156 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1957 



lator trying to regulate for a slightly higher cable current, would drive 

 its rectifier voltage to its stop or maximum output, unbalancing the 

 cable voltages. 



System Stability* 



A complete analysis of system stability represents an exceedingly 

 formidable, if not impossible, task. It has been established analytically 

 that for a linear network the two dc regulators in parallel and the system 

 as a whole are unconditionally stable. The details of this proof are too 

 long to be presented here but the line of reasoning with respect to the 

 overall system is as follows. The system of Fig. 1 is symmetrical about a 

 vertical plane through the middle of the figure. Under these conditions, 

 the system will be stable if, and only if, the following three simpler sys- 

 tems! are stable: 



(a) A power supply short-circuited; 



(b) A power supply feeding an impedance equal to twice that of the 

 half cable short-circuited ; and 



(c) A power supply feeding an impedance equal to twice that of the 

 half cable open-circuited. 



The transfer function of the servomechanism was measured over the 

 frequency range of principal interest, to 1 cps, the behavior near zero 

 frequency being determined from the asymptotic slope of the unit step 

 response. J In this frequency range the dc amplifier gain is a real constant, 

 flat gain and negligible delay as previously shown, therefore only the ac 

 servo feedback loop characteristic has to be known to predict the sta- 

 bility of condition (a). The Nyquist loop for this transfer function shows 

 that condition (a) above is satisfied. A similar examination of the Nyquist 

 plot, including the readily computed cable impedance shows that condi- 

 tions (b) and (c) above are satisfied. Thus the linear analysis indicates 

 stable operation for the system of Fig. 1. This result was confirmed by 

 tests of conditions (a), (b), and (c) individually and by the behavior of 

 the system as a whole, both in the laboratory with a simulated power 

 network for the cable and in the final installation. 



One of the most obscure aspects of the power system behavior is that 

 of equilibrium conditions after one or a series of large earth potential 



* The analysis briefly summarized here was made by C. A. Desoer. 



t In this discussion of simpler sj'stems a power supply consists of only the 

 elements shown in Fig. 2. 



J In the course of these time-domain measurements, it was quite apparent that 

 the ac control loop could be considered as a linear network only in an approximate 

 sense and thus that the analytical results were primarily useful in interpretation 

 of observed behavior of the system. 



