Electronic – How to calculate the Back-EMF energy on an Inductor

This is a deeper question than it sounds. Even physicists disagree over the exact meaning of storing energy in a field, or even whether that's a good description of what happens. It doesn't help that magnetic fields are a relativistic effect, and thus inherently weird.

I'm not a solid state physicist, but I'll try to answer your question about electrons. Let's look at this circuit:

^{simulate this circuit – Schematic created using CircuitLab}

To start with, there's no voltage across or current through the inductor. When the switch closes, current begins to flow. As the current flows, it creates a magnetic field. That takes energy, which comes from the electrons. There are two ways to look at this:

1. Circuit theory: In an inductor, a changing current creates a voltage across the inductor \$(V = L\frac{di}{dt})\$. Voltage times current is power. Thus, changing an inductor current takes energy.

2. Physics: A changing magnetic field creates an electric field. This electric field pushes back on the electrons, absorbing energy in the process. Thus, accelerating electrons takes energy, over and above what you'd expect from the electron's inertial mass alone.

Eventually, the current reaches 1 amp and stays there due to the resistor. With a constant current, there's no voltage across the inductor \$(V = L\frac{di}{dt} = 0)\$. With a constant magnetic field, there's no induced electric field.

Now, what if we reduce the voltage source to 0 volts? The electrons lose energy in the resistor and begin to slow down. As they do so, the magnetic field begins to collapse. This again creates an electric field in the inductor, but this time it pushes on the electrons to keep them going, giving them energy. The current finally stops once the magnetic field is gone.

What if we try opening the switch while current is flowing? The electrons all try to stop instantaneously. This causes the magnetic field to collapse all at once, which creates a massive electric field. This field is often big enough to push the electrons out of the metal and across the air gap in the switch, creating a spark. (The energy is finite but the power is very high.)

The back-EMF is the voltage created by the induced electric field when the magnetic field changes.

You might be wondering why this stuff doesn't happen in a resistor or a wire. The answer is that is does -- any current flow is going to produce a magnetic field. However, the inductance of these components is small -- a common estimate is 20 nH/inch for traces on a PCB, for example. This doesn't become a huge issue until you get into the megahertz range, at which point you start having to use special design techniques to minimize inductance.

If i remove both terminals rapidly enough to avoid any arc formation

The more rapidly you remove the terminals, the more an arc will want to form.

Ideal inductance is defined by:

$$ v(t)= L\frac{\mathrm di}{\mathrm dt} $$

If you remove the terminals "instantly", then the current must stop "instantly". That means the di/dt term will approach infinity, and consequently the voltage term will, also.

The faster you stop the current, the higher the voltage generated. That energy has to go somewhere, and the harder you try to stop it, the more it will try to get out. If considering this situation in a purely theoretical, ideal situation where you posit the energy can't go anywhere, it's simply not possible. There is no real mathematical solution.

In practice, there are places the energy can go besides arcing. If you could somehow magically eliminate arcing in an otherwise real circuit, you'd have to consider:

• capacitance in parallel with the inductor, including the parasitic capacitance of the inductor and the wires connecting to it, which makes a resonant circuit where the energy oscillates between the capacitance and inductance
• resistive, magnetic, and dielectric losses, creating heat
• electromagnetic radiation
• corona discharge

In practice several of these will be in play, in addition to arcing. Depending on the particular design of the circuit some may be more significant than others.