Control Engineering

Prof. S. D. Agashe
Department of Electrical Engineering
Indian Institute of Technology, Bombay

Lecture 07

The electrical rlc circuit, rlc and source of emf that circuit is said to be analogous to the mass
spring dashpot and force or displacement driver, mechanical system and the analogy comes out
when we write down the equations for the 2 systems component by component and then combine
the equations for each system separately into may be one single system equation that is when the
analogy or the similarity between the two different physical systems comes through.

We can spend a few minutes on the electrical system with the rlc and the electromotive force as
the 4 components. You will be all able to write down the equations for the 4components
separately. We had started of with one of the equations namely the equation for the resistor but
one can write down and I will do that the equations for the remaining 3 components. So here are
the equations V R equal to R i, V
L
equal to L di dt I C equal to C dV dt and V for the source V E
is equal to some function e actually I should be a little more careful here.

(Refer Slide Time: 02:50)



I should have written use the symbols which indicate what component or element of the circuit
or network they are associated with thus I have written V R indicating there is a voltage
associated with the resistor R of the resistor physical device resistor R and this voltage at any
moment of time is given by the product of the resistance of the resistor a positive number
expressed in units of ohms multiplied by the current in the resistor and therefore I should write
strictly speaking I R, I subscript R meaning it is the current in the resistor and again to be
absolutely correct I should write V R of t equal to R into I R of t. The voltage across the
resistance as we say at any moment of time t is equal to this parameter of the resistor called the

resistance multiplied by or multiplying the current in the resistor at time t, I Rt. So this is the
equation for the component of the network a component which is called a resistor. Now in this
point note one thing very clearly and I will go on emphasizing this a number of times because
otherwise a lot of confusion can occur.

(Refer Slide Time: 04:55)



(Refer Slide Time: 05:24)



Now first of all there is the actual physical thing or device called a resistor. Here for example is a
particular resistor of a particular kind this is what is looks like. Now you know that resistors can
have various appearances here are some more resistors which look different from the one that I
started with here is a one that is known as a wire wound rheostat it is a much bigger thing and

one can see that wiring more or less very clearly there is here is a very small resistance which
looks like a very small cylindrical thing it is not one does not know what is inside it and if you
did not know what it might be you would not even guess that it is a resistor or it is a resistor
certainly there are there is no wound wire inside that.

So there are those physical objects or things or components called resistors which one finds in
the laboratory on the shelf or one can go to the market and buy them, there are made or
manufacture also and as you know each one of these resistors comes with a value what we call
the value of the resistance of the resistor typically it will be expressed in ohms but in some cases
kilo ohms or even mega ohms and very rarely may be milli ohms.

(Refer Slide Time: 06:30)



So here for example it is a resistor whose resistance value is 1.3 kilo ohms and all of you should
be familiar with the way in which this value is marked on the resistor not numerically as 1.3 kilo
ohms but in terms of some bands of particular colors. So here is a 1.3 kilo ohm resistor which is
very small in size and as you can expect you cannot pass a very large current through it
otherwise the heat that is produced in it will burn it up. So here is the practical thing which has a
number called a resistance or the resistance associated with it expressed in some unit such a ohm
or kilo ohm and so on having a particular appearance it has some weight, size and also along
with it go some limitations like the current in it should not exceed may be 10 milli amperes or
100 milli amperes certainly the current in this tiny resistor cannot be maintained or kept at a level
of say nk ohm 10 amperes for even a moment of time the heat produced will be so large that it
will just heat it up and burn it.

So here is a physical device which has all these aspects it looks like this, it has weight, it has
volume and there are limitations associated with this use. Of course it cost something but for us
from the electrical engineering point of view what matters at the moment is only its resistance a
particular number denoted by R or R with a subscript like R 1, R 2 if there are 2 resistances and
so on and we are making the assumption here. This is the important fact that one should be aware

or note that this physical thing can be described or part of its behavior can be described by means
of an equation like V R of t equal to R I R of t, where in this case R would be the number 1.3 into
ten raised to 3.

(Refer Slide Time: 09:05)



If I is a number which represents the current in the resistor in amperes and V R is a number
which represents the voltage across it in volts and so we have volts equal to ohms into amperes I
should not have said one point I said 1.3 into 10 raised to 3 because the resistance was 1.3 kilo
ohm, if I have the current in milliamperes the resistance in kilo ohms then the products comes
out to be voltage in volts.

Now if this resistance is say heated up either because of heavy current flow in it or you keep it
near some other source of heat then will it continue to be described by this equation V equal to R
i t with R equal to 3 into 10 raised to 3 or if I cool it if I put it in the contact with a very cold
mass of ice let us say, will it continue to have a resistance of this same value. As you all know
this is not the case the resistance value of resistor will depend upon the temperature of the
resistor also.

