The Q designed for the amplifier

The Q the amplifier decides built a second-order section with an amplifier a hundred times its corner, the usual rule, and found the Q 1.97 per cent high at $Q = 2$ while the expression for it contained no amplifier at all. The ripple that is a temperature added that this error moves with temperature in a bias-set part, and where the Q comes from compared the two arrangements of one section against a slow amplifier. The error has been measured from three sides and it has been treated each time as a fixed cost of the amplifier, to be paid in more gain–bandwidth.

But the error is known. A circuit whose error can be calculated in advance can be designed at a value that puts the real result where it was wanted, and nothing in principle prevents it: if an amplifier’s lag turns 5.0 into 5.24, a section designed for 4.771 will come out at 5.000. Doing that is called predistortion, it has been practised for as long as active filters have been built, and it is often described in a sentence as a way of removing the amplifier’s effect. What it removes, and at what price, can be stated more exactly than that, and the exact statement is the interesting part. It removes the error the amplifier has in common with every other amplifier of its kind and it does nothing about the part that differs.

Asking for less than is wanted #

The section is the follower arrangement, with a capacitance ratio carrying the Q, on the one-pole amplifier the earlier essays used, built as a netlist with its own direct-current gain and output resistance. Solving for its natural frequencies gives a realised Q and a realised corner, without any approximation about the amplifier’s lag. Predistortion is then a fixed-point iteration on that netlist: scale the design Q by the ratio of what was wanted to what came out, scale the design corner the same way, and repeat. It converges in a few passes when a solution exists, to better than a part in ten million.

At a ratio of 100 and $Q = 5$ the answer is a design Q of 4.771, with a design corner 5.05 per cent above the one wanted. So both the capacitor ratio and the resistor values change, and the section as built does not have the Q its components suggest. A person checking the design from its parts would call it wrong, and a tolerance analysis that took 4.771 at face value would be analysing the wrong filter.

Against the actual amplifier the design’s Q is exact, to the iteration’s limit. The figure shows what happens when the part is not that amplifier. A part 30 per cent slow reads +1.64 per cent, and one 30 per cent fast reads −0.966. Comparing with the textbook design, whose errors run from +6.51 to +3.78, the two lines are nearly parallel, one shifted down onto zero at the design point and the other not. A data sheet’s typical gain–bandwidth is usually good to something like that ±30 per cent, and so that span is what a predistorted section can actually promise.

The design that does not exist #

The iteration can fail, and the failure has a meaning. A follower with a slow amplifier enhances its Q, and it enhances it more the higher the Q was. To reach a Q of 10 from a slow amplifier the design Q has to fall a long way, and a follower section’s design Q has a lower limit: a capacitance ratio with a Q of one half is equal capacitors with equal resistors, and there is no real pair below that with this structure.

There is therefore a floor in ratio for each Q: 2.9 times the corner at $Q = 2$, 7.2 times at $Q = 5$ and 14.2 times at $Q = 10$. These are close to being proportional to the Q, with about $1.4Q$ as the limit, and they are far below the hundred that the usual rule states. At 14.2 times the corner, a section can in principle be built with a Q of 10 from an amplifier that would ordinarily be regarded as hopelessly slow. It would be very sensitive to everything. But it is a real circuit, and it shows that the usual rule of a hundred times is not a limit on what can be built. It is a statement about what can be built without calculating anything.

Nearer that floor the line of design values turns over, since the corner has started to do a lot of the work: at $Q = 10$ the design fraction is 0.864 at twenty times the corner and 0.848 at thirty. The amplifier’s own pole is now a real part of the section, and “section plus amplifier” is a third-order system whose complex pair is being placed by choosing two values. The third place, where nothing happens at all found neither form oscillating at any gain–bandwidth. That is what keeps the iteration honest down here.

The amplifier a design needs #

For a designer the useful question is how much amplifier a stated accuracy takes, and each way of building has its own answer.

