Hi all,
While thinking about ways to speed up the settling of a single-supply single-op-amp RIAA amplifier, see https://www.diyaudio.com/community/...upply-phono-preamp-design.413571/post-7702435 , I found a way to include a second- or third-order Butterworth high-pass filter. As it may be useful outside the context of single-supply circuits, I give it a separate thread.
When you just look at the topology and ignore the component values, this is a rather conventional RIAA amplifier (you could make it even more conventional by connecting R7 in parallel with C5, that doesn't matter much for the principle):
Normally, C8 is used to cause roll-off in the subsonic region and the network R7...R9, C5, C6 realizes the RIAA poles and zero. In this case, however, I use C8 to realize the lowest RIAA pole at -1/(3.18 ms) and R7 to get subsonic roll-off.
Note that C8/C5 = 1000, meaning that without the subsonic roll-off, the DC gain would be 1001, a very ordinary value for a moving-magnet amplifier (1 kHz gain roughly 40 dB).
With everything ideal, at the value of the Laplace variable s where the impedance of C8 cancels the impedance of R12, the feedback disappears and the gain goes to infinity. This means that there is a pole at exactly -1/(R12C8), so if this has to be the first RIAA pole, one needs R12C8 = 3.18 ms. It's actually 3.196 ms in the schematic, pretty close.
The disadvantage of using C8 for the first RIAA pole is that C8, which has a relatively large value, needs to be accurate to get an accurate first RIAA pole. (C8 has practically no effect on the gain at frequencies much greater than 50 Hz, so its tolerance affects deep bass, but not channel balance.) The advantage is that you can include better subsonic filtering in the loop by adding two more resistors and a capacitor.
As an intermediate step, suppose you could add an ideal inductor with a huge value between the output and the negative input of the op-amp, chosen such that it resonates with C5 at the desired subsonic roll-off frequency, and that you chose R7 such that it damps the LC circuit to a quality factor of 1/2 √2. The subsonic response would then be very close to second-order Butterworth. That's because the gain of the RIAA correction amplifier is one plus the ratio of the feedback impedance to the impedance from the negative op-amp input to ground, and that "one" is quite negligible at low frequencies. Mind you, R8 and R9 contribute to the damping of the LC circuit, but not by much. You could also choose a quality factor of 1 and design the AC coupling at the input for the same cut-off frequency. The combined response is then third-order Butterworth.
Such an ideal inductor is totally impractical, but it can be approximated with a T network consisting of two resistors with values much smaller than R7 and a capacitor to ground at the point where they are connected, see this figure:
The transfer from the voltage going into R11 to the current coming out of R10 rolls off at a first-order rate from some very low frequency onwards, like would be the case with an inductor.
I haven't found any simple exact equations for any of the values except R12C8 = 3.18 ms. In fact, I've been very lazy and just calculated approximate values for the other components, and then used a pole-zero extraction program to fine-tune the values.
Regarding those approximate calculations:
R12C8 = 3.18 ms to get the first RIAA pole at the right spot.
The DC gain would be 1 + C8/C5 without subsonic roll-off, so C8/C5 = 1000 gives you a midband gain of roughly 40 dB.
At s = -1/((R8 + R9)(C5 + C6)), the impedance of the network R8, R9, C5, C6 goes to zero and the gain of the circuit becomes 1. As a gain of 1 is pretty close to 0, this must be close to the location of the RIAA zero. That is,
(R8 + R9)(C5 + C6) ≈ 318 μs
At s = -1/((R8 + R9)C6), the impedance of the parallel connection of C6 and R8 & R9 goes to infinity. The impedance of the whole feedback network remains finite due to the other branches R7 and R11, C7, R10, but it does get pretty large. That means the second RIAA pole must be close, so we get the extra criterion
(R8 + R9)C6 ≈ 75 μs
The (theoretical) inductance L is chosen to resonate with C5 at the required subsonic roll-off frequency and R7 is chosen to get the desired quality factor. R10 and R11 get convenient values much smaller than R7 with R10 also much greater than R12. We then have
C7 = L/(R10R11)
Best regards,
Marcel
Edits:
Input RC coupling
The RCR T-network that approximates an inductor actually approximates an inductor with inductance L = R10 R11 C7 and a series resistance of R10 + R11. At low frequencies, it stops behaving inductively, it just turns into the series connection of the two resistors.
As a result, one of the zeros of the high-pass filter that are supposed to lie at s = 0 actually lies somewhere around s = -(R10 + R11)/(R10 R11 C7). For the second-order cases, I have used the first-order high-pass at the input to cover this zero by making the input RC time constant approximately equal to R10 R11 C7/(R10 + R11), or actually to a more accurate value for the displaced zero found by the LINDA pole-zero extraction program.
