ground with resistivity greater than 10 to the second ohm-meters, the induced electric field at typical transmission-line altitudes differs from that at the ground surface by a few percent or less. The deviation grows at shorter periods and lower resistivities: at a period of 0.1 seconds, the departure from unity exceeds several percent for resistivities at or below 10 to the second ohm-meters at 100 meters altitude, marking the breakdown of the approximation. Notably, the study emphasizes that this result cannot be explained simply by the long wavelength in air, which exceeds roughly 3 times 10 to the fifth kilometers for periods longer than 1 second, being much larger than the line altitude; the skin depth is the controlling parameter.

To visualize the physics behind these results, the study employed one-dimensional finite-difference time-domain simulations of a simplified two-layer domain, with air above the surface and ground below it, using ground resistivities of 10 to the third ohm-meters and 1 ohm-meter to represent contrasting resistive and conductive conditions. A horizontally uniform current source with a period of 60 seconds was imposed at an altitude of 100 kilometers to excite downward-propagating plane waves. In the high-resistivity case, the simulated magnetic field, electric field and current density penetrate deeply into the ground and decay gradually with depth, with a skin depth of approximately 123.3 kilometers. In the low-resistivity case, the fields are strongly attenuated near the surface and confined to a shallow subsurface region, with a skin depth of about 3.9 kilometers. The simulations also reveal a polarity reversal at a certain depth, arising because the phase velocity is strongly reduced in the conducting Earth, so different phases of the wave appear at different depths at the same instant. For a 60-second period, the phase velocity is approximately 13 kilometers per second in the resistive ground and 0.4 kilometers per second in the conductive ground, with the travel time to the skin depth being 9.6 seconds regardless of resistivity.

The practical stakes are considerable. Within power transmission networks, GICs flowing through transformer neutrals can cause half-cycle saturation, elevate reactive power consumption and generate harmonics; in severe cases these currents can damage equipment or trigger widespread power system disturbances, making accurate representation of the voltage sources applied to network branches essential for mitigation and forecasting. The new formulation gives grid modelers a physically grounded way to incorporate ground electrostatic potentials, confirms that surface-field measurements can legitimately serve as the effective open-circuit voltage when they contain both field components, and quantifies when that shortcut fails. The results rest on the assumption of a one-dimensional plane wave over a uniform ground; the electrostatic contribution from lateral inhomogeneity is handled separately through the generalized formulation. The FDTD simulation data supporting the study are stored in the Zenodo repository, and the work stands as a rare example of a foundational assumption in an operational field being rederived, tested and vindicated from first principles.

Subject of Research: First-principles reinterpretation of the voltage source driving geomagnetically induced currents and a generalized Lehtinen–Pirjola formulation incorporating ground electrostatic potentials

Article Title: Where is the voltage source of geomagnetically induced currents? A first-principles reinterpretation and generalized Lehtinen–Pirjola formulation with ground electrostatic potentials

Article References: Ebihara, Y. (2026). Where is the voltage source of geomagnetically induced currents? A first-principles reinterpretation and generalized Lehtinen–Pirjola formulation with ground electrostatic potentials. Earth, Planets and Space, 78(1), Article 217. https://doi.org/10.1186/s40623-026-02536-2

Image Credits: AI Generated

DOI: 10.1186/s40623-026-02536-2

Keywords: geomagnetically induced currents, Lehtinen–Pirjola formulation, Faraday's law, electrostatic ground potentials, skin depth, transmission lines, space weather, geoelectric field, FDTD simulation, power grids, earthing electrodes, magnetotellurics