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Do we really need another design tool for Sallen-Key filters?

Do we really need another design tool for Sallen-Key filters? Aren’t there enough [i] [ii] [iii] [iv] [v] already? Obviously, you and I think that there’s something more to say on this topic, or you wouldn’t be reading this. Perhaps the simplest justification for my belief is to point out the differences in the answers, to the same request, from two of these tools. When asked for components in the highest Q section of a 1dB, 7^{th} order, 1kHz, DC gain of 3 Chebyshev filter, one tool offers 220pF and 100nF capacitors with a DC gain of 1, and another 10nF and 31.6(!)nF capacitors with a DC gain of 1.316. These sections exhibit noteworthy differences in noise, component tolerance sensitivities, and op amp gain-bandwidth product (GBP) requirements. It is the goal of this article to derive equations for calculating and managing these parameters and implementing that management in a provided spreadsheet.

Consider the second order topology of **Figure 1**. Assume for the moment an op amp of infinite GBP. Although it contains six passive components, the transfer function of the section is fully defined by only two parameters: the resonance frequency F_{0} Hz, and the quality factor Q. Therefore, there are four remaining degrees of freedom in the selection of passive components.

**Figure 1** A schematic of a second order Sallen-Key low pass filter.

A recent article in EDN [vi] described how to calculate R1 and R2 given C1, C2, Rf, Rg, Q and F_{0}. It also showed that the sensitivity of the response magnitude, at resonance F_{0}, can be expressed exclusively as a function of Q, C1/C2 and Rf/Rg.[1] These results are assumed here and incorporated into this article’s spreadsheet. The next step is to develop a means for evaluating noise performance. Application notes are available [vii] [viii] which provide an excellent overview of noise analysis. **Figure 2** provides a schematic showing the various noise sources that must be considered in the Figure 1 design. Going forward, we’ll focus on the broadband, white noise aspects of these sources.

**Figure 2** A Sallen-Key low pass filter showing all noise sources.

It’s convenient to think in terms of the signals in the figure as being volts and amperes per square root Hertz rather than of volts and amperes. The voltage noise source for an X ohm resistor at 20°C is equal to √ (4·k·T·X) = 1.272×10^{-10 }· √X. Here k is Boltzman’s constant and T = 293.15°K. And we can simply read the op amp noise source values from their data sheets. The following equations can be written by inspection of the Figure 2 schematic:

These equations can be solved for the portion of each individual noise source that contributes to the total op amp output voltage:

Generally, the individual noise contributions calculated above are uncorrelated with the others, which means that the total output voltage should be equal to the square root of the sum of their squares. However, there is an exception to this. The two noise currents from the bases of an input differential bipolar transistor pair contain components which are correlated with one another (hence the equation for i_{corr}) and ones which are not (i_{m} and I_{p}). The correlated parts come from equal splits of the emitter bias current and, if present, base bias cancellation circuit currents. The uncorrelated parts come from the bases of the individual transistors of the pair. These two current types must be handled in computationally different fashions.

The voltages resulting from the uncorrelated portions are among the sum’s squares. However, the difference of the voltages due to the correlated portions must be squared before being added to that sum. If both bases see identical impedances, the correlated currents produce voltages which cancel and do not contribute to the total noise, just as equal DC bias currents seeing equal resistances would yield cancelling results. And so, in the range of frequencies below resonance, matched impedances can offer this beneficial effect. Unfortunately, this is not possible at and above the resonance frequency of our filter due to impedance variations with frequency. Some manufacturers’ datasheets separately call out correlated and uncorrelated currents. For those that don’t, there is a measurement technique [ix] which allows each to be determined.

The op amp gain, A, requires some discussion. This is a frequency-dependent parameter equal (to a good approximation in most cases) to f_{gbw} / ( j·f + f_{p}). Here, f is the frequency in Hz, f_{p }is a low frequency pole, f_{gbw }is the GBP, and j = √-1. Refer to **Figure 3** for a graph of this expression using typical values. Data sheets don’t commonly specify f_{p}, but they do specify f_{gbw} and A_{DC}, where A_{DC} is the open loop DC gain. f_{p} is equal to, and will be replaced with, f_{gbw} /A_{DC} in later calculations.

**Figure 3** A graph of the magnitude of A = f

The application of this dimensionless parameter, A, to voltage feedback amplifiers is obvious. But Sallen-Key topologies can also employ current feedback amplifiers, whose gains have units of impedance. Fortunately, there is a way to use the equations that have been developed for these devices too [x] (**Figure 4**). With a little bit of algebra, we can see that for the current feedback amplifier, A(s) = T(s) / [ R_{O }+ R1·R2 / (R1 + R2) ]. Op amp datasheets give us T(s) and in some cases R_{O}. When R_{O }is not supplied, it can be easily determined through simulation or measurement to a reasonable accuracy. This can be done by removing R1 and R2, grounding V_{IN}, and applying a known current I (I is shown in the figure.) R_{O} = V/I, where V is the voltage that arises at the inverting input. Although R_{O }is typically constant over a broad range of frequencies, it couldn’t hurt to make this measurement near the resonance frequency of the intended filter.

**Figure 4** Closed loop gains of voltage (left) and current (right) feedback amplifiers, courtesy of Analog Devices.

Next, we consider op amp GBP concerns. Refer to the expression derived for the noise arising from resistor R1. Its voltage per root Hz noise source in Figure 2 is positioned exactly where a voltage source driving the section would be. From this, we can see that the expression for R1’s contribution to the output noise also provides the voltage transfer function of the entire filter section. Defining s_{0} = 2·π·√-1·F_{0}, evaluating that expression at resonance and applying a bit of algebra, we get

With an ideal, infinite gain op amp, the magnitude of the denominator is unity. But with finite gain, the denominator’s magnitude grows and that of H(s_{0}) falls. We can see that the following inequality constrains the error in the magnitude of H(s_{0}) to be less than E_{dB} decibels:

In the above expression, we have replaced f_{p} with f_{gbw} / A_{DC}. The equality of the two terms in the square brackets comes from constraining the filter to have the specified Q and F_{0} with an infinite gain op amp. The spreadsheet uses the entry of the parameter *max magnitude response error at resonance* (E_{db}) on the Design Parameters tab to evaluate the inequality and avoid displaying those portions of curves on the Noise and Sensitivity Graph tab which fail that constraint.

[Continue reading on EDN US: Spreadsheet features]

*Christopher Paul has worked in various engineering positions in the communications industry for over 40 years.*