For the third-order case, I have used the input RC coupling to make the real pole of the third-order Butterworth response, so I couldn't use it to cover the displaced zero. I used the output RC circuit in that case, or simply did not cover the zero. The effect of the zero not being in the origin is typically only seen below 1.something Hz anyway.
There is another zero not exactly in the origin, this is related to the + 1 term in the gain expression of a non-inverting op-amp amplifier. It is so close to 0 that I decided not to bother correcting for it.
16 Hz split-supply versions
This is a version for split supply and 16 Hz cut-off frequency, see post #58, https://www.diyaudio.com/community/...rworth-high-pass-included.413649/post-7927611
For the second-order version (values in parenthesis), the input coupling capacitor C2 can be replaced with a short circuit if you don't mind when the roll-off reduces to first order below 1.3 Hz.
This is a variant with 46 dB rather than 40 dB midband gain:
Thanks to having R6 split into R6 and R0, the time constant of the input RC coupling network can be set more accurately without needing awkward values for C2. This was implicitly suggested by hbtaudio on another thread. Because of the high midband gain, the op-amp needs to have a fairly high gain-bandwidth product to get accurate RIAA correction (16 MHz gain-bandwidth product will give about -2 % error of the location of the second RIAA pole).
Finite gain-bandwidth product
See post #100, https://www.diyaudio.com/community/...rworth-high-pass-included.413649/post-7964481 , for some rough calculations on the effect of finite gain-bandwidth product of the op-amp.
From post #101 onward, Nick Sukhov points out that an amplifier with a high open-loop output impedance would result in a loop gain that depends much less on the RIAA correction circuit impedance. That's something to keep in mind when designing a discrete amplifier, you don't have the ability to choose a high open-loop output impedance when using op-amps.
Applying the subsonic filter to a discrete preamplifier based on the Hoeffelman and Meys configuration
The discussion with Nick and Chris about open-loop output impedances made me realize that the subsonic filter of this thread could be combined with a low-noise ("electrically cold") input termination resistance realized with a special feedback configuration that Dual already used in the late 1960's (CV40 phono section, see https://www.diyaudio.com/community/...o-input-load-modification.424717/post-7947176 ) and that was advocated by Hoeffelman and Meys in a 1978 AES article (Jean M. Hoeffelman and René P. Meys, "Improvements of the noise characteristics of amplifiers for magnetic transducers", Journal of the Audio Engineering Society, vol. 26, no. 12, December 1978, pages 935...939, see also Ernst H. Nordholt, "Comments on "Improvement of the noise characteristics of amplifiers for magnetic transducers"", Journal of the Audio Engineering Society, vol. 27, no. 9, September 1979, pages 680...681). The very first electrically cold resistance was made by William Spencer Percival and W. L. Horwood in 1939 as far as I know, but they used a different configuration and did not apply it to phono preamplifiers. See W. S. Percival, "An electrically "cold" resistance", The wireless engineer, vol. 16, May 1939, pages 237...240.
The schematics below show the resulting configurations. They are identical, but the left schematic is for people familiar with nullators and norators, the right schematic for people who feel more comfortable with high-gain twoports and op-amps. At frequencies well above 50 Hz, the input impedance approaches (R13 + R14)/(1 + R13/R12 + R13/R10). You can make this equal to 47 kΩ while using an R14 that is much greater than 47 kΩ, thereby reducing the thermal noise current √(4kT∆f/R) that gets injected into the input.
You can't do this with op-amps (not without floating supplies anyway) because op-amps lack the negative output that conducts a (signal) current equal but opposite to the current through the positive output. That is, you can make electrically "cold" resistances with op-amps, but not as shown here.
Document about dimensioning the circuit
The attached zip file contains a pdf document that explains step-by-step how the component values were found (section 2) and that presents a more accurate method than I have used (one that doesn't need fine-tuning with a pole-zero extraction program, section 3). It also contains a spreadsheet for the more accurate way to calculate the component values.
Deriving the expressions was a nice exercise, but I'm not at all convinced that my more accurate calculation is of any practical use. It can very easily lead to negative or complex resistances.
While thinking about ways to speed up the settling of a single-supply single-op-amp RIAA amplifier, see https://www.diyaudio.com/community/...upply-phono-preamp-design.413571/post-7702435 , I found a way to include a second- or third-order Butterworth high-pass filter. As it may be useful outside the context of single-supply circuits, I give it a separate thread.
