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FUNDAMENTAL CONCEPTS OF ELECTRIC CIRCUITS

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FUNDAMENTAL CONCEPTS
OF
ELECTRIC CIRCUITS

Dr. Sudha Balagopalan &

EEE Department
Copyright
c 2014 Sudha Balagopalan

P UBLISHED BY VAST

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All rights reserved. No part of this publication may be reproduced, distributed, or transmitted
in any form or by any means, including photocopying, recording, or other electronic or me-
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For permission requests, write to the publisher, addressed “Attention: Permissions Coordina-
tor,” at the address below.

Dr. Sudha Balagopalan

Principal and Professor of Electrical Engineering
Vidya Academy of Science & Technology,
Thrissur, Kerala, 685 601
principal@vidyaacademy.ac.in

Layout and Cover Page set in LATEX

by Arun Xavier, EEE Dept., VAST

First Edition, June 2014

 A Booklet on

FUNDAMENTAL CONCEPTS OF

ELECTRIC CIRCUITS

Dr. Sudha Balagopalan and EEE Department

Principal and Professor of Electrical Engineering
Vidya Academy of Science & Technology,
Thrissur, Kerala, 685 601
principal@vidyaacademy.ac.in
Dedication

Dedicated to the Students

’Who ask for More’
Preface

This booklet is the result of my conversion to the doctrine of documentation.

It comes after two more conversions. A little hurt that students found my classes completely
away from the beaten book track, I had always consciously tried to convert my classes to ’Book
Based Lectures’. When a session for ’IEEE Early Career Faculty’ was offered to me by Mr.
Nitin Padmanabhan, my student, and conducted, the appreciation by teachers overwhelmed
me. One professor who sat through my class in Quilon, gave the same comment, "of being off
the track" but his words were big encouragement to bring my ideas ’into the Track’ by writing
them all out. A conversion of all the ’hurt’ to a positive outcome has resulted in the physical
presence of a booklet that contains those ideas.

It is a result of the learning I willed myself to do via ’LATEX’ documentation, not in a

piecemeal fashion, but at length, to follow in the footsteps of Professor Dr. KB Maheshwaran
Nambudiripad, and of course, to lead by example, the culture of ’Research and Documentation’
in Vidya Academy of Science & Technology.
 Contents

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Introduction 9
1.1.1 Poem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Answers to the Questions? 10

2 TENETS OF AN ELECTRIC CIRCUIT ? . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 The Three Concepts 13
2.2 Some Circuit Examples 14
2.2.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Simple Concepts in Duality 19

2.4 Dualism in Electric Circuits 21

3 DYNAMISM IN ELECTRIC CIRCUITS . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Understanding Inductance & Capacitance 23
3.2 Explaining Transients 25
3.2.1 Time Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Dynamism Initiated by the Inductor and Capacitor . . . . . . . . . . . . . . . . . . . . 26

3.3 Behaviour of Inductors and Capacitors in DC Circuits 27

4 TWO APPROACHES SUGGESTED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Source - Load Approach 31
4.1.1 Procedural aspects of the Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Some Case Studies 32

4.2.1 Speed Control of a DC machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.2 The Unit Commitment (UC) Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.3 The Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.4 The Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Signals and System Approach 34

4.3.1 Tools for analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.2 System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 CASE STUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1 Some interesting examples 37
5.1.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.1.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1.6 Example 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.1.7 Example 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1.8 Example 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1.9 Example 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 To Draw Some Conclusions 47

6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Introduction
Poem
Answers to the Questions?

1 — INTRODUCTION

1.1 Introduction
In this booklet is given, an account of some circuit ideas and concepts which are generally
not referred to in ordinary textbooks. Perhaps it conveys some interpretations and intuitions,
which may not auger well with students used to textbook oriented learning. The booklet is
the result of some classes the author has handled for faculty members who are in their early
career. Some examples, which may be called as case studies or expositors, are also covered.
They are included as they were declared "very interesting" by students, when some simple but
intuitive methods to solve a few problems in the GATE examinations was explained in the
GATE classes handled by the author.

The purpose behind its writing, is merely to urge the young readers, both teachers and
students, to think at length, to think in depth, to think in breadth and therebyto enjoy the
process of learning. Another motivation is to suggest to find a common thread that runs
through the different topics and connect between the differentsubjects and topics that are
taught or learnt. This will result in one getting exposed to the powerful impact of clarity in
the fundamentals. The direct benefit is the technical maturity one gains, and another is the
respect earned. No doubt, this is in addition to one’s increased capacity for comprehending
and coordinating ideas before they are presented to the class.

As a teacher, the author has always maintained that, a thorough grounding in basic ideas,
with no confusing or contradictory elements in the understanding of concepts by the teacher,
is the only process through which a teacher can empower himself or herself while preparing
to face a multi-dimensional composite, that a class is. And electrical engineering teaches
us that flow of energy or current or power is possible from a higher potential to a lower
potential, either in magnitude or with a phase difference. And that is the main aim of this
booklet: to attempt to increase the potential of the reader. Yet, this account is not meant to be
 1.2 Answers to the Questions? 10

comprehensive, nor does it purport benefits.

A question always arises as to what is fundamental. What are the fundamental ideas on
which we can build a topic? Where is the line of demarcation between fundamental knowledge
and advanced learning? How can you show the difference or do they really exist? Why is it
important to check with your fundamentals when in doubt or when too many ideas or methods
confuse or intersect? When can one outreach from the fundamentals to the topic of discussion?
Who? Who else, but you, the reader?

May I dedicate this segment to a teacher who in his ’chat’ with the first year students,
introduced me to the ’University’ where you and I can empower ourselves with fundamentals.
He gave me a poem, copied on a scrap of paper, which I treasure and carry with me. It is
quoted and given below.

