INDUCTANCES, CAPACITANCES AND RESISTANCES 



The voltage across the inductance may be found, as a function of frequency, 

 in a similar manner. The curves are of the same kind but are mirror images 

 (Figure 5.4). 



These curves have Q as parameter and Q — {LjCR^Y'^, that is, Q is large 

 when B^lL is small compared with 1/C. Referring back to the analysis for 

 series L, C and R connected to the direct voltage generator, we see that the 



o) — ^ 



Figure 5.4 



degree of damping depended on the relative sizes of 1/LCand R^jAL^, which is 

 the same as depending on the sizes of 1/C and R^I4L. It thus appears that Q 

 and the degree of damping are closely related, and in particular that at critical 

 damping Q = ^; series resonant circuits are below critical damping if Q > ^, 

 and more than critically damped if Q < |. 



We can now consider the relevance of Graphs 21 and 22 to the electro- 

 magnetic penwriter. The input current flows through a coil to produce a 

 force F on a moving mechanical system of mass M constrained by a spring 

 of stiffness X and viscous damping d. If the displacement produced is s, then 

 we have 



. d5 , d^j' 



and the actual force available for displacing the pen is Fpen = Xs, the remain- 

 der of F being used in overcoming the inertia and viscous damping forces. 

 Comparing this with 



dq q d^q 



^ = ^d7 + c + ^d^ 



where the capacitance voltage is v^ = qjC, we see that the pen deflection is 

 analogous to capacitance voltage. Hence, if a current be suddenly applied 

 to a penwriter, the pen movement has one of the forms of Graph 21, depend- 

 ing on the amount of viscous damping present. If an alternating current of 

 constant amplitude and variable frequency be fed to a penwriter the device 

 records, without distortion, up to a certain frequency, where there is a 

 greater or lesser resonance, depending upon the amount of damping, and 

 above this frequency the performance of the device falls off rapidly. From 

 Graph 22 it appears that a Q value of rather less than one will give the pen- 

 writer an optimum distortion-free band-width. 



Impedance of series resonant circuit 



The impedance of a series resonant circuit at frequencies away from reso- 

 nance is important in the design of certain tuned amphfiers. 



76 



