Embedded Systems Design Europe - May 2008 - (Page 33)

analog Using the general equation for sensitivity, the sensitivities of Q and ω are: n 1 R 3C 4 Q Q S R1 =- S R3 =- +Q 2 R1C 2 ⎛ RC ⎞ 1 Q Q 14 + R 3C 4 ⎟ S C2 =- S C4 =- +Q ⎜ 2 R1C 2 ⎠ ⎝ R 3C 2 (22) (23) S K =QK Q R1C 2 R 3C 4 R1C 2 R 3C 4 (24) S RA =- S RB =-Q(K -1) =-Q R B R1C 2 R A R 3C 4 Q Q (25) K V out = 2 V in S ( R1R 3C 2C 4 )+S( R1C 2 + R 3C 4 + R1C 4 - KR1C 2 )+1 R (19) n n n n S R1 = S R3 = S C2 = S C4 =- ω ω ω ω 1 2 (26) A lot of good information is in these plots, but we get all we need for most situations with the far simpler Q and ω sensitivities. Passive ﬁlters, like the RLC lowpass ﬁlter in Figure 3 always have Q and ω sensitivities in the –1 to +1 range, with ±0.5 being most common. Sensitivities in this range generally are considered as good as it gets. Active ﬁlters have more ﬂexibility in choices with increased complexity of sensitivities. We’ll use the venerable and ubiquitous Sallen-Key for our ﬁrst active ﬁln n ter. The Sallen-Key lowpass ﬁlter shown in Figure 5 ﬁrst described over 50 years ago is one of the most common ﬁlter topologies. The Sallen-Key lowpass ﬁlter transfer function is shown in equation 19 where K is the DC gain, K = 1+ R /R . The ﬁlter characteristic equations are: 4 b a n n S K n = S Ra = S Rb = 0 ω ω ω (27) ωn= 1 R1R 3C 2C 4 (20) 1 = R 3C 4 + R1C 4 +(1- K) R1C 2 (21) Q R1C 2 R 3C 2 R 3C 4 Note that the natural frequency sensitivities, as with the passive ﬁlter above, are either ± 0.5 or zero, whereas Q sensitivities are signiﬁcantly more complex. Natural frequency sensitivities to R and R (as well as to K) are zero. This means that the natural frequency is independent of the values of these resistors (and to the DC gain). These simpler sensitivities are constants. We can do nothing within this circuit topology to change these sensitivities. As discussed earlier, sensitivities of magnitude 0.5 and less are as good as it gets. So for this circuit, let’s concern ourselves with Q sensitivities. The example in Figure 6 is a 1 kHz low-pass ﬁlter with a nominal Q of 1. We’ll vary R to change the Q without changing the natural frequency. Let’s vary R from 1k to 19.9k. At R = 20k, the equation for Q goes to inﬁnity and should be avoided. Since R is a gain-determining component, the DC gain, shown in Figure 7, changes along with the Q. Monitor the feedback voltage at the op amp’s inverting terminal, Vfb in the schematic, to see the gain-normalized responses, Figure 8. a b b b b b www.embedded.com/europe | embedded systems design europe | MAY 2008 33 http://www.embedded.com/europe

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Embedded Systems Design Europe - May 2008 Contents Microsoft Provides Embedded Roadmap Enea Buys Developers Irish Start-Up Raises Funds for Telecom FPGAs Kontron Promotes COM Express Nano Mentor Nucleus Platform Provides UI for Atmel Small Form Factor Boards Head for the SUMIT Proffibus Advances IO-Link Integration Embedded Developers Cautious on Multicore Auto Cooperation Improves Test Altera Launches DO-254 Partner Network Building an ‘Instant-Up’ Real-Time Operating Systems An Architecture for Reusable Embedded Systems Software Free up Bandwidth in PCI Express Evaluating Software in Medical Devices Circuit Sensitivity in Analog Circuits Choosing Flash Memory New Products Advertising Contacts

Embedded Systems Design Europe - May 2008

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