WAVE SHAPES USED IN HARMONIC PRODUCERS 335 



with their tops chopped off. The analysis of a pulse of the dimensions 

 shown in Fig. 4 shows that the amplitude of the nth harmonics is given by 

 the expression 



4.4 r ^b nc~\ 



w-7r(c — o) [_ 2 2 J 



(4) 



In order to show better the relationship between a wave of rectangular 

 pulses and one of trapezoidal pulses, consider the ratio of the nth harmonic 

 for these two waves. From (2) and (4) 



hn for trap, pulse _ 2 (cos nh/2 — cos nc/2) ,-■. 



hn for rect. pulse n{c — h) sin nb/2 



Substituting c — h = b and expanding cos nc/2 = cos {nb/2 + n8/2), 

 the right hand side of (5) becomes 



2 fcos nb/2 cos nb/2 cos n8/2 , . . ,^"1 , ,>. 



-s -■ rr^ ~ ^ — ,/o + sm n8/2 (6) 



no |_sm nb/2 sm wo/2 J 



For small values of n8/2, that is for trapezoidal waves whose base is only 

 slightly wider than the top, cos n8/2 may be replaced by unity and sin n8/2 

 by n8/2. The first two terms then cancel and the approximation 



hn for trap, pulse ^^ . ^^v 



hn for rect. pulse 



is obtained showing that a slight slope in the sides of the pulse has only a 

 second order effect on the harmonic content of the wave. 



The curve in Fig. 4 shows the harmonic content of a rectangular wave 

 having a pulse width of 10° compared with that of a trapezoidal wave having 

 a pulse width of 10° at the top and 11° at the bottom. For lower harmonics 

 the amplitudes are nearly the same, but in the vicinity of the 36th harmonic 

 there is an essential difference. For the rectangular pulse, the 36th har- 

 monic vanishes, while the trapezoidal pulse has a minimum at a somewhat 

 lower value of n and all harmonics have finite values.^ This is shown in 

 Table 1 which tabulates the amplitude of the harmonics in this case. 



A second form of distortion in rectangular pulses is the rounding of the 

 corners at both the top and the bottoms of the pulse. This type of distor- 

 tion is more difficult to analyze and while no complete analysis has been 

 made the effect of such distortion is known to be, in general, to reduce the 

 amplitude of the higher harmonics. 



' In discussing the curves in Fig. 1 thru 5 it must be remembered that while these are 

 drawn as solid lines, the lines have a meaning only for integral values of n. Fractional 

 values of n are meaningless. 



