Razavi Basic Circuits Lec 23: RC Circuit Example; Intro. to Inductors

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[Music]

greetings welcome to lecture number 23

on basic circuit theory i am bezel zabi

today we will continue to look at one

more example of rc circuits before we

finish up this topic and then go on to

inductors and else are rl and rlc

circuits

so we'll take a look at inductors today

just to familiarize ourselves with their

properties

and maybe look at a very simple

rl circuit and then this will open up uh

our

discussion for other lectures in the

future

but before we go there let's take a look

at what we covered last time

we had

two examples of rc circuits we saw that

in the circuit

we first applied step and let this

capacitor charge to some voltage

so the voltage on the capacitor charges

up to some value it is headed towards

the v1 the maximum value

but before it reaches there at some

point t1 this switch begins to conduct

and introduces r2 in parallel with the

capacitor

as a result

now the circuit changes properties the

time constant changes from r1 c1

to r1 in parallel vr2 times c1 so time

constant goes down

and also the final value

that v out and va can reach

goes down because now we have a voltage

divider at time infinity

so the output voltage is zero up to t1

and then it jumps to this value and now

vr the out and va have the same value

and they go together towards

this final value whatever that might be

the second example

uh started with some simple capacitor

circuits and the response to a step

input and we saw that in these cases

the current has to be an impulse because

we're trying to change the voltage

across a capacitor or series of

capacitors

in zero time and that results in an

impulse

so we also saw that the

voltage in this case

also has a step behavior

so vl jumps in zero time but not as much

as the input because these two

capacitors act as a voltage divider

so we see that the output voltage is

given by the input step height times

this attenuation factor

so today we'll start with this example

and add a resistor here and see that

things become much more interesting

so let's go ahead and do that

so we have

this example

where

we again have a step so vn is equal to

v1 u of t and this step

is applied

to a

capacitor

c1

another capacitor

c2

and a resistor r1

and we are interested

in this voltage

here as a function of time

okay so with the evolution that we

followed last time

we can try to guess

what vr should look like and then of

course write the equations and solve it

so

we saw that without this resistor

this voltage jumped

not to the full amount of v1 but to a

fraction of it

the question is not that we had to have

this resistor

does the out still jump or not

so you have to think about that for a

moment and decide

okay

well

uh let's assume v out does not jump okay

we'll see what happens the circuit

doesn't bite we can always start with

some assumption and see what happens

right so we say

if

the out

does not

jump

it means that c2 at zero plus still has

zero volts on it right okay

so this has zero volts on it

how about c1

well a zero volt at the zero plus this

is equal to v1 volts right the input has

jumped to v1

this is equal to zero

so c1 experiences the voltage change of

v1 in zero time

so we say

c1

has

a

voltage

change

of

v1 in

zero time

okay that's the key

why because the input jumped

this hasn't jumped that's our assumption

so the voltage on this capacitor jumps

by an amount of v1

if the voltage on the capacitor jumps

what happens

well if the voltage on the capacitor

jumps

uh the current of the capacitor has to

have an impulse in it right that's what

we saw last time

so we say

that this current let's call it i n

i in has to have an impulse right has to

be proportional to an impulse there's no

other way

to charge a capacitor in zero time we

have to have an impulse all right so we

are drawing an impulsive current from

the voltage source through c1

but where does that current go

can that current go this way

no because an impulsive current through

a resistor gives us infinite voltage

but we don't have an infinite voltage

available right this is only v1 volts

there's another source of voltage so we

cannot possibly generate an infinite

voltage in the circuit

so we say

i in

cannot

entirely okay let me

clean this up a little bit

so we say

the impulse

cannot

flow

through

r1

otherwise

it would

create

an infinite voltage

okay so an infinite voltage is not

possible in the circuit

where all the voltages are finite the

only possibility to create an infinite

voltage in a circuit is to have a

current source whose terminals are left

open remember

so we have a current source that goes

