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d -Helix wire diameter, or helix tape width. p = Helix pitch. d' = Diameter of water-cooling hole in helix. a = Stefan's constant x emissivity [conductivity of helix material in eqn. (20)]. T = Helix temperature, °K. t = Ambient temperature, ° K. / = Effective fractional length on which heat is incident. / ' = Fraction of beam power reaching helix. a = Helix attenuation coefficient, i/r = Helix pitch angle. /e) 1 ' 2 = Impedance of free space. j3 = Geometrical contact factor for support bars. 8 = Surface contact factor for support bars, e = Permittivity of support bars. k' = Thermal conductivity of support bars. c = Radius of envelope in contact with support bars. •n = Viscosity of cooling fluid. 6 = Temperature difference across helix support bars. Q = Water flow rate, cm 3 /sec. SUMMARYScaling equations are developed for broad-band helix travellingwave amplifiers and are used to study the limitations due to cathode current density, beam interception and attenuation with cooling by radiation and conduction. The latter is more efficient but must be treated with reserve at the shorter wavelengths, owing to the added attenuation. By using the form of a theoretical equation to obtain correct dimensional behaviour and an experimentally determined factor, the attenuation calculations should be more reliable than hitherto. Limiting power/wavelength relationships for the various cases all show the allowable beam power to be proportional to a high power of the wavelength. Attenuation dissipation in particular imposes a rapid decrease in power level at wavelengths below the S-band.The possibilities of fluid-cooling small helices wound from tube are then explored, a peculiarity of which is that higher-power valves at a given wavelength are favoured. The power in the beam must exceed a given level which rises as the wavelength is reduced, because the increases in helix size brought about by higher voltages, at a fixed wavelength, lower the flow impedance more rapidly than that required to cope with the increased power. Eventually, at 1-2 cm wavelength, the helix size cannot be increased without backward-wave oscillation.This method of cooling thus makes possible c.w. broad-band amplifiers with outputs of carcinotron magnitude using beam powers of the order of kilowatts at wavelengths of a few centimetres. LIST OF PRINCIPALSYMBOLS G = Gain of travel ling-wave tube, dB. B = Pierce's gain parameter. C = Pierce's impedance parameter. A = Pierce's initial loss parameter. N = Number of wavelength in helix length (L'/X). k = Free-space phase coefficient, 2TT/A. y = Helix radial-propagation coefficient. a = Mean helix radius. b = Beam radius. A = Free-space wavelength, v = Helix phase velocity. / 0 = Beam current, amp. VQ = Beam voltage, kV. u 0 = Beam velocity. c = Velocity of light. L = Helix length. V = Stretched-out length of helix. &=• Beam micro-perveance, 31 -6/ 0 /K 0 3 ' 2 . P = R.F. power output, kW. P o = Beam power, / 0 V o , kW. P d = Power dissipated in helix by attenuation. P r = Power radiated from helix. P c = Powe...