Razavi Basic Circuits Lec 24: Inductor Circuits

[Music]

[Music]

greetings welcome to lecture number 24

on basic circuit theory i am bashar zavi

today we will continue to look at

inductors and their properties

and understand how

inductors respond to for example current

inputs or voltage inputs

as

examples of very simple circuits

and then

we will look at some other properties of

inductors that also are interesting as

far as circuit analysis and design are

concerned

and finally if we have time we will

begin to look at our first

simple rl circuit to see how it operates

and how we go about analyzing it

but before we go there let's take a look

at what we covered last time

so we started by introducing the concept

of inductance

and inductors and we saw that

the original observation by faraday was

that if we have a

wire

carrying a current

and hence generating a magnetic field

around it according to the right hand

rule

and if this current changes with time

which means the magnetic field changes

with time

then this time varying magnetic field

can induce a voltage

on a for example separate piece of wire

so if i place a wire here this red wire

i can measure voltage here even though

there is no current passing through the

red wire

so it's a fascinating effect and it's

certainly not ohm's law

but then we tried to extend this idea

and we said that even if you have a

single wire

carrying a time varying current and

experiencing its own time-varying

magnetic field then that wire also must

generate a voltage across it because

of faraday's law

so we would expect that the voltage will

be generated here even though this wire

has no resistance so again this is not

not ohm's law

so this is something that we call

self-inductance or just an inductor

and of course a simple piece of wire

will not be a very good inductor so we

typically wind it in some form

like what we saw last time in this form

and that allows us to have more

inductance for a given length or even

volume etc

so for such a device which we call an

inductor we have a current a voltage and

an inductance and these three are

related by this equation the voltage

that we generate across this inductor

is proportional to how fast

the current through the inductor is

changing with time

multiplied by l1 of course

so this equation

of course has to be observed with

respect to

the polarities that we have here so if

the current is flowing from high

potential to low potential

then there's a plus sign here if the

current is going from low potential to

high potential then there's a minus sign

here

all right but this equation also tells

us some interesting things about

the behavior of inductors

for example when there's a switching

event going on something is wants to

change suddenly the inductor has a

certain behavior

what we know is that if the voltage that

is available for this inductor from the

rest of the circuit

is less than infinity

then this derivative has to be less than

infinity that means that the current

through this inductor cannot jump

instantaneously

and that's the key property of inductors

they resist

the change of index and the change of

current they don't want the current to

change too fast

on the other hand if we go to an extreme

condition where

the

current through an inductor

has become constant so let's say this is

inside a big circuit and eventually all

the

transients have died away and we have a

constant current through the inductor

and then we know that the voltage across

the inductor will be zero

and a device that guarantees a zero volt

difference between these terminals is a

short circuit so that becomes a short

circuit

so these two cases are important to

remember because they often happen in

our transient analysis of circuits

okay and one last point was

that the initial condition

that we can specify for an inductor is a

current through it another voltage

across it fundamentally it's the current

through the inductor that can serve as

what gives the inductor energy and we

will see that today

remember for capacitors the initial

condition was a voltage because the

voltage that gives the capacitor energy

one-half cv squared so we will see

something similar in terms of current

and inductance and the energy stored

in the magnetic field of this inductor

all right one last point uh last time i

said that

this is an inductor right it's an

inductor and when i started out i said

this is not an inductor because just a

piece of wire and it satisfies ohm's law

and so on

now these this doesn't seem quite to

agree with what i just said here right

so is this an inductor is it not an

inductor we have to decide right this is

a piece of wire

well strictly speaking even a piece of

wire has inductance so the inductance

from here to is not zero even though it

looks like a straight piece of wire and

even though it may have no resistance at

all but for our analysis we assume that

you have a piece of wire is ideal it's

like a short circuit it has zeros

resistance and zero inductance

but if you want to build an inductor

then of course we wind it and all that

okay just to avoid any confusion

so in our studies we typically assume

that a piece of a straight piece of wire

does not have an inductance or its

inductance is negligible

all right so

let's go and

look at

some more examples

of

simple inductors

so here's a another example that we have

here

okay so

last time we

had a situation

where we had a

current source

feeding

a an inductor

like so

call this i1 and l1

and we're trying to plot this voltage

here

right if you remember and the current

source was

