I agree with Galu that 1 watt RMS is sufficient for quiet listening using efficient speakers, particularly if they are on a tabletop within a metre or less of one's ears, and if the bottom several octaves of bass - say everything below 200 Hz - doesn't need to be particularly audible.
The little germanium-transistor AM radios of my childhood had considerably less power output than that - and they were still loud enough to annoy your parents with, if you listened to music they didn't like.
I would much rather listen to 1 watt of power put through a nice flat-frequency-response speaker, than listen to ample power put through the typical consumer-grade speaker, with boomy upper bass, a muffled midrange, and a "hole" in the frequency response between woofer and tweeter.
Take a look at the attached image, which I grabbed from the full PAM8403 datasheet (also attached).
What the graph shows is that distortion starts to climb very steeply upwards just above 1 watt RMS output. Total harmonic distortion soars from a sweet 0.008% at 1 watt, to a marginal 1% at 1.5 watts, to an intolerable 15% at 2 watts. As for 3 watts, it's not even on the graph; I think the curve would hit 100% THD before it reached 3 watts.
The reason why the distortion curve takes off like a skyrocket and zooms almost vertically upwards, is that the output of the amplifier has run out of headroom, and has started to clip like crazy. Increases in output power beyond 1.1 watt are only possibly by clipping the output harder and harder.
I hadn't yet seen these graphs when I made my earlier prediction (just over 1 watt RMS into 4 ohms when run on 5 volts). So how did I know?
That's because a simple calculation tells us what we need to know. The output of the chip can't swing closer than 1 volt to either ground or Vdd, so when Vdd is 5 volts, the maximum output swing is 3 volts peak to peak. (The output can't swing below +1 volt, or above +4 volt.)
This chip (like all the low-voltage class D chips I've seen in recent years) actually contains four separate power amplifiers, two of them being used in bridge mode for each channel. That doubles the peak-to-peak voltage across each speaker to 6 volts.
Power is RMS voltage squared, divided by speaker resistance. Doing a little clean-up algebra, that formula simplifies to:
Output power (RMS watts) = (peak-to-peak speaker volts squared) / (eight times speaker resistance)
Assuming a 4 ohm speaker, eight times that gives you 32. We have 6 volts peak-to-peak, and squaring that gives you 36. So the maximum RMS power is 36/32, or just about 1.1 watts.
Which agrees perfectly with the measured results in the datasheet.
This particular calculation is so reliable, that I rarely bother looking up the datasheet graphs. Short of miracles or black magic (neither of which exist), the formula works for every amplifer that doesn't have an output transformer. That's pretty much 100% of all solid-state amplifiers made today.
-Gnobuddy
The little germanium-transistor AM radios of my childhood had considerably less power output than that - and they were still loud enough to annoy your parents with, if you listened to music they didn't like.
I would much rather listen to 1 watt of power put through a nice flat-frequency-response speaker, than listen to ample power put through the typical consumer-grade speaker, with boomy upper bass, a muffled midrange, and a "hole" in the frequency response between woofer and tweeter.
I figure that the people who write ad copy live in a different universe from the rest of us.
Take a look at the attached image, which I grabbed from the full PAM8403 datasheet (also attached).
What the graph shows is that distortion starts to climb very steeply upwards just above 1 watt RMS output. Total harmonic distortion soars from a sweet 0.008% at 1 watt, to a marginal 1% at 1.5 watts, to an intolerable 15% at 2 watts. As for 3 watts, it's not even on the graph; I think the curve would hit 100% THD before it reached 3 watts.
The reason why the distortion curve takes off like a skyrocket and zooms almost vertically upwards, is that the output of the amplifier has run out of headroom, and has started to clip like crazy. Increases in output power beyond 1.1 watt are only possibly by clipping the output harder and harder.
I hadn't yet seen these graphs when I made my earlier prediction (just over 1 watt RMS into 4 ohms when run on 5 volts). So how did I know?
That's because a simple calculation tells us what we need to know. The output of the chip can't swing closer than 1 volt to either ground or Vdd, so when Vdd is 5 volts, the maximum output swing is 3 volts peak to peak. (The output can't swing below +1 volt, or above +4 volt.)
This chip (like all the low-voltage class D chips I've seen in recent years) actually contains four separate power amplifiers, two of them being used in bridge mode for each channel. That doubles the peak-to-peak voltage across each speaker to 6 volts.
