Embedded Systems Design Europe - May 2008 - (Page 31)

analog Equation 1 is the mathematical deﬁnition of circuit sensitivity: ⎧ Δy ⎫ ⎪ lim ⎪ y ⎪ x ∂y ⎪ Sxy = ⎨ ⎬= Δ x → 0 ⎪ Δ x ⎪ y ∂x ⎪x⎪ ⎭ ⎩ (1) Where S is the sensitivity, X is the changing component, and Y is the circuit characteristic we wish to evaluate as X is varied. The middle part of this equation makes intuitive sense. It is the percentage that the dependent variable changes, Δy/y, relative to the percentage that the independent variable changes, Δx/x. Taking the limit as the change in x goes to zero evaluates this ratio for minute variations. This equation is so general that it can be used to evaluate the variation of any circuit parameter, relative to a change in any circuit component value. Endnotes 1, 2, and 3 have detailed treatments of sensitivity and derive many of the equations we will use. SIMPLE CIRCUIT EXAMPLE Consider the simple circuit shown in Figure 1—a voltage divider. Equation 2 is the DC transfer function: TDC = R2 Vout = Vin R1 + R2 (2) Use Equation 1 to calculate the sensitivity of the DC transfer function to R and R : 1 2 tion 3, the sensitivity to R is negative. As the negative sign implies, when R increases, the transfer function decreases. When R increases, the transfer function also increases, which is expected since Equation 4 (the sensitivity to R ) is positive. When R is substantially larger than R , the equations reduce to –R /R = –1 and R /R = 1. This implies that the transfer function should change by very nearly 1% for every 1% variation in either resistor under these conditions. Take the case where R = 1000*R . Here the transfer function is 1/1001 = 999e–3. If R is doubled, the transfer becomes 2/1002 = 1.996e–3, which is 1.998 times the earlier value, nearly double. Similarly, if R is doubled, the transfer function decreases by nearly a factor of two. Doubling R results in a transfer function of 1/2001 = 0.4998e–3, which is 0.498 times the earlier value—nearly a factor of two less. The other extreme, when R is substantially larger than R , results in the sensitivity equations reducing to zero for R = 0 and R = ∞. For values that can be realized, the sensitivities will be near zero. Thus, the transfer function should change very little as either resistor is varied. Where R = 1000*R , the transfer function is 1000/1001 = 0.999. If R is doubled, this becomes 2000/2001 = 0.9995, only a 0.05% change in the transfer function for a 100% change in the component value. 1 1 2 2 1 2 1 1 1 1 1 2 2 1 1 2 1 1 2 2 1 2 Similarly, if we double R instead, the transfer function becomes 1000/1002 = 0.998, only a 0.1% change in the transfer function for a 100% change in the component value. If R = R , the transfer function is 0.5 and the sensitivities are -0.5 and 0.5. You would expect the transfer function to change 0.5% for every 1% change in either resistor. Let’s increase R by 1%. Now the transfer function becomes 1/2.01 = 0.4975, which is a 0.5% reduction. Similarly increase R by 1% results in the transfer function being 1.01/2.01 = 5.025, or an increase of 0.5%. This is about as far as we can go using simple circuits with resistors. Now let’s include reactive elements, inductors and capacitors. This will create AC transfer functions that vary with frequency, such as ﬁlters. 1 1 2 2 1 ST 1 = − DC R R1 R1 + R2 (3) ST 2 = DC R R1 R1 + R2 (4) What do these equations mean? Recall that sensitivity is the percentage that the dependent variable, in this case the DC transfer function, changes relative to the independent variable, R for Equation 3 and R for Equation 4. These sensitivity equations are identical except for the sign. In Equa1 2 ANALYSIS WITH FILTERS As stated earlier, all circuits have characteristics that are functions of the component values of the circuit. Filter characteristics are dependent upon these component values. Some ﬁlters are more sensitive to these component value variations than others. Knowing how much a circuit’s behavior changes with a component variation, the sensitivity of the circuit to the component, is important for proper selection of components, as well as choice of circuit topology. These sensitivities should be evaluated in the paper-andsimulation design phase to ensure an adequate ﬁlter topology is used and that the components are chosen with the proper speciﬁcations. There are many ways to look at the ﬁlter’s sensitivity to the variations in its components. One way is to evaluate how the overall AC transfer function behaves as a component value is varied. Similarly, you can evaluate individual pole and zero sensitivities. A common way, especially when working with ﬁlters split into second order sections, is to look at the sensitivity of the natural frequency (ω ) and of the quality factory (Q) for the pole-pairs, and zero-pairs for each second order section. n www.embedded.com/europe | embedded systems design europe | MAY 2008 31 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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