through a very very very large resistor

then it's going to generate a very very

large voltage but in this circuit that's

not possible right so we cannot gen we

cannot have an impulsive current to r1

because there is no source of infinite

voltage in the circuit

okay so then the question is

where does that impulse go

we have an impulsive current coming in

you cannot go this way it has to go

through c2

so this line of thinking tells us that

impulse of current

must

flow through

c2

all right even though the resistor is

there

so right between zero minus and zero

plus the resistor doesn't really play

any role

because the impulse of current that's

flowing through c1 also flows through c2

so

how much is the voltage across c2

at time zero plus

well that should be the same as what we

found before right before without this

capacity without this resistor we found

that this voltage was a step with a

height that is less than this height due

to this capacitive divider

this time we should have the same thing

why exactly

well because

the charge that is delivered by this

voltage source

to the circuit

between zero minus and zero plus

entirely goes through this capacitor and

this capacitor

it cannot go through this resistor

why

well let's calculate

so we say charge

delivered

to

r1

from

0 minus 2 0 plus right

0 minus

2 0

plus so it's integral of

0 minus to 0 plus

of the current flowing through the

resistor sum i i r1

times dt

remember

charge charges integral of current with

respect to time

okay now we decide that the current to

r1 is not an impulse if there's any

current it has to be finite it cannot be

infinite

and because we have a finite amount of

current and we are integrating it across

a very very very short amount of time

the result will be close to zero so we

say this is zero

so the charge delivered by this voltage

source

has to go on to only c1 and c2 because

during that very short time period

no charge has been delivered to r1 so

all of it has to end up on these two

so during that very short time period r1

plays no role in the circuit whatsoever

as if it were not there as if it were

infinite

so we can say that the voltage on c2

still jumps by the same amount that we

calculated last time so we say

v

c2

at time

zero plus

is equal to v1

times c1 over c1 plus c2

okay so you can see that

this line of thought is necessary

to get to this result

right even if you sit down to write

equations equations won't tell us

everything that we have here right some

of this just has to come from thinking

there's no other way

so it all goes back to c dv over dt

right and all the conditions that we

found last time in this example

right so

so then we can try to sketch

vc2 of course vc2 is the same as v out

right

so we say

v out

and what we observe looks like this

so we had zero before times zero another

problem there

and at time zero

we decided that this voltage has to jump

just like before without r1 right

it jumps to this value here

another question is what happens after

this point

okay

all right so we're going to look at the

circuit a little while and understand

uh exactly what will happen

and this voltage is now v1 it's a step

right it's like a battery v1

this capacitor has some finite voltage

audit this much

and now we have this resistor

so our expectation is that this resistor

will discharge

these capacitors over time it wants to

draw some current right any finite

voltage here results in a finite current

so r1 begins to discharge these two

but we have to be careful here so let's

think about it as follows

what we know is that at time infinity

probably all the voltages have

stabilized they are constant

and we know that if the voltage on a

capacitor is constant the capacitor acts

as an open circuit

so if that is the case that this

capacitor is an open circuit

and so is this capacitor

so what can i say about v out at time

infinity

at time infinity this is not here

there's no conduction here there's not

here there's no conduction here you have

this resistor sitting dangling by itself

it has no current it has no voltage

so our prediction is

if

all

voltages

become

constant

at

t equals infinity

then c1 and c2 are open

right they're open circuits

and that means that all we have is a

resistor like this

r1

there's nothing connected to it so v out

has to be zero

so in the extreme case

this voltage has to approach 0 here

so that is our prediction

and we might say okay it probably goes

like this right

we just have to see if that's true

okay

so what the circuit does is it uh

takes part of the step

and brings it over here right this much

of the step

and then it exponentially decays

right the as the capacitors charge and

discharge eventually everything goes to

the output goes to 0.

