POWER OUTPUT 



611 



For a physical circuit the attenuation parameter d must be positive while, 

 for an increasing wave, x must be positive. We see that we may expect E 

 to be greatest for a given current when d and x are small, and when y is 

 nearly equal to the velocity parameter b. 



Suppose we use (12.4) in expressing the power 



P = 



^•"-{m/^'-p) 



t'v\{e'/^'p) 



(12.7) 



Here we identify /3 with —jTi . Further, we use (2.43), (12.5) and (12.6), 

 and assuming C to be small, neglect terms involving C compared with unity. 

 We will further let i have a value 



i = 2/o 



(12.8) 



5 

 4 

 3 



2 

 1 

 



Fig. 12.3 — An efficiency parameter k calculated by taking the power as that given by 

 near theory for an r-f beam current with a peak value twice the d-c beam current. 



We obtain 



P = kCIoVo 



(12.9) 



k = 



(b + yy + (x + dy 



(12.10) 



We will now investigate several cases. Let us consider first the case of a 

 lossless circuit (d = 0) and no space charge (QC = 0) and plot the efficiency 

 factor k vs. b. The values of x and y are those of Fig. 8.1. Such a plot is 

 shown in Fig. 12.3. 



If we compare the curve of Fig. 12.3 with the correct curve of Nordsieck, 

 we see that there is a striking qualitative agreement and, indeed, fair quanti- 

 tative agreement. We might have expected on the one hand that the electron 

 stream would never become completely bunched {i = 2/(i) and that, as it 

 approached complete bunching, behavior would already be non-linear. 

 This would tend to make (12.10) optimistic. On the other hand, even after i 



