Appendix A: Microwave Breakdown in Gases A very large fraction of the work done in the field of gas discharge has been concerned with breakdown. Measurements of breakdown electric fields and voltages have been relatively reproducible and have thus provided a good method of checking the predictions of theory and establishment of Paschen’s law accordingly. Electrical breakdown or dielectric breakdown is when current flows through an electrical insulator when the voltage applied across it exceeds the breakdown voltage. This results in the insulator becoming electrically conductive. Electrical breakdown may be a momentary event (as in an electrostatic discharge) or may lead to a continuous arc if protective devices fail to interrupt the current in a power circuit. Under sufficient electrical stress, electrical breakdown can occur within solids, liquids, gases, or vacuum. However, the specific breakdown mechanisms are different for each kind of dielectric medium. A.1 Introduction The problem of microwave breakdown near antennas at high altitudes could be considered in order to find out what is the limitation on transmission conditions [1]. Electrical breakdown is often associated with the failure of solid- or liquid- insulating materials used inside high-voltage transformers or capacitors in the electricity distribution grid, usually resulting in a short circuit or a blown fuse. Electrical breakdown can also occur across the insulators that suspend overhead powerlines, within underground power cables, or lines arcing to nearby branches of trees. Airborne radar system may initiate electrical discharges in front of the antennas at high altitudes because, at ultrahigh frequencies, the electric field required to break down air at low pressures is, in general, much less than that required at atmospheric pressure. The processes which determine ultrahigh-frequency (UHF) breakdown © Springer Nature Switzerland AG 2019 397 B. Zohuri, Directed-Energy Beam Weapons, https://doi.org/10.1007/978-3-030-20794-6 398 Appendix A: Microwave Breakdown in Gases Fig. A.1 Electrical breakdown driving electric discharge have been discovered and verified during the past decade. These have been applied to determining optimum transmission conditions for high-flying radar as an example. Breakdown criteria take place when an electric field is applied to a gas, and then the free electrons move in the direction of the field, constituting a current. There is a small number of electrons present in any collection of gas because of ionization by cosmic rays or some other phenomenon, such as photoelectric effect. If the electric field is gradually increased from zero, the gas first appears to obey Ohm’s law until the field becomes large enough to impart sufficient energy to some of the electrons to produce secondary electrons by collision. If the electric field is sufficiently great so that many secondary electrons are produced, there will come a point at which the gas will become highly conducting. For a very minute change in voltage or field near this value, the electron concentration and current will change by many orders of magni- tude, and the gas will start to glow as it can be seen in Fig. A.1 [1]. Note that this criterion was defined by Townsend many years ago and a criterion, which enabled him to describe breakdown quantitatively. Although the description was designed for direct current discharges, the Townsend criterion has proved useful in considering discharges produced by electric fields of any frequency, including microwaves. In a gas discharge, electrons are being produced by ionization and being lost by diffusion, recombination, and other ways we need not consider at the moment. See Fig. A.2. Dielectric breakdown is also important in the design of integrated circuits and other solid-state electronic devices. Insulating layers in such devices are designed to Appendix A: Microwave Breakdown in Gases 399 Fig. A.2 Variation of electron concentration with electric field withstand normal operating voltages, but higher voltage such as from static electric- ity may destroy these layers, rendering a device useless. The dielectric strength of capacitors limits how much energy can be stored and the safe working voltage for the device [2]. Breakdown mechanisms differ in solids, liquids, and gasses. Breakdown is influenced by electrode material, sharp curvature of conductor material (resulting in locally intensified electric fields), size of the gap between the electrodes, and density of the material in the gap (Fig. A.1). In solid materials (such as in power cables) a long-time partial discharge typically precedes breakdown, degrading the insulators and metals nearest the voltage gap. Ultimately the partial discharge chars through a channel of carbonized