1.1.1 Poem

SIX HONEST MEN. By Rudyard Kipling

"I KEEP six honest serving-men

(They taught me all I knew);
Their names are What and Why and When
And How and Where and Who.
I send them over land and sea,
I send them east and west;
But after they have worked for me,
I give them all a rest.". 

Thank you, Professor KBM Nambudiripad for being the biggest source of motivation in
my academic life.

1.2 Answers to the Questions?

In our context of ’Study of the Fundamentals’, let us attempt to use these teachers to obtain
some answers to the following questions.

(i) What are fundamentals? Any idea, which conforms perfectly with the fundamental laws of
’being’, of existence, can be expanded further, with confidence.
1.2 Answers to the Questions? 11

(ii) Where do we notice the line of deviation? I would like to explain, based on an idea that
was brought to me by a group of students a few years back. Any attempt to generate
power from where it is getting ’lost’ or from losses is of paramount interest in this era
of co-generation, tri-generation and further to harness energy from losses. Students
hit upon the idea of generating power from transmission lines which go through flux
changes or reversals in every cycle, when they carry current, rather power, to their
point of consumption. I had to explain that this is exactly what the transformer at the
destination point is doing and that tapping of power at any or every point, be it the
transformer or the transmission line, is accompanied by a conversion at the source point;
in short that the laws of conservation must be understood thoroughly and must not
be confused with Faraday’s law of electromagnetic induction, whenever generation is
thought of.

(iii) How do you explain what may be perceived as an anomaly? Again, my suggestion in
this specific case would be to advise ’never to use Lenz’s law for ’generation ideas’.
This is because Lenz’s law comes into play, when current starts to flow, i.e. when power
is drawn, as it is again a law of ’being’ or a different version of conservation laws, i.e.
simplify the concepts, and recast them differently

(iv) The question ’why’ ( why go to such lengths, instead of asserting that it is so, because I
tell you so!) is the reason for this booklet; the reason being that we, as teachers, being
entrusted with the minds of so many students must realize that it is a sin to lead students
astray with ’quack’ kind of treatment of technology or sciences.

(v) The remaining questions of ’who’ and ’when’ are yours to decide and find answers to.
More specifically, the ’who’ could be a teacher preparing for a class or a student, gifted
with curiosity, and having not lost it somewhere on the way, and ’when’ could be in the
class context or while you are doing a project, when small questions baffle and continue
to pester, seeking explanation.

As stated above, even an understanding on the basic tenets of a topic has several features.
It is perhaps difficult to explain them all; it is certainly not necessary to do so. In such a
context, perhaps the current scenario of the teaching- learning process orienting itself around
the text book is safer. In that case, this attempt will have significance if it can herd together
some concepts that can be used as a check-list when in doubt. More benefit may be derived,
if a couple of general approaches can be kept on one’s anvil to aid in the process of compre-
hension, interpretation, conviction and thereafter manipulation. Finally, some case studies as
corroboration to whatever is said, or for validation, can help the authors to take ’anticipatory
1.2 Answers to the Questions? 12

bail’. These are what are covered in the following chapters.

Yet there must be some criticism This is expected, else the connect to ground will be lost
for the authors. Though there is no doubt that knowledge or clarity of the fundamentals is very
useful and an almost indispensable tool for gaining confidence while facing students, the need
for higher learning cannot be discounted. "Of course, it is necessary to exercise moderation,
as indeed in everything in life !"

We shall now point out a few basic facts to get started with.
The Three Concepts
Some Circuit Examples
Example 1
Example 2
Example 3
Simple Concepts in Duality
Dualism in Electric Circuits

2 — TENETS OF AN ELECTRIC CIRCUIT ?

The most beautiful concept on which the entire theory of electrical engineering is built up, is
that of the circuit. It evolves directly from conservation laws and can serve us well, when in
doubt. It can be interpreted as a holistic Kirchhoff’s law, but in actuality it explains a source in
the best of terms. So, what is a circuit? I am fond of saying to my students that in a circuit,
a current emerges from a source with the ’promise to return’. If the circuit is complete, it
may take any route, again based on Kirchhoff’s laws, but whatever be the routes the current
takes, at the end, it has to return in its entirety to the source. ( My ’non-electrical’ mentor has
recommended a useful analogy in terms of a train, which does not leave the station, until the
next station is ready to receive to receive it; the train cannot stop midway. This explains three
concepts at the very least.

2.1 The Three Concepts

1. A source can be explained using the instantaneous direction of both current and voltage.
If both are in the same direction, the function of that device or element is of delivering
power. The load thus can be described as having opposing directions for the current
through it, as against the voltage across it, at any instant. This is one concept every
teacher must internalize and then pass on to all students because of the common mis-
understanding while detecting a source in a circuit. ( There are several instances of
students assigning ’zero voltage drop’ across current sources when Kirchoff’s mesh
law is applied because voltage is not given.) It needs to be realized that the source
acts as a load when it is charging or when current is flowing into the battery at the
positive terminal, where as it is functioning as a source when discharging. An interesting
example for a device which functions as a load, when utility supply charges it, and also
as a source when utility power is lost , is the chargeable battery.
 2.2 Some Circuit Examples 14

2. This unfurls another concept; that at all instants the status of the circuit can only be
explained, by simultaneous use of both voltage and current. Two ideas develop from this.
One is the concept of duality, to be dwelt on later. Another, that of power or in effect
energy consumed or converted and of energy stored in the different elements of the
circuit, based on this current- voltage duo. No wonder, that the open circuit concept with
its dual in the short circuit concept are in unison in terms of power dissipation being zero.