Power is RMS voltage squared, divided by speaker resistance. Doing a little clean-up algebra, that formula simplifies to:
Output power (RMS watts) = (peak-to-peak speaker volts squared) / (eight times speaker resistance)
Assuming a 4 ohm speaker, eight times that gives you 32. We have 6 volts peak-to-peak, and squaring that gives you 36. So the maximum RMS power is 36/32, or just about 1.1 watts.
Which agrees perfectly with the measured results in the datasheet.
This particular calculation is so reliable, that I rarely bother looking up the datasheet graphs. Short of miracles or black magic (neither of which exist), the formula works for every amplifer that doesn't have an output transformer. That's pretty much 100% of all solid-state amplifiers made today.
-Gnobuddy
Attachments
That is pretty much exactly the can of worms I visualized in my head when I typed my post #15. Except that I visualized two individual inductors, rather than two lumped into a single autotransformer, and a third one tacked on for no particular reason.
Since there is a third inductor tacked on, it looks like you could obtain a 4th order low-pass filter (instead of only a 2nd order) by adding just one more capacitor, from ground to the junction of the three inductors. (New cap in a fetching shade of pink in the diagram I just drew up.)
A little tinkering around in LTSpice would confirm or refute my idea.
Note that there are many serious flaws in this scheme as presented:
1) The subwoofer upper cut off frequency is not adjustable, making it unlikely to integrate well (crossover smoothly) with the woofers unless everything is measured with a good measurement microphone, and inductors are custom-wound by trial and error.
2) There are no loudness (sensitivity) adjustments for either subwoofer or main woofers, again, making it hard to crossover well from one to the other.
3) There will almost certainly be a big spike in the low frequency response of the satellite speakers ('satelit'), below their intended lowest frequency, because of the peak in their impedance curves at their fundamental resonance frequencies.
4) That spike in the satellite speaker acoustic response will, once again, prevent the satellites from integrating well with the subwoofer.
In my view, this is a big, expensive can of worms, and one that won't even work well when finished.
It may have made sense a few decades ago, when individual amplifiers were huge and expensive, and winding custom inductors was a much cheaper solution.
But in the era of small, light, cheap, clean class-D audio boards, even cheaper op-amps, and easy active filter design, this old passive 2.1 scheme is best left to gently gather dust in the archives.
-Gnobuddy
Attachments
I can understand if you have the amplifier modules already.
Otherwise you could just get a Wondom JAB 5 which are really cheap for what they are. BT, DSP & 2.0/2.1/4.0 capabilities.
So you can decide later if you want to make it 2x 2-way or 2.1
10V minimum supply voltage though.
Those tiny boost regulators (XL6009 etc.) are cheap as chips if supply voltage is important, although you loose some power for the conversion.
You also get buck-boost regulators of the same chip that should work with a wide range of input voltages and say step up/down to 12 or 24V for your amps.
You could design a passive crossover, maybe 1st order but that takes time, buying different parts, measurements etc.
I´d say going active is easier (its "only" a sub) but I know others will disagree.
Otherwise you could just get a Wondom JAB 5 which are really cheap for what they are. BT, DSP & 2.0/2.1/4.0 capabilities.
So you can decide later if you want to make it 2x 2-way or 2.1
10V minimum supply voltage though.
Those tiny boost regulators (XL6009 etc.) are cheap as chips if supply voltage is important, although you loose some power for the conversion.
You also get buck-boost regulators of the same chip that should work with a wide range of input voltages and say step up/down to 12 or 24V for your amps.
You could design a passive crossover, maybe 1st order but that takes time, buying different parts, measurements etc.
I´d say going active is easier (its "only" a sub) but I know others will disagree.
I know the power output is a LOT less at 5v.
The PAM8403 board is what I started this little journey with - purportedly 3w per channel from 5v (dont know if thats true or not, buy hey)
THIS IS ONE I USE A LOT
When I say "test at 5v", really what I mean is to test how the 5v-24v board compares to the little PAM8403 board when run at the same voltage.
Ive made a few boxes with the small PAM board, and when combined with Faital Pro 3" or 4" full range drivers (which are good quality and have high sensitivity) ive been amazed at the volume and quality thats achievable.