When you just look at the topology and ignore the component values, this is a rather conventional RIAA amplifier (you could make it even more conventional by connecting R7 in parallel with C5, that doesn't matter much for the principle):
Normally, C8 is used to cause roll-off in the subsonic region and the network R7...R9, C5, C6 realizes the RIAA poles and zero. In this case, however, I use C8 to realize the lowest RIAA pole at -1/(3.18 ms) and R7 to get subsonic roll-off.
Note that C8/C5 = 1000, meaning that without the subsonic roll-off, the DC gain would be 1001, a very ordinary value for a moving-magnet amplifier (1 kHz gain roughly 40 dB).
With everything ideal, at the value of the Laplace variable s where the impedance of C8 cancels the impedance of R12, the feedback disappears and the gain goes to infinity. This means that there is a pole at exactly -1/(R12C8), so if this has to be the first RIAA pole, one needs R12C8 = 3.18 ms. It's actually 3.196 ms in the schematic, pretty close.
The disadvantage of using C8 for the first RIAA pole is that C8, which has a relatively large value, needs to be accurate to get an accurate first RIAA pole. (C8 has practically no effect on the gain at frequencies much greater than 50 Hz, so its tolerance affects deep bass, but not channel balance.) The advantage is that you can include better subsonic filtering in the loop by adding two more resistors and a capacitor.
As an intermediate step, suppose you could add an ideal inductor with a huge value between the output and the negative input of the op-amp, chosen such that it resonates with C5 at the desired subsonic roll-off frequency, and that you chose R7 such that it damps the LC circuit to a quality factor of 1/2 √2. The subsonic response would then be very close to second-order Butterworth. That's because the gain of the RIAA correction amplifier is one plus the ratio of the feedback impedance to the impedance from the negative op-amp input to ground, and that "one" is quite negligible at low frequencies. Mind you, R8 and R9 contribute to the damping of the LC circuit, but not by much. You could also choose a quality factor of 1 and design the AC coupling at the input for the same cut-off frequency. The combined response is then third-order Butterworth.
Such an ideal inductor is totally impractical, but it can be approximated with a T network consisting of two resistors with values much smaller than R7 and a capacitor to ground at the point where they are connected, see this figure:
The transfer from the voltage going into R11 to the current coming out of R10 rolls off at a first-order rate from some very low frequency onwards, like would be the case with an inductor.
I haven't found any simple exact equations for any of the values except R12C8 = 3.18 ms. In fact, I've been very lazy and just calculated approximate values for the other components, and then used a pole-zero extraction program to fine-tune the values.
Regarding those approximate calculations:
R12C8 = 3.18 ms to get the first RIAA pole at the right spot.
The DC gain would be 1 + C8/C5 without subsonic roll-off, so C8/C5 = 1000 gives you a midband gain of roughly 40 dB.
At s = -1/((R8 + R9)(C5 + C6)), the impedance of the network R8, R9, C5, C6 goes to zero and the gain of the circuit becomes 1. As a gain of 1 is pretty close to 0, this must be close to the location of the RIAA zero. That is,
(R8 + R9)(C5 + C6) ≈ 318 μs
At s = -1/((R8 + R9)C6), the impedance of the parallel connection of C6 and R8 & R9 goes to infinity. The impedance of the whole feedback network remains finite due to the other branches R7 and R11, C7, R10, but it does get pretty large. That means the second RIAA pole must be close, so we get the extra criterion
(R8 + R9)C6 ≈ 75 μs
The (theoretical) inductance L is chosen to resonate with C5 at the required subsonic roll-off frequency and R7 is chosen to get the desired quality factor. R10 and R11 get convenient values much smaller than R7 with R10 also much greater than R12. We then have
C7 = L/(R10R11)
Best regards,
Marcel
Edits:
Input RC coupling
The RCR T-network that approximates an inductor actually approximates an inductor with inductance L = R10 R11 C7 and a series resistance of R10 + R11. At low frequencies, it stops behaving inductively, it just turns into the series connection of the two resistors.
As a result, one of the zeros of the high-pass filter that are supposed to lie at s = 0 actually lies somewhere around s = -(R10 + R11)/(R10 R11 C7). For the second-order cases, I have used the first-order high-pass at the input to cover this zero by making the input RC time constant approximately equal to R10 R11 C7/(R10 + R11), or actually to a more accurate value for the displaced zero found by the LINDA pole-zero extraction program.
For the third-order case, I have used the input RC coupling to make the real pole of the third-order Butterworth response, so I couldn't use it to cover the displaced zero. I used the output RC circuit in that case, or simply did not cover the zero. The effect of the zero not being in the origin is typically only seen below 1.something Hz anyway.