3. This explains the significance of a circuit, that impedance is the path seen by the current
as it progresses through the circuit on its way back to the source, if looked at it from the
pair of terminals associated with the source. What does that mean? It means that, the
path taken may be series in structure; the impedances may also be in parallel wherein
the current has options regarding the paths to take, as also in the ladder network. In the
Star (Wye or "T") or Delta (π) connected impedance networks the current continues
to diverge from the original path but in all these cases, the ’promise’ of re-unification,
makes the analysis of the circuit unique and interesting. So, if two network segments
are connected together by a single line representing one conductor only, there is no path
for return for the current. Hence two parts of the network are ’decoupled’ and so can be
cut apart as being independent of each other, with no interconnection whatsoever and
analysed accordingly. There is no one way ticket for the current to any network, as per
Kirchoff’s Nodal Law, commonly referred to as Kirchoff’s Current Law (KCL).

The concepts do not stop at that. Rather, there is an overflow and some caution is being
exercised to control the flow, with the reader in mind, to spread it over the booklet. It also
stands without saying that every concept learned and digested or internalized must not, at any
cost be contradicted by what comes next. The reader is hence advised to be equally cautious,
and be critical of any interpretations given in this booklet, and use the concepts given here to
clarify one’s own convictions. Learning also has a foundation, which should be very strong to
build atop it. Perhaps images are also needed to substantiate any idea offered here. Hence, the
attempt continues by using some circuits to show the application of some of these ideas and
also to demonstrate how they can be used for easy analysis.

2.2 Some Circuit Examples

Let us discuss a few examples to illustrative the ease with which circuits that look complicated
or complex, can be solved using these concepts. They are circuit problems, further used to
probe in some more directions in order to generalize and bring up more ideas. Some lessons
learnt from the examples are then used as stepping stones towards explaining more concepts.
The objective of these case studies is to evolve an approach for introducing a topic or to explain
a concept as the central theme of the topic.
 2.2 Some Circuit Examples 15

2.2.1 Example 1

Figure 2.1: Example 1

————————————–Space for Rough Work————————————–

Here is a case of two network segments connected together by two lines to the left of B
and D ( as identified by voltages at the B and D node marked as VB and VD respectively). It
may be noted that the current entering point B, from network segment left of BD, is equal
to 3A. The 3A has to return to the referred segment at node D, in full i.e. as 3A, come
what may, as per Kirchoff’s Nodal Law (poularly Kirchoff’s Current Law- KCL). Hence,
total 5A , again as per KCL, flows through the 1ohm resistor bringing the voltage drop
across it to −5V . The problem could have been confounded further by giving values to the
resistors R but the focus of the student has to be trained to be at the point where maximum
information is available. All the information to left of AC and to the right of BD is of
absolutely, no relevance, in this problem. 

If the first example sought to attract our attention to the central part of the network only,
next we will consider an example, where the peripheral part is gaining importance. Another
point of deviation is that the number of passive elements equals the number of active elements.
But are they all in effect active? Let us examine this. Let us also note that the upper part of the
circuit takes precedence over the lower half, according to the solution required.
 2.2 Some Circuit Examples 16

2.2.2 Example 2
As may be obvious, there are several methods of solving this. Let us enumerate some
observations that lead to an understanding of the simple underlying concepts.

Figure 2.2: Example 2

————————————–Space for Rough Work————————————–

1. The first point that should be noted is that, as four sources are seen, it is best to
apply superposition theorem to solve this problem. This is because:
• The circuit is definitely linear; there is no doubt about it.
• None of the sources are dependent so, they can all be represented by their
internal impedances, if any.

2. Now, looking closely at the voltage sources it becomes clear that:

Point or node D in the circuit is the same as node B.

The circuit can thus be folded axially, about the line AC. Now the 1Ω inductor is in
parallel with the 1Ω resistor just as the 1Ω capacitor is parallel to the other 1Ω
resistor.
2.2 Some Circuit Examples 17
Hurrah! Now, only a single 1V source remains for reckoning!

3. Applying this single voltage source and killing the current sources, it becomes evident
that no current flows from the voltage source because it gets no return path from it.
Now, this is an interesting point to ponder on. Why is the voltage source ineffective
in this case? Let us hope to get to the answer at a later stage.

4. Applying the current sources, it is obvious that the upper leg source alone is pushing
a current through the inductor and returns straight back to the current source through
the short provided by the voltage source, as per the analysis using superposition theo-
rem. Hence the inductor current can be calculated based on current division technique.

The whole computation then, boils down to a solution to this, and the answer is equal
to 1+1 j .


Before we proceed further let us note a few more interesting facets of this circuit. On
calculating current into point B as 1+1 j through the inductor, a similar reasoning gives us the
−j
current away from point B through the resistor as 1− j making the current through the voltage
source equal to zero. Hence what was portrayed as a short circuit for analysis, in reality
functions as an open circuit. How interesting! How do we explain that?
It is in such situations that the question arises whether the ’Wheatstone bridge ’ structure
of this network becomes intuitive or not. ( Note that impedance ratio of arm AD to arm DC
equals impedance ratio of arm AB to arm BC. ) This ’Wheatstone’ structure, very helpful and
easily noted by students is sometimes misleading. In this case it only says that an open circuit
exists between B and D which we had merged, while folding axially. It does not mean that no
current flows through the voltage sources serving to connect between them and also to point
O in between. Hence, if B and D had not merged on account of the equal voltage sources
connected to point O, it would have been a different matter altogether. In other words, the
’Wheatstone’ structure may not be useful in such cases.
Another interesting point to note in the above problem is that of the power delivered by
the sources. When the current sources were open circuited the circuit was dead, without doubt.
But with a short circuited voltage source we got a current through an inductor as given above.
This creates a voltage drop across it giving a voltage drop across the current source equal to
√1 ∠−450 .This is most surprising, as the consequent voltage direction opposes the current
2
direction and hence, the current source seemingly absorbs power. At least it indicates that
superposition cannot be made useful for the analysis of power absorption or delivered. Quite a
lot of food for thought however exists, in the context of the special topology of the circuit.
In the next circuit, the focus is on the power delivered or absorbed in a device.
 2.2 Some Circuit Examples 18

