It’s well known that the main job of a pulse width modulator’s filter is to limit the maximum peak-to-peak amplitude of the fPWM Hz-induced ripple. It attenuates this to a specified fraction—Frac of the full-scale PWM output—while passing PWMavg, the average value of the PWM signal.
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Although the duty cycle can change instantaneously, the filter’s response to that change takes time to settle. It’s convenient to define the settling time Tfrac to be that after which the transient response remains within ± Frac of PWMavg. (After fully settling, the response variations will be from the ripple only and will remain within ± Frac/2 of PWMavg.) And it’s generally true that the more the ripple is attenuated, the larger Tfrac is. But for a given filter with two or more poles, there is an infinite number of combinations of component values that will limit the maximum ripple to Frac. (Think of the number of poles as being the number of capacitors in an R – C filter.) And yet the value of Tfrac is typically different for each combination. So we have a filter optimization problem: find the component value combination that minimizes Tfrac while satisfying the ripple requirement Frac.
I’ve addressed this issue before in a Design Idea (DI), but the procedure’s complexity was perhaps off-putting and inadequately flexible. I’ve since revisited the problem, finding a somewhat improved and analytically optimal solution. But that improvement alone does not justify a new DI.
So, why this new DI?
What I think does justify this is a spreadsheet that offers greater flexibility in terms of filter requirements and automates all the work for you. Download the files from .
If you use OneDrive or something like it, you must install the files outside the OneDrive folder. (Safely ensconced there, OneDrive doesn’t “see” them and can’t interfere with the spreadsheet’s query of the paths to where certain files are stored locally.)
Open the spreadsheet. In the following, the yellow-highlighted parameters here and on the spreadsheet are inputs to be supplied by the user; the green-highlighted ones are spreadsheet outputs. Tell it your PWM frequency, in Hz, specify the required value of Frac, and press the “Calculate” button.
The Visual Basic Application (VBA)-driven spreadsheet takes that information and determines the values of the filter’s real and complex pole pairs ( the Q and ω0 of the latter ), which give you the optimal, smallest Tfrac, which it also displays.
To produce an implementable filter, it then combines this information with the (default) values of the filter’s capacitors c1, c2, and c3 . (These you can change and again press Calculate.) From all of this, it determines both the exact and the closest standard E96 values for the resistors r1, r2, and r3 needed to complete the filter. The filter itself is the third-order Sallen-Key low-pass depicted in the schematic portion of the spreadsheet screenshot seen in Figure 1.
Figure 1 A screenshot of the spreadsheet that runs the show. See the text.
And since we all like graphs, two have been provided. The one on top shows how ω0 and the real portion of all poles vary with Q. More importantly, it also shows that TFrac generally gets worse (larger) as Q is increased (not surprising, with the concomitant increase in oscillatory amplitudes).
The other graph shows the decay with time of PWMavg minus the absolute value of the transient response, with the voltage displayed on a logarithmic scale. The bumps are evidence of a damped oscillatory behavior.
But as they say in the late-night TV commercials (or at least they used to), “But wait! There’s more!”
How do I know this thing works?
You might ask how you can confirm that this filter will perform as advertised. The answer is easy if you’ve installed LTspice on your computer and you tell the spreadsheet the path starting from the root directory to the LTspice.exe file. Mine’s in C:\Users\chris\AppData\Local\Programs\ADI\LTspice\LTspice.exe.
Don’t worry if you can’t see the entire entry in the Excel cell provided. (NOTE – With the discussions surrounding the ongoing changes in LTspice versions 26.x.y, these files have been developed for use with the stable and still widely used LTspice 17.1.15. This version can still be downloaded and installed: I haven’t checked if the files work with the 26.x.y versions.)
Press the “LTspice: Exact…” button. It will automatically launch a simulation using the exact resistor values derived and plot the filter’s response to the two biggest transients: a “full” one from 0 to 100% duty cycle (no PWM ripple) and a “half” one from 0 to 50% duty cycle (maximum possible ripple). See Figure 2 for a sample LTspice run.
Figure 2 An LTspice run using Exact component values for a sample filter.
The responses have been offset to reach their final values at 0 V. Tfrac appears on the plot as a vertical line along with two horizontal lines, which are at ± Frac. You can zoom in to see that the value of Tfrac is indeed correct; it crosses a ± Frac line exactly at the point that the full transition response does. (The full transition always takes a little longer to settle than the half-step transition.)
But alas, alack; this assumes perfect components with 0% tolerances. So the “LTspice Standard…” button launches a simulation of 100 Monte Carlo runs with capacitor and resistor tolerances of 1% and 1% using the E96 resistor values. (You can change all three of these default values and re-run the simulation. In fact, it’s worth considering the overall reduced settling time that can be had with suitably chosen 0.1% resistors added in series with small 1% resistors to more closely approach the exactly calculated values. Better tolerance capacitors would also help, but they tend to be prohibitively expensive.)
As you’ll see, non-zero tolerance variations lead to settling times longer than Tfrac. But by performing an extended number of Monte Carlo runs, you’ll be able to determine the time beyond which even filters made out of real-world components will have settled to Frac.
Filter design constraints
The real portions of all poles in the filter have been constrained to be identical. The reason for this is that these values control the decay rates of the half- and full-step transients, either of which could dictate the overall settling time. Given that the total ripple attenuation is the product of the real parts of both poles, if one were smaller than the other, it would extend the overall settling time beyond that achieved with identical poles. This constraint also simplifies the optimization problem in that there is only one real and one imaginary value of poles to consider, rather than one imaginary and two real values.
Calculated resistor values
Depending on certain inputs to the spreadsheet, the derived values of the filter resistances might be smaller or larger than you’d like. In that case, the input values of the capacitors could be multiplied by a constant K of your choosing to obtain new resistor values divided by that K.
The spreadsheet’s default capacitor values are in the “Goldilocks” range—large enough that op amp input and PCB capacitances will affect them minimally, but small enough that NPO/COG type capacitors (whose stability with temperature and DC voltages are demanded in filter designs) are not prohibitively expensive. The ratios of one to the other of the default capacitor values have been shown to consistently result in realizable filters. Feel free to experiment with other values and ratios, but be aware that it might not be possible to realize filters with those changes.
Filter drivers
Do not drive the filter from a microprocessor directly. Its non-PWM functions draw currents that lead to small voltage drops across the IC-to-package-pin bonding wires. These induce errors by preventing signals from getting close to the ground and the supply rail. Instead, buffer the microprocessor with dedicated SN74AC04 logic inverters, which will swing to the rails, since they have no other currents to deal with and their outputs are minimally loaded. For a reasonably accurate reference voltage supplying the SN74AC04, consider the REF35.
SN74AC04-induced errors
It’s been pointed out that all digital drivers have different logic high and low resistances. These differences are sources of error that are worst at a 50% duty cycle. The part’s data sheet says that at a 3-V supply, the logic high voltage drop under a 12-mA load over the industrial temperature range could be as high as 560 mV, with a resistance of 45 ohms.
The logic low resistance maximum is a bit better, but there is no spec for the difference. The safe but admittedly ridiculous possibility is that the logic low resistance is 0 ohms, leaving us with a 45-ohm difference. This can be mitigated by paralleling G gates to reduce the drive resistance by that factor to produce a difference of Rdiff = 45/G.
Since no DC current can flow through the filter’s passive components, the fractional full-scale error at 50% is:
.5 · r1 / ( r1 + Rdiff) – .5 = – .5 · Rdiff / ( r1 + Rdiff)
For a b-bit PWM, you’d probably want the error to be less than half of one LSbit or 2-b-1. So you’d require that r1 > Rdiff · 2b.
Consider G = 5. For b = 8, r1 > 2300 ohms. For b = 12, it’s 37 kohms, and for b = 16, 590 kohms. But this brings up a second point: a large b means a relatively small fPWM and therefore a large TFrac. Fortunately, there’s a way around this.
Double up
Summing the contributions of two 8-bit PWMs, one of whose signals’ amplitude is 256 times that of the other, allows both to have an fPWM 256 times larger than that of a single 16-bit PWM. This yields a TFrac reduced by the same factor. Figure 3 shows one way to employ this approach.
Figure 3 Configuration with independent most significant (MSbit) and least significant (LSbit) 8-bit PWMs, the latter contributing 1/256 of the former, to replace a single 16-bit PWM. This arrangement reduces the settling time by a factor of 256.
Op-amp considerations
Figures 1 and 3 lead to the question of which op amp to use. A rail-to-rail input and output unit is warranted. The OPA376 family of singles, duals, and quads is a good answer.
It’s 25 µV at 25°C, ±1 µV/°C from -40 to +85°C, and barely disturbs the accuracy of even a 16-bit PWM. Its input bias current of 10 pA maximum at 25°C, and its typical (no maximum spec) of less than 50 pA at 85°C, introduces errors on par with its offset voltage. Consider the op amp’s output rail-to-rail limitations, however. Either avoid PWM duty cycles at the extremes, or extend the op amp’s supply rails a few tens of millivolts (see its data sheet) beyond those of the PWM.
Figure 4 The above Nomograph can aid in selecting the operating point of your design.
Problems, gripes, suggestions, requests, and accolades
The spreadsheet employs VBA numerical iteration routines to find the Q, ω0 pairs and the filter resistors. Although I’ve tested these routines extensively, it’s always possible that one or the other will fail to converge with some combination of input values.
In that case, please let me know by adding a note to the “Comments” section of this DI. This will generate an automatic email alert and will allow the inclusion in our conversation of others who might be interested. Please do not email me unless you have a comment that is truly meant to be private (a marriage proposal?) I encourage feedback of all kinds.
A grudging acknowledgement
I’d be remiss if I did not mention the help I got from a certain widely available AI program in developing this project. This ranged from deriving Inverse Laplace transforms and Newton-Raphson iteration algorithms to VBA coding.
But working with this AI wasn’t all lollipops and rainbows. In the course of the effort, I was reminded of Ronald Reagan’s admonition to “Trust, but verify.” But as things progressed, I dropped the “trust” part.
I found I had to break tasks down into sections, understand each that was provided, test assiduously, and make corrections before proceeding to the next step. Setting a multi-step task was a recipe for disaster. Still, AI is a valuable tool, and I find it even more valuable now that I better understand how to work with it.
I’d be interested in hearing about others’ experiences.
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