Electrodynamic Tether Fundamentals
The choice of the metal conductor to be used in an electrodynamic tether is determined by a variety of factors. Primary factors usually include high electrical conductivity, and low density. Secondary factors, depending on the application, include cost, strength, and melting point.
A motional electromotive force (EMF) is generated across a tether element as it moves relative to a magnetic field. The force is given by Faraday's Law of Induction:
Without loss of generality, it is assumed the tether system is in Earth orbit and it moves relative to Earth’s magnetic field. Similarly, if current flows in the tether element, a force can be generated in accordance with the Lorentz Force Equation
In self-powered mode (de-orbit mode), this EMF can be used by the tether system to drive the current through the tether and other electrical loads (e.g. resistors, batteries), emit electrons at the emitting end, or collect electrons at the opposite. In boost mode, on-board power supplies must overcome this motional EMF to drive current in the opposite direction, thus creating a force in the opposite direction, as seen in below figure, and boosting the system.
Take, for example, the NASA Propulsive Small Expendable Deployer System (ProSEDS) mission as seen in above figure. At 300-km altitude, the Earth’s magnetic field, in the north-south direction, is approximately 0.18 – 0.32 Gauss up to ~40º inclination, and the orbital velocity with respect to the local plasma is about 7500 m/s. This results in a Vemf range of 35 – 250 V/km along the 5-km length of tether. This EMF dictates the potential difference across the bare tether which controls where electrons are collected and / or repelled. Here, the ProSEDS de-boost tether system is configured to enable electron collection to the positively biased higher altitude section of the bare tether, and returned to the ionosphere at the lower altitude end. This flow of electrons through the length of the tether in the presence of the Earth’s magnetic field creates a force that produces a drag thrust that helps de-orbit the system, as given by the above equation. The boost mode is similar to the de-orbit mode, except for the fact that a High Voltage Power Supply (HVPS) is also inserted in series with the tether system between the tether and the higher positive potential end. The power supply voltage must be greater than the EMF and the polar opposite. This drives the current in the opposite direction, which in turn causes the higher altitude end to be negatively charged, while the lower altitude end is positively charged(Assuming a standard east to west orbit around Earth).
To further emphasize the de-boosting phenomenon, a schematic sketch of a bare tether system with no insulation (all bare) can be seen in below figure.
The top of the diagram, point A, represents the electron collection end. The bottom of the tether, point C, is the electron emission end. Similarly, and represent the potential difference from their respective tether ends to the plasma, and is the potential anywhere along the tether with respect to the plasma. Finally, point B is the point at which the potential of the tether is equal to the plasma. The location of point B will vary depending on the equilibrium state of the tether, which is determined by the solution of Kirchoff’s Voltage Law (KVL)
and Kirchoff’s Current Law (KCL)
along the tether. Here, and describe the current gain from point A to B, the current lost from point B to C, and the current lost at point C, respectively.
Since the current is continuously changing along the bare length of the tether, the potential loss due to the resistive nature of the wire is represented as . Along an infinitesimal section of tether, the resistance multiplied by the current traveling across that section is the resistive potential loss.
After evaluating KVL & KCL for the system, the results will yield a current and potential profile along the tether, as seen in above figure. This diagram shows that, from point A of the tether down to point B, there is a positive potential bias, which increases the collected current. Below that point, the becomes negative and the collection of ion current begins. Since it takes a much greater potential difference to collect an equivalent amount of ion current (for a given area), the total current in the tether is reduced by a smaller amount. Then, at point C, the remaining current in the system is drawn through the resistive load, and emitted from an electron emissive device, and finally across the plasma sheath . The KVL voltage loop is then closed in the ionosphere where the potential difference is effectively zero.
Due to the nature of the bare EDTs, it is often not optional to have the entire tether bare. In order to maximize the thrusting capability of the system a significant portion of the bare tether should be insulated. This insulation amount depends on a number of effects, some of which are plasma density, the tether length and width, the orbiting velocity, and the Earth’s magnetic flux density.
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