LINEAR SERVO THEORY 621 



of the speed, the expression for 6 may be obtained by dividing both sides of 

 equation (2.1) by p or, for the periodic case, hy jw. Thus 



joiZijoi) ' 



(2.2) 



The function juZ(jco) is the complex stiffness of the mechanical mesh. 

 The phase shift of 6 relative to ^ is —90 degrees, as seen from a comparison 

 of (2.1) and (2.2). 



For an electro-mechanical network consisting of a number of intercon- 

 nected meshes, a set of simultaneous differential equations of the type of 

 (1) may be written. If yw is substituted for p in these equations, there 

 results a set of simultaneous algebraic equations which lead directly to the 

 steady-state periodic solution. If a sinusoidal voltage or torque is applied 

 at some mesh of the network, the resulting sinusoidal current or speed 

 response in some other mesh is given by, using the notation of (2), 



/r, N (Cause) 



(Response) = „ . . . , 

 Ztijo}) 



where Ztijco) for the chosen pair of meshes is obtained from the solution 

 of the algebraic equations. Ztijoo) is called a transfer impedance, and may 

 express the ratio of a voltage to current or speed, or of a torque to current 

 or speed. The above relation also may be written as 



(Response) = Fi(yaj) • (Cause), 



where Ft(ico) = l/Ziijco) is called the transfer admittance between the two 

 chosen parts of the network. In this form the response amplitude is ob- 

 tained by multiplying the forcing sinusoid by | Yi(joj) |, while the phase 

 shift is given directly by the angle of Ytijo}). The transfer ratio between 

 like or analogous quantities at two parts of the network is similarly a 

 complex function of frequency, having the dimensions of a pure numeric. 



Servo systems usually consist largely of elementary networks isolated 

 by unilateral coupling devices (vacuum tubes, potentiometers, etc.). 

 Thus, over-all transfer ratios often may be evaluated by taking the product 

 of a number of simple "stage" transfer ratios, rather than by solving a large 

 array of simultaneous equations. If the absolute magnitudes of the transfer 

 ratios or "transmissions" are expressed in decibels'* of logarithmic gain, 

 both the over-all gain and phase shift of a number of tandem stages may be 

 obtained by simple addition of the individual stage gain and phase values. 



The transfer ratios of the conversion devices shown in Figs. 2 and 3 



' The gain in decibels for a given transfer ratio is taken to be 20 iogio of the absolute 

 value of the ratio. 