So we are making the assumption that the resistance is operated in such a way that its resistance
is given by 3 into 10 raised to 3 ohms. This is only an idealization in the sense, this is only in our
concept or idea about it that is we like to think that the resistor behaves like this V equal to R i
with R equal to 3 into 10 raised to 3, what we think will be correct of course or not too far wrong
hopefully most of the time and so there is another aspect of it which we must remember that is
this value 3 into 10 raised to 3 is to be taken with a pinch of salt in the sense it is not asserted that
the resistance value is exactly 3 into 10 raised to 3 ohms even if the bands on it may indicate that
it is say 3 into 10 raised 3 or rather 1.3 into 10 raised to 3 ohm, no matter how good a quality a
resistance one may or a resistor one may buy and this is associated with what is called the
precision. This value 1.3 into 10 raised to 3 by the very nature of it cannot be 100 percent precise
in a course on electrical measurements of course you would have looked at methods of

measurement of parameters like resistance, inductance, capacitance and also a physical quantities
like voltage current frequency of a alternating supply and so on a magnetic flux and what not?

(Refer Slide Time: 11:57)



So suppose we go to a laboratory and make actual measurement, so set it up. So that some
current flows in it measure the voltage across it measure the current in it will we find that the
voltage number in volts will be exactly equal to 1.3 into 10 raised to 3 multiplied by the current
number no and in fact for more than one reason because just as there is this problem of what is
exactly the resistance of the resistor is it that specified value there is a question of precision. In
fact when you buy a resistor you should ask about the degree of precision or you should buy it
knowing what will be the precision is the value going to be one point three into ten raised to 3
within plus minus 10 percent or 5 percent or one percent because this is all that can be
guaranteed of a readymade resistor.

Of course if you wanted to make a resistance of resistance 1.3 into 10 raised 3 ohms and want it
to be more precise of it you could chose wire and wind it yourself chose an appropriate length of
the wire and so on and so forth and therefore you may have more precision than is commercially
available in principle you can make a resistor of resistance value which is within plus minus .1
percent of what you think it could be. But commercial resistances or resistors do not come with
that kind of precision obviously it is going to cost you more if you want a more precise resistor.

So that is only one part of the difficultly the other part of the difficulty is that in order to find out
whether V is equal to 1.3 into 10 raised to 3 into I, I need to measure both current and voltage
also and there is precision associated with those measurements perhaps you can measure the
current within a milliampere the current may be one ampere but within one milli ampere this is
because of the nature of the instrument the dial reading on the dial of if it is a digital display, the
number digits that you have in the display. If you set it up in the ampere mode and it can go up
three places of decimals then the milliampere is the lower limit, you cannot go lower than that.


Of course you could use a more precise instrument or you, you could use a setup where as there
by you could measure the current within one micro ampere and again one can if one wants to go
further, go further even and talk about a nano amperes similarly, with voltage, volts or millivolts
or microvolts but ultimately there will be a practical limit where you will stop. So you will
measure voltage say within one millivolt a voltage will be of the order of a few volts but within a
few millivolts similarly current may be of the order of one ampere but within plus minus 1 milli
ampere.

(Refer Slide Time: 16:47)



As a result the resistance value you wanted to measure it this way by finding out the voltage and
the current and then dividing the voltage by the current will have its own precision. Of course the
voltage current method is not the only method of measuring the value of a resistance in fact it is
not necessarily the best method although it may be a simple method in principle there are in an
instrumentation course or measuring instruments course you would have studied some other
methods or some methods of measuring resistance other than simply measuring voltage and
current. But to come back to our point then an equation like V R of t equal to R into I R of t with
a particular value of R, R equal to 1.3 into 10 raised to 3 assuming V is in volts and I is in amps.

An equation that we write for this physical device here this particular resistor is only an
approximation or an idealization and that that is about all that we can say, we do not know of any
better relation but at the same time we will not insist that this hold exactly to 10 places of
decimals or even 3 places of decimal. One can sometimes use the word and this is the little better
word than modeling and that is the word representation. So instead of saying that this resistance
is modeled by V equal to R i or V R of t equal to R I R of t.

So this equation is said to be a model of the resistor we could say that this equation represents
the behavior of the resistor with the understanding that it is only under certain conditions and it is
only within certain limits that the equation represents the behavior of the actual resistor. In fact

right now, I use the word model in a third sense earlier we talked about a model of an aircraft a
scaled down model which is physically similar works on similar or the same principle but has
much smaller size weight etcetera, it is one kind of model. The other is what we just started with
that is the rlc circuit and the mass spring dashpot force or displacement source system, one of
them can be said to be a model of the other and in place of the rlc of course we can have an
operational amplifier or analog computer system setup which is the model of the mechanical
system.

