678 BELL SYSTEM TECHNICAL JOURNAL 



Upon substitution of {kl) in {ki) 



^Il=^ Tl[l - Tt\ (k3) 



which may be integrated to 

 C 



2^y,-3 [tan ' Tr + tanh ' 2%] = / + /o (k4) 



where /o is a constant of integration. Let 



h\{Tr) = [tan~'r, + tanh~'r,]. (k5^ 



This function is plotted in Fig. 79. In order to determine the cycHng time 

 for heating n assume 



At ^ = 0, Tr = TrC i.e. Trc = Tc/T^ 



t = Th, Tr — Trh Trh = Th/Tm 



where 



Tm is the value of the temperature Tg corresponding to the maximum 



power input Pm ■ 

 Th is the temperature corresponding to one band limit. 

 Tc is the temperature corresponding to the other band limit. 

 Tu > T,. 



The cycUng time for heating n, is then 



C _ 



Th = riT^rr^s [tau"' Trh " tau"' Trc + tanh"' Trh — tanh~^ Trc\ (k6) 



= 2"^ ^PiiTrh) - F,(Trc)] (k7) 



which gives the time required for the expanding element to rise in tempera- 

 ture from Trc to Trh , i-e. from Tc to Th . 



If we reduce the power input and wish to determine the cooling time the 

 analysis is similar. If the i)()wer input from electron bombardment is 

 reduced to zero there will still be i)ower input to the tuner. The residual 

 power which is kept to the minimum possible level comes from such sources 

 as heat radiated from the cathode and general heating of the envelope by 

 the oscillator section. 



Let Pa be the value of the reduced power inj)ut. 



