CHAPTER 6. OPERATIONAL AMPLIFIERS114and only drops 18 db/octave.For complex poles we must use either integrators or diﬀerentiators. Consider ﬁgure 6.14.CRRIVinVoutFigure 6.14: Active ﬁlter with complex poles.The closed-loop gain isG =−ZRCRI=−R/jωCRI (R +1/jωC)=−RRI (1 + jωRC).(6.27)By exchanging the input resistor for a capacitor we can change between a low-pass andhigh-pass ﬁlter.6.2.5General Feedback ElementsThe feedback elements in an operation ampliﬁer design can be more complicated than asimple resistor and capacitor. An interesting feedback element is the analog multiplier asdeﬁned in ﬁgure 6.15.Vy(t)Vx(t)Vz(t)=Vx(t)Vy(t)MULTFigure 6.15: Five-terminal network that performs the multiplication operation on two voltagesignals.The multiplier circuit itself can be thought of as another op-amp with a feedback resistorwhose value is determined by a second input voltage. Multiplication circuits with the abilityto handle input voltages of either sign (four-quadrant multipliers) are available as integratedcircuits and have a number of direct uses as multipliers. But when used in a feedback looparound an operational ampliﬁer, other useful functional forms result.The circuit of ﬁgure 6.16 gives an output that is the ratio of two signals, whereas thecircuit of ﬁgure 6.17 yields the analog square-root of the input voltages.