okay can i express the output as a

function of time

yes

this is a

first order system

and because the first order system still

satisfies the equation i showed you last

time and times before that right so y is

equal to y infinity plus y 0 minus y

infinity times the exponential so it's

the same thing here

but you may ask why is the system of

first order

even though it has two capacitors in it

all right so let's investigate that

okay so why

is

this system

of

first

order

okay well we talked about this a long

time ago we said that when we have a

linear system when we write the

differential equation for it

of order n

to solve this differential equation

completely we need n initial conditions

and those initial conditions have to be

independent meaning that you cannot take

you cannot get one initial condition

from the other ones right all of them

have to be independent we should be able

to enforce all of these n independent

initial conditions without any conflict

so here in this circuit

what we can see is that if i have an

initial condition

on c1

the initial condition on c2 is known it

cannot be independently set

why okay well let's see that's easy

let's change the color of our pen

okay

to this color here maybe

okay now that's

let me do this color okay

so suppose uh before i apply this step

right when this is zero i come along and

say i would like to have an initial

condition of

two volts across c1 okay

so i say this is two volts

so remember i'm trying to solve the

differential equation that differential

equation

requires n initial conditions so if i'm

thinking that this is a second order

system because it has two capacitors

then i should be able to enforce two

independent initial conditions

so i first want two volts across c1

can i impose

a new independential condition across c2

let's go ahead and do that so let's say

this is

5 volts

okay so i decided that i would like to

have two volts on this fibers on this

and if they agree if there's nothing

wrong with the circuit no problem we

have a second order system

but is that true

so before time zero this is zero

right the step is zero

so i should be able to write the kbl

here

zero

is equal to this voltage plus this

voltage right

so

kvl

maybe i write here

so kvl

so here's the situation

i have c1

uh

two volts on it

i have c2

five volts on it

right and i have zero

because we are talking about time zero

minus before the step comes in

so a kvl here says 0

this is equal to this plus this

2 volts

plus 5 volts

so you see that we have a problem

so we cannot set the initial conditions

of these two separately in other words

as soon as i call this two volts this

has to be minus two volts there's no

other way because this plus this has to

be zero right

okay so we see that only one initial

condition can be applied independently

so

only

one

initial

condition

can be

applied

independently

and that's why the system is of first

order

okay that's great all right so now i can

use my standard equation to solve the

circuit right

so we need the following mean y infinity

how much is y infinity

y infinity is the amount of this

quantity of interest at time infinity

we decided that has to be zero from this

little analysis here

so it's zero

y zero zero plus

how much is y 0 plus

how much is this voltage at time 0 plus

we found that we found that here is this

value here

so v1 c1 over c1 plus c2

how much is the time constant

well that's a little trickier right

because we have two capacitors

previously our computation was to take

the capacitor out

and find the resistance but now what do

we do

well not a big deal let's take our time

uh we follow the same procedure first we

set the independent sources to zero

so

tau

set

independent

sources

to zero

okay so you set this to zero what do we

get

we have

c1

c2

r1

right

can i find the time constant of the

circuit

sure what happens that when i go to this

computation topology

c1 and c2 actually appear in parallel

so i can just say that time constant is

given by

the sum of these two capacitors times

this resistor so we say tau

is equal to

r1

c1 plus c2

sounds like magic right it's interesting

that it happened that way

so does it mean that in all first order

systems when we set the independent

source to zero

we end up with something like this where

all the capacitors merge into one and we

have a single capacity single resistor

not really that's not true all the time

so in general if we want to find tau

we have to write the differential

equation

differential equation will be our first

order and i showed you a long time ago

how to compute tau from the differential

equation itself

but in cases like this we just go with

our method of finding time constant by

inspection okay but if your if you have

determined

from this type of analysis that the

system has

an order of one

but find the time constant seems tricky

you just have to write the differential

equation for the system

and then from there calculate time

constant

okay so we have found the time constant

now we can write v out right so the out

as a function of time

is given by

infinity value which is 0

then so it would be just

really this one which is

v1 c1 c1 plus c2

x above

minus t over tau

and u of t

right it's just a simple exponential

decay

all right i encourage you to compare

this circuit with another circuit that

we had before

