This is from a meta-stable oscillation of one of the switching power supplies. The switching speed is almost certainly much higher than this, which is why you don't hear it most of the time. Under just the right conditions of input voltage and load current, the control algorithm of the power supply oscillates at 16 kHz.
This is not necessarily a problem electrically. Most likely the resulting voltage is still within spec. The only real downside, as you have found, is that it makes an annoying noise.
There are three mechanism by which a circuit with varying currents or voltages can cause mechanical vibrations, which is what we hear as sound:
- Every little bit of wire wound around a magnetic core experiences a sideways force. Electric motors, generators, and loudspeakers work on this principle. Wire is usually held quite tightly in inductors, but nothing can be held absolutely rigid.
- By magnetorestriction. This is the property of a material that causes it to change size as a result of an applied magnetic field. The ferrite in the inductors used in the switching power supply was selected in part to minimize this, but some amount is always present anyway.
- By the piezoelectric effect. This is the property of a material that causes it to change size as a result of an applied electric field. Quartz crystals exhibit this effect enough to be used in some types of microphones and small earphones. In your case, it's material in the ceramic capacitors that is causing the problem. The ceramic changes size slightly as the voltage on the capacitor changes. Unfortunately, the types of ceramic that allow for smaller capacitors of the same capacitance also exhibit the piezoelectric effect more strongly. Whining is therefore a tradeoff with cost and board space, so shouldn't be much of a surprise in a consumer product optimized for low price.
The component itself may not actually be vibrating much. Due to it being rigidly mounted to the circuit board, it gets the whole board vibrating, which then couples much better to the air to produce the sound that you hear.
Try touching inductors and ceramic capacitors near any inductors on the main board when you hear the whining. You will probably find some places where touching changes the sound.
After finding the offending components, you might be tempted to dampen the sound by putting hot glue or epoxy around the component. Be careful with that. You can easily add thermal insulation to components that need to dissipate heat. That may cause the components to fail under high usage at high ambient temperature.
Do not poke at the large capacitors inside the PSU with your fingers! This is dangerous even though the power has been removed.
Right, which is why I said to touch near any inductors on the main board. Everything on the main board is limited to low voltages.
In case you aim to poke at components on a circuit board with your fingers, it's very important that you first touch some non-sensitive part of the board, such as the metal frame around the connectors. Otherwise ESD might break the board.