The third use of the word model is when we say that this equation is a model of that physical
thing the resistor. Sometimes one uses the word mathematical model implying for example that
in this model or in this representation what I have written V equal to R i V and I are numbers, R
is a number there is a product operation involved etcetera not much of mathematics here in V
equal to R i it is really simple arithmetic a multiplication and that is about all. We must
distinguish models in this case equations from the actual thing same thing holds for the model
aircraft. The model aircraft is only a model of the actual aircraft if you want fly from here to
Delhi, you cannot use the model aircraft for that purpose.

Similarly, if you want a spring to do something you cannot setup a rlc, emf circuit and hope that
it will do the work of a spring the model is different from the system, the model is model of the
system, it represents the system or the device. The model is mathematical in place of any
equation I may draw graph. So V equal to R, I could be shown by means of a graph of voltage
versus current in fact for some devices we may not be able to write down any equation so nicely,
it is easier and in fact that is what is you start with by giving graph, a graph of the behavior of V
and I such graphs or curves are known as device characteristics and probably you know we
already come across them in connection with diodes, transistors and the light.

The simplest and experimentally, the simplest to obtain representation of a diode will be by its
VI characteristic, a curve of V and I for different values of V and I. The equation for it it may not
be very easy to write in fact there is no nice single formula for the whole curve as you know of
course one can with some theory or from a purely practical point of view find out an equation
that describes some part of this characteristic and even use it. So in that sense the VI
characteristic is also a model but it is still more a mathematical or an arithmetical model in the
sense it involves physical variables like voltage and current and gives you the pairs of values of
voltage and current that you may find associated with a diode or with a resistor.

So when we use the word mathematical modelling then we can include such graphical methods
also in fact they are useful as we will see in control work very often one can get some idea of
what is going to happen by drawing graphs or pictures rough sketches even.
So you you do not have to really write down only equations and then find that you cannot do
much with them because you do not know what how to solve it.







(Refer Slide Time: 23:30)



So here is one device resistor for which there is this model V equal to R i or to be very precise
resistor voltage V Rt equal to R into resistor current I Rt, what about the second another
component in the circuit it is the inductor. Now the inductor has a parameter associated with it
namely the inductance the standard symbol for that is of course L just as the resistance is the
parameter associated with the physical device resistor, inductance is the parameter associated
with the physical device inductor and once again like the resistor and inductor or a coil can come
in different sizes and shapes and material and so on.

Here are a few examples of inductors of various shapes and of course various values of
inductance one can have an inductance with a small inductance value like few milli Henries or
one can have a much bigger inductor or coil with an inductance of the value of few Henries. Just
as you can have resistors with the few ohms resistance or resistors with mega ohms of resistance.
Now just as in the case of the resistor and here it is even a much stronger approximation or one
should say really a weaker approximation much cruder approximation namely the relationship
between the voltage across the inductor and current in the inductor is given differently V L and I
will write here.













(Refer Slide Time: 24:16)



(Refer Slide Time: 25:09)



Now V L of t once again V voltage L because L is the standard symbol for the parameter
inductance V L of t equal to L, the inductance value multiplied by now this time it is not the
inductor current as you know but the rate of change of the inductor current D I L dt or to
emphasize that the rate of change also may vary with time d I L dt at time t and later on we will
use a somewhat different symbol which is not used in many books but you may find it in a few
mathematics books.




(Refer Slide Time: 25:57)



We will write it also as L into D, D standing for the derivative operation derivative of the
current, so D I L, I L
i
s the current in the inductor as a function of time D I L is the derivative the
rate of change of it as a function of time and its value at some particular moment of time t. So the
representation of the model for the inductor will be D VL of t equal to L into D I L of t, D I L at
t. The voltage of the inductor is L times the rate of change of the inductor current and that same
moment of time.

(Refer Slide Time: 28:09)



Now this is not arithmetic anymore because it is not just multiplication but there is the idea of
derivative or rate of change and so to some exchange this means that we have gone beyond

arithmetic and algebra we have gone into calculus and in fact it is this kind of a thing that gives
rise to a differential equation finally when we begin to model a system. It is because there are
devices like inductor which can be represented approximately somewhat crudely but still it may
be adequate from many purposes by means of a relations like this which involves derivative in
electrical engineering of course one talks about this and the other relationship for the resistor as a
VI relationship or VI characteristic.