where we did not have this capacitor

do you remember we had a single

capacitor a single resistor go back and

compare the results of this circuit

there's also that circuit that's also

very interesting

okay

this concludes our study of

rc circuits

uh there are many more interesting

examples but in the interest of time

we'll have to move on to inductors and

explore

those devices as well

all right

so let's talk about inductors

of course you've seen inductors in

physics you have some idea as to how

they operate etc so we'll just review

some of those concepts first

and try to

frame their operation in our own

language in the language of circuits and

then try to build circuits out of

inductors and resistors and eventually

capacitors as well

but i wanted to show you something that

is very interesting

so here's a piece of wire right as you

can see that's a piece of wire it has

these two ends so this looks like a

pretty good short circuit right you can

try to connect two points of a circle

together right it's a short circuit

now if you want to be more precise you

can say that

this wire has some resistance associated

with it so it satisfies ohm's law right

so if you apply some voltage between

these two ends we're going to get a

current and that current is given by the

voltage divided by the resistance

all right so that's nothing surprising

but let's take this wire and do

something like this i'm going to take

this wire

and i'm going to

wind it around this pen

all right now we can see

what happens here

so we wind it around this pan

and this is what we get

what is this

this is an inductor

so what happened

we had a piece of wire it satisfied

ohm's law

but by just reshaping this wire we

create a device that doesn't satisfy

ohm's law

and that's the fascinating thing about

this and about magnetism because this

all relates to

magnetism

okay so

what really happened was up to the end

of the

19th century and the beginning at the

end to the end of 18th century and

beginning of 19th century

people knew about the electricity they

knew about batteries and so on they also

knew about magnetism you had magnets and

they repelled or attracted each other

and so on

but no one had connected electricity and

magnetism together

they seemed to completely separate and

independent phenomena

but then as around 1820

a

danish

scientist

named orsted

accidentally found out something

interesting

he saw that

1820

he saw that if we have a piece of wire

and we pass a current through it

he saw that there was a

compass

and the compass was pointing for example

the north and south like this right

i saw that

when there's a current to this wire

the compass deflects it changes its

direction

now they knew that the compass was made

of a magnet

so his conclusion was that

if there's a magnet

and we pass a current through a wire

this magnet is affected by this wire

and what that means is that this current

through this wire

generates a magnetic field around itself

and that magnetic field like the earth's

magnetic field

interacts with this compass

and deflects the needle on the camp

compass

all right so orson said that if you have

a piece of wire and it pass the current

through it we generate the magnetic

field and that's how our right hand rule

comes into picture

so we generate a magnetic field like

this

in the form of a circle around this

piece of wire if the current is passing

through the wire

so any object that has any magnetic

properties and is placed inside this

magnetic field will be affected by it

so that was the first connection between

electricity and magnetism that was very

interesting

all right but

that still doesn't come into picture as

far as inductors are concerned

right

at least not as far as circuits are

concerned so something else was

needed to be discovered and that was

discovered by faraday

so sometime later i don't remember when

faraday discover something

interesting

so far they said sure worsted is right

when we have a current passing through a

wire it does generate a magnetic field

around the wire

but if the current is changing with time

which means the magnetic field is

changing with time

then something even more interesting

happens

so here's our good old wire with the

current i but now i is a function of

time

and we have

a magnetic field around it that's also a

function of time

so

if i

what happens now with this magnetic

field that changing with time

is that if i place another wire in the

circuit

so let's bring

another wire

okay just a wire by itself it's not

connected to anything else right i bring

another wire and what i see is that this

wire begins to develop a voltage across

it

that's fascinating there's a piece of

wire it doesn't have any current through

it but it has a voltage on it

certainly not ohm's law right

so in summary it is what happens we have

a first wire through the first wire we

pass the current and we let the current

change with time

so the magnetic field around this wire

begins to change with time now be

remembering a second wire the red wire

bring the second wire just place it in

that field

that second wire begins to generate a

voltage across itself

right and that's what we call induction

we have induced

the changes in this current

through this magnetic field onto this

wire to the red wire so we have induced

a

current

induced voltage from this to to this