From the formula a section at $Q = 5$ needs 725 times the corner to be within 1 per cent across a ±30 per cent amplifier. Designed for the typical part it needs 196, a factor of 3.7. Those two numbers carry the core statement of this essay. Without predistortion the error at the worst part is 1.43 times the typical error, and that sets the requirement. With it, the error is the difference between the typical and the worst part, and its size relative to the first is about the spread itself, 0.3. So the exchange rate is the reciprocal of the amplifier’s fractional spread. A ±30 per cent part gets about a factor of three, and a ±10 per cent part would get about ten.

The third line is a different matter from the other two, and it needs reading carefully. With a ±10 per cent spread, the bisected ratio is 11 at $Q = 5$, not far above the floor of 7.2, and the requirement is no longer an accuracy requirement but the place where the design itself begins to run out. Error at that spread never reaches 1 per cent until the section has almost no design left. A designer with a part specified to ±10 per cent, or a batch measured to that, can build a Q of 5 from an amplifier eleven times the corner. A trimmed or measured amplifier makes a very large difference, far larger than the same effort spent on a faster part.

Which arrangement is better, and what decides it put both the follower and the gain stage on an equal footing against a slow amplifier. Predistortion changes the comparison, since the follower’s larger nominal error is now removed, and what is left depends on how that error’s size varies with the amplifier. On a follower it varies strongly with $Q$, and the gain stage’s dependence on the gain would give a different slope. That comparison is not made here.

A spread that is known in advance #

Not every spread is unknown. The temperature dependence of a bias-set amplifier’s gain–bandwidth has a known form. With a resistor-set tail current the transconductance falls as one over absolute temperature, as the earlier essay on temperature measured, and so the ratio is a known function of the temperature rather than a uniform unknown.

From the formula the error is +3.83 per cent at −40 °C and +6.15 at 125 °C, a slope with an offset. Designed at 25 °C the offset goes and the span is what remains, −0.912 to +1.30. Designed at the middle of the range, 42.5 °C, the worst case is 1.15 per cent rather than 1.30. So a section specified over a range has a slightly better design point than room temperature, and the improvement is small here because the range is nearly centred already.

That is a small refinement and the larger point is what predistortion leaves across temperature. It removes all of a known error except its change, and the change is proportional to the size of the original. So a temperature coefficient is reduced in exactly the same proportion as a nominal error, if at all, and this is why the ripple that is a temperature cannot be cured this way: the 2.3 per cent span here is about the same as the span the unmodified section has. A design point cannot remove a slope. To reduce the slope, the amplifier’s ratio has to be raised, or its bias arranged so that the transconductance holds still, which is the other result of that essay.

A whole passband #

A designer of a real filter cares about a response and not a single Q, and a whole filter is a test of whether section-by-section correction adds up as expected.

The half-decibel Chebyshev of order five has two complex sections and one real pole, with a highest Q of 4.55. Each section gets its own iteration, because an amplifier’s effect depends on that section’s Q and corner, and the highest-Q section’s correction is by far the largest. At a hundred times the corner, the plain design is 1.43 dB out on a typical part. That is nearly three times the half-decibel ripple the filter was designed for, so the filter as drawn does not exist. With the correction it is exact on a typical part by construction, and 0.547 dB on a part 30 per cent slow, against 2.13 dB without.

At thirty times the corner, the plain design is out by 5.82 dB and a slow part by 8.22: that is not a filter one would recognise. The corrected design’s 1.84 dB on a slow part is out of specification but it is a filter of about the right shape. The ladder filter built for the rest of itself gave another way to deal with the amplifier’s error in a complete filter, which is to reduce the sensitivity to it by structure. Predistortion is the design-time alternative, and it is the one that costs nothing in parts: it costs only knowledge of the amplifier.

And the exchange rate survives at the level of a passband. The ratio between 2.13 dB and 0.55 dB is 3.9, near the 3.3 that a ±30 per cent spread predicts, because a passband error at these sizes is almost a linear function of each section’s Q error.