So the VI relationship for an inductor is given by V equal to L Di dt, if I is in amperes, if the
time is measured in seconds. So the rate of change of current is measured as amperes per second
if the inductance L is in Henries then the product L Di dt will give you the voltage in volts. Now
what about the third component, the capacitor like the resistor and the inductor, capacitors also
come in various types sizes, shapes and capacitance value, here are the few of them, what is the
model or a model a useful model for a capacitor. Well it is given as iC of t current I in the
capacitor whose subscript C at time t equals C the capacitance multiplying the rate of change of
the voltage associated with the capacitor or d VC dt at t or using the D operator notation it will
be C times D V C (t)

(Refer Slide Time: 28:34)



Here again, we go beyond arithmetic and algebra, we go into derivative or calculus the current in
a capacitor or associated with the capacitor depends on the rate of change of voltage associated
with the capacitor and is obtained from the rate of change of voltage if the voltage is in volts, rate
of change is measured the time in seconds and capacitance in farads then the current will be
obtained in amperes. So as you know the values of capacitance or capacitors that one is likely to
meet in the laboratory is not of the order of farads but very likely of the order of microfarads
may be 100 microfarads or 1000 microfarads or even much smaller than that a Pico farad. But
the Faraday the farad named after Faraday is the unit and therefore volts per second into farads
only gives you current in amps but if you took volts per seconds multiplied it by microfarads,
you will get the current in micro amps.


(Refer Slide Time: 30:23)



Now I said that the approximation for an inductor or the model for an inductor VL equal to L di,
L dt is a crude approximation or I use the use the word strong and as I said that may be one
should say weak because in many cases it is not really a very good approximation, why, why
because the inductor or it is many inductors involve a long length of wire it is called a coil
simply because there is a wire which is coiled or wound around what is called a core?

(Refer Slide Time: 32:20)



Some magnetic material and so there is a good length of wire there now this wire is a piece of
resistor also it has resistance even if it is good conducting material like copper still it is not a
very thick wire and there is long length of it. So it has a good bit of resistance associated with it

by good bit of course I mean not meg ohms but it could be a few ohms. So this inductor is not
just L that is it is not that current in it produces a magnetic flux in the core and therefore the rate
of the change in the flux will produce a counter emf or an induced emf in the coil which is what
this V is about but it has also a resistive part of it as it were and if so, if one wanted a little better
a less crude approximation or model for it, one possibility is to think of it as series combination
of a ideal resistor with an ideal inductor and so write down an equation like V for the voltage as
not L di dt only but as consisting of 2 terms R i plus L di dt where the term R i is coming from or
taking around account of the fact that there is a lot of length of wire there and the L di dt parts
comes by taking account of the fact that the wire is wound round and magnetic core current flow
gives rise magnetic flux and change in magnetic flux leads to the phenomenon of a counter emf
or a back emf in the wire and so on.

But once again we have to remember that this is the model, nobody has connected actually a
resistor so called ideal resistor and an ideal inductor in series to make a practical inductor the
practical inductor is wire wound around a coil and how good this approximation is V equal the R
i plus L di dt will depend on how you are using this inductor. For example, if the current and the
voltage are not changing very rapidly and by very rapidly again one has to mean something very
specific. Let us say the voltage and varies sinusoidally may be at few hertz or a few kilohertz
perhaps then the representation like V equal to R i plus L di dt may be okay. But I want to
operate it at frequencies of the order of megahertz and if I want to be very ambitious gigahertz
then this model will be useless; it will just not be useful.

(Refer Slide Time: 35:13)



So this emphasizes the fact that the model is not this device of the system the model is only an
approximation it is crude unfortunately, we do not know exactly what the true mode is. So in that
sense it is an approximation to something that we do not know and we have to make use of it and
see whether it gives you good results or not if it gives good results, it is okay, if it is does not
then our modeling was not adequate. We will have to look for a better model that is the model

which will give you results which fit what one actually finds by experimentation the last of the
devices is the voltage source or source of emf or electromotive force for that the equation will be
V again voltage E a symbol for the electromotive force E like RL and C small e or capital E. It is
value at any moment of time t is equal to e of t and what do we mean by e of the t on the right
hand side what is meant is that e of t is going to be some known function of time. For example if
I have this cell freshly bought, what freshly prepared not very old then I can expect that this e of
t will be nearly 1.5 volts all the time.

So this e of t will be really remain constant, will remain constant as I operate the cell for a fairly
long amount of time. So if I can say confidently about this piece this cell that look the voltage
across it is going to be 1.5, no matter of how long you go, no matter what the t is then V E of t
equal to e t and in this case e t equal to 1.5 volts is a good model for the cell and once again we
know that this model can cease to be good. If you use a cell for the long time then it does not
behave like this anymore if you actually measure the open circuit voltage of the cell we will find
that its not 1.5 but it has come down to 1.4 volts or even less and not only that when you actually
use it in the circuit, when you have current in it so voltage may be even much less than 1.5 volts.

