Minimum sensitivity coupled resonators and biquads

Abstract

Summary 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.

Low‐noise and low‐sensitivity filters using current‐mode coupled biquads

“…For k = k 0 , the two resonators inside the feedback circuit are identical, and μ = 1/2 and α = 1. Comparing the denominators of (1) and (3), we obtain (see Jurisic and Moschytz): in which case and where ω p 1 , ω p 2 , ω p 0 , and q p 0 are given by (5) and (6).…”

“…As in Jurisic and Moschytz, transfer function in (1) is obtained from the low‐pass (LP) prototype filter by the common LP‐BP transformation s → ( s 2 + ω 2 p 0 )/( Bs ) (where B is the bandwidth and ω p 0 is the center frequency), containing the factor pairs as in (2) with equal pole Q‐factors: and pole frequency pairs ω p 1 , ω p 2 that are related by a factor α CA > 0 and defined by …”

“…Taking into account that (1) and (3) realize the same given transfer function, ie, T ( s ) = T CA ( s ) = T CO ( s ), and comparing the denominators, we obtain the open‐loop parameters Ω pi , and Q pi as a function of the feedback factor k , and the closed‐loop parameters ω pi , and q pi ( i = 1, 2) (see).…”

“…Parameters α (dashed) and μ (solid line) vs k . They replace the open‐loop parameters Ω pi and Q pi ( i = 1, 2) of the coupled structure; the latter are also shown vs k in Figure 3 in Jurisic and Moschytz [Colour figure can be viewed at wileyonlinelibrary.com]…”

“…It has already been demonstrated empirically (eg, Fotopoulos and Deliyannis) that a fourth‐order coupled biquad has not only lower sensitivity to component tolerances but also lower noise than the equivalent cascade of two second‐order biquads. In Jurisic and Moschytz, it is shown analytically that minimum sensitivity is obtained when the two biquads inside the coupled structure are identical. In this paper, it is shown analytically that this also applies to a considerable reduction in noise.…”