Chapter 30: Inductance

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Welcome back, everyone, to the Deep Dive.

Today, we are going to do a really in -depth deep dive.

We're going to be talking about inductance.

I know we talk about circuits a lot, this one's going to be a little different.

We're going to be looking at inductance, the whole thing.

A change in current and the magnetic fields that that creates and how those cause some pretty major effects in the world of electricity.

Yeah, for sure.

The whole chapter, we're going to try and break it all down into understandable pieces.

Hopefully, by the time we're done, you'll have a good idea of what mutual inductance and self -inductance are all about, and of course, the energy that gets stored in magnetic fields and how it all works in RL and LC and LRC circuits.

It always amazes me that inductance isn't just some theoretical thing.

It's everywhere, literally woven into so many technologies that we use every day.

Think about how we even get power delivered to our homes,

transformers, that's all inductance.

Or the little circuits in our phones and computers,

all of those tiny oscillations, it's mind -boggling how it's all connected.

Yeah, absolutely.

Okay, so I think a good place to start is with mutual inductance.

Yeah, sounds good.

So mutual inductance, it's kind of like this invisible connection between two coils when they're close to each other.

Imagine you have one coil, right, and you change the current flowing through it.

What happens is that creates a changing magnetic field that reaches out to a nearby coil.

And get this, that changing magnetic field, when it passes through the second coil, it actually creates a voltage, you know, an EMF in that second coil, even though they're not physically touching.

Wow, so it's like one circuit, you know, can actually mess with another one just through the magnetic field between them.

The book had this cool analogy, like think about dominoes, you know, when you knock one over, it hits the next one.

It's like that, but the invisible force is the magnetic field that's doing the pushing, kind of.

Yeah, that's a really good way to put it.

And the thing that really makes this work is something called magnetic flux.

Like imagine you're looking at the magnetic field going through a specific area.

So in our case, that area would be the second coil.

When the first coil's current changes, its magnetic field changes too, which then changes the magnetic flux through the second coil.

And that change in flux is what makes that voltage appear.

We measure how strong this connection is with something we call mutual inductance.

We use a capital M for that.

Right.

And there are those two equations that describe this relationship, right?

Like E2 equals negative M times the rate of change of the current in the first coil, which is like D1 and DDT.

And then you have the same thing for the first coil, you know, E1 equals negative M times DI2DT.

I always wonder what that negative sign means, you know?

Yeah.

So that negative sign is super important.

It actually tells us about Lenz's law, which basically says that the voltage we create, the induced EMF, it always tries to fight back against the change in current that created it.

It's like if the current in the first coil is going up, the voltage created in the second coil is going to try and make a current that pushes back, you know, to try and stop it from increasing.

So it's like trying to keep things stable.

Cool.

So what determines how strong this mutual inductance is, you know?

What makes that M bigger or smaller?

That's a really good question.

Okay, so the M isn't like set in stone.

It changes depending on how the two coils are set up, like their size and shape, you know?

The number of turns of wire they each have, which way they're facing each other and how far apart they are, all of that matters.

And here's another thing, anything between the coils affects it too.

Like, imagine you put a piece of iron in there.

Iron's magnetic, right?

So it can make that magnetic feel between the coils stronger, which means the M gets bigger.

So even with the same coils, just moving them around or putting something different near them, you could completely change how they interact.

Yeah, totally.

There's another way to define mutual inductance too.

It's all about the magnetic flux, but in a slightly different way.

It's like M equals the number of turns in the second coil.

We call that N2 times the magnetic flux through each turn of the second coil, which is like phi B2, all divided by the current in the first coil, which is I1.

And since it's like symmetrical, it also equals N1 times phi B1 divided by I2.

Wow.

So it's all linked together, right?

Like the changing current in one coil, you know, messes with the magnetic field and that changes the magnetic flux in the other one, which then creates that voltage.

And mutual inductance is like the key to how efficiently that whole thing works.

And just like with self -inductance, we use Henry's to measure it.

Exactly.

And, you know, mutual inductance is, well, it can be a good thing and a bad thing.

I mean, think about transformers, those things that let us change AC voltages, you know, that's like the whole foundation of our power grid.

And it totally relies on mutual inductance.

Without it, our electrical grid would be like totally different.

Yeah.

Transformers are super important, but you said it can be a problem too, right?

How does that happen?

So sometimes mutual inductance happens where you don't want it to.

Like you could have one circuit that's changing its current and it might accidentally create a voltage in another circuit that's close by.