material that conducts current across the gap. Electrical breakdown occurs within a gas when the dielectric strength of the gas is exceeded. Regions of intense voltage gradients can cause nearby gas to partially ionize and begin conducting. This is done deliberately in low-pressure discharges such as in fluorescent lights. The voltage that leads to electrical breakdown of a gas is approximated by Paschen’s law. See next section of this Appendix. A.2 Paschen’s Law Paschen’s law dates back to the nineteenth century, named after Friedrich Paschen, and is often referenced when in voltage breakdown calculations. For this effort you will find Friedrich Paschen in the Microwave Hall of Fame [3]. Paschen’s law is an equation that gives the breakdown voltage, that is, the voltage necessary to start a discharge or electric arc, between two electrodes in a gas as a function of pressure and gap length [4, 5]. It is named after Friedrich Paschen who discovered it empirically in 1889 [6]. 400 Appendix A: Microwave Breakdown in Gases Fig. A.3 Paschen curves [7] Paschen studied the breakdown voltage of various gases between parallel metal plates as the gas pressure and gap distance were varied: • With a constant gap length, the voltage necessary to arc across the gap decreased as the pressure was reduced and then increased gradually, exceeding its original value. • With a constant pressure, the voltage needed to cause an arc reduced as the gap size was reduced but only to a point. As the gap was reduced further, the voltage required to cause an arc began to rise and again exceeded its original value. For a given gas, the voltage is a function only of the product of the pressure and gap length [4, 5]. The curve he found of voltage versus the pressure-gap length product (right) is called Paschen’s curve, Fig. A.3. Figure A.3 is an illustration of Paschen curves obtained for helium, neon, argon, hydrogen, and nitrogen, using the expression for the breakdown voltage as a function of the parameters A and B that interpolate the first Townsend coefficient [7]. He found an equation that fit these curves, which is now called Paschen’s law, which is an empirical result as Eq. (A.1)[5]: Bpdhi V B ¼ ðEq:A:1Þ 1 ln ðÞÀApd ln ln 1 þ γ se Appendix A: Microwave Breakdown in Gases 401 In this equation, VB is the breakdown voltage in volts, p is the pressure in Pascals, d is the gap distance in meters, γse is the secondary electron emission coefficient, which is the number of secondary electrons produced per incident positive ion, A is the saturation ionization in the gas at a particular E/p (i.e., electron field/pressure), and B is related to the excitation and ionization energies. The constants A and B are determined experimentally and found to be roughly constant over a restricted range of E/p for any given gas. For example, air with an E/p in the range of 450–7500 V/(kPaÁcm), A ¼ 112.50 (kPa cm)À1 and B ¼ 2737.50 V/(kPa cm) [8]. The graph of this equation is the Paschen curve. By differentiating it with respect to and setting the derivative to zero, the minimal voltage can be found. This yields as illustrated in Eq. (A.2): 1 e Á ln 1 þ γ pd ¼ se ðA:2Þ A and predicts the occurrence of a minimal breakdown voltage for pd ¼ 7.5 Â 10À6 m. atm. This is 327 V in air at standard atmospheric pressure at a distance of 7.5 μm. The composition of the gas determines both the minimal arc voltage and the distance at which it occurs. For argon, the minimal arc voltage is 137 V at a larger 12 μm. For sulfur dioxide (SO2), the minimal arc voltage is 457 V at only 4.4 μm. Note that in the above scenario for the atmosphere condition, we first consider the nature of the atmosphere from ground level to 500,000 ft. It is now generally agreed that the composition of the atmosphere is constant up to approximately 250,000 ft [9]. In this range, nitrogen and oxygen compose 99% of the total gases and their relative amounts do not change [1]. At higher pressures and gap lengths, the breakdown voltage is approximately proportional to the product of pressure and gap length, and the term Paschen’s law is sometimes used to refer to this simpler relation [10]. However, this is only roughly true, over a limited range of the curve. A.2.1 Voltage-Current Relation Before gas breakdown, there is a nonlinear relation between voltage and current as shown in Fig. A.4. In region 1, there are free ions that can be accelerated by the field and induce a current. These will be saturated after a certain voltage and give a constant current, region 2. Regions 3 and 4 are caused by ion avalanche as explained by the Townsend discharge mechanism. As we have mentioned in the previous Appendix A.2, Friedrich Paschen established the relation between the breakdown condition and breakdown voltage.