Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

16-16 Passive, Active, and Digital FiltersIIIV–+g mV +V –+–g m–+V +V –+ +g m– –II(a)(b)(c)FIGURE 16.17 Simulated resistors: (a) positive single-ended of value V=I ¼ 1=g m ; (b) positive differential resistor ofvalue (V þ V )=I ¼ 1=g m ; and (c) negative differential resistor 1=g m .terminals results in a negative resistor as in Figure 16.17c.* Since transconductors and capacitors canbe used to build all components necessary for designing the filters, they are called transconductance-Cor g m -C filters. We discuss the remaining ‘‘composite’’ building blocks, integrators and gyrators ysubsequently in Section 16.3.1.In this section we will not go into the electronic circuit design methods for OTAs, but refer thereader to the literature, which contains a great number of useful transconductance designs in all currenttechnologies. References [5,7] contain numerous papers that discuss practical transconductance circuits.The most popular designs currently use CMOS, but bipolar and BiCMOS are also widely employed, andGaAs has been proposed for applications at the highest frequencies or under unusually severe environmentalconditions. Since transconductors are almost always used in open loop without local feedback,their input stages must handle the full amplitude of the signal to be processed. Typically, the OTA inputstage is a differential pair with quite limited signal swing before nonlinearities become unacceptable.Thus, much design expertise has gone into developing linearization schemes for transconductancecircuits. They have resulted in designs that can handle signals of the order of volts with nonlinearitiesof a fraction of 1%. Apart from simple source-degeneration techniques, the most commonly employedapproaches use variations of the principle of taking the difference between the drain currents of two MOSdevices in the saturation region but driven differentially, so that the difference current is linear in V gs :I þ d ¼ k(V gs V T ) 2 ¼ k Vgs 2 þ V2 T 2V gs V TI d ¼ k( V gs V T ) 2 ¼ k Vgs 2 þ V2 T þ 2V gsV T(16:27)DI d ¼ I þ dI d¼ 4V gs V TAnother approach reasons that the most linear (trans)conductance behavior should be obtainable fromthe current through a resistor. Thus, operating an MOS device in the resistive (triode) region,I d ¼ k (V gs V T )V ds 0:5Vds2 and taking the derivative with respect to V gs for constant V ds ¼ V DS results in a perfectly lineartransconductance,* Such ‘‘negative resistors’’ are often used to cancel losses, for example to increase the dc gain of transconductors or thequality factors of filter stages. Specifically, a negative resistor can be employed to increase the quality factor of the inductorin Figures 16.21 and 16.22.y A gyrator is a two-port circuit whose input impedance is inversely proportional to the load impedance: Z in (s) ¼ r 2 =Z load (s).If Z load ¼ 1=(sC), the input is inductive, Z in (s) ¼ sr 2 C ¼ sL. r is called the gyration resistance (see the discussion to follow).