from the first wire the green wire to

the red wire

so that eventually becomes a very

interesting concept it relates to

transformers you can build a transformer

like this i don't know if you've seen

that in the past or not but transformers

are using many different uh systems

today including power systems etc so you

see that there's an inductor on this

side there's an inductor on this side

they're not connected

and if i pass a current through this and

that kind of changing with time

i get a voltage here even though there's

no current through this part right

so that's the interesting and beautiful

result of

this relationship between electricity

and magnetism

we can create

magnetism from electricity

right passing a current gives us a

magnetic field but we can also generate

electricity from magnetism if i have a

magnetic field that's changing with time

i can generate a voltage i can generate

electricity

so this relationship between the two

became a an important and solid

foundation for everything that we do

today all right so but how does this

help us with circuits

well one thing that becomes interesting

is that if i have a piece of wire so

let's go back to the green wire

if i have a

piece of wire

and i'm passing a current through it as

a function of time

so this piece of wire is generating

magnetic field

so it's making me feel like this

and that magnetic field

is also changing with time

all right

so

if i have a piece of wire

that is

immersed in a magnetic field that is

changing with time

that piece of wire should generate a

voltage

right so in other words we don't have to

have a second wire to generate a voltage

even if you have a first wire

and even though that wire is creating

its own magnetic field

that magnetic field can induce a voltage

on the same wire

okay so here we have two wires we induce

the voltage from the first wire to the

second

here we have a single wire

and this single wire generates a voltage

across itself

provided

that the magnetic field is changing with

time

and that happens when this current is

changing with time

all right so let's make sure we

understand this if the current doesn't

change with time

the magnetic field doesn't change with

time

and there's no voltage here it's just a

simple piece of wire ideal piece of wire

uh we have a magnetic field everyone's

happy nothing is changing

but if the cut is changing with time

and as a result the magnetic field is

changing with time

then this wire develops its own voltage

here some voltage v

okay even though it seems like an ideal

wire so it definitely does not satisfy

ohm's law

so from other perspective

if we have a piece of wire

that carries a current

you have a piece of wire that carries

the current and the current is not

changing with time

then that piece of wire is just an ideal

piece of wire it's like this right just

a simple short circuit doesn't do

anything

but if the current rear wire changing

with time then that wire begins to have

inductance associated with it so even

though this piece of wire doesn't look

like an inductor it is actually an

inductor as far as the time

varying current is concerned

and that's what we want to study now

okay so let's go ahead

and go to

the next page here

all right

so

inductor

basics

all right so the symbol that we have for

inductor looks like this

and

we have some sort of voltage here

v1 some sort of current here

and this inductor has a certain

inductance value l1

and these three are related by a very

simple equation that comes from physics

we won't bother explaining it we won't

bother trying to make it as intuitive as

was the case for capacitors right it's a

little less intuitive has to do with the

magnetic flux

but we don't repeat all of that we just

accept the final result

the final result is that these three are

related as follows

the voltage across the inductor

so v1 is equal to

l1

di1 over dt

okay so this partially agrees with our

uh

basic understanding that i just

mentioned a minute ago right it says

that if the current through this

structure doesn't change with time

right necron doesn't change with time

there's no voltage the voltage is zero

so if there's no current through the

structure the structure just looks like

a piece of wire right

because only a piece of wire can assume

can guarantee a zero volt difference

between these two

all right okay

on the other hand if the current through

this structure through the inductor is

changing with time and we have some

finite derivative then we also have some

finite voltage across this device

as if it were a resistor even though it

doesn't satisfy ohm's law

so this will be the important equation

related to inductors just the way

i equals c dv over dt was the important

equation for capacitors

so pretty much everything we said there

applies here except that things are

turned around right here

what is proportion of the derivative of

the current

for capacitor it was the current that

was proportional to the derivative of

the voltage as long as we keep this in

mind we understand how inductors operate

all right so

two important results

so we say

if

v1 less than infinity

right

so if the circuit

containing this inductor cannot give it

an infinite voltage

then this derivative cannot be infinite

this means that i1 cannot jump

so

i1

cannot

jump

in

zero

time

meaning it cannot jump instantaneously

all right