a ramp last time so this time we'll make

it a little more interesting so suppose

i1 looks like this as a function of time

so i1 as a function of time

is 0 before time zero

and then it looks like a ramp so it goes

up

for some time

and then at

some time t1

it becomes constant

and then at some point t2

it uh

falls linearly back to zero

okay and it reaches zero at time t3

so this is a current waveform that i

have managed to generate by this current

source is pushing this current through

the inductor and i would like to find

the corresponding output voltage

now i know that the voltage across an

inductor is given by this equation so i

just need to take the derivative of this

waveform to obtain the voltage and of

course multiply by l

so i will say

that

v out

the voltage across the inductor is given

by the derivative of all of this so we

see that

before times zero

we have a constant value with zero

anyway so that's zero

at this point we have a certain slope so

let's call that slope alpha

so this jumps to

alpha times l

right the derivative times l

and this slope is constant up to t1

from t1 to t2 the slope is zero so the

voltage across the inductor is zero

so this voltage jumps to zero

and then at t2

we have

a negative slope so let's call that

beta

so

this slope is multiplied by l to give us

the voltage across the inductor

so now this will be like this

this value is beta l

and this goes on until

t3

okay so this is the overall

shape that we have now let's say that

this goes back to zero from here on just

for simplicity so this also

goes to zero like that

so the voltage waveform across the

inductor looks like this

all right it's just the derivative of

the current waveform that's passing

through the inductor

all right so that's simple enough but uh

let's try to see what happens as we go

to some extreme case

if i

make these transitions faster

what happens so suppose

this transition happens in a much

shorter time so something like that

and similarly this transition happens in

much shorter time like this

so then what happens the voltage

waveforms well they're proportional to

the slope

so this voltage waveform which is alpha

alpha is now much bigger

so this height will be greater so we'll

be for example here

and of course it lasts

let's say it lasts up to here to this

little time here so it lasts for a

shorter amount of time

right only from here to here past this

point the current is constant

and then this goes on

up to

right here at this point

and at this point we're dropping at a

faster rate so the voltage is larger but

more negative

so the voltage will be like this

and this lasts up to this point here

so it goes back and stays

at zero

so we see that as the slope

of these transitions increases

the

pulse that we have generated here

becomes narrower and taller both here

and here

so in the limit

if this transition is infinitely fast

what do we get here

this becomes infinitely narrow and

infinitely tall so it becomes an impulse

so in the limit if the current jumps

here and jumps here that requires that

we have an impulsive voltage here and an

impossible voltage here

if the circuit surrounded this inductor

over here is capable of

the

sustaining an infinite voltage then it's

fine we can do that

and in other words if this current

source really can do something like this

if it can jump from some zero to some

amount in zero time yes we do get a very

high voltage across the inductor and in

fact this is a problem

in some circuits that can damage some

circuits and we'll talk about that later

all right so that was one little example

let's go to another example

and see what we have here

so suppose i have a

voltage source

which we call v in

and i have directly

connected this to an inductor

l1

and i am interested in this current

as a function of time

if this voltage has the following shape

so

let's plot v in as a function of time

so our input or our stimulus

is uh given by something like this v in

initially is zero before times 0 then it

grows linearly with time with the slope

of

alpha

up to some point t1

t1 and then stays constant past that

point

all right

so this time i'm applying a voltage

across the inductor previously i was

applying a current through the inductor

and the voltage has a ramp section and

then a flat section and i would like to

find the current resulting from this

voltage input

okay so what we know is that the current

across this equation is the integral of

the voltage and then divided by the

inductance so we have to integrate this

so let's say this is alpha t here

so we say the current i in

is 1 over l

1

integral of

alpha t

dt

this is for zero to t1 right so from

zero to t

and for t

between zero

and t1

that's the equation that we have for the

current passing through the inductor in

response to this ramp voltage

okay so that comes out to

be uh alpha over l1

times t squared

our assumption is that there is no

initial current in the inductor if there

were then we'd have to add that at that

here because that would be

an integration constant that you would

have to include but right now we assume

there's no initial condition

okay so this is the current waveform

passing through the inductor

between zero and t1

so let's try to plot that for now and

then we need to see what happens after

t1

okay so here we go

i have i in

and then t

okay so this is a simple quadratic

so it goes hyperbolically parabolically

up to

this point t1

it goes up parabolically

like this

okay

so the current is growing relatively

fast right parabolic with time

okay then what happens after t1

after t1 the voltage has become constant

