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If I have a system of circuits like this: enter image description here

where Circuit 2 is providing power the Circuit 1, where does the energy come from? Do the inductors store energy? If not, how is power transferred from the Circuit 2 to Circuit 1?

I initially thought that inductors won't store any power and the power must come from the mutual inductance. Am I correct in thinking that?

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    \$\begingroup\$ "inductors won't store any power", read about inductor first. \$\endgroup\$ – Long Pham Aug 17 '18 at 3:48
  • \$\begingroup\$ flux in the core provides energy storage; thus 60Hz transformers are larger than 400Hz transformers, and heavier; airplanes use 400Hz for that reason. SwitchRegulators operating at 200,000Hz can be proportionally smaller in the magnetic materials. \$\endgroup\$ – analogsystemsrf Aug 17 '18 at 3:54
  • \$\begingroup\$ An ideal transformer is kind of an abstract thing. It has properties defined by mathematical equations. Since it is abstract, not physical, it doesn't transfer real energy. But in a real transformer, energy is transferred by way of the core. Energy is added on one side and removed on the other, more or less simultaneously. Inductors and capacitors do store energy. \$\endgroup\$ – mkeith Aug 17 '18 at 5:13
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    \$\begingroup\$ Are you OK with Faraday's law of induction? \$\endgroup\$ – Andy aka Aug 17 '18 at 7:12
  • \$\begingroup\$ Power is transferred via the coupling between the two coils, typically realised by a soft magnetic core. A changing current in one coil creates a changing flux in the core, which creates a changing voltage across the other coil ... and so the story unfolds. \$\endgroup\$ – Chu Aug 17 '18 at 10:07
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I initially thought that inductors won't store any power and the power must come from the mutual inductance. Am I correct in thinking that?

Yes, the equation is this for one inductor, \$\Phi \$ is the magnetic flux E is the EMF or voltage and N is the number of turns.

$$E= N\frac{d\Phi}{dt}$$

Since an ideal transformer has 100% mutual inductance connection from one coil to the next, it means that all of the magnetic flux is directly linked so \$\Phi \$ would be the same on both sides (in a real transformer some of the magnetic flux is lost and is not exactly the same on both sides)

$$E_{primary}= N_{primary}\frac{d\Phi}{dt} $$ $$E_{secondary}= N_{secondary}\frac{d\Phi}{dt} $$

Then you get this relationship $$\frac{E_{primary}}{E_{secondary}}= \frac{N_{primary}}{N_{secondary}} $$

The power is transferred through the magnetic flux (not the magnetic field, because energy transfer can only happen if the magnetic field is changing hence the rate \$\frac{d\Phi}{dt}\$)

Inductors do store power and do so for a given amount of time, they do this by creating a magnetic field around the inductor as a current is converted to a magnetic field. If the current is removed, they generate voltage or EMF.

Transformers have a 'load' on their coil so they don't store energy as well as an inductor because the energy is transferred to the secondary coil.

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    \$\begingroup\$ I think your last 3 paragraphs need some work. \$\endgroup\$ – Andy aka Aug 17 '18 at 17:15
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In most cases, transformers are not designed to store an appreciable amount of energy. The power is transferred directly from the primary to the secondary via the mutual inductance. An ideal transformer (with infinite primary inductance and unity coupling) would not store any energy. The flux from the primary and secondary would always perfectly cancel and the net flux in the core would be zero.

In a real transformer, if the secondary is open circuit there will still be some current flowing in the primary. That current is the "magnetizing current" and does result in some energy storage, but it is typically much less than the full load current. The core of a transformer is designed to maximize inductance but usually has a low saturation field, so if you try to store a lot of current the core will saturate and the inductance will drop dramatically.

This in contrast to inductors which are usually designed to store energy since they have no secondary to produce a cancelling flux.

Some transformers such as flyback transformers often used in switch mode power supplies combine both roles. They do store energy, but also act as transformers to provide isolation or large voltage change. In the first phase the primary is conducting and the secondary is blocked by a diode. During this phase the flux is increasing and the core stores energy. In the second phase the primary is shut off by a switch and the secondary supplies current to the output, releasing energy from the core.

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