Finished Project - 12AT7 Joint Stereo micro amplifier running of 555 SMPS

Started by M23Bomber, September 27, 2015, 04:06:27 PM

Previous topic - Next topic

M23Bomber

Hello All,

Here is my latest idea:



Here is a pic:



Using a 12AT7 tube, an old polish output transformer ( Zatra TG 2-20-666 » 16k : 8ohm ) and the 555 SMPS, I am able to run 2 x 4ohm speakers in series quite loud. It eats up 13mA at 235V.

Im wodering what happens when we run 2x555 SMPS from the same wallwart. Has anyone tried?

Regards,
M.


amptramp

Let me be the first to ask: is there a schematic?  It definitely looks good.

Transmogrifox

That's pretty sweet. 

There's no reason to run 2xSMPS unless you need 2 different voltages.  To get more power you just need an inductor and FET rated for the amount of power output...and tweak the switching frequency and/or inductor value to be able to intake and output more juice per cycle.

The real question is how much power does your wallwart supply?  13 mA @ 235V says it's eating 3 watts.  Is this while outputting max volume to a speaker or is this measured while quiet?

With an ideal 100% efficient SMPS, then power out = power in, so
Vin*Iin = Vout*Iout

Since you will waste power in the SMPS, then it's really
Vin*Iin = Eff*Vout*Iout

or to find out how much power you need from a wall-wart,

Vin*In/Eff = Vout*Iout

For example, if your wall wart is 12V at 1A, then you have 12 Watts available.  I wouldn't expect your SMPS to do better than about 70% unless you put some R&D effort into optimizing it.  In that case you will have about 8.4 Watts available for the tube preamp.

Then suppose you lose 3 Watts just to have it up and running, that leaves 5.4 Watts available for output power.

Unfortunately, a push-pull output amp is probably 75% efficient in terms of final audio power delivered to the speaker leaving you with a maximum of about
4 watts actually going into the speaker (which is actually pretty loud).

Probably it is best to work your way backward from the maximum power you can deliver to the speakers into an 8 ohm load.  For your transformer it looks like about turns ratio n=45.  With a 235V square wave, max voltage output at the load would be 5.2 V amplitude, and into 8 ohms is 5.2^2/8 = 3.4 Watts.

Knowing that you have
(3.4W (output) +3W (quiescent))/75%(pushpull eff)/70%(smps eff) = 12.2 Watts.

It looks like you can do it just fine with a 12V @ 1A wall wart.
2 SMPS's would be fine on one such wall wart to give you a second (stereo) channel OR just beef up your SMPS and run left and right off the same B+.


trans·mog·ri·fy
tr.v. trans·mog·ri·fied, trans·mog·ri·fy·ing, trans·mog·ri·fies To change into a different shape or form, especially one that is fantastic or bizarre.

J0K3RX

Doesn't matter what you did to get it... If it sounds good, then it is good!

M23Bomber

Hello All,

I tweaked the following schematic for the amp:



For the SMPS, I used the following schematic:




It runs well.

Dear Transmogrifox, Do you know where I can find more info about the SMPS design to make it more beefy? And Yes I google it but most info is too theoretical, Im looking for a practical application guide. If I could tweak it more and get 220V -35mA then this thing would be proper stereo amp,I still have a couple of transformer laying around.

Regards,
M.

Transmogrifox

Quote from: M23Bomber on September 28, 2015, 03:30:07 AM
Im looking for a practical application guide. If I could tweak it more and get 220V -35mA then this thing would be proper stereo amp,I still have a couple of transformer laying around.

This is a good primer, but maybe it is too theoretical...maybe more of what you have already found.
https://www.maximintegrated.com/en/app-notes/index.mvp/id/2031

I'll give a quick overview:
A boost converter is like a slingshot.  The inductor is like the bands. 

You take a rock and pull it back with a long, slow draw, then when you let it go the rock flies off at a high velocity.  In this case voltage is like the distance the rock goes.  Really short distance pull on the input (you pull it arm's length), and the rock goes much much further than the draw length going out.

Now suppose you have a really large cart you want to move, and for some strange reason you want to make it stay a certain distance up an incline by showering it with a steady stream of flying rocks.

The size/weight of your cart and how much elevation change you want to maintain (high voltage load) will dictate how large of rocks you need to launch at it, how fast they are going, and how frequently they need to hit the cart to make it stay at a certain distance up the incline.