And that can mess things up, you know, like imagine it's a really sensitive circuit, like in some medical equipment or something, that extra voltage could cause noise and mess with the readings.

That's why engineers have to like design things to minimize those unwanted effects.

They use shielding and clever layouts to keep things separated.

So it's like electromagnetic crosstalk, kind of.

Okay, so we've covered mutual inductance.

Let's move on to self -inductance.

Is that like when a single circuit messes with itself?

Yeah, that's a great way to put it.

It's like instead of two separate coils, we're just looking at one loop or coil of wire.

And the crazy thing is a circuit can actually create a voltage within itself.

Wow, how does that even happen?

Like how can it affect itself?

So anytime current flows through a wire, it makes a magnetic field around it.

And that magnetic field then creates a magnetic flux through the area of the circuit itself.

Now if the current changes, that magnetic field changes too, which means the magnetic flux changes.

And that change in flux, according to Faraday's law, makes a voltage appear in EMF.

But here's the really cool part.

This voltage, this self -induced EMF, it actually tries to stop the current from changing in the first place.

It's like it has built -in resistance to change.

Like if the current tries to go up, this self -induced thing pushes back.

And if the current tries to go down, it tries to keep it going.

Yeah, that's a really good way to think about it.

It's like electrical inertia.

And this resistance to change in the current, we call that self -inductance, and we use a capital L for it.

The equation that connects the self -induced EMF E and the rate of change of the current, di dt, is E equals negative L times di dt.

Again, that negative sign is Lentz's law showing up.

So it's like mutual inductance, where self -inductance isn't a fixed number.

It depends on how the circuit's made, right?

You got it.

The shape and size of the circuit matter.

And anything inside the loop can affect it too, especially magnetic materials.

Like if you have a coil with n turns and a current i is making a magnetic flux phi b through each turn,

then the self -inductance L is calculated as n times phi b divided by i.

So the more turns you have, the bigger the magnetic flux for a certain current.

And that makes the self -inductance bigger too.

So we have these things called inductors, right?

Those are components specifically designed to have a lot of self -inductance.

They're usually coils of wire sometimes wrapped around something magnetic to make the effect stronger.

You got it.

And because inductors naturally fight against changes in current, they're incredibly useful in lots of circuits.

Like they can smooth out the bumps in a DC power supply, acting like a little reservoir of energy that keeps the current steady.

And they're key for filter circuits too, which can block or allow easy signals based on their frequency.

Okay, so we have mutual inductance between two coils and self -inductance in one.

Both of these come from the interplay of changing currents and magnetic fields.

Now the chapter talks about energy being stored in these magnetic fields.

This sounds like a big deal.

Yeah, it's kind of like how it takes energy to charge up a capacitor and store it in its electric field, right?

Well, with inductors, it takes energy to get the current flowing and that energy ends up being stored in the magnetic field around the inductor.

So is there an equation for that?

Like how much energy is stored?

There is.

The energy, which we call U, stored in an inductor with inductance L that's carrying current I, is U equals one half times L times I squared.

It's kind of similar to the energy stored in a capacitor, which is one half times C times V squared.

See how inductance L is like capacitance C and current I is like voltage V in those equations.

It's a cool parallel.

And this ability to store energy in a magnetic field rather than an electric field like a capacitor is really useful when you need high currents or like in those switched mode power supplies.

Inductors are like little temporary batteries that quickly store and release energy to move power around.

Oh, that's interesting.

So where is this energy actually stored though?

Is it like tracked inside the inductor itself?

Not really.

It's actually stored in the magnetic field, which spreads out around the inductor.

To better understand this, we have something called magnetic energy density, which is just a fancy way of saying how much energy is stored in each little bit of space where the magnetic field is.

So what's the equation for that density?

In a vacuum, the energy density, which we call U, is U equals B squared divided by two times mu naught.

Here, B is how strong the magnetic field is, and mu naught is a special constant that tells us how easily a magnetic field can form an empty space.

Now, if there's something other than a vacuum, like say a piece of iron in that magnetic field, the equation changes a bit.

Then it's U equals B squared divided by two times mu.

And mu is connected to something called relative permeability, which is represented by KM.

The equation for that is mu equals KM times mu naught.

KM tells us how much better a material is at concentrating magnetic field lines compared to a vacuum.

Iron, for example, has a really high relative permeability.

So it's like a magnetic field superhighway.

Oh, wow.

So material like iron can store a lot more energy in the same space than air or vacuum for the same magnetic field strength.

Exactly.

And the book has some awesome real world examples to show how important this magnetic energy is.

