208 BELL SYSTEM TECHNICAL JOURNAL 



and how these are related to each other and to the concepts already men- 

 tioned. For this purpose it is convenient to consider the circuit shown in 

 Fig. 31 (c). Across the terminals AB is connected the L-C circuit in which 

 the resistive losses are represented by the circuit conductance Gc. The 

 circuit is loaded by the admittance Y ^ = Gi, + JBl. 



Looking into the circuit at the terminals AB one sees the admittance: 



F, = G,+jBc + yI = Gc + rr +J^C + Y'i 



juL 



which may be rewritten: 



Y. = Gc+j a/^(---)^Y] 



A /^ { — — — \ 



\ coo / 



(19) 



^ a + 2i ^" Foe + F 



i J 



where coo = 



i/i. 



The expression k/ — , having the dimensions of an admittance, is by definition 



the characteristic admittance of the circuit, Yfic- Its relation to the energy 

 stored in the circuit, and through this to the various Qs defined above, may 

 be seen from the following: Using the root mean square value of voltage, the 

 energy stored in the circuit is CV rf. This can be reduced by the use of the 

 definition of the resonant frequency and by differentiation of the expression 

 (19) for the admittance, thus: 



CY RF — Y UF — — 



0)0 ^ 



dB. 



dec 



rFL. (20) 



resonant frequency, 



Thus, at a given frequency coo, the energy stored in the circuit for unit voltage 

 applied to it may be specified either by the characteristic admittance of the 

 circuit, Yoc, or by the rate of change of susceptance with frequency at the 

 d^c 

 dw 



The loss of energy per cycle in the circuit itself, that is, in the shunt con- 

 ductance Gc, is the power loss in the circuit divided by the frequency, 



V^ G 



^^ ' . From this and equation (20) the unloaded Q is seen to be: 

 <oo/27r 



V RF 



e.= 2.^^ = i^. (21) 