if the voltage is constant we have to

integrate properly

so now for

t greater than t one

we say

i in

is equal to

one over l one

so let's call this voltage something

call this a v1

so

that would be v1 times t

because i am integrating

v

out of v times dt and v is given by v1 a

constant value so it comes out of the

integral and it just just gives me t

so beyond t1 the current still grows but

now linearly with time

so we have an equation of this form so

this is a straight line

and has a certain slope which is v1 over

l1

that is the behavior of the current that

flows through the inductor

so one thing that is particularly

interesting here is that the current

through the inductor

grows unbounded right it just goes to

infinity

and that sort of makes sense does it

make sense

well

when the vote is constant it's like you

have applied a battery across an

inductor

and what should happen the inductor

wants to draw more and more current so

the current just keeps going up and

going up as if the inductor wanted to

act as a short circuit right it's not

entirely true but intuitively that's

what would happen

all right

so

let's go on to another example

to see what happens here

so let's change the

color of our pen

to green

look at another example here

okay so in this case we will consider

something very simple

we have

a battery

a resistor r1

an inductor l1

a voltage source vb

and we are interested to find this

current here

call it i1

all right very simple

our assumption is that this circuit has

been connected like so for a very long

time so all of the transients have died

away

there's no

time varying component anymore

so if there are any currents on any

voltages in the circuit they're all

constant

and we would like to find the current

that flows through the loop under that

condition

all right so what do we do

okay

so

we say that if everything has become

constant for example i1 has become

constant

so we say

i1

is constant

that means

that

the inductor voltage vl1 is equal to

zero

right

the current doesn't change with time so

the voltage across the inductor is zero

if the voltage across the inductor is

zero

how much is the voltage across the

resistor

okay so this voltage is equal to this

voltage plus this voltage right so this

voltage is zero the voltage across r1 is

just the battery voltage

so this voltage here

is just vb

and because i have the voltage across

the resistor i can find this current so

that means that i1

is equal to

vb

over r1

okay very simple

all right the purpose of this example is

to show you

how we can create

an initial condition in an inductor

before we start doing something with

that inductor

so if you wanted to create an initial

current through the inductor you would

hook it up like this to a resistor to a

battery you adjust the value of the

resistor to give us the current that we

need or the battery and the resistor

right and then this inductor is ready it

has certain current in it now we can

switch it out of here switch into

something else etc and those are things

that we'll see later

but this is a very simple example of

creating

a known current through an inductor

all right

let's go to another example

and

make it a little more interesting

so let me do this

all right so we found this current in

the circuit right

now i'm going to add

a resistor here r2

okay

so we'll still keep this current call

this current i1 i like before

and the question is

is the red i1 different from the green

i1 now that i have added this resistor

r2

so i'll think about it and see again

we're assuming that a circuit was built

a long time ago all the transients have

died away everything is constant

okay well

what we see is that because the current

through the inductor is constant

whatever it is it may not be i1 but

whatever it is constant right

so we say

a current

through

l1 is constant

and again this is because the circuit

was built a long time ago and all these

transients have decayed have gone away

right

okay so if the current through l1 is

constant

we know that the voltage has to be zero

across the inductor no other way right

so we say v l1 is equal to zero

if the voltage across the inductor is

zero how much is the voltage across r2

well that's also zero they're in

parallel

so how much is the current through r2

that's also zero

so we say vr2

equals zero which means

current

through

r2 is zero

so this current is zero this connection

this current is zero right this current

is zero

okay if the current through r2 is zero

and this current i1 is coming in where

would it go

it cannot go this way it has to go

through the inductor right

so because the current through r2 is

zero

all of

i1

must

flow

through

l1

all right

how much is i1 the red i1 well this

voltage is still zero right if this

voltage is zero and this side of r1 is

zero this side of r1 is vb so the

current to r1 is still vb over r1

so we say i1

is vb

minus 0 over r1 right

vb minus 0 over r1

okay so the current didn't change

so even though i brought another

resistor and placed it in parallel with

the inductor nothing really changing the

circuit as far as these initial

conditions are concerned

so

this is just because

if we have waited a long time and

the current run inductor is constant

it forces a zero volt difference across

it so it acts like a short circuit

so this is a short circuit and we have a

short circuit here and place a resistor

in parallel with it the short circuit

dominates right the resistor doesn't

have any role in the current division

between this and that

okay so that's

an extension of that example

[Music]

