Embedded Systems Design Europe - May 2008 - (Page 32)

analog A second-order low pass function with two poles has a generic transfer function (Equation 5). Vout 1 = 2 Vin S 2 ω n + S Q ω n + 1 (5) Some designers use the “damping factor,” ζ, rather than Q. These are directly related by Equation 6: ζ= 1 2Q (6) SIMPLE PASSIVE FILTER Consider a basic L-C lowpass ﬁlter as a starting point shown in Figure 2. Equations 7 through 9 give the transfer function, natural frequency, and Q: 1 V out = 2 V in S ( C 1L1)+S(C1 L1 )+1 stants, which means there is nothing we can do within this circuit topology to improve sensitivities. Also, note that the natural frequency sensitivity to R is zero. This conﬁrms our earlier statement that R has no effect on the natural frequency. These results are useful in understanding what sensitivity equations mean. For example, if C is increased by a factor of four, the natural frequency decreases by a factor of two (Equation 14): 1 1 1 H SC 1( s ) = − s 2 L1C1 S L1C1 + S L1 R1 + 1 2 (16) 4 −1 2 = 1 4 1 2 = 1 2 (14) Similarly, the transfer function sensitivities to L and R are: 1 1 (7) ωn= 1 L 1C 1 C1 L1 (8) Q = R1 (9) Q, on the other hand, increases by a factor of two when C increases by four. As discussed earlier, we can also calculate the sensitivity of the transfer function to each of the component values. Using Equation 1, which deﬁnes sensitivity, the sensitivity of the transfer function for this circuit to C is: 1 1 H SL1( s ) = − s 2 L1C1 + s L1 R1 S 2 L1C1 + S L1 R1 + 1 (17) and: H SR1( s ) = s L1 R1 S 2 L1C1 + S L1 R1 + 1 (18) By inspection we see that the natural frequency is independent of the resistor value. Therefore, use the resistor to independently vary the Q, resulting in a family of curves shown in Figure 3. These will be familiar to anyone who has studied second-order systems. Using the resistor value speciﬁed in Figure 3 gives us the “critically damped” case with a Q of 0.707. Using the general equation for sensitivity, the sensitivities of Q and ω are: n H SC 1( s ) = C1 ∂H (s) H (s) ∂C1 (15) Without showing all the intermediate steps (I had to pull out my old calculus text book to get this right), the end result is: These results are far more complex than the Q and natural frequency sensitivities, and thus difﬁcult to put to use, especially since they are functions of both the components and frequency. We can plot these sensitivity functions using the nominal component values from the schematic. The plot is shown in Figure 4. S C1 =- S L1 = S R1 = 1 n S C1 = S L1n =- Q Q 1 2 (10) Q (11) ω ω 1 2 (12) n S R1 = 0 ω (13) These sensitivity equations are simply the exponents of the components in Equations 8 and 9. They are all con32 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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