Inductance, Inductors and Mutual Induction

Mutual Induction

Variant of Faraday’s Law

$${ϵ}_2=-M\frac{d{I}_1}{dt}$$

ϵ = EMF (V) M = Mutual Inductance (H, Henrys) I = Current (A)

• This is how wireless charging works! The magnetic field can transfer power!
• Two coils interacting with each other

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$$\frac{{ϵ}_2}{{ϵ}_1}=\frac{N_2}{N_1}$$

ϵ = EMF (V) N = Number of Turns

• The ratio in between the two EMF’s is the same as the ratio in between the amount of turns of looped wire!
• These are how transformers work. You can step higher voltage down to lower voltage. Power brick transformers step 120V down to 12V for our devices!
• Going from low voltage to high voltage works this way, (12V-120V), but there will always be a trade off.
• (eg, more loops equals more resistance, (V=IR), so your voltage may be higher, but the current will drop)

Transformers!

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Self Inductance

$$ϵ=-L\frac{dI}{dt}$$

ϵ = EMF (V) L= Self Inductance (H, Henrys) I = Current (A)

• L, the curly thing in the circuit, is an INDUCTOR

Current change over time due to this EMF

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Closing the Switch (Current growing to max)

Current at a time:

$$I(t)=I_{max}(1-e^{-\frac{1}{\tau }})$$

Time Constant:

$$\tau =\frac{L}{R}$$

Half Time:

$$t_{\frac{1}{2}}=0.69\tau$$

Opening the Switch (Current going from max to 0A)

Current at a time:

$$I(t)=I_{max}{e}^{-\frac{1}{\tau }}$$

Time Constant:

$$\tau =\frac{L}{R}$$

• This is cool, because this is how surge protectors work!
• The exponential growth slows surging current, and a high inductance can slow surges.
• The slope is nicer to electronics

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Inductors can store and release energy!

• This can be defined by the following equation

$$U=\frac{1}{2}L{I}^2$$

U = Energy stored/released L = Self Inductance (H, Henrys) I = Current (A)