2.2.3 Example 3
The next circuit emphasize that total power delivered is equal to the total power absorbed.

Figure 2.3: Example 3

————————————–Space for Rough Work————————————–

Simplistically stated, as the 60V source is absorbing power, current direction is downward;
and this leads to the solution that current I of the current source has to be less than 12A.
But it would be more useful, if we give some thought to the 12A source and the 20V source,
both of which are delivering power of a total of 720W . Here is a case where current and
voltage directions are same for both these sources. Clearly, all this power must be expended
in the other two batteries, both of which function as loads since they are charging, as
indicated by the current and voltage directions. 

In a circuit, neat with power sources and only resistors, the principle of conservation is
nicely demonstrated. The Joule’s law takes care of the fact that all energy supplied by the
sources are dissipated in the resistors or else used for charging the depleted sources. Thus the
origin of the resistor from Ohm’s law as an element ordained to function according to Conser-
vation laws by opposing flow of current is a concept that can be given physical interpretation
in the relation R = ρ al . However, opposition to flow of current alone will not keep in check the
build up of magnetic energy. Here comes the concept of inductance which is a manifestation of
opposition to change of current with respect to time. Thus dynamism enters the circuit. Also,
the dual concept of opposition to change in voltage, and the consequent build up of electric
field energy, demands attention and explanation in the form of a capacitor. The energy in these
elements hence is only stored temporarily, as is admissible by conservation laws. Thus, with
 2.3 Simple Concepts in Duality 19

a 180 degree phase difference between the currents in an inductor and a capacitor with the
voltage as reference, the shuttle of energy between these two elements becomes a foregone
conclusion. This is so, once the 180 degree phase difference is interpreted as an inward flow
of current in one element while it is simultaneously outward in the other element, with respect
to the same voltage direction across both. Power absorption followed by delivery is the natural
outcome. Thus, in one cycle while the capacitor is discharging into the inductor for half a
cycle, it is at the receiving end when this inductor, returns this energy stored, back to the capac-
itor. Thus, a very special understanding develops between the two elements exhibiting dualism.

Next, we move to the concept of duality which has been showing itself up at several, nay,
all occasions.

2.3 Simple Concepts in Duality

In the above section it has been observed that electrical engineering quantities have duals
because of the concept of power. We had already said that if we speak of voltage, in the same
breath, the mention about current needs to follow. This can be interpreted again, as happening
because of the bounds placed by conservation laws.

Whenever charges are separated, a field of ’attraction or repulsion’ develops. This is

how a potential or voltage is developed and this concept can be equated to a stored potential
energy. The duality in respect of magnetic field developed around a current flowing in a
conductor, implying a kinetic energy is implicit. While the former conceptualizes a capacitor,
the latter describes an inductor. The conservation laws are inherent in the associated theorems
as compartmentalized and given in the table below.

Table 2.1: Duality in voltage and current

Electric Field Magnetic Field
Voltage Current
Potential Energy Kinetic Energy
Capacitor Inductor
Constant Charge Theorem Constant Flux Linkage Theorem
CV = Q LI = Nφ
Open Circuit to DC Short Circuit to DC

(Constant Flux Linkage Theorem: The constant flux linkage theorem states that flux
linkages in an inductor cannot change abruptly.
Proof: Let flux linkages change abruptly. Consequently, in keeping with Faraday’s laws,
infinite voltage has to be induced. As infinity is not practical, the assumption of abrupt
change is wrong. (This logic is called the logic of “reductio-ad-absurdum or the logic of
contradiction.”) If L is constant, this theorem is called the constant current theorem.It has
practical application in the voltage-second balance equations widely used in Power Electronics.
2.3 Simple Concepts in Duality 20

Constant Charge or Constant Voltage Theorem: These theorems state that the charge or voltage
in a capacitor cannot change abruptly. Both of these theorems can be proved using the same
logic described for the previous theorem.)
Another useful theorem which has disappeared from the syllabus of circuit theory for
students of electrical engineering is the Tellegen’s theorem. This theorem is a beautiful
exponent of duality and is given below.
onsider an arbitrary lumped network whose graph G has bbranches and nt nodes. In an
electrical network, the branches are two-terminal components and the nodes are points of
interconnection. Suppose that to each branch of the graph we assign arbitrarily a branch
potential difference Wk and a branch current Fk for k = 1, 2, . . . , b, and suppose that they are
measured with respect to arbitrarily picked associated reference directions. If the branch
potential differences W1 ,W2 , . . . ,Wb satisfy all the constraints imposed by KVL and if the
branch currents F1 , F2 , . . . , Fb satisfy all the constraints imposed by KCL, then ∑bk=1 Wk Fk = 0.
This is a very general theorem, and is applicable to all types of networks. It is fully in tune
with conservation laws and contains within it all types of duality- voltage-current, charge-flux
linkage, capacitance-inductance, source-load etc. - all such dual quantities miscible in the
concept of power.
The reaction of an inductor as a short circuit to dc can be viewed from the perspective
of the conservation laws as well. Since energy cannot be stored indefinitely in an inductor
nor in a capacitor, in the absence of recycling typical of an ac circuit, the power has to be
zero. Hence the voltage across a pure inductor carrying its short-circuited current becomes
zero while the current through an open-circuited capacitor becomes zero. The latter can be
visualized physically in a parallel plate condenser, as a consequence of the charges no more
being required to assemble on one plate, then the other, cyclically as in ac.
Though it is suggested that the constant charge theorem dictates that at steady state, the
capacitor is an open circuit, this idea in itself is a fountain-head of several dual features as
follows:
1. The transient condition of the capacitor in that case is that it acts as a short circuit,
i.e. it either charges or discharges.
2. When a capacitor is fully charged, i.e. at steady state, it becomes an open circuit.
3. The fully discharged status of a capacitor is that of a short circuit.
4. Reverse is the case with an inductor: an open circuit at t = 0+ and a short circuit at
steady state.


