# Circuit Sensitivity Analysis–An Important Tool for Analog Circuit Design: Part 2

In part one, we learned to calculate circuit sensitivities on simple passive circuits. We then did the same for two common active filters, the Sallen-Key lowpass filter, and the multiple feedback (MFB) lowpass filter.

Our analysis showed that the MFB filter provided circuit sensitivities limited to the range of –1 to +1, with most sensitivities being of magnitude less than half. This is similar to that of passive networks and is about “as good as it gets.” The Sallen-Key filter, however, had no such limits on sensitivity. In fact, it can result in very sensitive circuits, as we shall see.

**Lessons learned from sensitivity analysis **

Let's go back to our Sallen-Key filter and see if we can find ways to lower the sensitivities into the same range as the MFB filter. **Figure 1** is the same filter schematic for the Sallen-Key as Figure 5 in Part 1.

The transfer function for this filter, as we learned in Part 1, is:

(1)

where *K* is the DC gain, *K* =1+ *R* _{b } /*R* _{a } .

The filter characteristic equations are:

(2)

(3)

The sensitivities are:

(4)

(5)

(6)

(7)

(8)

(9)

Consider the *Q* sensitivity to *R* _{a } and *R* _{b } (Equation 7). Note that at *K* =1 the sensitivity goes to zero. This makes sense because a gain of one means that *R* _{b } is a short, and *R* _{a } is an open–there are no resistances to vary.

With the simplification of limiting the gain to 1, the equation for *Q* reduces to:

(10)

As a result of this simplification, our sensitivity equations for this circuit with *K* =1 degenerate to:

(11)

(12)

(13)

(14)

(15)

This looks quite promising. We only have three sensitivities that aren't a small constant. Two of these sensitivities, *Q* 's sensitivity to* R* _{1} and* R* _{3} , are complementary. If these two resistors are made to track closely, for example as a matched set of resistors on a substrate, then the two sensitivities would tend to cancel each other out. Better yet, if we choose* R* _{3} *=R* _{1} then the *Q* sensitivities to* R* _{1} and* R* _{3} go to zero.

Of course, these resistors will not be identical and, therefore, the sensitivities will not be exactly zero. Let's look at a case where these are five percent tolerance resistors (the real cheap resistors), and push them to worst case extremes. In this case the resulting sensitivity is:

(16)

This is small enough that we can consider it to be zero for most practical applications. Our sensitivity equations are now:

(17)

(18)

(19)

(20)

(21)

Note that the* Q* sensitivity to* R* _{1} and* R* _{3} is zero, but the sensitivity to *K* is not. This may seem odd as it is* R* _{1} and* R* _{3} that set the gain. But that is only in the ideal case of high loop gain. As the loop gain of the circuit decreases with frequency, the gain eventually deviates from that set by the two resistors.

With an ideal operational amplifier (op amp) connected as a follower, *K* will always be precisely one. But a real op amp has finite open-loop gain. At frequencies with sufficient loop gain, reduction of *K* will be negligible. At higher frequencies the loop gain will be low enough that reduction in *K* will noticeably affect *Q* .

So, as long has we have chosen op amps with sufficient open-loop gain over our frequency range, we can assume *K* =1, and the equations further simplify to:

(22)

(23)

(24)

(25)

(26)

We have reduced all sensitivities to half or zero, except for* Q* sensitivity to* K* –the parameter that varies the least. You can't do much better than that!

What we have done, in a sense, is similar to noise shaping in a sigma-delta data converter where we move most of the noise out of the band of interest. Here we move the sensitivity into the inherently most stable parameter.

Note that the equation for* Q* has also simplified to:

(27)

We can rearrange this equation to get* C* _{2} in terms of* C* _{4} :

(28)

We can substitute this expression into the equation for w_{n } and solve for *C* _{4} :

(29)

where* R* _{1} =* R* _{3} =* R* .

Substituting this result into the equation for *C* _{2} (Equation 28), we get:

(30)

These last two equations are useful when designing a filter of this type. One just selects a resistor value and then uses these two equations to calculate the capacitor values.

**Practical examples**

Let's consider a couple of practical examples. The first is a simple second order, lowpass filter that we will implement using a “cookbook” approach^{2} where the resistors are of equal value, the capacitors are made equal, and *Q* is set with the gain. For our example the design parameters are: *f* _{n } = 4.8kHz and *Q* =1.

Choosing 1nF (.001uF) for the capacitors, we can calculate that the resistors should be 33.2k using the closest standard one percent value. Now we must adjust the DC gain to set *Q* =1. This requires a gain of two. The resulting circuit is shown in **Figure 2** .

Now let's implement a filter with the same design goals using our simplified circuit starting with the same 33.2k resistor value. The capacitors are calculated to be; *C* _{2} = 2nF and *C* _{4} = 500pF. The schematic for this circuit is shown in **Figure 3** :

The nominal responses for these two circuits, **Figure 4** , are the same except for the expected gain difference of two (6dB).

Now that we have established the component values, let's calculate the sensitivities associated with both circuits (Table 1).

All the natural frequency sensitivities match between the two methods but, as expected, *Q* sensitivities to each of the components are better with the new method. Only the sensitivity of *Q* to *K* is worse with the new method but, as discussed earlier, *K* does not vary as long as we are using an adequate op amp.

But how much better in practice will this new method be over the cookbook method? An easy way to evaluate the two circuits is to run AC Monte Carlo analyses on both circuits. A Monte Carlo analysis runs the same analysis multiple times (AC analysis in our case), but randomizes each component value within it's tolerance for each run. The result is similar to a production run.

Let's use one percent resistors and five percent capacitors, and a flat (uniform) distribution.

**Figure 5** shows that the cookbook version of the filter has about 1.5dB variation near the natural frequency, while the unity-gain version has only 0.5dB of variation.

As one might expect, we don't get “something for nothing” with this low-sensitivity version of the Sallen-Key filter. We cannot get gain out of this circuit. Secondly, there is a practical limit to the magnitude of *Q* . To get high *Q* s, the capacitor value ratio gets very high making it hard to implement with practical capacitor values. We can determine this relationship very easily from the equation for *Q* , which has been greatly simplified to:

(31a)

or:

(31b)

Thus, the capacitors are related as follows:

(32a)

or:

(32b)

The capacitor ratio increases as the square of *Q* .

While there are many types of capacitors providing values from a picoFarad to many Farads, most types of capacitors have voltage coefficients that cause excessive distortion in most signal processing applications. Many types also vary too much over temperature, time and other conditions. For decent precision over environmental conditions along with good linearity, the most cost-effective and volumetrically efficient choice is COG (or NPO) ceramic. Many film capacitors also can be used if you have the space and dollar budget for them. While larger values are available today, there is a substantial price premium for them. A good limit for keeping costs down is 0.01uF (10nF).

At the lower end, the limitation is circuit stray capacitances. A good rule of thumb is to not stray too much below 100pF, maybe 50pF. Using equation 32b, with 10nF for* C* _{2} and 100pF for* C* _{4} we come up with a practical limit of five for* Q* . This is not a hard limit; just a rule of thumb. Of course, if you can get good capacitors larger than 10nF within budget, then this limit on* Q* increases.

Can we reduce the capacitor ratio in this topology? If we remove the restriction of* R* _{1} *=R* _{3} , can we improve the capacitor ratios? Recall that we will not take a large sensitivity hit by removing this restriction on the resistor value, and we will suffer no increase in sensitivity if the resistors track each other.

Recall that the equation for* Q* is:

(33)

Let's call* R* _{3} /R_{1} = *n* and* C* _{4} */C* _{2} = *m* . Then the equation for* Q* becomes:

(34)

If* R* _{3} *=R* _{1} gives us the highest* Q* , then the derivative of this equation should be zero when* R* _{3} *=R* _{1} or *n* =1. Since there are no terms in the numerator, the denominator would be at a minimum in this case and, thus, its derivative would also be zero. So let's take the derivative of the denominator with respect to *n* , which is:

(35)

We have either a maximum or minimum* Q* at* R* _{1} *=R* _{3} . Simply plug in ratios of *n* =2 and then *n* =1/2 to show that this is a maximum, indicating that keeping the resistors of equal value is the optimal choice.

**Back to multiple feedback filters**

Recall that in Part 1, we looked at both the Sallen-Key and MFB lowpass filters and found that the MFB filter always had natural frequency and* Q* sensitivities to the components with magnitude less than one. Now that we know how to make Sallen-Key filters with similar or perhaps even lower sensitivities, we compare an MFB implementation of the same nominal filter characteristics as the Sallen-Key filters. **Figure 6** is a unity gain MFB lowpass filter with the same natural frequency, and* Q* as the Sallen-Key filters we designed earlier.

We would expect the nominal frequency response to be the same as for the Sallen-Key filter unity gain version. **Figure 7a** shows the frequency responses of both filters. **Figure 7b** is a close-up at the peak showing only .02dB difference.

As we did when comparing the two different Sallen-Key implementations, let's run Monte Carlo simulations on the MFB filter and compare it to the better of the Sallen-Key filters. **Figures 8a and b.**

Notice that at low frequencies the Sallen-Key has significantly less variation. This is because at DC the Sallen-Key filter has virtually no variation. Its gain is one, while the MFB filter has a gain dependent of two real resistors. If we were to cancel out the low frequency variation, we would find that the two circuits have very similar high frequency variation.

**Implementing a higher order filter**

Now let's implement a significantly more complicated filter using the topologies previously discussed using the following specifications:

1. Seventh-order Chebychev

2. 0.05dB nominal in-band ripple

3. 10kHz nominal passband (highest frequency with at most 0.05dB attenuation):

4. Gain = 10

The poles required to implement such a filter are listed in Table 2:

The schematic in **Figure 9** shows this filter implemented in three different ways:

1. Multiple Feedback (MFB)

2. Sallen-Key with unity gain and equal-value resistors: low sensitivity variation.

3. Sallen-Key with* R=R* and* C* =*C* : cookbook variation.

The cookbook Sallen-Key version (at the bottom of **Figure 9** ) has capacitor values from 1nF to 2nF, while the low-sensitivity Sallen-Key version capacitors vary from 82pF to 10nF. The MFB capacitors vary from 68pF to 12nF, a bit more than the low-sensitivity Sallen-Key. As with the low-sensitivity Sallen-Key, this topology also requires that the capacitor values spread further apart as* Q* increases. We can get the capacitor spread to be similar to the low-sensitivity Sallen-Key. To do this requires that we decrease the DC gain of the high*-Q* stage to about –20dB. This gain is impractical for most applications.

**Figures 10 through 12** show the variations in AC Frequency response that we get using one percent resistors and five percent capacitors for all three filter circuits by running Monte Carlo analyses with 300 runs.

The low-sensitivity Sallen-Key and MFB versions exhibit about 3dB of gain variation around the natural frequency, or about ± 15 percent. The cookbook version has over 14dB gain variation meaning the gain of any two units in production would could differ by as much as a factor of five!

**Op amp open loop gain considerations**

We made several statements like “…as long as we have chosen op amps with sufficient open-loop gain over our frequency range….” What happens if you use an op amp with insufficient loop gain? What loop gain is sufficient? Even a follower's gain will decrease with insufficient loop gain.

Let's model an otherwise ideal op amp with 120dB low frequency open-loop gain and a single dominant pole to role off the open-loop gain. We make the model, **Figure 13** such that the GBWP can be varied for our experiment.

Implementing the high*-Q* stage of the low-sensitivity Sallen-Key filter with this simplified op amp model, the frequency response changes with the different GBWP versions (**Figure 14** ).

Both 10M and 100M GBWP versions seem to be identical to the ideal op amp, while the 1M op amp seems close and the 100k GBWP is clearly inadequate. Let's take a closer look in **Figure 15** .

This close-up shows that the 10M GBWP op amp deviates slightly from the ideal, while 100M still seems identical to the ideal.

As with passive component variations, we can use Monte Carlo analysis to vary the GBWP. This model allows us to readily modify the GBWP simply by varying* R* _{3} . Real op amp GBWPs tend to vary ±30 percent. With the tolerance of* R* _{3} set to ±30 percent, we run Monte Carlo analyses on the 1M, 10M and 100M GBWP op amps.

The 100MHz GBWP op amp shows no perceivable difference from the ideal op amp. The 10MHz op amp deviates from the ideal by about 0.25dB with about 0.05dB variation. The 1MHz op amp peak deviates in frequency from the ideal by about six percent, and about 3dB in amplitude with about 0.7dB variation in amplitude.

Let's also look at the high*-Q* stage of the MFB implementation in this same way. We'll skip right to the Monte Carlo analysis. **Figure 17** .

As with the Sallen-Key circuit, the 100MHz GBWP op amp shows no perceivable difference from the ideal op amp. The 10MHz op amp deviates from the ideal by about 0.35 dB with about 0.1 dB variation, slightly more than with the Sallen-Key. The 1MHz op amp peak deviates in frequency from ideal by about nine percent and about 3dB in amplitude, with about 0.9dB in amplitude variation, again, slightly more than with the Sallen-Key.

The earlier Monte Carlo analyses show that the MFB and low-sensitivity Sallen-Key filters frequency responses varied with their passive components by about 3dB. Therefore, the 1MHz op amp would cause a substantial shift in the response and add moderately to the variations. The 10MHz op amp would not significantly modify the response over the ideal op amp. While some applications may be able to allow the increased variations caused by the 1MHz op amps, a 10MHz GBWP should be adequate for any application using this filter.

**Overcoming capacitor value limitations for high-** **Q** ** stages**

As discussed before, the low-sensitivity Sallen-Key and the MFB topologies have allowed us to implement this filter with low sensitivities. The main trade-off is that the capacitor values for the high-*Q* stages vary over more than a 100:1 range, from 82pF to 10nF for the Sallen-Key and 68pF to 12nF for the MFB. If we needed a higher* Q* stage, the capacitor values for both topologies would spread even further, making it difficult to implement using capacitors with reasonable properties at reasonable prices.

In such cases, the high-*Q* stages (usually only one stage) can be implemented with other, more complex topologies, which can provide higher* Q* stages while using reasonable capacitor values. There are several three-op amp topologies that can do the job. These include the KHN (Kerwin-Huelsman-Newcomb or State-Variable), the Tow-Thomas and Akerberg-Mossberg topologies.^{1,2,3,4}

Interestingly, despite their increased complexity and ability to implement higher* Q* stages with practical component values, they are no better in terms of component sensitivity than the MFB and low-sensitivity Sallen-Key. While tripling the number of op amps for a single stage is not desirable, typically this only has to be done for one stage. The rest can be done with the topologies we have been discussing.

By being aware of component sensitivity for filter circuits we can at least chose to tighten up on tolerances for components to which the circuit response is most sensitive. By using this sensitivity information intelligently we can often configure circuits to be inherently less sensitive. Understanding the limitations of these less sensitive circuits, we can chose to use more complex circuit topologies only when necessary while using simpler, but more limited in usability, insensitive circuits for the majority of needs.

The definition of and methods for using circuit sensitivity presented here have uses beyond that of analog filters. Similar insights and improvements in designs can be made for virtually any circuit.

*For the last five years, Mark Fortunato has been the Southwest Analog Field Applications Manager for Texas Instruments. When not working with customers, Mark enjoys reading, coaching youth sports and listening to his son perform live Jazz and Latin music. Mr. Fortunato has not written a line of code since the fall of 1992.*

**Endnotes:**

1. 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.

2. Huelsman, L.P. and Allen, P.E., *Introduction to the Theory and Design of Active Filters* , McGraw-Hill, New York, 1980.

3. Budak, Aram, *Passive and Active Network Analysis and Synthesis* , Houghton Mifflin company, Boston, 1974.

4. Ghausi, M.S. and Laker, K.R., *Modern filter Design: Active RC and Switched Capacitor* , Prentice-Hall, Englewood Cliffs, N.J., 1981.

5. 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_07presentations