Second lecture on circuit theory and the topic
is review of signals and systems continued.
We did some in the last lecture and we ended
up in the definitions of passivity, a passive
network and then we also discussed what we
mean by a linear network. The next concept
that we talk about is bilateralness, bilateral
or unilateral. This refers, this term refers
to a 2 terminal element, a 2 terminal element,
a network element which has only 2 terminals
or 1 port and therefore, all you can do is
connect a voltage source, measure the current,
or connect a current source and measure the
voltage all right. Now a device, a 2 terminal
element is called bilateral if it can pass
current equally well in both directions. That
is, if you connect a voltage source here,
with this polarity, current passes like this
and let this current be capital I then, if
you interchange the polarities that is, if
you change the polarities to, let us say,
this is minus and this is plus then the current
will pass in the other direction and the same
current should pass, then it is called a bilateral
element, bilateral, the term bilateral refers
to a 2 terminal element. Sir, this is applicable
for both active and passive circuits?
both active and passive circuits. Yes. And
this is for when you connect a voltage source.
Or a current source. If you connect a current
source here then there is a polarity of voltage.
If the current source is reversed the polarity
of the voltage should reverse the magnitude
should remain the same. This is called a bilateral
element. On the other hand, if it does not
happen, if on changing the polarity of the
voltage, the current polarity changes but
the magnitude also changes then, it is called
a unilateral element and the most common example
of a unilateral element is a diode. A diode
as you know, conducts very well in the forward
direction, that is, if there is a voltage
source connected like this, it conducts as
if it is a short circuit, an ideal diode.
On the other hand, if this polarity of the
voltage source is reversed then it acts like
an open circuit, the current almost reduces
to 0. The diode is an example of a unilateral
element. If I now make a statement whether,
if I now ask you a question, whether a transistor
is unilateral or bilateral, your answer should
be the question is, question is Ill-posed,
because it is a 3 terminal element, all right?
But if I say, between emitter and base, is
it unilateral or bilateral? Yes, it means,
it makes sense it is unilateral. What about
between collector and base? Unilateral. Okay,
collector and emitter? unilateral.
Unilateral, and what is the reason? Because
the surface area of the emitter and the surface
area of the collector are quite different
and therefore, the amount of current that
flows when the battery is connected in one
polarity will be quite different from the
amount that happens if the battery polarity
is reversed. This is because of the change
of surface area. Therefore, you understand
what is unilateral and what is bilateral.
A network, a multi terminal network instead
of 2, 3, 4, 5, 6 or n number of terminals
having a number of cores, if it consists of
unilateral elements only, then such a network
is called a reciprocal network. If a network
consists of unilateral elements only, then
it is called a reciprocal network and the
reciprocity is formally defined like this.
We will come to the fine details of reciprocity
a little later but reciprocity is formally
defined like this; that if the cause and effect
are interchanged, a reciprocal network is
one in which, if the cause and effect are
interchanged, the relationship between cause
and effect does not change. Let me repeat,
a reciprocal network is one in which if the
cause and effect, we have given different
terms to cause and effect we have said cause
is an excitation and effect is a response.
So let us redefine it in terms of excitation
and response. A reciprocal network N is one
in which, if the excitation and the response
are interchanged, the relationship between
them does not change. For example, if you
connect a voltage source here and measure,
let us say, the short circuit current I. This
is port 1, this is port 2 and then, if you
take the same network and interchange the
excitation and the response, that is, connect
the voltage source here V and measure the
short circuit current here. Okay? Same voltage,
the network would be reciprocal if I prime
is equal to I. The relationship between them
does not change, that is, if it is the same
source then the response should be the same.
Now, if the source is doubled, let say the
source is not the same, if it is doubled,
then, the current should also be doubled.
In other words, the ratio of the response
to excitation remains the same and this is
what we mean by the relationship between the
excitation and the response. Yes? What is
the question? Sir, the network should consist
of unilateral elements or bilateral elements?
I am coming to that question later, but let
us make a definition first of reciprocity.