as you can imagine circuits and

electronics also have wide application

and robotics so here's an example that

is quite interesting

in this paper they deal with flapping

wing robotic insects so these are very

small insects that are robotic and they

can travel to different places and

collect information for example they can

fly over a

fire and collect information they can go

into the battlefield and collect

information etc

so they are very small and very

light in weight so here's the hand and

here's

the

the

flapping wing

a micro robot so you can imagine this

robot of course has to have electronics

in it to be able to fly and to be able

to collect information

in particular for these wings to flap

they need a

piezoelectric device that creates this

movement

and for that they need to generate a

voltage

and that's how the

they come up with this circuit

that takes a

battery voltage from here

and generates a high voltage that is

necessary for the piezoelectric device

on the far end

now there are a whole bunch of devices

in here we know some of them we know

switches and capacitors and inductors

when resistors we don't know these

devices these are called diodes these

are what you would study in electronics

courses

but we can see that there's a

great deal of what we have seen in this

course in terms of the devices and the

concepts that are also useful in an

application like this

when we are trying to generate for

example a high voltage very low voltage

or the other way around

so that is one application of

electronics and circuits in robotics

isn't that beautiful

now let's go

and find look at another example here

so i have a battery

vb

i have a switch

and then i have an inductor l one

and this which turns on at time zero

and would like to

plot this current i1

all right so as you can see the circuits

that we are considering right now are

very simple because we need to learn how

to walk before we start running and

that's the objective of all these very

simple situations

all right so

um the switch is not conducting is

open is off before time zero so the four

times zero there's nothing here

this inductor has no current no there's

no life everything is zero right so the

current is zero

this current is zero so we say

for t less than zero

i1 is equal to zero

right and then we turn the switch on it

starts conducting

okay

so how much is i1 at 0 plus

so

i1 at

0 plus

what we have to decide can i 1 jump

instantaneously from 0 minus to 0 plus

or not

if the only way for i want to jump

is

for this derivative to be infinite is

for the voltage available for the

inductor to be infinite

can the circuit deliver infinite

voltages inductors

no because this battery has some finite

amount 1.5 volts 5 volts something like

that so an infinite voltage is not

available here which means that the

current of the inductor cannot jump

instantaneously

which means the current through this

loop at time zero plus is still the same

as at at time zero minus

so that still has to be

zero

all right okay

so and then what happens

all right so then there's a current

flowing we have a battery a piece of

wire inductor and some current starts

flowing how do i calculate the current

not a problem i say from here right i

say

i1

is equal to 1 over l1

integral of vb

dt

vb is constant

so that's just vb

over l1 times

this so what we see is the following

you see that i one

as a function of time

was zero before time zero

right after zero and zero plus still has

to be zero according to the analysis we

just did and then past that point it's

just going up linearly so it's going to

go like this right

and this is of course not that much

different from this situation here we

had a constant voltage across an

inductor

from now on from t1 on and of course we

saw

a ramp

for the current generated through it

okay so that's a nice little example

that we have here

all right so

i think it's good to

summarize our findings so far

for

some of the properties of inductors

before we look at more properties

so we say

summary

we say with

a0

initial

actually let me

erase this

i have to put it somewhere else

okay so here's the summary and it is a

particular particular comparison i want

to

have between inductors and capacitors

so

all right so let's see here

okay so we say

inductors

and capacitors

so there are

uh four different cases depending on

whether we have an inductor or a

capacitor and whether

the voltages or currents are changing

rapidly or have stopped changing they're

constant okay so those are four cases

you want to consider

all right so we say

for inductors

we say if

v is less than infinity

right if the voltage that we can give to

the inductor from outside whatever it is

