ULTRA-HIGH-FREQUENCY OSCILLATIONS 283 



stream. Reckoned per square centimeter of area this may be written,** 



rp = • ^Q, ° [2(1 - cos d) - d sin 0] ohms for cm.^, (2) 



where /o is the direct current density in amperes per square centimeter 

 flowing to the anode and 6 is the electron transit angle, given by 



Ax 

 d= — p^ radians, (3) 



xVFo 



Here Vo is the constant potential difference in volts between the 

 cathode and anode and A is a numerical factor which depends upon 

 the amount of space charge within the electron discharge, being equal 

 to 6300 for negligible space charge and to 9500 for complete space 

 charge with intermediate values for intermediate space charge. As an 

 alternative the resistance (2) may be written 



12 ro 



rp = - 



[2(1 - cos d) - d sin 0] ohms for cm.^, (4) 



where ro is the low-frequency series resistance of the device. With 

 space charge, ro is the slope of the static characteristic derived from 

 Child's equation 



, 2.33 Vo'" , 2 ,,. 



""Toe" — 5— amperes/cm.'^. (5) 



More generally ro is given by the expression 



1 48 re* 

 To = -TTT^ /oyi^^ ohms for cm.^, (6) 



where A is the same as was defined under (3). 



Figure 2 shows a graph of the electron stream resistance as a function 

 of transit angle and is repeated from previous papers.^ However, it 

 may not have been emphasized in the literature that the graph as well 

 as equations (2) and (4) apply not only with complete space charge 

 but with intermediate values when interpreted correctly, namely in 

 terms of the d-c. current density /o rather than in terms of the applied 

 potentials. 



TT 



Whenever the transit angle is equal to 2Trn + - where w is 1, 2, 3, 



* Equation 41 in this reference applies where the initial velocities are very small. 

 With complete space charge q = J and Oo = whereas without space charge g = 0. 

 Either condition gives the same series resistance in terms of Iq. 