Think about old car engines, the ones with spark plugs.

You know how they need a huge jolt of electricity to create that spark?

Well, that energy comes from the magnetic field collapsing in an inductor.

It's like releasing a ton of energy all at once.

And I remember you mentioning another example, something about the sun.

Oh, yeah.

Solar coronal mass ejections.

Those are massive bursts of stuff from the sun's atmosphere.

They can carry a ridiculous amount of energy, and it all comes from the sun's crazy strong magnetic fields.

When those fields change suddenly, a lot of that magnetic energy turns into kinetic and thermal energy, blasting that stuff out into space.

Wow.

So we see this principle of magnetic energy storage happening on tiny scales like in car engines and on absolutely massive scales like with the sun.

Okay, now let's dive into RL circuits.

What happens when we introduce a resistor into the mix with an inductor?

So when you have a resistor and an inductor connected in series to a power source, like a battery,

the inductor doesn't let the current jump up to its final value instantly.

It wants to resist that change, remember.

So instead, the current starts at zero and gradually increases over time.

So the inductor acts like a speed bump, basically, making sure the current doesn't go too fast too soon.

How fast does it actually build up, though?

Well, there's an equation for that.

It's I of t equals E over R times one minus E to the power of negative R times t over L.

So at the very beginning, when time t is zero, the current is also zero.

But as time goes on and gets much bigger than L over R, that exponential term with the E in it gets really small, almost zero.

And that's when the current gets close to its final value, which is just E over R.

That's good old Ohm's law.

I see.

So that part of the equation, negative R times t over L, that seems really important in determining how fast things change.

Absolutely.

That ratio of L over R is called a time constant, and we usually use the Greek letter tau for it.

So we can rewrite that part as negative t over tau.

The time constant tells us how quickly the current reaches its final value.

After one time constant, which is when t equals tau or L over R, the current will have reached about sixty three point two percent of its final value.

And after about five time constants, it's basically reached a steady state.

A larger inductance, L, means a bigger time constant, which means slower current growth because the inductor is putting up more resistance.

On the other hand, a bigger resistance hour means a smaller time constant and faster growth, but towards a smaller final current, E over R.

Makes sense.

It's like the inductor is trying to slow things down while the resistor limits how high it can ultimately go.

Now what happens when we have a steady current in an RL circuit and then suddenly take away the power source, like cutting the battery out?

What happens to the current then?

In that situation, the inductor tries its best to keep the current going because, well, it hates change.

The equation for this current decay is I of t equals I naught times E to the power of negative R times t over L.

Here, I naught is the current that was flowing just before we took away the power source.

And once again, the time constant, tau equals L over R, controls how quickly the current drops.

After one time constant, the current will be down to about thirty six point eight percent of its initial value.

So all that energy that was stored in the inductor's magnetic field gets turned into heat in the resistor as the current slowly fades away.

Exactly.

And remember Kirchhoff's loop rule is really important when we're figuring out what's going on in these RL circuits, especially when we have voltages that are changing over time, like the voltage across the inductor, E equals negative L times di dt, and the resistor V equals I times R.

At any point in time, if you add up all the voltage drops around the circuit, it has to equal the voltage from the power source, if there is one, or zero if there's no power source.

Right.

That's because the voltage across the inductor isn't constant and depends on how quickly the current's changing.

Okay, we've covered RL circuits.

Let's move on to LC circuits.

Those are circuits with just an inductor and a capacitor, right?

What kind of interesting stuff happens in these circuits?

Oh, LC circuits are really cool.

They introduce something called electrical oscillations.

Imagine you have a all the energy is stored in the capacitor's electric field.

But as soon as you connect them, the capacitor starts to discharge, sending a current through the inductor.

So as the current goes through the inductor, it builds up a magnetic field, right?

So then the energy starts to get stored in the inductor's magnetic field instead of the capacitor's electric field.

You got it.

Now, when the capacitor is totally discharged, all the energy is in the inductor's magnetic field.

But remember, the inductor hates for the current to change suddenly.

So what happens is that magnetic field starts to collapse.

And as it collapses, it creates a voltage that charges the capacitor back up, but this time with the opposite polarity.

So it's like the energy is bouncing back and forth between the capacitor and the inductor, creating an oscillating current and voltage in the circuit.

Exactly.

If we had a perfect LC circuit with no resistance, this energy transfer would just keep going forever.

The equation for how fast these oscillations happen called the angular frequency is O equals one over the square root of L times C.

