620 BELL SYSTEM TECHNICAL JOURNAL 



it is often unnecessary to perform these precise operations in order to obtain 

 a broad picture of the network behavior. 



The method for determining the network response to a sinusoidal input is 

 developed as follows. It is assumed that the circuit parameters are con- 

 stant, independent of signal amplitude. Then, as indicated in Fig. 1, a 

 single R-L-C or R-J-C mesh may be represented by a constant-coefficient 

 linear differential equation. Choosing the electrical mesh for illustration, 



{Lp+ R+ l/Cp)i{t) = e(0, 



where p^ = d^/dt"^, \/p = / dt, and e{t), i{t) are the mesh voltage and cur- 

 rent respectively. This may be further abbreviated as 



Z{p)i{t) ^e{t), (1) 



where Z{p) = Lp -{- R -{- 1/Cp. For purposes of frequency analysis we 

 are interested only in the forced or steady-state solution of (1), where e{t) 

 is a sinusoidal voltage E sin co/. This steady-state solution is 



*'W = I ^. ■ X I sin (cot -f (t>), 



where 7co has replaced p in the function Z(p), and the phase shift (f> is the 

 negativ^e of the phase angle of the complex number Z{iu>)} This result is 

 conventionally abbreviated as 



/ = ^, , (2) 



where 7 is a complex number whose magnitude equals the peak amplitude 

 of the current, and whose phase angle gives the associated phase shift. 

 The function Z{jix>) is called the impedance of the mesh. 



The relationship between torque, angular speed, and mechanical im- 

 I)edance is of course the same as expressed by (2). That is, 



e= yf^,, (2.1) 



where 6 is the complex peak amplitude of the sinusoidally varying speed, T 

 is the peak amplitude of the applied sinusoidal torque, and Z(jcx}) is the 

 mechanical impedance obtained by substituting ju: for p in the operator 

 Z{p) = Jp-{- R-\- 1/Cp. Since the angular displacement is the time integral 



^ CO is used to represent frequency in radians/sec, or Itt times frequency in cycles/sec. 



