63d BELL SYSTEM TECHNICAL JOL'RNAL 



As we have already assumed r is small, we may as well write 



8V = {V[a/2w)2.3 logm (j^^ (b35) 



This emphasizes the sign of 6 \' . 



According to (b28), the grid [)lane appears from a distance to be at zero 

 potential. Thus, 



Vui - 28V = V (b36) 



and from (h.^2) 



f = il + (a/w (I) 2 J logu, (a/27rr)}"' (b37) 



If we go back over our results, we have for lined-up singly-wound grids, 

 from (b31), (b34) and (b37), the average modulation coefficient 



^„ = /Isin {ya/2)/iya/2)]G(ya,n) (b38) 



The quantity sin (ya/2) can be obtained from P'ig. 119, G(ya,n) is plotted 

 in Fig. 125, and/ can be calculated from (b37) above. 



It must be emphasized again that these expressions are good only for 

 very line wires (;- « a), and get worse the closer the spacing compared 

 with the w'ire separation. It is also important to note that G{ya,n) indi- 

 cates little reduction of (3„ even for quite wide wire separation. Now yd 

 will be less than 27r, as /3„ = at 7^ = 27r. As a approaches d in magnitude, 

 the assumptions underlying the analysis, in which the integration around 

 each grid was carried from — ac to 00, become invalid and the analysis is 

 not to be trusted. 



It is very important to bear one point in mind. If we design a resonator 

 assuming parallel conducting planes a distance L apart at the gap, and then 

 desire to replace these planes with grids without altering the resonant fre- 

 quency, we should space the grids not L apart but 



d - t'L (hm 



apart to get the same capacitance and hence the same resonant frequency. 

 Mesh grids are sometimes used. To get a rough idea of what is expected, 

 we may assume the potential about a grid to be 



V = V[x/2 + (aT'(/87r) In \2 (^cosh — - cos — 'M 

 + (aT'(/87r) In [2 (cosh ~''--' - cos """M 



(1)40) 