The idea of an open circuit when pitted against a short circuit itself needs to be seen as a
reconfirmation of this duality. Only if a current does flow through a portion of the network,
but still with only zero voltage developed across it, we proclaim that segment a short circuit.
In contrast, without the accompanying description of a finite voltage across a pair of terminals,
that nil current flows through it, does not qualify it to be declared an open circuit.
 2.4 Dualism in Electric Circuits 21

The transient and steady state behaviour of the two elements, the inductor and the capacitor,
is in line with their functional obligations, as can be seen below and as guessed in lieu of
Lenz’s law. The concepts gathered from above can be consolidated to compare the physical,
functional, relational and locational explanation for the path of current, which was explained
as circuit impedances earlier. The comparisons are made between the capacitor and inductor
but the equivalence in terms of the resistor is implied in all interpretations.

Table 2.2: Capacitor Vs Inductor

Capacitor Inductor
Geometric Interpretation :
ε0 εr A N 2 µ0 µr A
d l
Lenz’s law interpretation :
C dv
dt = i
di
L dt =v
Input -output interpretation:
C= q
v L = N dφ
di
Energy Storage Interpretation :
CV 2 LI 2
2 2

2.4 Dualism in Electric Circuits

We would like to pen down a few more concepts or ideas here which are propelled by the
dualism experienced in electric circuits.
1. Any impedance strictly in series with a current source is ordained to carry that
current. In one sense, it can be said that this impedance has no relevance, adds no
new meaning to the circuit and can often be considered as non-existent in solving the
circuit. Perhaps they serve the purpose of power balance or energy balance only.
2. In dual terms, the impedance in parallel to a voltage source is accused of a similar
sense of ’Purposelessness’.


A poser was placed in example 2. Let us try to explain the ’ineffectiveness or redundancy’
of the voltage sources that we witnessed then. If the circuits are split into two, it can be seen in
both the sub-circuits formed of a voltage source, a current source and an impedance (a parallel
combination of a resistor and a reactance), fully separated or disconnected from each other,
that the series circuit destines that only 1A can flow through the full circuit, including the
voltage source. Hence neither the impedance nor the voltage source can influence the current
2.4 Dualism in Electric Circuits 22

flowing through them at all. On the other hand, the voltage source maintains a voltage balance
across the impedance and current source, such that the total is 1V and without any phase angle
difference between the current and voltage. However, the voltage across the current source has
a phase angle with respect to the current through it. Interesting! In this case, one can see that
the ’ineffective’ voltage source supplies the energy, while the 1∠0A battery is charging and
hence functions as a load.

An inductor or a capacitor cannot continue to store energy when powered by a DC source

in order not to violate conservation laws. Hence in circuits supplied by DC sources, the
inductor becomes a short circuit with no power demand and its dual the capacitor functions
as an open circuit at steady state. But transient behaviour is the reverse of the steady state
positions. The inductor (the capacitor) progresses from open circuit to short circuit ( vice
versa), from switch on or off to steady state. Here again, a time dependent phenomenon has
occurred. How does this happen? In the case discussed immediately above, there is a phase
difference concept emerging. A phase difference is also explained in time frame. In other
words, the circuit is dynamic and not static, as ’wrongly’ expected and assumed, when DC
sources are used and when both current and voltage sources are in phase. Whenever the energy
storage devices enter the scene, i.e. the circuit, immediately dynamism comes into being.

There are obviously many little details which have to be understood and followed scrupu-
lously, when the circuit becomes dynamic. Students are told to identify dynamic systems, nay
circuits, by the presence of differential equations in the model. If that point is taken, now let us
proceed to see how differential equations explain the concept of conservation laws in dynamic
circuits.
Understanding Inductance &
Capacitance
Explaining Transients
Time Constant
Dynamism Initiated by the Inductor
and Capacitor
Behaviour of Inductors and Capacitors
in DC Circuits

3 — DYNAMISM IN ELECTRIC CIRCUITS

3.1 Understanding Inductance & Capacitance

Let us revisit the functional understanding of the inductor and the capacitor.

vL
di
=L (3.1)
dt

iC
dv
=C (3.2)
dt


These effect-cause kinds of relations lead us to the implications of such an interpretation of

the energy storage devices.

1. The value or impact of an inductance is equal to the amount of voltage set up across it
for every unit of change in current in unit time. (Dual statement for the capacitor)
2. The larger is the value of the capacitance, ie. its capacity to store charge, larger is its
opposition to change of voltage across it. This means that for every unit of rate of
change of voltage, it can charge or discharge, larger amounts of currents to oppose this
change of voltage across it. (Duality applies)
3. The Lenz’s law is seen in working mode here, since the voltage vL set up across the
inductor, opposes the applied voltage in an attempt to decrease the rate of change of
current through it. Similarly for the capacitor, iC is an indication of the capability of the
capacitor to oppose change in voltage across it.
4. The reaction of both these devices corroborates the conservation laws inherent in
constant charge/flux linkage theorems.
5. The differential equations connect the devices to conservation laws and also to the real
3.1 Understanding Inductance & Capacitance 24

world or the limited world where nothing can go to infinity. In other words differential
equations are indicative of the ’non-permanent nature’ of their coefficients.
6. Another outcome of a differential equation model for a circuit containing inductor and
capacitor is the understanding that the circuit will now have modes of oscillations, also
called Eigen Values.