right the voltage that we can give the

inductor is not infinite and then the

current of the inductor cannot jump

instantaneously

so we say

i

can't

jump

right

so if

i is equal to 0

as 0 minus it will want to stay at 0 as

0 plus

so i equals zero

and if the current is zero

uh what does the inductor represent

we have an inductor

the inductor says my current shall be

zero

here and it shall be zero here

right that's what we decided

a car a device that wants to maintain a

zero current

what is it called

it's called an open circuit so we see

that the inductor is equivalent to an

open circuit

all right so just right around here the

inductor is acting as an open circuit

because the current through it cannot

change right away of course past that

point is different right because now we

are giving it time

for it for it to change its current but

right around that transient that

switching event whatever transition we

have the inductor acts as an open

circuit provided that it has a zero

current at time zero minus

okay

for capacitors what was the situation

well for capacitors was the other way

around we said that if the current

available

for a capacitor is not infinite

right so we have a capacitor we can

disconnect the rest of the circuit the

rest of circuit is not capable of giving

it infinite current then the voltage

across the capacitor cannot jump

so we said

v

can't

jump

so that means that if the voltage on the

capacitor is zero

at zero minus it has to be zero as zero

plus

so we say if the voltage is zero

it wants to maintain that zero voltage

for a little while

so it acts as a device that guarantees a

zero volt difference and such a device

is called a short circuit

so the capacitor

is equivalent to a short circuit

all right

so

so these

two cases correspond to situations where

we have some sort of

jump some sort of transient some sort of

switching event in a circuit and we're

looking at the circuit just before that

event and just after that event and

we're trying to decide what happens in

the circuit so these two are extremely

important for those cases

all right and then we have the other

extreme case where we have waited for

the circuit to settle all the transients

have died away and we would like to see

what role an inductor plays and what

role a capacitor plays

so for that lecture we decided that if

i is constant

right y is constant so we have waited a

long time like in here and the current

has to become constant if the current is

constant then the voltage is zero and

the only device that guarantees a zero

volt difference is a short circuit

so we say l

is equivalent to a

short circuit

and for capacitors we saw that it was

the other way around if the voltage on

the capacitor has become constant so we

have weighted all the transients have

died away and then c dv over dt is zero

so i is zero

all right when i is zero what kind of

device can guarantee a zero current

that would be an open circuit

so we say if

v is constant

then

the capacitor acts

as an open

okay so the first row is important when

there's a switching event some sort of

transient

and

the second row is important when we have

weighted we're trying to look for final

values in the circuit remember we always

want to look for y infinity right

there's a final value of a voltage or a

current or something so the second row

is handy for those cases

all right very good so

with this now we are ready to

go over some more

inductor properties so let's go

to the next page

see what we should do here

okay now

change the color

all right so the next property of

inductors that we're going to look at

is the energy

so energy in

inductors

so just a way we can store energy in the

electric field inside the capacitor so

we put some charge here some charge here

we got electric field we have energy

we can also store energy

in the magnetic field of an inductor so

if you have an inductor and i pass a

current through it it generates a

magnetic field that magnetic field

actually has energy in it

so to calculate the energy in an

inductor what we do is we start with the

definition of power remember we

calculate the power

as

v of v of t times

i of t this was true for any device

right it doesn't matter what it is

for any

any device any circuit we have a voltage

and we have a corresponding current the

product of these two tells us how much

power is going to that circuit

okay so let's do that

and here's

our good old inductor

i of t here

v of t here and l

you may be wondering why i'm using

lowercase here it doesn't matter

lowercase uppercase

and then i'm interested in energy and i

know that

energy

is integral of power

with respect to time

from sum 0 to some t

right so we'll go ahead and integrate

this

integral of 0 to t