Thats sort of my benchmark
Jim
Well, the board that claims to be able to run at 5v - 24v has come and ive tested it. I can only assume that at 5v the Bluetooth is MARGINAL and thus the sound isnt great, and too quiet. NOT as good as the basic PAM8403 boards. And thus, as I said, this board becomes pointless - may as well use a board that needs 12v or more, seeing as i'm going to need to use a power supply of some type.
Ive got another one on the way
Are you finding this to be the case? Is Bluetooth not working well on your boards?jimcroisdale said:
Most newer computer chips are designed to run on 3.3 volts DC. Previous generation chips run on 5 volts. It's been a long time since I can remember any digital chips that needed more than 5 volts.
So it's likely that the onboard Bluetooth chips on your amplifier boards are quite happy with a 5V power supply.
Delivering power to a speaker is a different kettle of fish. The less voltage is available, the less power you can deliver before the inevitable clipping starts.
There are only two ways around that restriction, and one of them is impractical.
The practical solution is to use a speaker with unusually low impedance, less than the widely available 4 ohms.
I'm attaching the datasheet for an inexpensive little speaker with a 1-ohm impedance. The manufacturer-provided frequency response for this little mini-speaker looks very good from roughly 120 Hz to 12 kHz. Sensitivity is not great, but stringing two in series will bump up sensitivity 3 dB, and the resulting two-ohm load might be easier to drive.
Incidentally, a chip designed to drive 50 watts RMS into a 4 ohm load, has to be capable of driving peak currents of 5 amps. That same chip can therefore drive a one-ohm speaker with up to 5 volts peak, or 10 volts peak-to-peak.
In other words, a 50-watt@4 ohms, bridge-mode class D module should be capable of driving a 1-ohm speaker as long as the power supply voltage is kept to no more than 7 volts DC.
On the other hand, if powered by 5 volts, a bridge-mode chip amp can deliver around 4.5 clean RMS watts to a 1-ohm speaker before it runs out of headroom and starts to clip. This goes quite well with the 3.5 watt maximum continuous / 8 watt transient power rating of this particular little 1-ohm speaker.
Oh yeah, peak speaker currents will be around 4 amps in this case (5V, 1 ohm speaker, bridge-mode amp.)
-Gnobuddy
You get a 3dB increase in efficiency (defined in the usual way as dB SPL@1m@1 watt), because of purely acoustic effects. Basically, because you doubled the vibrating area.wolf_teeth said:
This happens because the speaker diameter is far less than the wavelength of the sound waves it produces. Under these conditions, the efficiency of coupling vibration from speaker to air is proportional to speaker area. Double the vibrating speaker area , and you get +3dB increase in coupling efficiency to the air.
If you instead define efficiency in terms of *volts* of drive voltage, and not watts, then you are correct. The +3dB increase in acoustic sensitivity is cancelled by the -3dB change in power into the speaker.
In this particular case of the very low impedance speaker, depending on the amplifier chip used, maximum available peak speaker drive current might be the limitation (not drive voltage). In that case, putting two speakers in series gives you +3dB sensitivity increase due to acoustic effects, and also allows you to drive twice as much power into the pair, i.e., the same amount of power as before into each individual speaker.
In this current-limited situation, the end result is to give you a net +6dB increase in maximum SPL.
-Gnobuddy
You said sensitivity in your earlier post, not efficiency. Yes, efficiency is +3dB with a pair in series. Voltage sensitivity nets no gain in total.
I understand that a lower impedance speaker will likely have a higher sensitivity rating, but likely also at 4W/2.83V in this case due to the lower impedance. 4 ohms is 2W/2.83V.
Even so- this is power based, not sensitivity, where you are getting that +3dB. So, it is efficiency, not sensitivity, and I agree with the +3dB addition in that mindset.
People use sensitivity and efficiency interchangeably, and they are not the same thing.
Wolf
I understand that a lower impedance speaker will likely have a higher sensitivity rating, but likely also at 4W/2.83V in this case due to the lower impedance. 4 ohms is 2W/2.83V.
Even so- this is power based, not sensitivity, where you are getting that +3dB. So, it is efficiency, not sensitivity, and I agree with the +3dB addition in that mindset.
People use sensitivity and efficiency interchangeably, and they are not the same thing.
