CHARACTERISTICS OF TUNED LC CIRCUITS.16[CHAP.1Next,calculate thebandwidth.foBW =—2ckt2MHz17.6=113.6kHzAs you can see,the addition of aloadresistor RL increasesthe bandwidth. Without aload presentin Fig.l-10(a),the bandwidth wasonly 20kHz. However, with thepresence of a27kft loadin Fig. 1-10(h), thebandwidth wasbroadened to113.6 kHz.1.3TRANSFORMER COUPLINGA commoncoupling arrangementencounteredincommunications (RF)equipment is acouplingtransformer.Figure 1-11showsvarious tuned coupling arrangementsin which acapacitor or capacitorsareused in conjunction with atransformer toallow only thedesired band of RFsignals topass. Theconnection of acapacitor toeither the primary orsecondary ofthe transformer forms atuned LCcircuit.Figure 1-11(a)shows acoupling transformerwith atunedprimary whereasFig. 1-11(h) shows atransformerwith atuned secondary.Figure 1-11(c)shows adouble-tuned transformerin which both theprimary andsecondary aretuned.Thecoefficient ofcoupling (k) of atransformeris ameasureofhow muchof the magnetic fluxoriginated by theprimary links thesecondary ofthe transformer. Although for audio equipment it is notunusual tofind coefficients ofcoupling on the order of0.90 andhigher, transformersin RFservice havecoefficients ofcoupling onthe orderof0.01to0.05. RF transformercoupling arrangements have frequencyresponse curvesasshowninFigs. 1-12 and 1-13. Refer toFig. 1-13,the response curvefor thedouble-tunedtransformer, and notethe reference toovercoupled, undercoupled, andcritically coupled conditions.Thecritically coupled situation isthe onethatprovides maximum output andmaximum bandwidthwithout adip in output atthe resonantfrequency. For adoubled-tuned transformer,criticalcoupling resultswhen the coefficient ofcoupling k isequal tothereciprocal ofthe square rootofthe product ofthe Q oftheprimary and the Q ofthe secondary:1(1.18)VQpQsIn Eq. (1.18), QP representsthe Q of the primary and Qs represents the Q ofthe secondary.Theundercoupled situation where k<kc provides neither maximum output normaximumbandwidth.Intheovercoupled case where k>kc adip appears in the response curve atthe resonantfrequency. Theslightly overcoupled condition is sometimes desired because itprovides steeper sides to the response curveand thussharpens rejection ofundesiredsignals. Acoefficient ofcoupling k of 1.5times the criticalcoefficient ofcoupling isfrequently used andconsidered desirable.The bandwidth of adouble-tunedcoupling transformerisequal totheproduct ofthe coefficient ofcoupling k and the resonantfrequency f0. This is shown as:BW =#o(1.19)EXAMPLE 1.9. Determinethe necessary value ofk toprovide critical coupling for adouble-tuned transformer inwhich QP =60and Qs =90.Ans.Using Eq. (1.18),thecalculations are asfollows:1K =yIQpQsIX90=0.0136