I repeat, a network is reciprocal if an interchange
of excitation and response does not change
the relationship between the excitation and
the response. If it does not happen, then
it is non reciprocal. The question that was
asked is; if the network N consist of unilateral
elements only then it is definitely reciprocal.
On the other hand, if it contains bilateral
elements, if it contains, I am sorry I made
a mistake. If the network contains only bilateral
elements, then it is definitely reciprocal.
If the network contains only bilateral elements
there is no question of non reciprocity, the
network shall be reciprocal. On the other
hand, if the network contains some unilateral
elements, there is a possibility that the
network may be non reciprocal. It is not necessarily
non reciprocal. For example, if you have 2
diodes connected back to back; identical diodes,
this contains 2 unilateral elements, but as
a whole, this is a bilateral network because
it passes current equally well in both directions.
Sir but as a whole if we see it, it is a unilateral,
bilateral. That is what I am saying. So, then,
suppose it is a network in a black box, then
we cannot say that whether it contains any
unilateral part or not. Correct. If it is
a black box like this, the only test that
you have to do is to connect an excitation,
measure the response, then interchange the
2. But this is the way of testing also. Say
so for multi terminal if, suppose, we will
be saying that, for it being a bilateral,
there if 1 terminal must be bilateral with
respect to all other terminals? For a multi
terminal network the question is, for a multi
terminal network how do you establish reciprocity?
You shall have to test all the ports. It is
not sufficient to test 1 excitation and 1
response. You will have to go round and test
for all of them. The network at, is at a couple
of ports may be reciprocal. It may not be
so at another couple of ports. Therefore,
the question has to be answered very carefully.
Let me repeat, if a network consists of bilateral
elements only, it is definitely reciprocal.
If a network contains in addition unilateral
elements also, it may be reciprocal, it may
not be reciprocal. It is only by carrying
out the tests or by looking at the internal
construction. Suppose we have 2 identical
diodes back to, back to back, like this, not
back to back, connected in parallel in opposite
direction then, definitely, it is a bilateral
network, bilateral network, okay? Bilateral
element and if a network contains such a pair,
you can say yes, it is a reciprocal network
so one has to be careful about the definition
of reciprocity.
Excuse me sir, in a multi port network say,
if it is true for, if it is reciprocal for
only 1 pair of ports but not for the others,
then can we call it as reciprocal? We have
to make it conditional. You have to say N
is reciprocal with respect to ports i and
j otherwise, we cannot say. We have to be
very guarded. We cannot say anything otherwise,
all right? We have to we have to qualify this.
Next term that we apply, this is the kind
of network that we shall study causal. Causal
is a word which has been defined in signals
and systems and the other name for causal,
the common name for causal is non anticipatory.
That is, it is a network which cannot anticipate,
in mathematical terms if e, e of t, of the
excitation, this excitation could be a single
excitation or an excitation vector; if it
is a multiport. Let us say an N port M of
which are excited, then small e can stand
for a vector, e 1 t, e 2 t up to e n t. So
if the excitation is equal to 0 for t less
than capital T, if the excitation is 0 for
up to an instant of term capital T, then a
causal network is one in which the response,
which could also be a scalar or a vector response,
is also 0 t less than T. This is called a
causal network or an non anticipatory network.
In other words, the output cannot precede
the input. If the input is 0, the output has
also to be 0, output cannot precede the input
or the network cannot anticipate what shall
be applied to its input in future.
Sir, cant it even be a constant, the response?
The response can be a constant but if the
excitation is 0 then, it is not a causal network.
Okay?
Sir, would not this be in the case only of
linear networks?
Only in the case of? No. Non linear networks
can anticipate, can also be subjected to such
definition; non linear networks can also be
causal. Yes sir, but then do we have the restriction
that if e t is 0 then r t also has to be 0?
It has to be 0 even if it is non linear; this
is the strictest, strict definition.
Then the other qualification of our network
shall be time invariant. Time invariant network
simply means that if e of t, by application
to the network leads to a response r of t,
then the delaying the input by an amount tau
should delay the output by the same amount
r of t minus tau. Okay? That is, that is,
in popular words, it means that the shape
of the wave form, shape of the signal clotted
versus time, remains the same. Only thing
that changes is the location of the response.
