Minimum sensitivity coupled resonators and biquads

The effect of coupling on the overall sensitivity to component tolerances of two second-order resonators is compared with the sensitivity of a non-coupled cascade of two second-order resonators. Coupled resonators consist of two second-order resonators “coupled” within a negative feedback loop. The resulting overall fourth-order transfer function of the two circuits, coupled and non-coupled, is identical. The “cascaded” poles, ie, the poles of the two cascaded resonators, are therefore identical to the poles of the coupled circuit, the coupled poles. The poles within the negative-feedback loop, the “open-loop” poles, will be different. We assume that the manufacturing technology used to realize the open-loop poles of the coupled circuit is the same as that of the cascaded, non-coupled circuit. The open-loop poles will therefore be subject to the same component tolerances as those of the cascaded non-coupled circuit. Our analysis leads to the optimum location in the s-plane with regard to minimum sensitivity, for the open-loop poles of the coupled circuit. Since resonators are essentially the equivalent of second-order bandpass filters, the results obtained are applied to coupled second-order active-RC filters, or biquads, for which insensitivity to component tolerances is critical. The examples given pertain to the coupling of biquads.