so that is in contrast to behavior of

the capacitors for capacitors the

voltage could not jump if the available

current was less than infinity here's

the other way around right the current

cannot jump if the voltage available for

that inductor is less than infinity

okay the other extreme is if the current

is constant

so if

i1

is constant

then v1 is zero

so this inductor guarantees a zero volt

difference between here and here

and what is that type of device that

type of device is a short circuit right

just like this so

we say that l1

is equivalent to a short circuit

so those are the two extremes for

capacitors we also saw some something

similar except that of course the

situation is different

so in all of the rl circuits and other

systems that we'll study in in the next

few lectures we have to keep both of

these in mind

the this typically happens when there's

a switching event something is switching

in something switching out or maybe

there's a step somewhere and this

typically happens near the end when

things have settled no current is

changing with time so the voltage across

the inductor is zero so but we have to

we always have to remember these again

if you don't remember them you go back

to the original equation for the

inductors

all right so just the way there was room

for confusion

for the polarity of currents and

voltages of capacitors this confusion

here as well so we have to make sure

that is clear

so again we are assuming that the

current

the current goes from high potential to

low potential

all right so that's how this equation

comes about

if we have for example something like

this

and the current is like this

and the voltage

we have

assumed like that right if i choose

these arrows and directions as shown on

this diagram and then i should say that

i1

is going from low potential to high

potential

and that means that v1 is equal to minus

l1

di1 over dt

so again you have to go through all the

exercises that i mentioned before for

capacitors to make sure that you master

this polarity and you don't get confused

all right

and one last point is that if we have

the voltage of an inductor we can find

this current right

so we say

to find

the current

of an inductor

we just say

we need this current so we say i1

is equal to 1 over l1

integral of v1 dt

so the current random vector is given

by the time integral of the voltage or

the area under that curve divided by the

inductance value

okay so that's how we can find that we

can always go back and forth between the

current and the voltage according to

this equation or that equation

all right

so

let's see what else we have here

[Applause]

okay let's look at some examples uh

actually one more point

so one

more point

so we saw that an initial condition on a

capacitor was clear right we have a

capacitor there's some voltage

difference or some charge here positive

charging negative charge here that was

the initial condition

for an inductor is a little different

for inductor initial condition

means the current through the capacitor

through the inductor is not zero so

the initial condition

is like this

we have an inductor

l1 and just happens that at the moment

we look at it

there's a current i0 here

and that would be the initial condition

of the inductor

so in any circuit that we are solving

for example you have a differential

equation

into trying to find initial conditions

the initial conditions to look for are

the voltage on capacitors and the

current three inductors right not the

other way around that doesn't work

okay so with that

we can just look at some simple examples

to warm up

as far as inductors are concerned

so here's an example

that i have here

suppose uh

i have a current

flow internet inductor

that looks like this

call it alpha t

is a ramp

that is applied to an inductor

okay i1

and i am interested in the voltage

across the inductor

because the current is changing with

time

there must be a voltage right so let's

try to find the voltage

okay so we say

the voltage

is equal to

l1

di1 over dt

and that's equal to l1 times alpha

so i can plot the voltage

the voltage as a function of

time before time zero the current was

constant it wasn't changing it was zero

anyway so the voltage is zero

and now we have a constant derivative

the derivative is alpha and then

multiplied by l1 so v1 is constant so we

generate a constant voltage across the

inductors

all right so in response to a ramp

a an inductor a ramp current an inductor

generates a voltage that's constant

strange but true

so that's what it does

so we see that the voltage across the

inductor can jump nothing wrong with

that but the current run inductor cannot

jump provided that the the

voltage available for the inductor is

not infinity

okay

let's see here

all right one more point what happens if

i make this current slope higher and

higher if this becomes sharper and

sharper

we can see that this voltage goes up

right because this is given by alpha l1

times alpha so the sharper this current

changes the more sharpness current

changes the higher this voltage will be

we generate a larger voltage across the

inductor and that again makes sense

so the faster we change the current the

larger the voltage drop across the

inductor

and that sometimes is useful sometimes

it's harmful and we'll see examples of

that in the future

i will see you next time

foreign