At this point, another GATE question in Control Systems is replayed here to corroborate
some arguments given above. In this question describing a control system as given below, the
response is required as t → ∞.

d2x dx
2
+ 6 + 5x = 12(1 − e−2t ) (3.3)
dt dt


Though it is very easy to take Laplace Transform and apply Final Value Theorem, we do not
attempt that. With very little work, the answer will be obvious as equal to 12
5 . Instead we try
to give an explanation on response and meaning to a differential equation, which is important
to get a practical view of an electric circuit.

But before we generalize, let us seek the solution for the equation 3.3. There is a zero
input and a zero state response. First, we need to clarify the meaning of response. Often I
have seen students confusing between solution to an equation and response, which at this
point of time is extended to mean, state response. In other words, when a system is expressed
as a differential equation, the differential is with respect to time for a control system. Also,
the dependant variable is a state variable, being excited by both internal and external en-
ergy input. In this case we have the internal dynamics ( How? Due to which elements?)
expressed on the left hand side (LHS) and the external input on the right hand side. On exam-
ining the LHS, it becomes clear that the characteristic equation has two roots (eigen values),
and they are of the decaying type. The forced response has two components, of which, one
decays. So all said and done, only one term 125 u(t) does not decay and is the response as t → ∞.

We are all familiar with the second order differential equation representing an RLC cir-
cuit. First, we equate or visualize the given equation to fit the form of an RLC circuit, not
autonomous, but fed by a source with some DC and some exponentially decaying content.
Realizing that steady state condition of any circuit is that the response has settled, all terms
containing dx −2t can be set as equal to zero. Thus the steady state
dt or its higher orders and e
response or x(∞) = 12 5.

One word of caution is regarding the internal energy, which if is purely oscillatory or
underdamped, then obviously, a final value may not exist.
 3.2 Explaining Transients 25

3.2 Explaining Transients

It may be borne in mind, that several natural phenomena rise or decay exponentially. Thus the
transition from an open circuit to a short circuit or, vice versa, of an inductor or a capacitor is
also exponential. Thus, very few equations need to be remembered to explain the response
function of these circuits to DC. With only a fundamental understanding of the rate of this
progress of charging, analysis is more or less intuitive. As may be expected the progress is
generally quick at the initial phase and slows down as the final steady value for the response is
approached. This initial phase and its duration are characterized by the natural content of the
circuit and the characteristic time is called the time constant τ.

3.2.1 Time Constant

Though time constant refers to the circuit content, it may be noted that it refers to the
exponential build up of voltage across a capacitor or current through an inductor or the
exponential decay in dual terms. From what is already learnt, we understand this quantity of
time as

1. the reciprocal of the slope at the initial portion of the exponential build up or decay
response curve; and
2. it is the time required for 100% decay or growth if the initial slope of the response is
maintained all through.
3. It is equal to RL for an RL circuit and equal to RC in the case of an RC circuit.


Each energy storage element is associated with a specific growth or decay; time constants
are also accordingly applicable to specific elements. Hence, Thevenin’s equivalent impedance
from the point of view of the relevant inductor or capacitor is a useful quantity to compute and
analyse the time constant and the transient speed. An example may be the best medium to
validate this perspective.

Figure 3.1: Example 4

 3.2 Explaining Transients 26

————————————–Space for Rough Work————————————–

With the transient generating behaviour of the inductor, it can be surmised that the current
increases exponentially in the inductor circuit. Here the resistors are in series in the Thevenin’s
L
equivalent, as observation is from the perspective of the inductor. Hence τ = R+R S
. So, the
IS
initial rise of current is τ

3.2.2 Dynamism Initiated by the Inductor and Capacitor

The three important quantities that define the dynamic behaviour of both the Inductor and the
Capacitor, when a change comes over the network are

1. Initial condition
2. Time Constant
3. Steady state response


At first, let us consider, what makes a system dynamic, or for the limited purpose of ex-
plaining dc circuits, what causes transients. Though, no doubt, it is the inductor and capacitor,
either in singles or in duo, that give rise to this dynamism, there are some network conditions
that bring about the transition in the response of a circuit.

• Some causative factors

– Switching on or off
– Changes in source values
– Sudden change in network configuration
– Faults


Some of the interesting comparisons we can make based on the study of transient & Steady
state behaviour of such circuits include the accountability to the response generated.
The manner in which the responses are classified is quite intuitive, especially in terms of
internal energy and external stimuli. They are:
• ZSR- Zero State Response
• ZIR- Zero Input Response

 3.3 Behaviour of Inductors and Capacitors in DC Circuits 27

Here, what is brought to the fore is the importance of initial conditions. If the inductor or
capacitor is initially charged, the prospects are that this stored energy is capable of eliciting a
response. This is what is called as Zero Input Response. This response is transient in nature,
as the energy associated with the initial conditions die rapidly, expended in the resistance
of the circuit. What is lasting is the input associated response. The energy put in externally
motivates a Zero State Response from the network, and based on the network configuration.
Such aspects are discussed in the next section.

3.3 Behaviour of Inductors and Capacitors in DC Circuits

Though dual in nature, and propelled by two different sources and nature of energy, the
essential behaviour of both the inductor and the capacitor when connected to or disconnected
from a DC source remains the same. Some relevant aspects are coordinated and given below.

