Embedded Systems Design Europe - May 2008 - (Page 34)

analog The results in Figure 8 are much the same as with the passive ﬁlter in Figure 7. Moving forward from here, we know that we can calculate sensitivities for this and other circuits. We can also use this information to determine which components are contributing most signiﬁcantly to response variation so we can specify more precise (and more expensive) parts in these locations. Sometimes this is all that we can do. THE MULTIPLE FEEDBACK FILTER Another very common ﬁlter is the multiple feedback ﬁlter (MFB) shown in Figure 9. This ﬁlter has two feedback paths, thus, its name. It is an inverting circuit, whereas the Sallen-Key utilized a non-inverting ampliﬁer. The transfer function of this circuit is given by: R Vout =− 2 Vin R1 1 S 2 R2 R3C 2C1 + SC1 R2 R3 ( 111 + + )+ 1 R1 R2 R3 ⎛ 1 R RC R3C1 ⎞ 231 + ⎟ ⎜ R1 C2 R2C2 ⎟ Q⎜ Q SR = − ⎜ ⎟ 3 2 ⎟ ⎜ − R2C1 ⎟ ⎜ R3C 2 ⎠ ⎝ Q SR = 1 R2 R3 R / / R2 / / R3 =1 R2 R3 + R3 R1 + R2 R1 R1 (36) Q SR = 2 (37) 1 R1 R3 − R2 R3 − R1 R2 2 R1 R2 + R1 R3 + R2 R3 1 R1 R2 − R2 R3 − R1 R3 2 R1 R2 + R1 R3 + R2 R3 (28) The natural frequency and Q are given by: ωn = 1 R2 R3C 2C1 (29) R3 C1 ⎛ R2 R3 R2 ⎞ 1 = + + ⎜ ⎟ Q C 2 ⎜ R1 R2 R3 ⎟ ⎝ ⎠ (30) Equation 30 tells us that, once again we have a component (R ) that can be varied to change the Q without affecting the natural frequency. Equations 31 and 32 indicate that all sensitivities to the capacitors are –0.5 or 0.5. Equations 34, 35, and 36 are quite complex. Unfortunately, many textbooks leave these equations in this form. However, if we simply plug in Q from Equation 29 to each of these, we greatly simplify these equations: 1 1 (38) Q SR = 3 (39) where “//” means “in parallel with.” By inspection we can tell that these Q sensitivities to the resistors are all less than one. The numerator in Equation 36 must be less than the denominator as R paralleled with any other resistors necessarily is smaller than R itself. Similarly, numerators in Equations 37 and 38 must 1 1 The sensitivity equations are: S S ωn R1 ωn R2 =0 =S ωn R3 (31) =S ωn C2 =S ωn C1 = −1/ 2 (32) (33) S = −S = 1 / 2 ⎛1 Q SR = Q ⎜ 1 ⎝ R1 R2 R3C1 ⎞ )⎟ C2 ⎠ Q C2 Q C1 (34) ⎛ 1 R RC R3C1 ⎞ 231 − ⎟ ⎜ C2 R2C 2 ⎟ Q ⎜ R1 Q SR = − ⎜ ⎟ 2 2 ⎟ ⎜ + R2C1 ⎟ ⎜ R3C 2 ⎠ ⎝ 34 (35) MAY 2008 | embedded systems design europe | www.embedded.com/europe 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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