There is another zero not exactly in the origin, this is related to the + 1 term in the gain expression of a non-inverting op-amp amplifier. It is so close to 0 that I decided not to bother correcting for it.
16 Hz split-supply versions
This is a version for split supply and 16 Hz cut-off frequency, see post #58, https://www.diyaudio.com/community/...rworth-high-pass-included.413649/post-7927611
For the second-order version (values in parenthesis), the input coupling capacitor C2 can be replaced with a short circuit if you don't mind when the roll-off reduces to first order below 1.3 Hz.
This is a variant with 46 dB rather than 40 dB midband gain:
Thanks to having R6 split into R6 and R0, the time constant of the input RC coupling network can be set more accurately without needing awkward values for C2. This was implicitly suggested by hbtaudio on another thread. Because of the high midband gain, the op-amp needs to have a fairly high gain-bandwidth product to get accurate RIAA correction (16 MHz gain-bandwidth product will give about -2 % error of the location of the second RIAA pole).
Finite gain-bandwidth product
See post #100, https://www.diyaudio.com/community/...rworth-high-pass-included.413649/post-7964481 , for some rough calculations on the effect of finite gain-bandwidth product of the op-amp.
From post #101 onward, Nick Sukhov points out that an amplifier with a high open-loop output impedance would result in a loop gain that depends much less on the RIAA correction circuit impedance. That's something to keep in mind when designing a discrete amplifier, you don't have the ability to choose a high open-loop output impedance when using op-amps.
Applying the subsonic filter to a discrete preamplifier based on the Hoeffelman and Meys configuration
The discussion with Nick and Chris about open-loop output impedances made me realize that the subsonic filter of this thread could be combined with a low-noise ("electrically cold") input termination resistance realized with a special feedback configuration that Dual already used in the late 1960's (CV40 phono section, see https://www.diyaudio.com/community/...o-input-load-modification.424717/post-7947176 ) and that was advocated by Hoeffelman and Meys in a 1978 AES article (Jean M. Hoeffelman and René P. Meys, "Improvements of the noise characteristics of amplifiers for magnetic transducers", Journal of the Audio Engineering Society, vol. 26, no. 12, December 1978, pages 935...939, see also Ernst H. Nordholt, "Comments on "Improvement of the noise characteristics of amplifiers for magnetic transducers"", Journal of the Audio Engineering Society, vol. 27, no. 9, September 1979, pages 680...681). The very first electrically cold resistance was made by William Spencer Percival and W. L. Horwood in 1939 as far as I know, but they used a different configuration and did not apply it to phono preamplifiers. See W. S. Percival, "An electrically "cold" resistance", The wireless engineer, vol. 16, May 1939, pages 237...240.
The schematics below show the resulting configurations. They are identical, but the left schematic is for people familiar with nullators and norators, the right schematic for people who feel more comfortable with high-gain twoports and op-amps. At frequencies well above 50 Hz, the input impedance approaches (R13 + R14)/(1 + R13/R12 + R13/R10). You can make this equal to 47 kΩ while using an R14 that is much greater than 47 kΩ, thereby reducing the thermal noise current √(4kT∆f/R) that gets injected into the input.
You can't do this with op-amps (not without floating supplies anyway) because op-amps lack the negative output that conducts a (signal) current equal but opposite to the current through the positive output. That is, you can make electrically "cold" resistances with op-amps, but not as shown here.
Document about dimensioning the circuit
The attached zip file contains a pdf document that explains step-by-step how the component values were found (section 2) and that presents a more accurate method than I have used (one that doesn't need fine-tuning with a pole-zero extraction program, section 3). It also contains a spreadsheet for the more accurate way to calculate the component values.
Deriving the expressions was a nice exercise, but I'm not at all convinced that my more accurate calculation is of any practical use. It can very easily lead to negative or complex resistances.
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Using C8 for the 50Hz pole is an idea that I have not seen before. The tight tolerance may have been a showstopper.
Ed
Ed
Electrolytics are typically very slack tolerances when new and change as they age significantly - so aren't useful for setting time-constants in an analog filter, as you'd need a trimmer resistance to go with it and regularly re-calibrate...
Is it possible to use several ceramic capacitors in parallel for C8 instead of electrolytic? I have just tried using the box of cheap Chinese capacitors. I got to 6.8 uF with just three of them in parallel. As a rule of thumb, their values may jump around in the same batch, especially for higher ones. Their relatively high-temperature dependence is not very significant in this application: warming them up to a body temperature from the regular room conditions moved the capacitance from 6.83 down to 6.79, which should be acceptable, I believe.