So that is something which is not written here when I write V E of t equal to e of t equal to 1.5
the claim is that this cell or battery will be described by this no matter, what current flows in it
what value of current and whether any current flows in it or not. Now that is certainly not true of
a practical cell from this small cell you should not draw perhaps a very large amount of current
its voltage will no longer be then 1.5 volts not only that the cell can get even ruined. So may be if
you draw a current or of the order of a few milli amperes or 10s of milli amperes as when you
put it in a transistorized radio receiver not one cell but may be 2 or 4 of them, the current that the
cell supplies could be of the order of few 10s of milli amperes under that condition if the cell is
fresh it is not been used for too long a time the cell will be described by an equation like this.
This is of course the simplest in the sense there is no arithmetic here the voltage is just some
number in this case 1.5.



















(Refer Slide Time: 38:47)



(Refer Slide Time: 39:00)



Now like the rlc voltage source says also come in various forms a cell of this kind or dry cell is
one in your physics course earlier we have heard about the wet cell, the cell that goes back to
volta in way or the cell with zinc and copper dipped in hydro sulphuric acid or hydrochloric acid
and various other kinds of cells that you have studied earlier but it could look quite different a dc
generator is also an example of this kind of a thing. Since we are talking about an electric dc
motor which requires power supply for the field winding as well as the for the armature winding
therefore a dc power supply.


Now this dc power supply itself can be modeled under certain conditions by simple equation like
V of t, V of the source at time t is equal to not may be 1.5 volts dc but 230 volts dc. Now like the
rlc this kind of an equation is referred to as an ideal characteristic of course it is ideal in a
difference sense also that is if you could have something like this then perhaps some things other
thing would be very simple. So in that sense this is something which is also desirable and it is
true that you when you talk about the power supply, the power supply should be such that no
matter what current you draw from it the voltage should be guaranteed to remain at some definite
value in the case of the ac power supply let say 230 volts it should remain in a RMS value, the
wave form should remains sinusoidal and the frequency should remain at 50 hertz or 50 cycles
per second.

(Refer Slide Time: 40:54)



So in this sense when talks about an ideal voltage source but here the word ideal also means that
the voltage of the source does not depend on the current. The source that we have looked at is a
cell in which the voltage is constant as a function of time that is it does not change with time but
an ideal ac voltage source, we will have a voltage which is given by a different equation or
relationship namely V E of t equal to may be 230 root 2 into sin 2 phi into 50 into t, where as you
know the root 2 come because 230 is the RMS value and 2 phi into 50 comes because the
frequency is 50 hertz.

So in order for it to look like A sin omega t, where A is the amplitude and omega could be called
the angular frequency of the sinusoidal wave form, A is RMS value into root 2 and omega is 2
phi into the frequency as cycles per second or frequency in hertz. Now of course on the other
hand I can look at the mechanical system look at the various devices write down the
mathematical models for each one of them and of course I expect that all of you will be able to
do it because the 3 particular devices that are involved here are all familiar to us that piece of
matter or a block which already one has called mass implying that in its description a number
will play a role that number will be denoted by the symbol m for mass such that a particular
equation or a relationship hold between two variables associated with the mass and what is this

equation just as V E equal to R I, you know some sense goes back to Ohm although he did not
really write equations like this.

(Refer Slide Time: 42:56)



(Refer Slide Time: 43:37)



Similarly, the equation that one writes today for a mass goes back to Newton and as the familiar
equation F equal to M into A, but since the acceleration may vary with time the force may also
vary with time one will have to write F of t, the force associated with the mass or acting on the
mass equal to the mass of the body multiplied by the acceleration of the body at that moment of
the time t and because acceleration itself is a rate of change of velocity and velocity is a rate of
change of displacement or position therefore ultimately we may write this as M dv dt at t and

therefore with the D notation m into Dv at time t and because the velocity itself is a rate of
change of displacement or position, we could write this as m d square s dt square at t or we could
write this as m into d square s of t, where d square indicates that the differentiation operation is
carried out twice successively.


First you differentiate s a displacement function or the position function, we get a new function
that is the velocity function then you differentiate that to get the acceleration as a function of
time that you multiply by the mass number m that will give you the force acting on the mass of
the body at any moment of time t. So here is a model a mathematical model of the mass unlike
the resistor perhaps and even much more so unlike the inductor we believe that this is a much
better model, this is not as crude as the other models are. We believe that everybody has a mass
and there is this kind of a equation associated with that body but then as you know perhaps if you
have studied or heard about the theory of relativity and so on that one has to be careful when
using this equation.

In other words there are situations when this equation no longer may continue to hold because I
am not talking about the situation where the mass of a body so to speak changes as time proceeds
because one could actually argue that is not the same body, that is the body itself is changing as
time goes on and so no wonder its mass changes because it is not the same body, what could be a
good example of what do you think of as a body whose mass is changing with time. In fact any
device which consumes or uses fuel is a device of this sort. As you ride your motor bike or your
car fuel is getting consumed. So, if you think only car as single body and if you include the
weight of the fuel.