p is v of t i of t now i know that for

an inductor

it just happens

that we have this

so why don't we replace for a v

from this equation into this equation

into this equation so i have

l d i over dt

times i times dt

okay and now you can see that what we

have here is

this l comes out you can cross this dt

with the cte and you have idi so that

equals one half of i squared

so we have one half of l

i squared of t

so the amount of energy stored in a an

inductor

is given by one half times the

inductance times the current at that

moment through the inductor

squared

for the capacitor we had one half cv

squared was one half times the

capacitance times the voltage across the

capacitor at that moment squared

okay so that's the simple equation that

we have

okay

so that's one little point that we need

to know

and the next property relates to

inductors in

series

so we have found the series combination

of resistors and capacitors etc so now

it makes sense to look at the series and

parallel combinations of inductors as

well

all right so another problem let's draw

a bunch of inductors here

l1 and 2

l3 it doesn't matter how many you draw

and i would like to find

the equivalent inductance right

this whole thing still has two terminals

so i can put in the black box and i can

say that

this black box has probably some

equivalent inductance

and the question is how much is that in

terms of l1 l2 l3 etc

all right another problem so ordinarily

what we do is we apply a voltage between

these two terminals we call it vx

and we have a current ix

if i can find vx in terms of ix or x

into the vx

and i can simplify that to

an equation that that resembles that of

inductors i can find the equivalent

inductance

okay

so

i have a current ix flowing through all

of these these are in series so by

definition they carry the same current

how much is this voltage from here to

here

the voltage across l1

well we know that the voltage russell

inductor is given by l

di over dt

and i is indeed going from high

potential to low potential so i do not

need a negative sign

so the voltage

across

l1 is

l1

dix

over dt

about the voltage across l2

same thing right so here you have l2

dix over dt and so on right

so i found this voltage i found this

voltage i found this voltage and so on

now i can write the kvl i can say this

voltage

is equal to this plus this plus this etc

so i can say

around the kdl

we have vx

is equal to

l1 dix

over dt

plus l2 dix

over dt

etc

these are all ix because the current

flows through all of them

in the same form if they are in series

they have to have the same current

now you can see something going on here

right so we can

factor i x over d t so we have

l one plus l two

et cetera

d i x

over d t

does this equation look familiar we have

a two-terminal device

this box here

whose voltage

is given by

the derivative of the current that flows

through it

multiplied by some amount so what would

you call that amount

that would be the equivalent inductance

of the circuit right so we say l

eq

is equal to l1 plus l2

etc

so we see that when inductors are placed

in series they add

and that's similar to the case of

resistors right when resistors are

placed in series they also add

okay

how about inductors in parallel

so let's

change the color

and look at

inductors

in

parallel

okay so it's the situation we can draw

it for as many inductors as you want

let's just try it for two for simplicity

so two devices in parallel by definition

must share both of these terminals right

these terminals are shared oh these

terminals are shared

how do i find this inductance same

method right so we apply a voltage

vx

and find this ix

all right our hope is that the

relationship between vx and ix it

resembles that of an inductor

all right so

how much is the current to l1

this current here

so that's the quiz of the day i will

give you one minute to think about it

all right

[Music]

all right so what did we get

well we see that the voltage across l1

happens to be vx right these are all in

parallel so if i have the voltage across

the induction i can find these currents

right so this current would be

1 over l1

integral of vx dt

similarly this is 1 over l2

integral of vx dt

and i know that this current plus this

current will be equal to i x so i can

say

i x

is equal to

one over l one

plus one over l two

integral of v x d t

does this equation look familiar

sure we said that the current inductor

is equal to one over the inductance

times the integral of the voltage so

this has to be one over the inductance

so we say one over l eq

is one over l one plus one over l two

the same equation that we have for

resistors in parallel so it's similar

i will see you next time

[Music]

so

you