All circuits, analog or digital, have characteristics dependent on the values of their component parts. Invariably, these parts are non-ideal in many ways. Even their most fundamental characteristics, for example a resistor's resistance, will be at least slightly different from the specified value. These characteristics also can vary over time and environmental conditions.
The wise circuit designer learns how to analyze the sensitivities of circuits' critical characteristics to the variations in the components. With this knowledge, the designer can better decide how to balance any trade offs between performance and cost. Deeper use of sensitivity analysis can lead designers to modify circuit topologies, or make completely different choices in order to optimize these trade offs.
Generally, circuit sensitivity analysis is not part of the bachelors curriculum in most universities' electrical engineering programs. Fortunately, this important topic is easy to learn and use and is covered in circuit design books, especially those focusing on filter design.
Circuit sensitivity defined
A simple, if imprecise, definition of circuit sensitivity is how much a particular circuit characteristic changes as a particular component value varies. This can be any circuit characteristic: amplifier gain; bus receiver input impedance; an RF port's voltage standing wave ratio; or a digital gate threshold voltage. In this article, we use analog filters to explore sensitivity analysis as this information is often critical to good filter design. We will see that it is fairly easy to extend sensitivity analysis to other types of circuits.
Equation 1 is the mathematical definition of circuit sensitivity:
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:
Use Equation 1 to calculate the sensitivity of the DC transfer function to R 1 and R 2 :
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 1 for Equation 3 and R 2 for Equation 4.
These sensitivity equations are identical except for the sign. In Equation 3, the sensitivity to R 1 is negative. As the negative sign implies, when R 1 increases, the transfer function decreases. When R 2 increases, the transfer function also increases, which is expected since Equation 4 (the sensitivity to R 2 ) is positive.
When R 1 is substantially larger than R 2 , the equations reduce to –R 1 /R 1 = –1 and R 1 /R 1 = 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 1 = 1000*R 2 . Here the transfer function is 1/1001 = 999e–3. If R 2 is doubled, the transfer becomes 2/1002 = 1.996e–3, which is 1.998 times the earlier value, nearly double.
Similarly, if R 1 is doubled, the transfer function decreases by nearly a factor of two. Doubling R 1 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 2 is substantially larger than R 1 , results in the sensitivity equations reducing to zero for R 1 = 0 and R 2 = ∞. 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 2 = 1000*R 1 , the transfer function is 1000/1001 = 0.999. If R 2 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.
Similarly, if we double R 1 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 1 = R 2 , 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 2 by 1%. Now the transfer function becomes 1/2.01 = 0.4975, which is a 0.5% reduction. Similarly increase R 1 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.
Sensitivity 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-and-simulation 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 (wn ) and of the quality factory (Q ) for the pole-pairs, and zero-pairs for each second order section.
A second-order lowpass function with two poles has a generic transfer function (Equation 5).
Some designers use the “damping factor,” ζ, rather than Q . These are directly related by Equation 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 :
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 ωn are:
These sensitivity equations are simply the exponents of the components in Equations 8 and 9. They are all constants, which means there is nothing we can do within this circuit topology to improve sensitivities. Also, note that the natural frequency sensitivity to R 1 is zero. This confirms our earlier statement that R 1 has no effect on the natural frequency.
These results are useful in understanding what sensitivity equations mean. For example, if C 1 is increased by a factor of four, the natural frequency decreases by a factor of two (Equation 14):
Q , on the other hand, increases by a factor of two when C 1 increases by four.
Transfer function sensitivity
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 1 is:
Without showing all the intermediate steps (I had to pull out my old calculus text book to get this right), the end result is:
Similarly, the transfer function sensitivities to L 1 and R 1 are:
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.
A lot of good information is in these plots, but we get all we need for most situations with the far simpler Q and ωn sensitivities.
Passive filters, like the RLC lowpass filter in Figure 3 always have Q and ωn 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.
Active filter example: The Sallen-Key Filter
We'll use the venerable and ubiquitous Sallen-Key for our first active filter. The Sallen-Key lowpass filter shown in Figure 5 first described over 50 years ago4 is one of the most common filter topologies.
The Sallen-Key lowpass filter transfer function is:
where K is the DC gain, K = 1+ R b /R a .
The filter characteristic equations are:
Using the general equation for sensitivity, the sensitivities of Q and ωn are:
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 a and R b (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 b to change the Q without changing the natural frequency.
Let's vary R b from 1k to 19.9k. At R b = 20k, the equation for Q goes to infinity and should be avoided.
Since R b 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.
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:
The natural frequency and Q are given by:
The sensitivity equations are:
Equation 30 tells us that, once again we have a component (R 1 ) that can be varied to change the Q without affecting the natural frequency.
Equations 31 and 32indicate that all sensitivities to the capacitors are –0.5 or 0.5.
Equations 34, 35,and 36are quite complex. Unfortunately, many textbooks leave these equations in this form.1 However, if we simply plug in Q from Equation 29to each of these, we greatly simplify these equations:
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 36must be less than the denominator as R 1 paralleled with any other resistors necessarily is smaller than R 1 itself. Similarly, numerators in Equations 37and 38must be smaller than the denominators. All the terms are the same but, while a difference is taken in the numerator, a sum is taken in the denominator. The half term out front means that Q sensitivities to R 2 and R 3 are actually less than 0.5.
As discussed, having all sensitivities with magnitudes of one or less is as good as it usually gets. Does this mean that the MFB filter is always less sensitive than the Sallen-Key? In many instances it does. Digging deeper into the math, however, we may find some surprises.
In Part 2, which will be posted online at www.embedded.com, we will delve deeper into the mathematics of the Sallen-Key filter, and find that we can implement highly insensitive filters with this topology. We will also look at the effects of op amp bandwidth, as well as how to use Monte Carlo analysis in determining how much a filter's transfer function will vary in production.
For the last five years, Mark Fortunato has been an analog field applications manager for Texas Instruments where his staff covers Southwest USA. He has a BS in electrical engineering from CalTech and works primarily on DC to 200 GHz.
1. Huelsman, L.P. and Allen, P.E. Introduction to the Theory and Design of Active Filters . McGraw-Hill, New York, 1980.
2. Budak, Aram. Passive and Active Network Analysis and Synthesis . Houghton Mifflin Company, Boston, 1974.
3. Ghausi, M.S. and Laker, K.R. Modern filter Design: Active RC and Switched Capacitor . Prentice-Hall, Englewood Cliffs, N.J., 1981.
4. Sallen, R.P. and Key, E.L. “A Practical Method of Designing Active Filters,” IRE Transactions on Circuit Theory , vol. CT-2, pp.74-85, March 1955.
Fortunato, M. “Circuit Sensitivity; With Emphasis On Analog Filters,” Texas Instruments Developer Conference 2007, March 2007: http://focus.ti.com/general/docs/tidc/general.tsp?templateId=6180&navigationId=12622&path=templatedata/cm/tidcgeneral/data/am_landing/ww_07 presentations.