LINEAR SERVO THEORY 617 



sen'o systems usually are combinations of electrical, mechanical, thermal, 

 or hydraulic circuits. In order to describe the behavior of these various 

 circuits in homogeneous terms, it is desirable to recognize the analogous 

 relationships established by similarity of the underlying differential equa- 

 tions. Before proceeding to a discussion of frequency analysis, a typical 

 analogy between electrical and mechanical systems will be described. 



2.1 Electrical-Mechanical Analogy 



Confining the discussion to rotating mechanical systems, the analogy 

 which will be chosen here puts voltage equivalent to torque, and current to 

 rotational speed. This choice leads to the array of equivalents shown in 

 Fig. 1; inductance, capacity, and resistance corresponding to inertia, com- 

 pliance, and mechanical resistance, respectively. Charge is equivalent to 

 angular displacement, and both kinetic and potential energy are self- 

 analogous. The ratio of voltage to current, or torque to speed has the 

 dimensions of resistance. In an interconnected electro-mechanical system, 

 the ratio of voltage to speed or torque to current may be called a transfer 

 resistance. Similarly, the ratio of voltage to angular displacement, or of 

 torque to charge, is a transfer stiffness (reciprocal of capacity or compliance). 



Some commonly used devices for coupling between electrical and me- 

 chanical circuits are shown in Figs. 2 and 3. The motor. Fig. 2a, is used to 

 convert an electrical current i into a mechanical speed or "current" d ( = 

 dd/dt) . The electrical control current i is produced by the voltage difference 

 between an applied emf e and a counter-rotational emf (not indicated), 

 acting upon the total electrical mesh resistance Re* In the mechanical 

 circuit, a torque proportional to i forces a "current" B through the mechanical 

 load Rm , /. 



.\x\ equivalent mechanical mesh directly relating shaft speed to the applied 

 emf is shown in Fig. 2b. A fictitious generated torque y.ie acts upon the 

 mechanical load through an apparent mechanical resistance Rm ■ Mi 

 is a transfer constant determined both by the motor properties and the 

 electrical mesh resistance Re . Rm is similarly governed and is inversely 

 proportional to Re . 



The motor may be compared to a vacuum tube having an amplification 

 factor nt and a plate resistance Rm . However, the motor is usually much 

 more a bilateral coupling element than the vacuum tube, due to the effect 

 of the counter emf upon the electrical mesh. 



The potentiometer, tachometer, and synchro circuit shown in Fig. 3 are 

 all means for converting a mechanical quantity to an electrical one. .-Vll 

 three are substantially unilateral coupling elements. The potentiometer 



* Re includes both the source resistance and the motor winding resistance. 