As for the slingshot band, you can either design it to stretch a long way back, or you can make it so it takes a lot of force to pull it back.  In both cases it needs to be strong enough to handle the forces and large enough to hold the rocks you're launching at the cart.

Now I don't know whether that analogy helped, but maybe it will connect some more with below:

***********************************
Some Theory
(Skip to "The More Practical Guide" if you don't do well with math/algebra)
************************************

Energy stored in the inductor is:
WL = 0.5*L*i^2
WL = Energy (Joules)
L = inductance
i = current in inductor

Also, current in the inductor is related to voltage applied and time:
i = V*T/L

i = current in inductor
V = DC voltage applied
T = Time DC voltage is applied
L = inductance

This helps you go both ways -- for an input of 12V and inductance of, say, 100 uH for a time duration of 10us, the current is:
i=12V*10us/100 uH = 1.2 A

Now the energy is:
0.5*100uH*1.2A^2 = 72 uJ

How long will it take to discharge all of this energy at 235V?
WL = 0.5*L*i^2, and,
i = V*T/L, so with a little algebra,

T=sqrt(2*WL*L)/V = sqrt(2*72uJ*100uH)/235V = 510 ns

So it takes 10 us to to pull back the slingshot and about 1/2 microsecond to let it go.  Adding these 2 periods together gives us a switching frequency:
Fsw = 1/(Tcharge + Tdischarge) = 95.14 kHz

Now how much power is going out of this at 235V?  Well, every cycle we're putting 72 uJ into the inductor and transferring it to the other side (neglecting power losses).  Energy is Power times Time, so if
WL = P*Time,
P = WL/Time
Fsw = 1/Time, so
P = WL*fs = 72uJ*95.14 kHz = 6.8 Watts

How much current is this at 235V?
P=V*I, so
I= P/V = 6.8/235 = 28 mA

That's not quite enough for your purposes, but I did a bit of algebra to help get where you need to be.

********************************************************************************
The More Practical Guide
********************************************************************************
You want 220V@35mA, so let's first come up with the required power:
P=V*I=220V*0.035A=7.7 Watts
Now let's assume efficiency is 70% for good measure:
Pin = 7.7/0.7 = 11 Watts


To make the numbers easier, suppose you have a 12V @ 1A wall wart, you can do 12 Watts.  Here's a help that I came to with a little algebra from above, and this is more of the simple "guide" to smps design:

Ts = 2*P*L*(1/Vi+1/Vo)^2
Ts = switching period (1/Fsw)
P = Power input, which is Pout/efficiency
L = Inductance
Vi = input voltage
Vo = boosted output voltage

Now we can arbitrarily pick an inductance value and if you don't like the switching frequency then make the inductor smaller to go faster or larger to go slower.
Example:

L = 47 uH
Vi = 12V
Vo = 220V
P = 12 Watts

Ts = 8.7 us
Fsw = 1/Ts = 115 kHz



Ok, this is all well and good, but we need some components to be able to do this whole thing, so what is the worst-case current in the inductor?


Because duty cycle is so long,
Approximate Time = Ts = 8.7 us

Worst case current will be on the 12V charge stroke, so
I = V*T/L = 12V*8.7us/47uH = 2.22A

You need an inductor that can handle more than 2.22A without saturating.
You also need a MOSFET that can handle 2.22A peaks, and about 1A average.  Also the MOSFET needs to be rated for significantly more than 220V (best like 350V or 400V).

The rectifier diode has a much less severe set of constraints.  You just need to make sure the voltage rating is met and also that it's package is relatively good-sized to handle the extra stress during startup (charging the 220V side capacitor bank). For example, don't use 1N4148's stacked in series.  Probably a good 1A 400V switching diode is adequate.

The IRF740 I have seen referenced in this 555 SMPS design will easily handle this.  Something to keep in mind, though is its RdsON = 0.55ohms, so 1A*.55ohms = 0.55 Watts.  You also have some switching losses, so probably suppose about 1W loss in the FET.  This means it really should have a good slug of metal attached (heat sink).

If the inductor you're using can handle >2.2A without saturating you're in great shape to simply modify the switching frequency according to the above.  If you're not quite getting enough juice, then lower the switching frequency a bit more until you get the needed output power.