Wolf
You know, in fact they are exactly the same thing, but measured and described in different units.wolf_teeth said:
A loudspeaker is a transducer that converts electrical power into acoustic power.
The efficiency of the loudspeaker is simply (acoustic power from speaker)/(electrical power input to speaker), expressed as a percentage.
For modern speakers, efficiency is typically only around 1%, because our loudspeakers are extremely inefficient. Its the price we pay for getting a relatively flat frequency response, extended low frequency response, and good power handling out of them.
The sensitivity of the loudspeaker is the efficiency of a loudspeaker, but it is expressed in different units, in terms of sound pressure level (SPL) in decibels at one metre distance at one watt of power into the speaker.
This is easier to measure than acoustic power delivered the air in watts, so sensitivity is a convenient way to measure a speaker's efficiency.
The two things are not equal, but they are proportional. It's like expressing the power output of your car's engine either in kilowatts or in horsepower. They are just different units used to describe exactly the same thing.
In the case of the loudspeaker, double the efficiency, and you get +3dB increase in sensitivity - because +3 dB equals a doubling of acoustic power.
The converse is equally true. If you do something that causes a +3dB increase in sensitivity, it follows that the speaker is now delivering twice as much acoustic power into the air, with the same electrical power into the voice coil. By definition, this means the efficiency has also doubled.
But we are now wandering off-topic.
-Gnobuddy
No. Sensitivity is defined with relation to voltage, mainly 2.83VAC, so that all speaker drivers are measured and compared on an even scale. You cannot compare drivers when based on wattage as the voltage will be different depending on the impedance. When comparing and measuring with voltage standards, it removes the impedance issues from the comparison.
To go by usage of dB @ 1W/1m is actually efficiency expressed in dB instead of a percentage, because it is based off of wattage.
If you double the efficiency, you get +3dB, but this is based on power input. When you double the woofers to 2 in parallel over a single, you get +6dB in voltage sensitivity and only +3dB in power efficiency.
I repeat, they are not the same thing.
Wolf
Of course it isn't. Sensitivity is measured in dB SPL at 1 watt at 1 metre. Notice the unit in the middle is "watt", not "volt". A watt is a unit of power, not of voltage.wolf_teeth said:
Some manufacturers may do this out of ignorance, but applying 2.83 volts to speakers of different impedance guarantees that you are NOT measuring them on "an even scale". It is quite incorrect to do the measurement this way.wolf_teeth said:
Just think about it for a minute. If you apply the same drive voltage to every speaker, a 16-ohm speaker would receive one-quarter as much power as a 4 ohm speaker. The 1-ohm speaker I mentioned earlier on this thread would receive four times as much power as a nominally 4 ohm speaker. And what about those 600-ohm Phillips speakers used with their valve OTL amplifiers?
It's the same story with headphones. They come in a wide variety of impedances, but sensitivity is always specified in terms of dB SPL @ 1 milliwatt of applied *power*.
Again, the unit of speaker sensitivity is SPL@1metre@1 watt of input power. It's trivial to calculate the necessary drive voltage needed to supply 1 watt of power to whatever impedance the speaker happens to have. And that is the proper way to do the measurement.
-Gnobuddy
It's not about the power received, it's about the output level emitted.
It appears you have come from the car-audio arena, or you would not think about speakers with regards to power. Loudspeaker drivers are voltage-sensitive devices.
When measuring drivers, 2.83V is the standard, and no conversion is required like in wattage measurements.
You really are mistaken. I've been doing this for almost 20 years, and you cannot base output levels from wattage when they are voltage-sensitive devices.
Wolf
It appears you have come from the car-audio arena, or you would not think about speakers with regards to power. Loudspeaker drivers are voltage-sensitive devices.
When measuring drivers, 2.83V is the standard, and no conversion is required like in wattage measurements.
You really are mistaken. I've been doing this for almost 20 years, and you cannot base output levels from wattage when they are voltage-sensitive devices.
Wolf
More information:
Wattage is a HUGE misnomer in the design of loudspeakers. Where it's useful is in terms of thermal handling, which at most listening levels is not even remotely infringed upon. A fried speaker is a misused speaker, and usually results from mechanical power handling failure which leads to thermal failure.
Mechanical power handling is always much less than that of the thermal rating, and has to do with the box the driver is placed in.