Whether you see if e of t leads to r of t
and e of t is a square pulse, r of t is, let
us say triangular, then r of t minus tau is
the response. If the square pulse is delayed,
then it is the same triangle which shall be
delayed. It shall be moved to the right by
an amount tau, so the wave form remains the
same; it is only the location that changes
and one of the results of time invariance
is that if e t leads to r of t then e prime
t, that is, the differential coefficient of
e t should lead to r prime t. This can be
very easily proved, very easily proved and
I would encourage you to undertake this proof
if you can. So the kind of networks that we
are going to discuss in this course will be
linear, L for linear, this L we have not yet
discussed, F we have not discussed, P we have
discussed; P stands for passive and B stands
for bilateral, bilateral.
Now, it is in historical context that what
B is there, it should actually be replaced
by R, that is, we are going to talk of linear,
passive and reciprocal network. This B stands
for bilateral; it should actually be replaced
by R because bilateral or unilateral refers
to a 2 terminal element only. Now let us look
at, let us look at these two. This L stands
for lumped, lumped, this L stands for lumped,
a lumped network is one in which the electrical
effects can be thought to be concentrated
at a certain point in space. A lumped network
is one in which the space variable x, y, z
are of no importance. The effect that is occurring
is occurring at a certain point in space,
space variable is not important. On the other
hand, if the space variable is important then,
it is called a distributed network, a distributed
network. For example, the telephone line or
the power transmission line, the power that
comes from a grid to a substation. Well, it
is a distributed network, the definitions
of lumped and distributed strictly refers
to the dependence or independence of, dependence
on or independence from space variables. The
other definition is, the other physical explanation
is, that a network is lumped if the wave length
of excitation 
is large compared to the dimensions of the
network. For example, 50 hertz, if the frequency
is 50 hertz; what is the wave length? 3 times
10 to the 8 divided by 50. So many meters
and any circuit any power circuit, power supply
or an electronic amplifier, whatever you make,
is going to be very small compared to this,
this dimension, and therefore the circuit
can be thought of as lumped.
On the other hand, for a power line going
from here to Karnataka, the distance may be
comparable, will be definitely comparable
to the wave length and therefore the space
variable is also important. So, lumped and
distributed means, a lump network is one in
which the dimensions, physical dimensions
of the network are negligible compared to
the wave length of excitation. That is, the
definition of lumped and then this F stands
for finite networks. Being finite beings,
we cannot think of infinite but there are
infinite networks which, for which a different
kind of theory is to be applied. For example,
the ionosphere, the ionosphere over the earth,
it is a network of charges. There are charges
over the earth; ionosphere is responsible
for reflection of radio waves and for long
distance communication. Ionosphere for all
practical purposes is an infinite network
for us. We will not be concerned with infinite
networks. We will be concerned only with finite
network. So this is what we are going to discuss
in this course L L F T B
Now let us look at the elements, the circuit
elements that we shall be considering, network
elements or circuit elements. First we consider
sources. As you already know, we can have
voltage source or we can have a current source,
that there are only two kinds of sources that
we shall consider in this course and the description
of voltage or current source always comes
with an adjective, that is, ideal or non ideal
and it is important to understand what is
an ideal source.
First let us consider an ideal voltage source.
An ideal voltage source is one represented
like this, with a polarity. An ideal voltage
source is one which maintains its terminal
voltage, terminal voltage V independent of
what is connected as the load, independent
of what is connected as the load. In other
words, whatever be the current demand from
the source, the terminal voltage remains the
same. This is an ideal voltage source. An
ideal voltage source, in theory, can supply
any amount of current whatever the load, except
for open circuit. Under open circuit of course,
no current can flow. Except for open circuit,
an ideal voltage source maintains its terminal
voltage constant irrespective of what is connected
to it.
On the other hand, if this voltage varies
with load, if the terminal voltage varies
with what current you are drawing from the
source, then it is called a non ideal voltage
source and a non ideal voltage source can
always be taken care of by a series impedance,
that is, an ideal voltage source in series
with an impedance can account for a non ideal
voltage source. In the case of a battery,
it is an internal resistance of the battery.