************************
One final consideration:  Input supply ripple.
You will want a beefy capacitor bank on the input to support 2A peak ripple currents.  Now these average out to about 1A because it's more of a linear ramp.

With a steady 1A current applied to a capacitor it will charge or discharge according to this formula:
V=I*T/C
I= current
T= time
C = Capacitance


Let's aim for
V= 100 mV ripple
T = Ts = 8.7 us
I = 1A (averaged over Ts)
C = We want to find out what this should be, so

C = I*T/V = 1*8.7u/100mV = 87 uF


I would suggest (2) parallel 47 uF caps and (2) 0.1 uF caps.  The 0.1 uF caps will help suppress high frequency content in the switching edges.

****************************************
The Bottom Line
****************************************
The working components are the inductor, MOSFET and output Diode.
The amount of power delivered to the load depends entirely on these 3 components.

Required switching frequency is determined by
A) Value of the Inductor
B) Output power

Smaller Inductor --> Faster Switching frequency but higher peak currents
Larger Inductor --> Slower Switching frequency but lower peak currents
Higher power always means lower switching frequency AND higher peak currents given the same inductor.

Where does the 555 timer fit in?  The 555 timer helps you time how long you turn on the FET to charge up the inductor (duty cycle), then how long before it starts a new cycle.  After the 555 turns off the FET, it's entirely on the inductor and diode to let the charge go out into the load.

Feedback, what does it do?
The feedback modifies the duty cycle so if the voltage on the output is too high, it turns down the charging time relative to the discharge time. 

There are three ways this can be done:
A) Define a fixed charging time and use feedback to decrease switching frequency to decrease power output.
B) Define a fixed switching frequency and use feedback to decrease charging time per cycle to decrease power output.
C) Fixed duty cycle, variable frequency.  This one is where the driving circuit senses when output cycle has discharged all the energy and feedback limits the peak current allowed on the charging cycle.  Switching frequency tends to go lower with increased power demand due to the longer required charge and discharge times.

In (A) my calculation for switching frequency above would be the maximum frequency of the converter, so for lighter power loads the frequency would go even lower.

In (B) my calculation for switching frequency would be the fixed switching frequency for the converter based on the maximum power output you expect to deliver.

In (C) my calculation for switching frequency would be the minimum frequency at which the system operates, but the more useful number is the peak input current at this frequency so you know where to set the maximum input peak current reset threshold on the switcher.  Here is an example of this method:

And a slightly different approach to calculating the inductor:
http://cackleberrypines.net/transmogrifox/BoostConverter/SMPS_Boost_Equations.pdf
trans·mog·ri·fy
tr.v. trans·mog·ri·fied, trans·mog·ri·fy·ing, trans·mog·ri·fies To change into a different shape or form, especially one that is fantastic or bizarre.

M23Bomber

Transmogrifox my man :) You know your SMPSs , Im really impressed and happy that you are sharing it :) I will spend some time trying to work it out :) I will post my questions soon :) I added a video in the first link :)

Transmogrifox

Quote from: M23Bomber on September 28, 2015, 02:45:46 PM
You know your SMPSs
Designing and testing these beasts has been a major part of my day job for several years, so I have learned a few things. 

It's also fun to see if I can explain it in a way that an average person can understand it well enough to tweak it on the bench.  If you had an oscilloscope it would be easy to tweak one of these things to whatever parameters you like.  You would learn a lot and (I think) it would be a ton of fun too.  I always enjoy my job best when I get to go play in the lab ;)

These things are so simple yet so complex at the same time.  The basic idea of repetitively charging an inductor and let it discharge into a load is pretty simple, but implementing a circuit that actually does so in a controlled and predictable manner without burning stuff up is where it gets interesting. 

Please be very careful.  235 VDC is 235 VDC whether it's pumped up there from a 12V wall wart or whether it's coming from a battery or outlet, or if it's just sitting on a capacitor like a coiled snake after you remove power from the circuit.  This can cause you some real harm so take it slow and double check everything before you apply power.  If something seems strange, unplug the wall wart and check the circuit with a meter to make sure all voltages are safe before touching anything.
trans·mog·ri·fy
tr.v. trans·mog·ri·fied, trans·mog·ri·fy·ing, trans·mog·ri·fies To change into a different shape or form, especially one that is fantastic or bizarre.

thomasha

Very interesting!

about the fet, I'm using IRF644 for the MAX1771 SMPS, as suggested by Desmith, but I found another one:
IRFB4137PbF

It's a little expenciver, but if it would make things work cooler it would be nice.

