Why Does My Toaster Hum Due to AC-Induced Magnetic Forces?

[snoopies622]

TL;DR

I know that the heating coils must be vibrating to make sound, and that the frequency of vibration must be related to that of the alternating electrical current coming through the wall. But i don't know the details.

For instance, a current through a wire makes a magnetic field around it, and the changing magnetic field creates an electrical field, which then creates a magnetic field, and so on. Are these fields pulling on the wire? If so, how exactly? Isn't every piece of the wire dx electrically neutral at any moment? Is it because the wire becomes a magnetic dipole and the created magnetic fields appear in such an orientation as to act on this dipole moment?

 

[Baluncore]

Is the hum at the mains frequency, or twice the mains frequency?

 

[snoopies622]

Is the hum at the mains frequency, or twice the mains frequency?

Unfortunately i lost my tuning fork long ago, so I'm only making an assumption about the relationship between these two frequencies.

 

[Ibix]

Because it doesn't know the words! (Sorry.)

You can download instrument tuner apps that might give you an idea what frequency you're hearing.

 

[Baluncore]

The resistance wire in the heating element, will have parallel currents flowing, that have magnetic fields. If the currents are running in the same direction, the wires will attract, if the currents are in opposite directions, the wires will repel. As there are two current peaks per cycle, that will make a vibration at twice the mains frequency.

If there is a permanent magnet near an AC conductor, the magnet or the wire may move, causing a vibration at the mains frequency.

So the frequency of the possible hums will be one octave apart. That should be easy to identify.

 

[snoopies622]

Thanks, Baluncore!

 

[russ_watters]

The type of app I would look for is called a "spectrum analyzer".

 

[weirdoguy]

Because it doesn't know the words!

But it may know that @snoopies622 is humgry!

 

[sophiecentaur]

TL;DR Summary: I know that the heating coils must be vibrating to make sound, and that the frequency of vibration must be related to that of the alternating electrical current coming through the wall. But i don't know the details.

because the wire becomes a magnetic dipole

Take any pair of conductors in the toaster. The current through each wire will be proportional to the (alternating frequency f) voltage. In the absence of any nearby permanent magnets the force between them will be proportional to the product of the currents. That is proportional to the square of the supply volts so you would expect the vibrations to be basically at 2f. In the presence of a significant permanent magnetism the force will be proportional to f.
It will be much more complicated than that because the mechanics (displacement related to volts) will not be linear but a raggedy squareish wave so the sound will be full of harmonics in addition to the fundamental or twice fundamental frequency driving force.

My dad used to be an EE, working for the Central Electricity Generating Board (CEGB in UK). He just wouldn't commit himself to saying if I could hear 50Hz or 150Hz when walking around a substation. Actually, we didn't have the discussion about 100Hz vs 300Hz. Those big transformers used to vibrate noticeably to the touch so I guess there was a lot of 100Hz about. I never saw a three phase toaster . . . .

 

[Baluncore]

Those big transformers used to vibrate noticeably to the touch so I guess there was a lot of 100Hz about.

The distinctive transformer buzz, is due to magnetostriction within the iron core. That generates a second harmonic.

 

[snoopies622]

Thanks, everyone! I've been asking this site questions for almost twenty years now - it's been priceless.

 

[sophiecentaur]

The distinctive transformer buzz, is due to magnetostriction within the iron core. That generates a second harmonic.

. . . . and a three phase transformer? These were three m high and a long walk round, iirc.

 

[snoopies622]

Interesting: using my musical keyboard and my ears, it sounds like to me the most prominent frequencies from my toaster are the ones 2 and 3 octaves above the wall 60 hertz alternating current. (Of course, that may be where my ears are more sensitive.)

 

[Baluncore]

Interesting: using my musical keyboard and my ears, it sounds like to me the most prominent frequencies from my toaster are the ones 2 and 3 octaves above the wall 60 hertz alternating current.

You are correct, it is interesting. And your hearing is good, the hum will be rich in components, actually 1 and 2 octaves above the 60 Hz fundamental. (Zero is not 1, so I think you counted wrong, so NOT 2 and 3 octaves above the 60 hertz).

My explanation goes like this.

For a 60 Hz fundamental, the magnetic force between wires would be in the form of pulses, at the second harmonic, 120 Hz, one octave above the fundamental. It is that pulse train, at 120 pulses per second, that you are hearing as the hum.

Those pulses are one-sided, asymmetric about zero force. So the even harmonics of the pulses will be present. The first even harmonic is the second harmonic of the 120 Hz pulse train, which is at 240 Hz, the fourth harmonic of the 60 Hz mains fundamental, two octaves above the 60 Hz mains fundamental.

If the pulses were produced by a 60 Hz sinusoid and a permanent magnet, then the 60 Hz fundamental would dominate. Because those pulses are symmetrical about zero force, the odd harmonics may also be present. The first of those will be at 180 Hz, which is 3/2 octaves above the 60 Hz mains fundamental. That is not the case with your toaster.