In the case of a general voltage source, it
can consist of inductance, capacitance and
resistance and therefore, we talk of an internal
impedance instead of an internal resistance.
Similarly, an ideal current source is one
which maintains the current delivered to a
load constant irrespective of what the load
is, that is, irrespective of what the potential
drop or the voltage across its terminals is,
it maintains a constant current. This is called
an ideal current generator.
On the other hand, if this voltage changes
with the load, that can be accounted for by
a shunt admittance, by a shunt admittance
which in the case of a resistive current source,
will be a resistance and it can in general
be an impedance. The blue colour takes care
of non idealness, non ideal current source.
A non ideal current source is an ideal current
source, in parallel with an impedance. In
the case of non ideal voltage source, it was
in series and the red colour stands for ideal
situation. You must be able to distinguish
between ideal and non ideal sources. In reality
there are no ideal sources. All sources have
certain internal impedance or internal admittance.
But for the purpose of theory many a times
we shall talk only of ideal sources.
Then we talk of the three elements: the resistance,
capacitance and inductance. Resistance as
you know, a linear element, a linear bilateral
resistance obeys Ohm's law, that is, V equal
to I R or instead of resistance, it can be
described in terms of a conductance, its reciprocal,
and you can write I equal to G times V. Where
capital G stands for one by R and it is called
the conductance. A resistance has no memory,
it cannot memorize, it cannot hold charge,
it cannot hold energy, it is in fact a dissipative
element. In other words, if current passes
through it, the resistance dissipates or absorbs
energy. It is a dissipative element. On the
other hand, if you take an inductance L and
our notation would be, if this is V then,
I beg your pardon, the notation is, if this
is V with this polarity, then the current
I is taken in this direction and the relationship
between voltage and current, as you know,
is V equals to L d i by d t, V equals to L
d i by d t. Forget about the right hand, left
hand and all those rules. They will determine
in the case of an actual inductor, the sign
of this voltage.
If you change the direction of sign of this
voltage can be changed, but this is a schematic
description. I am not saying current flows
clockwise or anticlockwise. No. What i am
saying is, the terminal currents is, it enters
in this at this terminal goes out at this
terminal and this is the polarity of the voltage
and V is equal to L d I by d t. This is Lenz's
law and this is what determines the terminal
relationship for an inductor and we naturally
see. Yes?
Sir, V is the induced voltage or the external
applied voltage?
V is the total voltage across the element
and it has to be equal to the external applied
voltage. Whatever happens inside the inductor
is a different story. We are only taking the
terminal description. V equals to L d I by
d t and therefore if I want to find the current
I at time t, then obviously this will be 1
over L integral i tau, i beg your pardon,
V tau d tau. I have changed the integrand,
i am sorry, the variable of integration to
tau. The question is, from what limit to what
limit? Well, since I am finding out a t therefore,
the upper limit must be t then lower limit
can be, it has to be minus infinity but usually
you see in electrical engineering or in any
other engineering. We cannot go to minus infinity,
we limit our range of vision to t equal to
0 and therefore, this is to be 0 to t and
if it is 0 to t, then the previous current
at t equal to 0 must also be taken account
of. In other words, what we have to do is
0 plus i of 0 plus i of 0 because we cannot
integrate from minus infinity to t, minus
infinity to 0. We are accounting for an i
of 0 which is called the initial condition
and i integrate from 0 to t.
Now, in this in this, 0 has to be qualified,
0 has to be qualified. Is it 0 minus or 0
plus? 0 minus is before the excitation is
applied, and 0 plus is after the excitation
is applied. There is a difference between
the two concepts. Although physically they
mean the same point, 0 is a single point but
0 minus is a point which is infinitesimally
close to 0, but not quite equal to it. Therefore,
what we have to do is 0 minus to t and then
this i has to be 0 minus. This is how we shall
indicate i of t is 1 by L 0 minus to t V tau
d tau plus i of 0 minus. Let me write it down.