• Response of an inductor to DC

– Initial condition to steady state

∗ Functions as an open circuit (initial condition) at switch on or when con-

nected to a DC source
∗ Charges to the status of a short circuit as Isc (1 − e−t/τ) where Isc refers
to the final current flowing through the inductor, once the transients have
died.
∗ Functions as a current source Isc in parallel with sL at switch off or when
disconnected from dc. This is the initial condition of the inductor when a
change occurs in the circuit where the inductor was carrying Isc
∗ Discharges to nil current as Isc e−t/τ

Let us garner some minimum understanding of the time constant.

• Time constant
– Thevenin’s equivalent from the perspective of the inductor is taken to compute
the time constant.
L +L
– Time constant is given by τ = thevenin
Rthevenin

At this juncture we list some thought provoking points to ponder on by the reader, from the
basic ideas so far conveyed.
3.3 Behaviour of Inductors and Capacitors in DC Circuits 28

1. The direction of the current through the inductor does not change. Why?
2. But the direction of the voltage across the inductor changes. Why?

What happens in a capacitor is very similar and is listed as given below.

• Reaction of a Capacitor to DC
– Initial condition to Steady state
∗ Functions as a short circuit at switch on initially or when connected to any
DC source
∗ Charges to a open circuit as V0 (1 − e−t/τ) where V0 is the voltage to which
the capacitor can charge.
∗ As a voltage source in V0 , series with an impedance at switch off or discon-
nection
∗ Discharges to nil voltage as V0 e−t/τ
– Time constant and some fundamental questions
∗ Thevenin’s equivalent from the capacitor’s perspective
∗ Voltage direction across the capacitor does not change. Why?
∗ Current direction changes. Why?


So far, we have discussed the inductor and capacitor, separately, acting alone. Dynamism is
there, as revealed by the exponential curves for the responses. All the same, the most interest-
ing combination is when both these energy storage devices come together in a circuit. There are
very enchanting ’battles’ that occur within the circuit. For one thing, the inductor and capacitor
are tied together in a literally eternal ’give and take’ game. When the inductor charges, it is by
virtue of the discharge from the capacitor and vice versa, in each cycle. For another, there is
the resistor, all set to quelling this exuberance, a real damper! The oscillations are thus con-
trolled by the resistive components in the circuit. Based on the predominance of the oscillating
initiative or the damper mechanism, the oscillatory response is over, under or critically damped.
Such details are given below along with some insight into the system that Eigen values provide.

• Reaction of a Capacitor and Inductor to DC

– Second order system

∗ Damping Ratio ξ & Natural Frequency of Oscillation ωn

∗ Over, under & critically damped
3.3 Behaviour of Inductors and Capacitors in DC Circuits 29
∗ Oscillations depend on the eigenvalues λ
∗ Lyapunov’s 1st Theorem on Stability?

– System data through λ

∗ Lyapunov’s 1st theorem on Stability?
∗ Characteristic Equation
∗ Cayley-Hamilton Theorem
∗ State Transition Matrix. Controllability? Observability?


As is intended, we hope the readers have been initiated into the basic workings of circuits
which lead to the decisions of applicability. But applications are oriented around the figures of
merit of such circuits or systems as revealed by controllability, stability and observability. If,
considered as higher learning, we will restrict our discussion to transient performance only
here. As before, a circuit subjected to transients is analyzed below to get a feel for handling
such situations.

Figure 3.2: Example 5

————————————–Space for Rough Work————————————–

3.3 Behaviour of Inductors and Capacitors in DC Circuits 30

Let us try to solve this circuit without paper or pen, but in different phases.

1. Phase 1: The phase before the switch S is on is also a steady state. The inductor is a
short and so the current through the inductor is half of 1.5A = 0.75A
2. Phase 2: At the moment of switch on the current trapped in the inductor starts to
discharge through the available circuit with the applicable time constant
−3
3. Time constant τ = 15x10
10+ 10
= 10−3
2
4. Discharge of current through the 10Ω resistor i1 (t) = − 0.75 −1000t
2 xe
5. Charging of the inductor iL = 1.5 −1000t )
3 x(1 − e
6. The remaining current from the battery source of 1.5A flowing through the 10Ω
resistor is i2 (t) = 12 x(1.5 − 1.5 −1000t )
3 x(1 − e
7. Thus the current i(t) = i1 (t) + 12 (t) = 0.5 − 0.125e−1000t


Dynamism even in the context of ac sources is also relevant to introduce the concepts of
resonance and filters and associated topics. Without such an understanding, the space-cum
time dynamism as evinced in the travelling wave phenomenon lacks the soul any topic must
necessarily have. Perhaps, an approach to deal with any topic may be a good idea for helping
with the thought process a teacher must go through before presenting a topic. The next chapter
attempts two such approaches.
Source - Load Approach
Procedural aspects of the Approach
Some Case Studies
Speed Control of a DC machine
The Unit Commitment (UC) Problem
The Optimization Problem
The Control Problem
Signals and System Approach
Tools for analysis
System Models

4 — TWO APPROACHES SUGGESTED

Some kind of compartmentalization of ideas helps analysis, and later on a synthesis of the
results aid comprehension and presentation of topics. The only condition is that there are
no contradictions when presenting related topics and that the ideas are in harmony. The
classification of the topic being taught or understood into two, the source and the load, has
helped the authors to a large extent. The approach is described in brief here.