(Refer Slide Time: 46:14)




As a part of that body then the mass of that whole body is changing as you drive the car or as you
drive the scooter the mass of the scooter as a whole is changing. Now whether we want to say

that the mass of the scooter is changing or we want to say the scooter itself is changing, although
some parts of it remains the same and strictly speaking we are we are one would be right in
saying that the scooter itself is changing because after all if one looks at the fuel tank one can see
that fuel level is going down, if you look at the fuel gauge, the fuel gauge pointer is going down
this is a home example but a much more important example is provided by a rocket because
there the amount of fuel that the rocket requires to get itself up to a particular height is a can be a
substantial part of the total weight of the rocket that is rocket with fuel and therefore as the
rocket ascends there is less and less fuel therefore one has to think twice before applying the
equation F equal to m a, how to change this equation that is a different story. But here is a
situation when you are considering the ascent of a rocket and you want to be realistic about it we
have to take account of the fact that the rocket is no longer a body with a constant mass and this
is not because of any relativistic effect that the body is moving at a high velocity therefore its
mass is not the same as rest mass and all that stuff it is simply that fuel is getting consume and
fuel is a substantial part of the total mass of the whole package.

Now if we look at the fuel tank separately and the rocket separately as 2 bodies may be
connected to either mechanically because after all the fuel tank is housed in the rocket. Then the
rest of the rocket the mass will remain unchanged but the fuel tank mass of the fuel and the tank
mass that will go on changing that will be a different way of looking at it but the equation F
equal to ma will no longer be applicable. So even a time order equation like a F equal to ma there
can be some reservations about it. The question of precision off course arises here you can
measure mass within a gram the mass may be of few kilograms within a gram or within a
milligram or a microgram but somewhere you stop.

We cannot be sure about the last milligram or the last microgram and the two quantities on the
other on the two sides force and acceleration both of them also cannot be measured with any
arbitrary amount of precision. Acceleration through displacement let us say displacement in
meters could be measured to within one centimeter millimeter or even micron or a micrometer
and therefore rate of change of position or rate of change of velocity acceleration will be
measured within a certain accuracy. On the other hand the force a Newton will also have
precision associated with it.

I should not have used the word accuracy one should use the word precision in this case.
Accuracy arises when the instrument that you use is purely designed or it has some faults in it or
is not very carefully designed or it could be replaced by another instrument which gives you a
result which is closer to what you think is the correct one where as precision is something quite
different, precision is almost inherent in all measurement simply because you know we know
that earlier we could not measure things to within centimeter or within a millimeter because if
one uses the scale then you can barely see a millimeter on a scale but then some person by name
Vernier came with an idea with an idea that resulted in a vernier scale and using vernier scale
now you can go down to 10th of a millimeter something which you cannot see directly but
indirectly you say that your measurement length measurement is now precise within one 10th of
a millimeter and if I combine it with a microscope and so on I may be able to increase the
precision even more which makes us think that we have not reached the absolute end of it. We
have increased the precision of our measurement presently this is where we stand but we will not
claim that this is the length is exactly equal to this up to 10 or 20 places of decimal.

(Refer Slide Time: 51:55)



So F equal to ma therefore is an approximation or a representation of the body which is good for
some purposes and it may not be good for some other purposes but for our mechanical system let
us say F equal to ma is a acceptable description or representation or model, what about the other
two components the spring and the dashpot, well here is a spring and here is a variety of spring
of various sizes of various natural or free lengths various material and so on, what about their
model it is interesting that like the resistors which differ from each other in appearance the
inductors the capacitors the bodies or masses springs also will look so different from one
another.

All of them can be described for many purposes by an single common equation and what is that
that is related to a law, which is named after Hooke just as your Newtons law which is
embodied in F equal to ma, you have Hookes law which will relate force to extension that is
there is the idea of what is called the natural length or the free length of the spring and actual
length at any moment of time because the spring, the spring is being stretched or is being
compressed and therefore with some appropriate sign system F will given as k, small k into the
extension.












(Refer Slide Time: 53:12)



The extension could be written a length of the spring at any moment of time say l of t minus l, 0,
0 is to remind you that this is the length of the spring when there is no force acting on it it is free
or the natural length of the spring. So if l (t) is a length at any moment of time l 0 is the free
length then this difference is positive if the length of the spring is longer is greater that its natural
length which means the spring is being stretched and therefore this extension which is positive
multiplied by a constant which is the elastance of the spring it is connected with a property of
elasticity of the spring.