In terms of box modelling:
I always model for loudspeakers to achieve a 100-105dB output level at the worst-case scenario, because that is rarely going to be breached unless the room is larger and the distance longer to the listening position.
In subwoofer modelling, I prefer to shoot for 110dB at worst, as most typically want +6dB from the main pair, and the mechanical power handling has to suffice at that level. Checking for Xmax breach here is a must.
Wolf
Wattage is a HUGE misnomer in the design of loudspeakers. Where it's useful is in terms of thermal handling, which at most listening levels is not even remotely infringed upon. A fried speaker is a misused speaker, and usually results from mechanical power handling failure which leads to thermal failure.
Mechanical power handling is always much less than that of the thermal rating, and has to do with the box the driver is placed in.
In terms of box modelling:
I always model for loudspeakers to achieve a 100-105dB output level at the worst-case scenario, because that is rarely going to be breached unless the room is larger and the distance longer to the listening position.
In subwoofer modelling, I prefer to shoot for 110dB at worst, as most typically want +6dB from the main pair, and the mechanical power handling has to suffice at that level. Checking for Xmax breach here is a must.
Wolf
Wolf, I'm sorry, but if you've been measuring sensitivity at 2.38 volts RMS regardless of speaker impedance, well, you've been doing the wrong measurement for 20 years.
Loudspeakers are electrical-to-acoustic transducers. Here is a technical definition of a transducer: What is a Transducer? - Definition from WhatIs.com
Notice the definition of transducer efficiency on that page:
Speaker sensitivity is electrical-to-acoustic transducer efficiency, which is (electrical watts in / acoustic watts out).
Both input and outputs are powers. Measured in watts.
Volts don't enter into it at all. Sensitivity is just efficiency expressed in different units. Both are measures of watts in / watts out. One is direct, the other indirect.
The indirect way is to measure SPL at a 1-metre distance in an anechoic environment. SPL@1 metre - in an anechoic environment - is a simplified way to measure acoustic output power. It is a stand-in for acoustic watts output by the speaker.
For the measured SPL to be meaningful, you have to drive the speaker with 1 watt of electrical power, no matter what the impedance of the speaker might be.
To do this, you have to adjust speaker drive voltage to whatever is necessary to achieve one watt into the speaker.
For example, you need 24.495 volts RMS to drive a 600-ohm Phillips speaker to 1 watt.
You need 4 volts RMS to drive a 16-ohm speaker to 1 watt.
You need 2.83 volts to drive an 8-ohm speaker to 1 watt.
You need 2.0 volts RMS to drive a 4-ohm speaker to 1 watt.
You need 1.0 volts RMS to drive a 1-ohm speaker to 1 watt.
Speaker efficiency and sensitivity are engineering terms. Like all technical terms, they are tightly defined, and there is no latitude for interpretation. Sensitivity is SPL at 1 metre at 1 watt. That's all there is to it. Sensitivity is not SPL@1 m at 24.5 volts RMS of drive voltage, or 2.83 volts, 2.0 volts, or 1.0 volts.
When I worked in loudspeaker R&D, we did encounter some speaker manufacturers who quoted SPL at 2.83 volts, instead of at 1 watt. These were the manufacturers who did not know what they were doing. They did not have any trained engineers in-house who understood that watts change with speaker impedance, or perhaps they were trying to make their 4-ohm speakers look better than they actually were.
If you drive a 4-ohm speaker with 2.83 volts RMS, you are actually pushing two watts into it, not one watt. This raises the SPL everywhere by 3 dB. If the person looking at the curves does not understand this, she/he will be fooled into thinking the speaker is more efficient than it actually is.
If someone gives you a 4-ohm speaker response measured at 2.83V (instead of the proper 2.0 V), you can correct the error by subtracting 3 dB at every point on the graph; lower the published frequency response by 3dB, in other words.
We used to throw away these sorts of badly measured curves, because they obviously were wrong and could not be trusted, since they couldn't even get the input power right. Then we would measure all the speaker parameters ourselves, including frequency response curves at the proper 1 watt of input power (not necessarily 2.83 VRMS unless it happened to be an 8 ohm speaker.)
I don't make up stuff out of thin air - I'm using the definitions you'll find in any reputable engineering textbook, used in any reputable engineering course at any reputable university.
I think it is unlikely you and I will find any agreement on this, so there's no point in continuing to discuss it. And this entire discussion is way off-topic, anyway.
