POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 327 



where the sine or cosine is used according as the number of oscillation 

 constants is odd or even. 



There are two sources of error in the finite representation (29) : The first 

 is due to the finite extent of the charged plates, and may be called the 

 "truncation error." Its effect will be important only near the ends of the 

 plates, which explains why it is advisable to prolong the charges beyond 

 the upper frequency bound coo . Its magnitude is exactly determined by 

 integrating the effect of uniform charge density, of magnitude 1/&, over 

 the region beyond the finite plates: 



d^ 2x [2^-1 « .2,-1 a 1 

 - -/i = — - - tan — — - + - tan , 



(31) 



where ±coe are the real frequencies at the ends of the plates. The bracketed 

 expression represents the non-constant part of the phase delay, due to the 

 finite extent of the plates. Note that 2coe = nb = total extent of natural 

 mode intervals = plate width. The correction term becomes smaller as co, 

 increases. 



The second source of error Hes in the use of lumped charges instead of 

 a continuous charge distribution, and may be called the ^'granularity 

 error." Its magnitude may be approximately determined from (30) if we 

 replace the sines and cosines by their exponential equivalents, differentiate 

 with respect to co, and assume that the error is small. We find. 



d^ 



do3 



27r , 4x / liraA 2x0) /^^n 



-^±-^exp(^-— jcos — , (32) 



where the plus and minus signs refer respectively to odd and even numbers 

 of modes. 



We may assume that both errors are small, and that they act inde- 

 pendently, so that the total error is given approximately by the sum of the 

 non-constant factors in (31) and (32). We note that if we increase the plate 

 spacing, a, the granularity error becomes smaller while the truncation error 

 increases. This increase may be offset by increasing a>e , but this means 

 extending the condenser plates and therefore adding additional lumped 

 charges, with consequent increase in network complexity. Thus the choice 

 of specific spacing and dimensions is hkely to represent a balance between 

 granularity errors, truncation errors and network complexity. 



The truncation errors may be somewhat reduced, with no increase in 

 network complexity, by increasing the charge densities near the edges of 

 the plates. Later we shall discuss a systematic method of adjusting the 

 charge distribution. 