This product will give you the force in this case the force of tension somebody is pulling at the
two ends or at one end of the spring the force with which the spring is being is pulled is given by
an equation like F equal to k into l (t) minus l 0. If l (t) the length is less than l, 0 which means
this spring length is actually less than its free length therefore the spring is being compressed. So
this difference has a negative sign and as a result the force also has a opposite sign and therefore
it in the first case it was tension in this case it is the force of compression somebody is pushing,
squeezing the spring with this force and of course I should once again put F of t is the force
acting on the spring, the force of tension or the force of compression is given by k, the
coefficient k into l (t) minus l 0 and the spring has 2 parameters associated with it one is the free
length of the spring and the other is the coefficient k which is also sometimes called the spring
constant because it is associated with the spring, the spring constant or the coefficient of
elastance or elasticity of the spring or elastance of the spring.









(Refer Slide Time: 55:49)



(Refer Slide Time: 56:43)



So that is the spring of the mechanical system the third thing is the dashpot with the symbol that
we are seen already. Now one may not come across a dashpot very commonly as an explicit
component simply because what the dashpot does is to be found without really asking for it the
dashpot is based on the idea of resistance to motion but of a different kind from F equal to ma,
the force of inertia so to speak. F equal to ma says that in order to accelerate a body you need
force, you do not need force to keep it in constant velocity at a constant velocity, you need it if
you want to change the velocity either in magnitude or directional or both.


In the case of a dashpot, dashpot will oppose any motion but here is a practical dashpot which we
can think of really as a cylinder not this one necessarily but think of cylinder with a piston at the
one end of it and with air under the piston and assumed that the piston moves in an air tight
manner then obviously as the piston moves in the air is getting compressed. So it offers some
resistance to the movement of the piston in place of, of course the air one have oil and one has to
make some proper arrangement and so on. For our purposes of modeling in this particular
example the equation relating two variables associated with the dashpot will be the force or what
may called the plunger of the dashpot the other part of it is fixed.

(Refer Slide Time: 57:35)



So if the force acting on the plunger this is given by a coefficient multiplying the velocity of
motion of the plunger. So v of t into a symbol that will denote this resistance to motion of the
dashpot I will insert the statement giving the actual name of this particular constant is I do not
remember this standard name for it. So this is something which we will add in the video, so this
is the equation for the dashpot F at any moment of time t is equal to this coefficient into vt and as
before in place vt we may write D the derivative of the differentiation operation acting on the
displacement function Ds of t.

So these are the 3 components of the mechanical system the mass the spring and the dashpot
with their models which are all the equations it may be good for some purposes may not be some
good so good and there is one more namely a force, source or a displacement source. Some
external force may be acting on the mass of the body or one end of the of the mass itself of the
body may be moved physically by some other device and undergoes some displacement.

So like the ideal voltage source we may be told that somehow the position of the mass say, x of t
varies in this sinusoidal way with time that is the mass is made to move somehow in this fashion
move back and forth along the line or alternately on the mass may be exerted force somehow
may be by something attached to it or some remote force of attraction may be exerted on that
mass whatever way that is the force source that is there is an external force acting on the mass

which is guaranteed to behave in such and such manner may be a constant force or again an
oscillatory force may be sinusoidal or whatever.

So that will be the fourth component of the mass spring dashpot system and in fact one can have
a special situation and the corresponding situation in the rlc circuit also namely there is no force
source or displacement source except that at what one may call the beginning that is when you
start observing things or start looking at the system typically we call our t equal to 0 initial time
start your watch or clock and starts looking what is going on and this initial movement of time let
say the spring has been compressed or somebody is holding the spring with his hands
compressing it and then at t equal to zero as you see the watch strike 0 he lets the spring go
removes the force which is holding the spring in a compressed state then as we can expect the
spring will vibrate it was compressed.

(Refer Slide Time: 01:01:33)



So it will expand as its property of elasticity but unfortunately it will over do it or it may overdo
it. So it gets pressed on its own but that gives rise to a force of restoring force of tension so it
becomes compressed again and so it may go on vibrating like this for quite sometime therefore
after sometime the comes to stand still with its natural length. So in this case one would have put
the external force, source, force or displace the force source equal to 0.

Similarly, in the rlc circuit one may have the circuit initially and their will be one more element
may namely a switch and I keep telling my students that this is one very important part of
electrical system the switch, without the switch life would be difficult if you could not switch off
the fan we did did not want it or you could not switch it on when you wanted it life would be
difficult, the switch is very important device.





(Refer Slide Time: 01:03:51)



So in the rlc circuit suppose I had a switch initially and the switch was closed and as I see my
clock strike 0 or here its strike 0, 12 oclock or whatever t equal to 0, I call it, I open the switch
or rather things will not happen if I do it or something will happen but I would like to think about
what will happen but instead of this kind of a thing we have this alternate arrangement where
now what will happen is there will actually 2 switches there will be one switch and there will be
another switch and this one switch S 1 will be opened at t equal to 0 and almost simultaneously
with it this other switch S 2 will be closed, S 1 will be open so that the battery or the cell is
removed and S 2 is closed so that the circuit remains close, rlc circuit remains closed.

