. The Bell System technical journal . n the assumption that the parameter a equals 2, is shown at the bottomof the figure. At first sight, this may appear to be a trivial illustration, since ^oand ip are merely constants, and the structure thus has only the proper-ties of an ordinary gain control. It is possible, however, to introduceauxiliary networks by means of which Qo and ip can be made prescribedfunctions of frequency. For example, 0o can be altered by adding anordinary equalizer in tandem with either terminating resistance. Themodification which allows us to vary ip may be somewhat less
. The Bell System technical journal . n the assumption that the parameter a equals 2, is shown at the bottomof the figure. At first sight, this may appear to be a trivial illustration, since ^oand ip are merely constants, and the structure thus has only the proper-ties of an ordinary gain control. It is possible, however, to introduceauxiliary networks by means of which Qo and ip can be made prescribedfunctions of frequency. For example, 0o can be altered by adding anordinary equalizer in tandem with either terminating resistance. Themodification which allows us to vary ip may be somewhat less consists of the introduction of a symmetrical four-terminal networkhaving the image impedance Ro, between the variable resistance andthe terminals to which it was previously connected, as shown byFig. 4. VARIABLE EQUALIZERS 235 -y^ aRp2(a2-,)< -wv- aRo aRf (a2-i) e^ = a 12 ©0 + * 10 - a =2 8 - 6 ©0 4 - 2 - 0 1 1 1 r 1 , ,^0-^ ,,,,,, Pig 3_The simplest type of symmetrical variable equalizer Fig. 4—Adjustment of the variable by the additionof an auxiliary network. 236 BELL SYSTEM TECHNICAL JOURNAL The effect of the added network is easily understood from the pre-ceding equations. It will be noticed that although these equationswere written under the assumption that i? is a real quantity, they willstill be valid if R is complex. We need therefore merely to replace R bythe impedance of the auxiliary network terminated by the variableresistance. If we represent this impedance by Zr, the appropriateexpression is X + tanh xP /Lr — Aoi—j TTT \i \ -\- X tanh \p where i^ is the transfer constant of the added network and x is, asbefore, the ratio of the variable resistance to i?o. Since reciprocal val- Z ues of X still correspond to reciprocal values of ^ , all of the preceding conditions of symmetry in the resulting family of characteristics aremaintained. The simplest formulation for the new (p is secured from equation
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