4.1 Source - Load Approach

Classification always helps comprehension and analysis. In the case of engineering systems
or studies, we realize that there is always two sides to a coin. On one side there is always a
giver, a supplier, a nourisher, a repertoire to choose from, i.e. as a source. Duality dictates
that for every giver there is a taker. This is the source-load concept with a skeletal structural
distinction as given below.
• Source - Load Approach
– Source
∗ Chemical; Electro-Mechanical
∗ Based on Potential Energy or Kinetic Energy
∗ Stimuli; Excitation; cause- effect or input- output equations
∗ Both current and voltage have same directions
– Load
∗ R-L-C; Not necessarily
∗ Reaction; Response; cause- effect or input- output
∗ The directions of voltage and current are opposite to each other
• A feel for the circuitry is the immediate result
• Such an approach is in line with the Laws of Conservation and obeys the Principles
of duality.
 4.2 Some Case Studies 32

Regarding classification being conducive to better comprehension, perhaps there is no

contention. But, the question may arise as to what good the approach is for analysis. For this,
a few more ideas are outlined as a procedural technique.

4.1.1 Procedural aspects of the Approach

The first step is to realize that a topological picture or the interconnection of the source(s)
and load(s) is necessary to start with. Then we come to the procedural aspects of using this
approach. The interconnection decides the application of governing equations, using funda-
mental laws like Kirchhoff’s laws or D’Alembert’s Principle. Theorems and transformation
techniques help in solving or analysis. The minimal details are given below.

• Laws, Principles, Theorems and Techniques

– Laws of Conservation, Ohm’s, Joule’s, Lenz’s, Faraday’s Law of EMI .....
Kirchhoff’s
– Principles of duality, D’Alembert, Dimensional Homogeneity, ...
– Thevenin’s, Norton’s, Superposition, Millman’s
• Applications
– Laws and Principles to model; theorems to solve?
– Mesh and Nodal Analysis- a conceptual bulk solution technique
– Similarly source Transformation- an aid to solve


Let us have a brief look at some case studies to suggest the possible demarcation. No
example from circuit theory is taken because they are self-evident.

4.2 Some Case Studies

A wide range of cases are brought into the purview of such a classification for generating some
ideas and later for discarding if not conforming with one’s own perspective.

4.2.1 Speed Control of a DC machine

A simple DC machine which accommodates both armature and flux control is examined for
classifying.

1. In a speed control problem, of say a simple DC machine, the load is the speed require-
ments that need to be fulfilled.
2. Another load on this problem or system is the constraints like open loop or closed loop
controls possible
 4.2 Some Case Studies 33

3. The source is the available methods for speed control: Armature or flux control as the
case may be

4.2.2 The Unit Commitment (UC) Problem

This is a general decision making problem in power systems. It epitomizes most power system
problems.
1. In the UC and comparable problems, the load is the system load curve requirements
that need to be fulfilled.
2. The source is the available combinations of the generators that can be scheduled
3. Another source is the available programming methods for UC (Priority ordering,
Dynamic programming, Neural networks etc.)

4.2.3 The Optimization Problem

Here, we have the most common problem of maximizing benefits while minimizing the cost
or efforts.

1. In any optimization problem, the load is the Objective function that need to be
fulfilled.
2. The source is the available knowledge on the nature of constraints, including on the
design variables
3. Another source is the available programming methods for optimization, based on the
nature of variables

4.2.4 The Control Problem

A very general and difficult problem is implied by a control problem. The perspective of the
reader is most important here.

1. In any control problem, the load is the objective or plant function to be fulfilled.
The state variables may project the load.
2. The source may be initial state of the system and the State Transition Matrix, in that
case.
3. Another source is of course the input vector, and the distribution matrix, as the case
may be

An analogy in this manner for control problems is difficult. Yet, specific problems like
pole placement, controller design etc. may draw benefit from such an approach. However, the
signals and system approach is more suitable for control problems.
 4.3 Signals and System Approach 34

4.3 Signals and System Approach

In this approach, the system is understood using a pair of signals, the input and the output
signals. The first outcome of this approach is that, the system is classified and explained via
comparing these two signals. This is useful for modelling the system, solving it and also for
selecting the applicable tools for analysis. The transfer function approach for analysing the
system throws up a wealth of information. Yet, it is applicable to linear systems only. Hence,
at several occasions linearization is resorted to. Perhaps the usefulness of Taylor’s theorem
for linearization is unfortunately not much appreciated by the students. The impulse response
model has its own use, especially for design, since it reveals the system characteristics. There-
after convolving with any input signal we can obtain the realizable output. To reiterate the
above, the rudimentary points are given below.

• Understand the System using a pair of signals

– Properties of the system

∗ Static/ Dynamic, Linear/ Non-linear, LTI/ LTV, Causal, CTS/ DTS, Stochas-
tic/ Deterministic
∗ Transfer Function Approach
∗ Laplace, Fourier, Z Transforms and other transformation tools

– Modelling the System

∗ Applicable only to linear systems
∗ Transfer Function Model
∗ Impulse Response Model
∗ How is convolution helpful


The idea of modeling is not to get a specific response to a specific excitation. On the other
hand, once the system is modeled it can serve many purposes. The transfer function model
is itself a good example. The characteristic equations and their analysis, eigen values and
their implications, synthesis possibilities, the impulse response and convolution applications
etc. are only some that are named. For modeling a system, while using the time dependent
equations as it happens when inductors, capacitors and their analogous elements are involved,
a host of other tools are also available. The advantage of such a transformation is that the
frequency response can be assessed.

4.3.1 Tools for analysis

Study the response to any input signal. Normally elementary signals are used as input. This
is because a linear system permits, superposition of outputs to all the elementary signals
to get the response to any complex signal constructed from the elementary signals. Hence
4.3 Signals and System Approach 35

modeling via transfer functions or impulse model are applicable to linear systems. And it is
achieved through transformation tools like Laplace Transform or Fourier Transforms which
are applicable to all types of signals, even if they are non-linear with respect to time. