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A new set of Sallen-Key filter equations

New design equations let you pick equal capacitor values and solve for the resistors to achieve any desired gain in one stage.

By Martin Cano, National Semiconductor -- EDN, 10/1/2009

Most references for calculating the components of a second-order Sallen-Key section given the complex conjugate pole locations do not present a simple closed form of equations to realize any gain with equal capacitor values. The literature suggests you either set the section gain to 2 or try iterative numerical computations given a target gain and capacitor ratio. These iterative guess methods not only are time-consuming but also can lead to marginal, erroneous, and unstable solutions. Another approach requires that the capacitor ratio be sized according to the gain value. This aggravates the problem of capacitor sizing and matching.

Here, simple equations for arbitrary gain realizations with equafl capacitors given the complex conjugate pole pair locations are derived and presented and the underlying dependence between gain and resistor ratio is pointed out, along with a simple equation for the evaluation of the corresponding section feedback-loop stability. By having the ability to assign any gain per stage, section gain assignment can be optimized in multistage filters and gain stages in the signal path can be eliminated. Simple equations for the resistors subject to equal capacitors can make integrated programmable realizations easier to compute. Filter realizations that require the use of current feedback amplifiers can benefit from this added degree of freedom. Having a simple metric for the loop stability can help in the design of high-reliability systems.

Introduction

When designing a filter as part of a system, the choice of filter characteristics is a key decision that may affect many things. This has led to the development of optimized approximations to the ideal filter, such as Tchebysheff, Butterworth, or Bessel, that engineers consider classical filter shapes. Each shape is a mathematical optimization for a given performance characteristic. You would design in these filter types because you know the dynamic response of the filter.

When you implement these filter functions (or any prescribed shape that is characterized by unique complex conjugate pole pair locations), it is important that each filter section reproduces the transfer function specified by the pole locations. Many texts, cookbooks, handbooks, and application notes offer circuit topologies and accompanying equations that allow you to either create or use relationships between the pole pair locations and the circuit passive component values.

The Sallen-Key filter is also known as the (VCVS) voltage-controlled voltage-source filter. It is a popular active filter topology; it also has drawbacks. One of those drawbacks is that the math needed to create relationships between pole locations, section gain, and circuit component values is not straightforward. Usually you set the gain to a particular value in order to simplify the math. Since any change in the stage gain will ruin the filter response shape, you must provide any additional gain with another stage. You can use iterative guess processes for finding values, as some literature recommends. These steps can be time-consuming and often yield impractical solutions, sometimes even unstable ones. However, in all of these references there is no closed-form solution that will hold true for arbitrary gain.

The whole filter response issue involves your creating a relationship between the filter parameters A, α, and β from the transfer function side of the equation:

(1)

and

(2)

ω_n²=α²+β² and A, R₁, R₂, C₁, and C₂ for any value of A, such that the denominators became equal in equations 1 and 2.

This article presents explicit design equations that guarantee a valid solution the first time, free of iterative guesswork and restrictions on gain. The design equations also offer the gain margin of a passive noninverting loop. This is a condition for stability if you use the circuit to implement a particular section gain and pole location combination.

The design equations

The pole locations of a second-order section are given by a complex conjugate pole pair with normalized pole locations α±β and section gain A (Figure 1). The derivation of the equations that solve for R₁, R₂, C₁, and C₂ is presented as follows:

For a section with pole locations at α±jβ and a section gain A,

1)      Select C=C₁=C₂ and calculate parameter

(3)

where A_MINis the minimum gain for which a solution with equal capacitors exists and

(4)

2)      For the case where A≥A_MIN, calculate the parameter M, the resistor ratio:

(5)

With M, calculate R₁, R₂, and G_MARGIN:

(6)

(7)

(8)

3)    For the case where the desired gain A is less than the parameter A_MIN, several options can be used. A simple method is to set the gain to A≥2 and proceed with equations 3 through 8. Setting the gain to 2 reduces to the VCVS uniform capacitor structure solution proposed by Williams and Taylor (Reference 1). We will show later in the text that the solution proposed by them is simply a special case of the derivation here. To accommodate the desired overall gain (now designated by the variable A' since A, the amplifier gain, was set to 2), you split resistor R₁ into two resistors, R_1A and R_1B (Figure 2), and compute the values with equations 10 and 11:

(9)

(10)

(11)

Note there is an explicit closed form to find component values that relate any section gain A to any complex conjugate pole pair (all-pole second-order section) given by α±jβ for the Sallen-Key filter. It is no longer necessary to use tabulated gain/section relationships or single values of gain for the Sallen-Key filter. This gives you greater flexibility in assigning gain in filter sections for low-noise design and improves the dynamic range in multistage filters. You no longer need to use iterative guess methods to arrive at the values. The derivation of the resistor ratio M is carried out in the section below titled "Derivation of the resistor ratio M."