So the bigger the inductance or capacitance, the slower the oscillations, meaning the energy takes longer to move back and forth.

So both the charge on the capacitor and the current in the inductor are changing over time, like a sine wave with this specific angular frequency.

Yep.

It's a perfect example of energy conservation in an electrical circuit.

The chapter makes a really cool comparison to a mass on a spring.

The capacitor with its current and thus has energy from moving charges is like the mass that resists changes in velocity.

That analogy is really helpful.

Okay.

So next up we have LRC series circuits.

This is where we bring back the resistor and have all three components, an inductor, a resistor, and a capacitor all connected together in series.

What happens when we add the resistor back in?

The resistor basically acts like a break on the oscillations in the LC circuit.

You know, as current flows to the resistor, some of that electrical energy turns into heat.

So the oscillations slowly lose energy and eventually die out.

So it's like adding friction to that mass on a spring system.

The oscillations gradually get smaller until everything stops.

Exactly.

There are actually three different ways these oscillations can die out in an LRC circuit, and they depend on how big the resistance is compared to the inductance and capacitance.

Okay.

Let's hear about these three different scenarios.

First, we have under damping.

This happens when the resistance is pretty small compared to the inductance and capacitance.

The circuit will still oscillate, but the size of those oscillations shrinks over time until they eventually disappear completely.

The angular frequency of these damped oscillations is given by equals the square root of one over L times C minus R squared over four times L squared.

You can see that the resistance term R squared over four times L squared makes the frequency a little lower than in a perfect LC circuit with no resistance.

So it's like a pendulum swinging back and forth, but each swing gets smaller because of a little bit of friction until it eventually comes to a stop.

That's a great way to visualize it.

The next case is called critical damping.

This happens when the resistance is just the right size, specifically when R squared equals four times L divided by C.

In this special case, the system settles down to its steady state as fast as possible without any oscillations at all.

It's like the perfect balance, just enough resistance to prevent overshooting, but not so much that it slows things down too much.

So no more swinging back and forth.

It just smoothly and quickly settles down.

You got it.

And the last case is over damping.

This when the resistance is really big, specifically when R squared is bigger than four times L divided by C.

In this case, the system still settles down without any oscillations, but it takes much longer than with critical damping.

The big resistance basically makes it hard for energy to move between the inductor and capacitor quickly,

so it prevents those oscillations from happening.

It's kind of like a door with a really strong closer.

It takes its time closing without any bouncing back and forth.

Perfect analogy.

And just like with the RL circuits, Kirchhoff's loop rule is our best friend for understanding LRC circuits.

It helps us relate the voltages across each component of the resistor, inductor, and capacitor to the voltage of the power source if there is one at any given moment.

Wow.

We've covered so much in this deep dive into inductance.

We started with the fundamentals of mutual and self -inductance, then learned about the energy stored in magnetic fields, and explored the fascinating behaviors of RL, LC, and LRC circuits.

I think it would be helpful to quickly go over some of the key terms from the chapter glossary just to make sure we're all on the same page.

Yeah, definitely.

So the heart of it all is inductance, both mutual and self, which is basically the ability of a changing current to create a voltage.

And we measure that in henries.

Inductors are those special components that have a lot of self -inductance, and they act like the guardians of current, making sure it doesn't change too quickly.

Right.

And then there's magnetic energy density, which tells us how much energy is packed into the magnetic field around an inductor.

And in RL circuits, we have the time constant L over R, which dictates how quickly the current rises or falls when we connect or disconnect the power source.

And in LC circuits, the inductor and capacitor work together to create those electrical oscillations with an angular frequency that's equal to one divided by the square root of L times C.

And when we throw a resistor into the mix, making it an LRC circuit, we see different damping behaviors, under damped oscillations that slowly decay, critically damped where it settles down as fast as possible without oscillating, and overdamped where it slowly settles without oscillating.

It's amazing to see how these concepts that might seem abstract at first are actually behind so many everyday phenomena.

Yeah, it really shows how intertwined electricity and magnetism are.

And thinking about how these basic principles of inductance are essential for so much technology.

We use the power adapters for our phones and laptops, the tuning circuits in our radios, and even the advanced energy storage systems of the future.

It makes you wonder what other amazing roles inductance plays in the world around us that we haven't even discovered yet.

That's a great point.

There's still so much to learn and explore.

But I think we've done a pretty good job of covering all the major points,

theories, findings, and examples from this chapter on inductance.

Agreed.

And with that, we'll wrap up this deep dive into the world of inductance.

Thanks for joining us on the deep dive.