A film capacitor would be the best option, and it would be quite feasible at 6.8 μF. With a 5 % tolerance, capacitor inaccuracies could cause an error of about 0.4 dB well below 50 Hz, if you have the bad luck that you get the worst capacitors that pass the production test. Example of a suitable film capacitor: https://www.reichelt.de/mks4-pet-kondensator-6-8-f-5-63-vdc-rm-15-mks4-63-6-8--p172806.html This is MKT; polypropylene would be even better, but much larger.
Bipolar electrolytics are used all over the place in loudspeaker crossover filters, so they should be reasonably stable - loudspeaker crossover filters cross over in the middle of the audio band, where deviations are probably more noticable than at 50 Hz and below. They are generally less accurate than good film capacitors, though.
I wouldn't use ceramic capacitors, as they are bound to be class 2 at these values. Maybe you could get away with the nonlinearity because of the low signal level across C8, but there is also microphony, long-term drift (ageing) and temperature dependence, and the initial tolerance isn't very good either.
C7 is less critical, as it mostly affects the subsonic response, and C3 is not critical at all, as it only affects the deep subsonic response.
Bipolar electrolytics are used all over the place in loudspeaker crossover filters, so they should be reasonably stable - loudspeaker crossover filters cross over in the middle of the audio band, where deviations are probably more noticable than at 50 Hz and below. They are generally less accurate than good film capacitors, though.
I wouldn't use ceramic capacitors, as they are bound to be class 2 at these values. Maybe you could get away with the nonlinearity because of the low signal level across C8, but there is also microphony, long-term drift (ageing) and temperature dependence, and the initial tolerance isn't very good either.
C7 is less critical, as it mostly affects the subsonic response, and C3 is not critical at all, as it only affects the deep subsonic response.
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I believe production tolerance is not a big issue unless the goal is batch production. For a single-device design, one may pick the capacitor with a value slightly below the nominal and use one or two film capacitors of a smaller value to get the exact match.
To give everyone some context, the thread I designed it for started with: "It's nothing fancy, but I need a working preamp for a pretty average system." Otherwise I would have drawn a film capacitor straight away for C8.
Yes, of course. If you have the equipment to accurately measure capacitance and resistance, you can measure R12, calculate C8 = 3.18 ms/ R12 and aim for that value, rather than the rounded value of 6.8 μF.
Forgot one thing: there is a DC bias voltage across C8, which would reduce its capacitance if it were a ceramic class 2 capacitor.
That could very well be, I tend to unintentionally reinvent things. For example, I came up with a distortion cancellation scheme for differential pairs in the 1990's and read a few months ago on this forum that the basic idea was already used in valve amplifiers in the 1950's.
The resistor-capacitor-resistor T-network was quite common in discrete transistorized RIAA amplifiers of 50 or so years ago, but I don't know if realizing the 3.18 ms time constant with R12 and C8 was ever done back then. Is that how it was done in the Radford (?) amplifier?
The resistor-capacitor-resistor T-network was quite common in discrete transistorized RIAA amplifiers of 50 or so years ago, but I don't know if realizing the 3.18 ms time constant with R12 and C8 was ever done back then. Is that how it was done in the Radford (?) amplifier?
T6 is the final zero introduced by the series-feedback configuration due to the gain never falling below 1, so T5=75uS cannot continue downwards at 6dB/octave forever (Lipshitz numbering). In this circuit it is given by C5C6/(C5+C6)*R12 = 0.781uS ~= 200kHz, which is high enough not to worry about. If it gets below 100kHz you need to add an extra passive pole at the end to compensate for it. See Lipshitz passim, especially Table 1(b).
Your scheme models pretty well under the Lipshitz calculations. I'm unclear how exactly interchanging the lower LP and HP poles and frequencies can work, but it does.
http://ukhhsoc.torrens.org/makers/R...aintenance_Manual_C25_Issue1_1274/Page14.jpeg
I wonder why they use 47 ohm and 80 μF rather than 47 ohm and 68 μF. Maybe some correction for finite loop gain? Or due to the fact that the T-network is connected to the other side of the 47 ohm?
I wonder why they use 47 ohm and 80 μF rather than 47 ohm and 68 μF. Maybe some correction for finite loop gain? Or due to the fact that the T-network is connected to the other side of the 47 ohm?
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The RIAA part of the Radford HD-250 circuit is +2.25 dB at 10Hz reference RIAA, because T3=3030uS, should be 3180uS. Maybe compensating for that.
I've renovated a few of those and never understood what the other bits of that circuit were doing before: thanks guys.
I've renovated a few of those and never understood what the other bits of that circuit were doing before: thanks guys.