Equation 8 provides a unique way to relate A, α, and β to R₁, R₂, and a chosen C=C₁=C₂. As the values and therefore the relationship between these parameters change, so will a very important aspect of this circuit, its relative frequency stability. As will be shown in the next section on loop stability, the gain margin is the most convenient way to evaluate the relative stability, and for these constraints, it depends on A and M. Therefore, this change in relative stability is both the penalty and advantage of having a solution in A and M.

Loop stability criterion and gain margin formula

The loop stability is important because it is a measure of how far the transfer function is from instability. If you change the gain, stability is something that needs to be tracked. Solutions presented without quantifying stability are incomplete. With iterative calculations, you can arrive at solutions that satisfy the iteration steps yet fail to satisfy stability requirements.

It is important to clarify that the stability discussion here refers to the loop formed by the passive feedback in series with the amplifier, where the amplifier is assumed as a closed-loop circuit. The stability discussion does not refer to the feedback loop within the amplifier A formed by using the classic noninverting op amp. It is assumed that this loop is of an adequate stability figure and can be studied separately, first assuming the passives associated with the filter are absent, and then modeling the destabilizing effects as the filter passives are added. A detailed evaluation of the factors affecting the amplifier stability is beyond the scope of this treatment. It should suffice to say that load capacitance tends to destabilize amplifiers, while series resistance tends to stabilize them in the presence of those load capacitances. As long as the resultant passive network doesn't present a large capacitive load to the amplifier, then stability effects due to the amplifier loop will be minimal.

The overall transfer function of the filter H_F(s) can be decomposed into the components H_FW(s) representing the forward path and H_LOOP(s) representing the feedback path from the amplifier (Figure 3). The transfer functions H_LOOP(s) and H_FW(s) are

(12)

(13)

The overall filter transfer function H_F(s) is given by

(14)

Indeed, if we substitute equations 12 and 13 into Equation 14, we get the familiar transfer function for the Sallen Key:

(15)

The circuit to determine H_FW(s) represents the voltage transfer function assuming no feedback (Figure 4).

To get the transfer function H_LOOP(s), the feedback loop is broken (Figure 5), with the input grounded and the new input being the point where the feedback was connected (Figure 6). The transfer function H_LOOP(s) is the voltage transfer function from V_INLOOP to the input of the amplifier.

Understanding the gain and phase of H_LOOP(s) is key (Figure 7). As can be seen from the closed-loop expression for the filter H_F(s),

(16)

If the condition is reached such that |AH_LOOP(s)|=1 while /H_LOOP(s)=0, the denominator of Equation 16 will blow up and the circuit will be or will become unstable. By looking at Figure 6, we can see that H_LOOP(s) is a passive bandpass filter that peaks at the center frequency ω_n, which is the same as the cutoff frequency of the closed-loop lowpass filter. At this frequency,

(17)

while the magnitude

(18)

So now the condition for stability is

(19)

As long as the peak at ω_n is less than 0 dB, the circuit will be stable. The distance to 0 dB from |AH_LOOP(ω_n)| is the gain margin.

If the pole location and/or gain of a filter stage changes, the gain margin will also change, and this should be kept track of through the use of the design equations. You can calculate the gain margin given the constraint of equal capacitors and the given resistor ratio M. This will help you decide if a different gain will yield a better gain margin or whether or not to use this topology if the stability figure is inadequate.

The gain margin can be thought of as the amount of additional gain needed to shift the magnitude curve such that it hits 0 dB when phase equals 0 degrees, which happens at ω_n². When the circuit undergoes that gain change, you get an unstable situation as shown in Equation 16. Therefore, it becomes important to limit the factors that affect gain change. For example, resistor tolerances and temperature coefficients become important in guaranteeing a certain level of stability.

Development/derivation of the resistor ratio M

The determination of the right combinations of α, β, and A from Equation 1 as well as A, R₁, R₂, C₁, and C₂ from Equation 2 that will make the denominators of equations 1 and 2 become equal is not a straightforward task. This computational difficulty has led to a relatively limited use of gain values in filter-stage design. One popular configuration is the use of a gain of 2, with all capacitors being equal (Reference 1). Another method uses tabulation, where the gain is a particular value in order for the transfer function to be realized with equal capacitors. The development in this section will show that both of these cases are specific instances of the general relationship that will be derived here. We will define

·         The resistor ratio M=R₂/R₁

·         The capacitor ratio K=C₂/C₁

·         The case of equal capacitors K=1

Equation 1 is the standard equation for a second-order transfer function (complex conjugate pole pair).