Again, i of t equals to 1 over L 0 minus to
t V of tau d tau plus i of 0 minus.
Sir, even zero minus is evident because we
should not expect anything before zero we
could just say zero plus and let the zero
minus come in
There is a problem, the problem is if V of
t, if V of t contains the impulse and things
are quite different, if V of t contains an
impulse or a derivative of an impulse, well,
if V of t does not contain an impulse, that
is, that is, V of 0 minus is the same as V
of 0 plus. Then i of 0 plus would be equal
to 0 minus to 0 plus integral of V tau d t
that would be 0, if V of t does not contain
an impulse and therefore, this would be i
of 0 minus if V of t does not contain a delta
t or a derivative of it. If that is the case,
then i of 0 plus is equal to i of 0 minus
which is another way of saying that the current
in an inductor cannot change suddenly.
The current in an inductor at 0 minus and
0 plus should be the same, unless it is forced,
unless it is forced by an avalanche of voltage.
You see, integral V tau d tau if V tau is
delta tau, then the integral is 1, from 0
minus to 0 plus, then of course it has to
change. So, unless that happens, i of 0 plus
is equal to i of 0 minus and as you know,
the inductor stores energy in the magnetic
field and the energy is half L i square and
therefore, this is the another way of saying
the conservation of magnetic energy, magnetic
energy is an inductor or magnetic flux in
an inductor cannot change suddenly, unless
forced by an avalanche an infinite amplitude
pulse, which is an impulse. If you look at
this relation once more and suppose we want
to be general, that is, we want to admit the
existence, admit the possible existence of
an impulse or its derivative, then you have
to use this total relation and if you look
at this relation carefully, you see that i
of t is the sum of two currents. One is i
of 0 minus and other is this integral.
So one can represent an inductor which carries
an initial current, let us say i of 0 minus.
One can represent at t by means of an equivalent
circuit, that is, I have i of t, I have V
of t plus, minus. Equivalent circuit is it
contains two terms, one is a current source
i of 0 minus. So I indicate this as i of 0
minus and then an inductor which is initially
relaxed, initially relaxed, that is, it carries
no current. So a current carrying inductor
can be represented by an inductor which has
no initial flux in parallel with a current
source. This is the equivalent circuit of
an inductor.
Sir does that imply that inductor is a non
linear device? No. It does not imply that
inductor is a non linear device. Inductor
is a linear device provided it is initially
relaxed. If it is not initially relaxed, this
constant may cause the inductor to be non
linear. Yes, in the strict sense of the term
but before I pass, before I pass to capacitor,
can I also mention to you that an equivalent
description of this relationship, in the frequency
domain, shall be capital I of S equal to,
as you know, integral, Laplace of an integral
is division by S and therefore, it would be
V of s divided by s L plus, what is the Laplace
of a constant that divided by s, i of 0 minus
divided by s.
So this would be the frequency domain description
of an inductor, where if this current, if
the initial current is 0, this is the term,
this is the relationship that you are familiar,
that is, current is equal to voltage divided
by impedance of the inductor. But if the inductor
carries an initial current, then this term
is a must. We cannot ignore this and you see,
in the equivalent circuit now, in the frequency
domain, you shall have indeed a current source
but the source is not i of 0 minus. It is
i of 0 minus divided by s. So if this circuit
is to be modified to Laplace transform domain,
then this is V of s, this is I of s, this
would be i of 0 minus divided by s and this
shall be written as s L.
So in one stroke, we have done the time domain
equivalent and also the frequency domain equivalent
without any approximation, without any loss
of generality. i can represent an inductor
carrying an initial current, either the time
domain or in the frequency domain by means
of a parallel circuit and the parallel circuit
consists of an ideal current generator. This
current generated is ideal and an inductor
which is initially relaxed.