Another problem I face everytime I try to build one of these SMPS is the Inductor.

The inductor get's too hot IMHO, but as I'm using cheap chinese made 100uH 3A inductors, maybe that's the problem.
The thing is that other inductors are difficult to find on ebay, and Mouser shipping prices outside the USA are a little too high.
But if you can it is always better to use a low ESR inductor.

In fact if you can use low ESR capacitors too I guess it would be better.

Cheers,
Thomas

Transmogrifox

Quote from: thomasha on September 28, 2015, 05:28:13 PM
The inductor get's too hot IMHO...it is always better to use a low ESR inductor.
The magnetic material core losses are often a big contributor as well.  However, ESR and bad magnetics material often move together.  If you have a poor magnetic material then it takes more turns of wire (higher ESR) to get the same inductance.  Cheaper inductors use cheaper cores, and perversely, it is cheapest to make cores that waste power and saturate more quickly.

The best way to measure "too hot" is with a thermocouple and compare the temperature to the datasheet.  The other concern is keeping it away from parts you can touch and things that can combust.  For the boost inductor you want to keep it far from touch and inside an enclosure simply from an electrical shock point of view.

trans·mog·ri·fy
tr.v. trans·mog·ri·fied, trans·mog·ri·fy·ing, trans·mog·ri·fies To change into a different shape or form, especially one that is fantastic or bizarre.

amptramp

You also want to make sure you use a fast switching diode rather than a 1N4007 or something like that.  A slow diode will make this very inefficient.  If you have a small enough current that you can use a 1N4148, do so.  Your voltages are way too high for a Schottky diode.

M23Bomber

Hello all,

So from what I understood if I exchange the a few values in the 555 timer then I can get those those characteristics you calculated:

- Frequency : 95.14 kHz with a Duty Cycle of 95% ( Total T of 10us plus 0.5us),

- Inductor of 100uH with 2.5A for example,

- Wallwart of 12VDC with 2.2A

- The output capacitor should be 2x 47uF and 2xnF


Heres the caculation I got from an online 555 calculator:




I kind grasped the working operation principle,thanks to your analogy and practical guide ,Im not fully in it. But for example the NE555 timer has a limit around 50kHz, I need the beefy LM555 which has 500kHz I think.

If you would have the time to design and share, that would be the most amazing new audio SMPS.

Regards,
M.

duck_arse

that's a cracker of a post, transmog.

I have an inductor question. the app notes for the old mc34063 switcher says to use a toroid core for the inductor for lower EMI and hash, which seems appropriate in these circuits. yet mostly I see people recommend against using toroids. can you offer some pointers?
don't make me draw another line.

Transmogrifox

Quote from: M23Bomber on September 29, 2015, 08:17:39 AM
- Frequency : 95.14 kHz with a Duty Cycle of 95% ( Total T of 10us plus 0.5us),

- Inductor of 100uH with 2.5A for example,
In my example above I think I ended up with 47 uH and 115 kHz switching frequency for a converter that will do 35 mA @ 220V.  With 100 uH I think it would be about 54 kHz.
Quote
I need the beefy LM555 which has 500kHz I think.

You can still do the simple 555 timer-based controller and just use a bigger inductor to bring down the switching frequency.  The downside with lower switching frequency is the size of everthing goes up, including high voltage capacitors to filter out the switching noise.

The discrete component switcher I posted is another cheap flexible way to do it.  It simulates up to 1 MHz ok, but efficiency is best below about 300 kHz.  I try to keep switching frequency between 100 kHz and 200 kHz.  It's far away from the audio band, but still slow enough efficiency doesn't suffer severely.  [EDIT] Added 220V @ 40 mA version [/EDIT]

This simulates well even down to light loads of 100k (2.2 mA) and up to a 4k load (55 mA), able to deliver about 12 watts.  Switching frequency is about 95 kHz full load and light load goes up to about 350 kHz.  My simulation estimates it to be about 84% efficient at full load with an ideal inductor so it's actually not a bad power waster.  Adding some losses to the inductor to the simulation model brought it down to about 78% efficiency.

The inductor sees about 1.25A RMS during steady state operation and about 2.5A peaks during startup.  Basically you need an inductor that doesn't saturate until something more than 2.5A.

Either way with a 12V at 2.2A wall wart you're in good shape for the desired 35 mA @ 220V and still have some power available on the 12V supply for the heaters.
trans·mog·ri·fy
tr.v. trans·mog·ri·fied, trans·mog·ri·fy·ing, trans·mog·ri·fies To change into a different shape or form, especially one that is fantastic or bizarre.

M23Bomber

Dear All,

Transmogrifox, I would like to share my enourmous gratitude for spending some your time calculating and sharing an improved SMPS for my needs.  ;) Its really great and Im sure that a lot of people will use it the future :) If you are in Poland let me know, we will have a  few beers on me. :)

When I look at the schematic you created and simulated , some of the parts in it are not in my inventory or my local electronic dealers, is it possible  for you to run a simulation with a few diferent parts:

- Transistor for the oscilating circuit 2N5088 or BC547B instead of the 2N5089

- The mosfet maybe the IRF740 or the IRF644

- The diode instead of the UPSC600 maybe the common UF4007 or BYV27-600

- Instead of that 350mOhm Resistor is it possible to make that all values will be in the ohms range

Is it possible to run that simulation and share back the results?

Regards,
M.



Also would like to thank thomasha because when I wanted to build a submini guitar amp he was also really helpfull, he shared and advised me during the process. And believe me my neighboors were also thankfull.  ;)


thomasha

Hi,
thanks M23Bomber, when it's about this small amplifiers I'm always interested!

I was just going to ask something similar, I guess there is no way of using a higher resistance at the mosfet, but would it be possible to short it? I guess the result would be no current control...but I'm no expert.

I'm also looking for better mosfets (lower Rdson, lower capacitances, higher current/voltages).
The IRF644 is currently my best option because it's cheap and easy to find, but the 250V limitation could be a problem.

With the SMPS here presented it would be possible to run a 12ax7 - EL95 amplifier with a 3W output, very interesting!





Transmogrifox

Quote from: M23Bomber on September 30, 2015, 05:51:37 AM
Transmogrifox, I would like to share my enourmous gratitude for spending some your time calculating and sharing an improved SMPS for my needs.  ;) Its really great and Im sure that a lot of people will use it the future :) If you are in Poland let me know, we will have a  few beers on me. :)
A few beers would be wonderful.  May be a while before I ever cross the Atlantic but I'll take note there are a few beers over there with my name on it.  For now I think this is the best we can do:
Cheers!

Quote
When I look at the schematic you created and simulated , some of the parts in it are not in my inventory or my local electronic dealers...
Good news is this isn't very sensitive to part selection (within reason), so most substitutions will work as long as you have FET with RDSon <1 ohm, rated to >250V, general purpose transistors ok, 1A fast switching diode rated to >250V.

Quote
is it possible  for you to run a simulation with a few diferent parts:

- Transistor for the oscilating circuit 2N5088 or BC547B instead of the 2N5089
BC547B works fine in simulation.  Any general purpose transistor ok (2222, 3904, 5088, BC546x/BC547x...)
Quote
- The mosfet maybe the IRF740 or the IRF644
IRF644 would probably be better than IRF740. I don't have SPICE models for these specifically but I selected devices with similar properties and they both work fine.   
Quote
- The diode instead of the UPSC600 maybe the common UF4007 or BYV27-600
I would say either of those diodes will work here.  Mostly you just need the voltage rating and a decent sized package to handle the transient peaks.

Quote
- Instead of that 350mOhm Resistor is it possible to make that all values will be in the ohms range

Yes, put 1-ohm resistors in parallel.  A little more simulation revealed voltage regulation will be better at full load using 250m instead so I suggest (4) 1-ohm resistors in parallel.


******************************
A little theory of operation
******************************
The resistance there in the MOSFET source (R2) sets the peak current for each cycle.  It gets compared against the Vbe drop of transistor Q1, which then resets the switching cycle.  The drain of M2 flies off to 220V and conducts through R4 to drive Q1 on harder.  When finally L1 has discharged its energy it relaxes and lets the voltage at Q1 base decay below Vbe, shutting it down and then letting Q2 conduct, turning on the FET, restarting the inductor charging cycle.

As the inductor current increases, the voltage drop across R2 increases until it gets high enough to turn Q1 on again and it repeats over and over...

Now once the output voltage gets to 220V DC, you want a way to limit the energy going over there.  That's where current conducts through the zeners in the feedback network and they increase the offset at Q1 base.  This means the voltage at R2 doesn't need to get as high to turn on Q1, so this means the peak currents can't get as high before the cycle restarts.

In really light load situations it may enter a burst mode operation where it bursts on, overcharges the output until Q1 is stuck on and the system has to discharge before it will start again.

A final word is startup can be a little bit finicky.  If you turn on the 12V too slowly it's likely to not start up.  The best way to start it up is to have the wall wart already plugged into its power source, then plug it in so the circuit gets a good fast jolt when power is applied. 

You may also want to employ a "Standby" and Power switch.  Standby turns on the heaters to get the tubes warm, then the power switch will apply 12V to the SMPS converter.  Switching the power switch will always give it a sharp jolt for startup. 

On the tube preamp I am building, I have one of these providing 320VDC and startup has not been a problem.



trans·mog·ri·fy
tr.v. trans·mog·ri·fied, trans·mog·ri·fy·ing, trans·mog·ri·fies To change into a different shape or form, especially one that is fantastic or bizarre.

M23Bomber

Hey all,

Going to breadboard this idea this weekend,Will post the result here :)

Also about the 555 SMPS, I read a bit about and I got a few conclusions from looking arround the web:

- Its better to use a 100uH 2A inductor becuase the maximum current that can be acumulted in that inductor is around 1.8A if the frequency of the 555 is 31kHz.

- The PSU needs to be at least 12V 2A.

- Instead of one 4.7uF capacitor, its better to use three 2.2uF and two 47nF capacitors i parallel.

- In the DC input its better to replace the 100uF with two 470uF low ESR capcitors and one 100nF capacitor in parallel.

- Change the MOSFET to one with a RDSon of 2 to 3 ohm at 400V 125W or higher.
Regards,
M.

thomasha

Hi M23bomber,

could you explain why use a higher Rdson mosfet, or where you found this information?

To me it would increase heat losses, but I only have  a small experimental experience with SMPS and there could be something I'm ignoring.

Another info I found in a Japanese blog:
Use two 2.2uF film capacitors instead of one 4.7uF, because film capacitors have lower ESR and better tolerances than electrolytic capacitors. (I don't know if this is true, but I read it here using google translator> http://lucythesoloist.blog33.fc2.com/blog-category-64.html)

Thanks,
Thomas

M23Bomber

Hey Tomasha,

Yes, film capacitors are slighty better than electrolitic in ESR but much bigger in size.

Also thanks to the japanese link you added , found out that the output tranformer I used the , Zatra TG 2-20-666 :(8k to 4ohm or 16k to 8ohm), has an american equivalent, the Fender TF103-48, it has 8k to 4 ohm, wich means that with an 8ohm speaker it will be 16k. The link is here: http://triodeelectronics.com/tfchxfwi48oh.html

I kinda found the RDS ON here: http://www.edaboard.com/thread154097.html

With lower the RDSon, a mosfet will saturate more easily and there for more current can be obtained , above all you are right , heat is a consequence, but a kinda natural part of it. I drill holes in my enclosures to improve heat handling.

Usually I drill 16x4mm holes right under the SMPS, in a 2 by 8 partern, and after I make the holes for the transformer slightly bigger to allow exaustion on the top.

I usally do it like this: Acircle=PixRxR

Pi=3.14
R=Radius and the radius is half of the drill size ,  in this case a 4mm drill,the radius is 2mm.

The math is : A=3.14x2x2
                    A=12,5 sqare mm

                   16x12,5= 200 square mm

                   200=3.14xRsqrt
               
                  ( 200/3.14)2=R
                   
                  7.9 = R


Multiply R by 2 and this gives me the drill size / hole in the top in this case 16mm, I dont calculate the cables yet  :icon_mrgreen: . Since heat moves from hot to cold, in this case from inside to outside, convection of heat says that hot air moves up, causes a presure drop inside the box wich will pull cooller air from the bottom. This means that you should have rubber feet in the bottom.

But  Im considering a 30x30x10mm fan soon.

Best regards,
M.