(20)

where

(21)

The equation for the circuit of Figure 1 is given by

(22)

Equating coefficients of like powers of s in the denominator of Equation 22 with those of the denominator of Equation 20 yields the following relations:

(23)

(24)

The components are expressed in terms of resistor and capacitor ratios:

(25)

(26)

Plugging Equation 25 and Equation 26 into Equation 23 and then rearranging yields Equation 27:

(27)

Plugging equations 25 and 26 into Equation 24 and then rearranging yields Equation 28:

(28)

Equating equations 27 and 28 yields Equation 29 below:

(29)

Equation 29 has been derived from first principles and is in terms of pole locations and component ratios. For this problem we will concern ourselves with the case of K=1, implying that

(30)

Substituting K=1 into Equation 29 yields

(31)

At this point the equation has taken on the form of a quadratic in M, the resistor ratio. This quadratic is of the form shown below:

(32)

We recall that a quadratic of the form of Equation 32 has roots (solutions; values of x that are valid for the equation) given by

(33)

Drawing on the above analogy, we make the following substitutions:

(34)

(35)

(36)

(37)

Plugging equations 34, 35, 36, and 37 into Equation 33, we finally arrive at the roots of the quadratic in M (the resistor ratio) given the convenient constraint that K=1, or that C₂=C₁=C:

(38)

This expresses the dependence of the resistor ratio to the pole zero filter parameters and the gain of the section. Before proceeding further, it's worthwhile to make a few observations.

Observations and simplifications of the equation for M

For values of A where the quantity within the square root in Equation 38 is negative, the roots of M will be complex and therefore unrealizable. The threshold value of gain where this happens is when the term under the square root is 0, expressed below:

(39)

The value of gain when this occurs is when A=A_MIN. Substituting this into Equation 39 yields:

(40)

(41)

The resultant value of M is given below:

(42)

When the gain hits another critical value of A=2, there are two roots of M, so we must introduce notation to that effect:

(43)

Plugging A=2 into Equation 38 yields an M₁ and M₂ of

(44)

and

(45)

It is possible to express M in terms of A_MIN by substituting Equation 40 into a rearranged form of Equation 38. This yields

(46)

(47)

(48)

Since there are two possible values of M, we must determine if M₁or M₂ is the correct one. To do this, we recall the stability criterion, which was that

(49)

This occurs with the solution yielded by the M=M₁ solution, which corresponds to the positive radical. To see this, we note that from Equation 49,

(50)

(51)

Substituting the original equation for M (Equation 43),

(52)

Simplifying

(53)

Evaluating the first option of the plus-minus inequality, we note that with increasing gain A, the inequality below would be true:

(54)

However, if we evaluate the inequality for the negative term of the inequality, we note that for increasing gain above A=2, the term under the radical would result in a big negative number that would violate the inequality below:

(55)

In the range A_MIN<A<2, the inequality does hold but represents an undesirable situation because the solution M₂ approaches 0 in the vicinity of A=2, which corresponds to a higher component spread than the solution yielded by M₁ (Figure 8).

Therefore, we conclude that the positive term in the plus-minus term of the resistor ratio M is used. The behavior of the resistor ratio with gain is shown in Figure 9.

Experimental setup

The practical implications of these equations is that it is now possible, using the Sallen-Key/VCVS filter section, to assign any gain and be able to reproduce the intended shape of the section pole pair while keeping both capacitors at the same value. We can do this with a predictable and quantifiably stable solution, just by plugging in to simple equations. To demonstrate this, the equations were used to design two second-order filters, a Bessel and a Chebyshev, with a frequency cutoff of 10 kHz. Both filters were designed at four arbitrary gains. The gain values chosen are 1, 2, 3, and 5. The circuit used to evaluate the filters is shown in Figure 10 for the gain values of 2, 3, and 5. Tables 1 and 2 show the calculated resistor values for each gain setting. Using this method to design filters for gains below the corresponding A_MIN requires a slightly modified version of the circuit of Figure 10. This is the case for the gain of 1 case in this example. The modified circuit is shown in Figure 11 and the corresponding resistor values are shown in tables 3 and 4.

The two capacitors, C₁ and C₂, were set equal to a value of 1 nF, while the feedback resistor was set to 1 kΩ. All the filters were generated by changing R₁, R₂, and R_G only, while C₁, C₂, and R_F were left unchanged. In the case of the gain of 1 circuit, R₁ is substituted by R_1A and R_1B. The op amp used was the LMH6645, which has a 55-MHz gain bandwidth product. The supply rails were set to ±5V. Each filter was tested using the HP/Agilent 3562A dynamic signal analyzer (Figure 12). The results are shown in Figure 13.

Conclusion

A closed-form solution for the components of the Sallen Key given any value of gain with equal capacitor capacitor values has not been available in the literature accessible to the practicing engineer. This work provides simple design equations to let you determine resistor values with any filter section gain, and equal capacitor values. It also adds insight into the topology's relative stability, as well as a simple equation to evaluate gain margin. The gain margin tradeoff can be a very useful figure to circuit and system designers because it relates stability to gain changes. For example, you can improve high-reliability applications by figuring out how much gain change the design will tolerate due to aging, radiation, and other factors before it will start oscillating. It could also help you decide to use a certain Sallen-Key filter based on the relative stability implied by the gain margin.

References

Williams and Taylor, Electronic Filter Design Handbook, “VCVS uniform capacitor structure.”

Author Information

Martin Cano most recently was a principal applications engineer at National Semiconductor, where he developed Webench tools and provided application support. He holds bachelor's and master's degrees in electrical engineering from Boston University's College of Engineering.