So that a current can flow in the circuit if you do this you do find that after you have opened S 1
and closed S 2 the activity does not stop immediately and if one connects an ammeter in the
circuit to see whether there is a any activity going on or if one connects this to an oscilloscope
one will find that indeed activity goes on in the circuit even after the source has been removed,
current continues to flow for quite some time and under certain conditions the current may be in
fact oscillatory that is it will go through positive and negative values and keep on oscillating for
a sometime before the current becomes almost 0. So the force source, emf source could be set to
0 but the current the circuit will have some initial current, the capacitor will have some initial
voltage and so on.











(Refer Slide Time: 01:04:13)



(Refer Slide Time: 01:06:39)



Whenever it as if that time t equal to 0 prior to that you had an excitation or a force or something
acting on it after t equal to 0, you have removed it but this earlier activity of the force put the
system in some initial condition or initial state where by the system can cool on for some time,
before it comes to rest once again before it becomes dead whether it is the electrical circuit there
I closed and open switches in a particular way but the system the circuit was earlier connected
and carrying current or in the mechanical case I hold the spring in this fashion with my hands
and then I let it go at some moment of time and watch it oscillate before finally it becomes still
once again. So a system may have a force source or an emf source or a displacement source or
the system may not have and still it may have a some activity associated with it.


Now going back to the analogy then I have setup or I have written down we have looked at the
equations for the electrical components, the equations for the mechanical components. Do they
look similar? May be yes, may be no for example the inductor equation v equal to L di dt looks
very much like the dashpot equation perhaps because F is equal to the coefficient into derivative
of displacement.

(Refer Slide Time: 01:07:14)



(Refer Slide Time: 01:07:25)



So if I take current to be analogous to displacement then the voltage across the inductor is
analogous to the force on the dashpot, this is one analogy but may be one can look at it

differently for example if I look at the mass equation then off course in F equal to ma there is no
derivative. So it looks like v equal to R i but if I look at F equal to m Dv dt then it becomes the
look more like an inductor because I have a derivative but if I look at the second model m D
square s (t) then it is looking like something else altogether it is second derivative now instead of
a first derivative, what about the electrical side? Do I have anything like that on the electrical
side.

(Refer Slide Time: 01:08:32)



(Refer Slide Time: 01:08:51)



So the answer is yes because of I if I look at v equal to L di dt once again and if I say that
ultimately the current is a result of motion of some charge or some charge at some place in the

circuit is changing or charge is flowing across some cross section in the circuit then R becomes
the rate of change of flow of charge across the circuit of the rate of flow of charge across the
cross section, it charge for charger use the symbol q then current itself is the derivative of charge
flow and therefore voltage of a inductor will be L times the second derivative of the charge flow
rate.


Now this begins to look like the mass equation F equal to m d square s dt square. So if I take
displacement to be analogous to charge then the mass becomes analogous to an inductor and the
force on the mass becomes analogous to the voltage across the inductor. So this analogy can be
studied in several different ways there are several analogies therefore that are possible. Now of
course we have to thing of putting these 4 devices electrical devices together or the mechanical
components together to get the electrical circuit or network or the mass spring dashpot with a
force or a displacement source.

On the other hand and continue with these equations of course but write down some more
equations where by one will see that the analogy will extend further and finally we may be able
to write down one equation for the electrical network and one equation for the mechanical
system. The two equations will look different, one will have voltages current charge may be rlc,
emf, the other will have m elastance friction or coefficient of viscus friction and what not and of
course displacement, acceleration, velocity force but except for these symbols the two equations
will look very similar to one another and that is what will make the two systems analogous to
one another and therefore one we will say that one is a model of the other and in this case there is
another word which has becomes very fashionable now and that is the word simulation,
simulation.


So one will say that the electrical network or circuit can be used to simulate the mechanical
system or the other way around the mechanical system can be used to simulate the electric circuit
or the electric network and this works simply because both of then under certain conditions are
described by equations which look very much alike except for a change of symbolism instead of
v, I will have F may be instead of q, I will have s the displacement time t fortunately is the same
in both. But apart from that the two equations are similar or almost identical of course parameter
values will be different instead of 1.3 into 10 raised to 3 I may have 10 their and so on and so
forth but the but the 2 equations are similar.


In fact they are of the same kind known as a second order linear differential equation with
constant coefficient and a forcing function. So in both cases we are led to this common model, a
second order linear differential equation with constant coefficient and a forcing function or an
input function.