In a similar manner, I can consider a capacitor
C, which has an initial voltage. Let us say
V c 0 minus and then I apply either a voltage
source or a current source to charge or discharge
the capacitor. Let the currents and voltages
be V of t and i of t, then as you know, i
of t is simply equal to C d v by d t. This
is independent of initial conditions. However,
if you want to find out V of t, then of course
you shall have to integrate, that is, one
by C integral i of tau d tau and the limits
have now to be put carefully. This will be,
upper limit would be t
Should not you denote it by V c of t because
V of t you are using for
If the, if the voltage is the same, is the
same across the capacitor, I can use, if you
are so, for say, I will distinguish between
them and write V of t equal to V c of t. Okay?
So the lower limit should be 0 minus and then
we must add V sub c of 0 minus, which as you
can see, is the initial charge in the inductor
q 0 minus divided by C. Now, if V sub c of
0 minus is 0, if this is 0 and i of t does
not contain any delta function, you understand
the conditions, then what would be V of 0
plus? Would be 0 which, amounts to the effect
that a capacitor, an initially relaxed capacitor
acts as a short circuit, voltage across it
is 0 means it is a short circuit. Also, if
i of t does not contain a delta function,
then you see V c 0 plus shall be equal to
V c 0 minus, which amounts to the law of conservation
of charge, that is, the charge in a capacitor
cannot change instantaneously, unless forced
by an avalanche of charging, that is, if i
of t, the current contains a delta or its
derivatives, then this relationship shall
not be valid. But in general if the capacitor
is initially, if the capacitor is initially
charged and charged or discharged by a current
which contains no impulses, then V c 0 plus
is equal to V c 0 minus and in this case,
the integral relationship can be represented
by an equivalent circuit in which, there is
a source. You see, V t is the sum of V c 0
minus and the voltage across the capacitor
initially relaxed.
The equivalent circuit of a capacitor is therefore,
the series connection of an ideal voltage
source and an initially relaxed capacitor,
I hope this is clear, by the same arguments,
and if I want to change this to, let us say,
the Laplace domain, then we have V of s. This
will be V c 0 minus divided by s and this
capacitor shall behave as an impedance of
1 by S C where S is the complex variable.
This is the equivalent circuit of a capacitor.
Any questions so far? A circuit containing
initial conditions is an interesting circuit,
is an interesting proposal but it does create
complications unless you are very careful,
unless you are very careful in solving. For
example, in a typical situation, in a typical
situation, you shall be given a network, some
network and let us say, a voltage source V
with this polarity is switched on at t equal
to 0 which means that it is being excited
by means of a step function.
The excitation, the switch can be replaced
by u of t and therefore the excitation in
effect is V times u of t and you might be
asked to find the response at some port, at
some port and the response to be either a
voltage or a current. Then what you will have
to do is, you will have to write the internal
dynamics of the network N and the dynamics
is represented by a differential equation
or an integral differential equation, that
is, an equation which contains integrals as
well as differential coefficients. But in
integral differential equation can only be
reduced to a differential equation. Which
is, go on taking the derivatives, differentiating
till there are no integrals. So we simply
say, write down the differential equation.
Now, once you write down the differential
equation, you can solve it by finding the
complementary function and the particular
integral and as you know the complementary
function shall consist of undetermined constants
and these constants have to be determined
from initial conditions.
Now, in the context of network, since we are
admitting an impulse function, we have to
distinguish between conditions at 0 minus
and 0 plus. For the network N, the solution
have to be found out for t greater than equal
to 0 plus, that is, after the application
in the excitations and therefore, the initial
conditions are to be taken as the conditions
at 0 plus, not minus, 0 plus. How does one
determine the initial conditions? If you,
obviously for the network to be solvable,
you will be given the conditions at 0 minus,
that is, the condition of the network before
the excitation is applied. From that either
from the differential equation or from physical
reasoning, you will have to find out what
are the conditions at 0 plus. This is one
of the problems of network analysis. The first
problem is, is to find out the conditions
at 0 plus. These are the initial conditions,
what you shall be given is the conditions
at 0 minus. Now, if there are no delta functions
in the excitation, then of course your initial
conditions are very easy. The inductor currents
at 0 minus shall be the same as inductor currents
at 0 plus.