 

[sophiecentaur]

so I think you counted wrong, so NOT 2 and 3 octaves above the 60 hertz).

The popular pitfall of confusing Harmonics with Overtones. PF should sort this out, imo.

 

[Baluncore]

PF should sort this out, imo.

I did my best to avoid the term overtones.

An octave is a factor of two, on the exponential musical scale, where the twelve notes per octave are separated by frequency ratios of ≈2^(1/12).
The second harmonic is one octave above the fundamental.
The fourth harmonic is two octaves above the fundamental.

It gets confusing here because the hum is already one octave above the 60 Hz fundamental. Then the analysis is referenced to harmonics of the audio hum, not the 60 Hz AC.

 

[sophiecentaur]

I did my best to avoid the term overtones.

Why? Pointing out the alternative term avoids confusion and many PFers need some help with that. The resulting frequencies from a 'squared' wave will be even harmonics of the (zero or much reduced) fundamental.

The 'Overtones' from an instrument could well have been easier to identify by a low tech instrument or bell maker without any electronics to help them (except maybe some tuning forks). It would be interesting to know where, why and when the two different quantities were named and used.

 

[snoopies622]

The frequencies i heard were the ones just above the 440 A and the one an octave below that, which are two and three octaves above the 60 Hrz.

 

[Baluncore]

The frequencies i heard were the ones just above the 440 A and the one an octave below that, which were two and three octaves above the 60 Hz.

Yes.
I know it is a bit mind-boggling, but we are analysing the harmonics of the hum that repeats at 120 Hz, not the AC mains voltage at 60 Hz.

Two and three octaves above the 60 Hz AC, (240 Hz, 480 Hz) is the same as, one and two octaves above the 120 Hz hum, (240 Hz, 480 Hz).

Harmonic presence, or prediction, is based on amplitude symmetry about the x-axis. Symmetrical signals have only odd numbered harmonics. Asymmetrical signals include even numbered harmonics. For that reason, we must consider and analyse the hum to have a fundamental frequency of 120 Hz. Then we consider the EVEN harmonics of the 120 Hz hum.
The second harmonic will then be at 240 Hz. One octave above the 120 Hz hum.
The fourth harmonic will then be at 480 Hz. Two octaves above the 120 Hz hum.
So the hum that you hear at 120 Hz, contains the dominant harmonics predicted by the analysis.

 

[sophiecentaur]

So the hum that you hear at 120 Hz, contains the dominant harmonics predicted by the analysis.

The mechanical induced motion will have both odd and even components and you can't rely on the excitation waveform to be anything like symmetrical. The coils of elements will clatter around in the mounts and wave shape will depend on the actual temperature - you (I) can hear the difference as the cooking proceeds.

 

[Baluncore]

The mechanical induced motion will have both odd and even components and you can't rely on the excitation waveform to be anything like symmetrical.

Exactly.

The excitation voltage comes from the AC mains, and appears across a resistive element, so the current flow is very close to symmetrical. Since that force ∝ i², rectification process is symmetrical, the alternate pulses at 120 Hz will be very similar, so there should be almost no 60 Hz fundamental present in the audio. We are therefore analysing the dominant 120 Hz pulse stream that we hear as a hum.

If the 120 Hz hum was perfectly symmetrical about zero force, then the odd harmonics of the fundamental 120 Hz would be present; 3f = 360 Hz; 5f = 600 Hz.

But we know it is strongly asymmetrical, because it is generated between a one-sided force and a zero force, (at the voltage zero crossing), so I must expect the even harmonics of the fundamental 120 Hz hum to dominate; 2f = 240 Hz; 4f = 480 Hz.

Yes, I know the amplitude of the fundamental, and the harmonic makeup of the hum will change with the albedo of the toast, as the flat resistance wire expands and contracts, but the underlying principles of harmonic generation, and the mathematics of Fourier analysis are reliable.

 

[snoopies622]

So to go back to the thoughts that led me to my original post - just to be certain : a single, isolated wire with an alternating current does not vibrate, correct? There's no self induced mechanical force happening? (even assuming it's coiled so there's a clear non-zero inductance)

 

[Baluncore]

: a single, isolated wire with an alternating current does not vibrate, correct?

False.
Anything with an AC current will be subjected to cyclic magnetic forces. The filaments of current that flow in parallel, on opposite sides of a conductor, interact to change the dimension and alignment of the conductor.

An isolated wire will also vibrate against itself, because one end of the wire, or solenoid, will react against the other end. Each turn of an inductive winding, reacts with forces against every other turn.

The amplitude of the vibration will depend on how well the conductor is attached to its environment. Young's modulus is not infinite, so there will always be vibration.

Power lines, hanging in the Earth's magnetic field, vibrate at the AC frequency. Isolated wires, that react against themselves, vibrate at twice the AC frequency.