What the parts then look like #

A worked section makes the size of the change concrete. A follower section at $Q = 5$ has a capacitance ratio of $4Q^2$, or 100. Predistorted for an amplifier a hundred times its corner, it has a ratio of $4(4.771)^2$, or 91.0, and its resistors are moved as well to put the corner 5.05 per cent above the frequency wanted. So the capacitors differ from the textbook’s by 9 per cent. That is a much larger change than the 1 per cent tolerance a designer would specify for them, and it moves the design across several values in a standard series. The design’s change is therefore easy to build and easy to get wrong. A technician replacing what looks like the wrong capacitor with what the formula says would raise the Q by nearly 5 per cent, and an inspection of the schematic has no way to tell that a correction is what it is seeing. Predistortion is a calculation that has to travel with the drawing.

What knowledge of the amplifier means #

Everything here depends on what the phrase the typical part can be relied on to mean. A data sheet’s typical gain–bandwidth is usually given at one supply and temperature and with a footnote about its load. A second source’s part may be 20 per cent away from it while staying within its own typical. So a filter predistorted for one vendor’s amplifier is not an equivalent design for another, and its bill of materials has become part of its specification in a way a filter designed from the formula’s rule of a hundred times is not. A manufacturer’s change of process, invisible in the data sheet’s limits, moves the response.

There is an in-circuit answer for high-volume production: measuring a section’s actual corner and Q and trimming to them removes the amplifier’s part-to-part spread directly. The design-time correction here and a production trim are the same operation at two different moments, and the second removes the unit’s own spread. The capacitance that is not one number adds a warning for the capacitors, whose values at the operating point are usually what is really moved by the trim.

Tolerance is the other matter. A predistorted section still has its component tolerance, and why the sensitivity that is enormous does not produce an enormous error says a follower’s elements carry a half each. A 1 per cent capacitor is a 0.5 per cent Q error at any design. That is half of the budget this essay’s 1 per cent allows, and at some point it stops being sensible to spend effort on the amplifier’s share.

What the model leaves out #

The amplifier is a single pole with a fixed direct-current gain and a resistive output. A second pole would change the section’s real part, as the gain that stops making a stage faster shows for a closed loop, and a follower at low ratios is a place where that matters. Slew is also absent: a section designed against an amplifier eleven times its corner is running with very little margin of output current on any large signal. The correction is computed at small signal. So all of the low-ratio numbers here are small-signal statements. And the cascade’s sections are treated as unloaded, because the amplifier’s low output impedance permits that; at low ratios that impedance rises with frequency and the assumption weakens.

The number to carry #

A follower section designed from its formula is 4.79 per cent high at $Q = 5$ and a hundred times the corner. Asking for 4.771 lands on 5.000, and leaves +1.64 and −0.966 on parts 30 per cent slow and fast. So predistortion’s exchange rate is the amplifier’s fractional spread: 725 times the corner becomes 196 for one per cent across ±30 per cent, and about 11 across ±10. Below about $1.4Q$ times the corner no design reaches the Q wanted at all. It cannot remove a temperature’s slope, only its offset. On a Chebyshev 5 at a hundred times the corner, the passband error falls from 1.43 dB to nothing on a typical part, and from 2.13 to 0.55 on a slow one. A rule of a hundred times the corner is what building without calculating requires.

Still open: the gain-stage form, a second pole, and the trim that knows the part #

The gain stage’s correction. A section whose Q comes from an amplifier’s gain has an error that goes as the square of that gain, and a different dependence on the amplifier. Whether predistortion favours it or the follower, once the common error has been removed from both, reverses or confirms the comparison at $Q = 3.08$ that the two forms were compared at.

A second pole in the amplifier. A section designed from a one-pole model and built on an amplifier with a second pole has a residual that does not scale with the spread. How far above the design floor a second pole starts to matter, as a function of where it is, says how low in ratio any of this is believable.

A trim that measures the part. A section can be trimmed in production, against its actual response. Trimming the Q at one temperature removes the unit’s spread and leaves the slope; whether trimming the corner alone, which is a single resistor, gets most of the benefit in a whole filter is a question of which section’s error dominates the passband, and it has a simple answer on this netlist.