So please have a nice day, and so will I!
-Gnobuddy
Loudspeakers are electrical-to-acoustic transducers. Here is a technical definition of a transducer: What is a Transducer? - Definition from WhatIs.com
Notice the definition of transducer efficiency on that page:
Power is measured in watts, and transducer efficiency is defined as (watts in / watts out).
Speaker sensitivity is electrical-to-acoustic transducer efficiency, which is (electrical watts in / acoustic watts out).
Both input and outputs are powers. Measured in watts.
Volts don't enter into it at all. Sensitivity is just efficiency expressed in different units. Both are measures of watts in / watts out. One is direct, the other indirect.
The indirect way is to measure SPL at a 1-metre distance in an anechoic environment. SPL@1 metre - in an anechoic environment - is a simplified way to measure acoustic output power. It is a stand-in for acoustic watts output by the speaker.
For the measured SPL to be meaningful, you have to drive the speaker with 1 watt of electrical power, no matter what the impedance of the speaker might be.
To do this, you have to adjust speaker drive voltage to whatever is necessary to achieve one watt into the speaker.
For example, you need 24.495 volts RMS to drive a 600-ohm Phillips speaker to 1 watt.
You need 4 volts RMS to drive a 16-ohm speaker to 1 watt.
You need 2.83 volts to drive an 8-ohm speaker to 1 watt.
You need 2.0 volts RMS to drive a 4-ohm speaker to 1 watt.
You need 1.0 volts RMS to drive a 1-ohm speaker to 1 watt.
Speaker efficiency and sensitivity are engineering terms. Like all technical terms, they are tightly defined, and there is no latitude for interpretation. Sensitivity is SPL at 1 metre at 1 watt. That's all there is to it. Sensitivity is not SPL@1 m at 24.5 volts RMS of drive voltage, or 2.83 volts, 2.0 volts, or 1.0 volts.
When I worked in loudspeaker R&D, we did encounter some speaker manufacturers who quoted SPL at 2.83 volts, instead of at 1 watt. These were the manufacturers who did not know what they were doing. They did not have any trained engineers in-house who understood that watts change with speaker impedance, or perhaps they were trying to make their 4-ohm speakers look better than they actually were.
If you drive a 4-ohm speaker with 2.83 volts RMS, you are actually pushing two watts into it, not one watt. This raises the SPL everywhere by 3 dB. If the person looking at the curves does not understand this, she/he will be fooled into thinking the speaker is more efficient than it actually is.
If someone gives you a 4-ohm speaker response measured at 2.83V (instead of the proper 2.0 V), you can correct the error by subtracting 3 dB at every point on the graph; lower the published frequency response by 3dB, in other words.
We used to throw away these sorts of badly measured curves, because they obviously were wrong and could not be trusted, since they couldn't even get the input power right. Then we would measure all the speaker parameters ourselves, including frequency response curves at the proper 1 watt of input power (not necessarily 2.83 VRMS unless it happened to be an 8 ohm speaker.)
I don't make up stuff out of thin air - I'm using the definitions you'll find in any reputable engineering textbook, used in any reputable engineering course at any reputable university.
I think it is unlikely you and I will find any agreement on this, so there's no point in continuing to discuss it. And this entire discussion is way off-topic, anyway.
So please have a nice day, and so will I!
-Gnobuddy
Excuse me for walking into this but in addition to these facts, I see there is another aspect that has its own application. When comparing sensitivities it is useful to note that voltage source amplifiers are the de facto standard. When measuring speakers' response, pink noise is used which is specified as a voltage.
What he's not seeing is that the 2.83 voltage spec makes everything on an even-keel field comparing their outputs. Going by wattage designing a 3-way will get you nowhere.
I guess I'm done trying to correct this misinformation he's spewing, as he's not going to get it right and refuses to.
Point blank, you have voltage-sensitive drivers being driven by a voltage-source amplifier in 98% of applications.
It is how a majority of loudspeakers are designed.
Thanks for your input, Allen,
Wolf
I guess I'm done trying to correct this misinformation he's spewing, as he's not going to get it right and refuses to.
Point blank, you have voltage-sensitive drivers being driven by a voltage-source amplifier in 98% of applications.
It is how a majority of loudspeakers are designed.
Thanks for your input, Allen,
Wolf