628 BELL SYSTEM TECHNICAL JOURNAL 



to determine the stability of a completed serv^o design by obtaining the 

 transient solution of its differential equation. Though often very tedious, 

 this is fairly straightforward. However, this method of procedure often is 

 of little help either in guiding the initial design or in predicting the necessary 

 changes, should the trial design be found unstable. The addition of even 

 one circuit element to a design will generally create an entirely new differen- 

 tial equation whose solution must be found. 



An alternative method for determining servo system stability, based on 

 frequency analysis, furnishes the necessary information in a form which 

 greatly facilitates the design procedure. This method is relatively simple 

 to apply, even when the system has a large number of meshes and a high 

 order differential equation, and the additive effects of minor circuit modi- 

 fications are easily evaluated. 



The stability of a single loop servo system — or of a primary loop, when 

 the subsidiary loops are individually stable — may be investigated by plot- 

 ting the negative of the loop transmission, —y.^{joi), on a complex plane 

 for real values of co ranging from minus infinity to plus infinity. (The 

 negative sign is introduced because the loop transmission ju/3 is generally 

 arranged to have an implicit negative sign, apart from network phase shifts. 

 Thus — /i/3 is a positive real number when the network phase shift is zero.) 

 Then the necessary and sufficient criterion for system stability is that the result- 

 ing closed curve must not encircle or intersect the point —1,0* This type of 

 plot is commonly called a Nyquist diagram, and is widely used in the design 

 of electrical feedback amplifiers. An added stipulation is necessary- if fxlSijo:) 

 becomes infinite at a real value of co, say w'. In this case an infinitesimal 

 positive real quantity e must be added to jco; that is, the function to be 

 plotted is M/3(.7co + e). This has no effect upon the plot except in the neigh- 

 borhood of the singularity, where fx0{jo) + e) is caused to traverse an arc of 

 infinite radius as co is varied through the value co'. 



As seen from (3.1), inclusion of a motor in a servo loop of the type shown 

 in Figs. 5 and 6 will cause an infinite loop transmission at a- — 0, assuming 

 there is transmission around the remainder of the loop at zero frequency. 

 The motor is the only commonly encountered circuit element capable of 

 producing an infinite loop transmission at real frequencies. 



In order to illustrate the use of the Nyquist diagram, a motor servo sys- 

 tem of the type shown in Fig. 6 will be chosen. Again referring to equation 

 (3.1), it may be seen that the transfer ratio of the motor and potentiometer 

 is, 



* This criterion is due to H. Nyquist, "Regeneration Theory,". B. S. T. J., January 1932. 

 Detailed descriptions of stability criteria for single and multiple loop systems are given by 

 Bode, loc. cit., and by L. A. MacColl, "Fundamental Theor}^ of Servomechanisms," D. 

 Van Nostrand Co., 1945. 