The capacitor voltages at 0 minus, shall be
same as capacitor voltages at 0 plus. So there
is no problem. But if there are impulses,
then you will have to be careful. You will
have to find them out either from the differential
equation and we shall show, we shall demonstrate
how to do this or from physical reasoning
physical arguments. So after you find out
the initial condition, then you put down in
the solution, determine the undetermined constants
and that, that does the job. This is the time
domain solution. Even if you want to determine
things in the frequency domain, the initial
conditions are important because you see,
in the equivalent circuit; you do have current
sources or voltage sources in terms of the
initial condition. So that is absolutely important.
Now in the rest of the few minutes that are
left, we will discuss two other terms which
shall be important in the later part of the
course.
I have told you that the synthesis problem
network, synthesis problem is the reverse
of analysis problem. In the analysis problem,
the excitation e of t and the network N are
given. You have to find out the response r
of t. In the synthesis problem, the excitation
and the response or a relationship between
the two and typically the relationship that
is given, is the ratio of the Laplace transform
of the response to the Laplace transform of
the of the excitation. This is what is given
and this is usually denoted by H of s, usually
denoted by H of s. H of s, in general terminology,
is called a transfer function, in control,
in many, in signals and systems and many other.
But in the case of networks, we have to make
a qualification; we have to make a qualification.
We cannot blindly call it a transfer function.
For example, if the network is a 2 terminal
one, if the network is a 2 terminal network,
well, then all you can do is; if the network
is a 2 terminal one, all you can do is you
apply a voltage and determine the current.
So the function would be, the Laplace of current
divided by Laplace of voltage or its reciprocal
and it will have the dimension of impedance,
it will have the dimension of impedance or
admittance and as you, as you perhaps know,
impedance and admittance are collected together
in a single term, immittance. This part impedance
or admittance is called, together is called
an immittance and therefore, if the network
is a 2 terminal network and there is only
one port, then all you can do is find out
the immittance function F of s which is either
an impedance or an admittance.
Suppose F of s is given. Well, this for reasons
to be made clear in a few, in a few minutes.
F of s, the immittance function, is called
the driving point immittance DPI. Driving
point immittance because it is the impedance
or admittance measured at the port, at which
you drive the network, at which you excite
the network. So it is called a DPI, driving
point immittance and if driving point immittance
is given, how to find the network? As you
know, the first task is the given, F of s.
Is it realizable? Can I find the network?
I have told you on the first occasion that
it may be or may not be realizable. So the
first question that one asks in synthesis
is the given function. Is it realizable? If
the answer is yes, then you go ahead with
the synthesis.
Now in the case of DPI, we shall show at the
in the later part of the course that it would
be realizable if F of s is a special kind
of function which goes in the mathematical
literature by the name positive real functions,
that is, if capital F of s is positive real,
it is only then F of s is realizable. Now
I used the word driving point, which means
that there must be other types of network
functions and this is the so called, so called
transfer. As I said in network functions,
have to be careful whether the, the term transfer
must be used to denote difference between
excitation and response ports, that is, excitation
at one port and response at some other port.
Only then the word transfer shall be used.
Now, suppose the currents and voltages at
the excitation are V of s, I of s and the
currents and voltages here are, let us say,
V 0 of S and I 0 of S. This is port 1 and
this is port 2. Now, the network function
can be the ratio of a voltage to current or
its reciprocal or it can be the ratio of,
let us say, V 0 s to V of s. It can be ratio
of voltages, it can be ratio of currents,
and therefore, a transfer function can be
either dimensionless or dimensioned. If it
is a ratio of voltages then obviously it is
dimensionless, if it is a ratio of currents
it is dimensionless. But if it is a ratio
of, let us say, V 0 by I, then obviously the
dimension is impedance and then it is called
transfer impedance. This is why we call the
previous one driving point, to distinguish
between transfer impedance and driving point
impedance. Similarly, if it is, let us say,
I 0 divide by V, then it is a transfer admittance
and in this case, we define what is known
as transfer immittance. It can be either an
impedance or admittance, to distinguish it
from driving point immittance. These terms
are not important in general systems but in
circuits, they are very important it, as usual,
see the driving point synthesis problem is
much easier than transfer function synthesis
problem and this is where we close.
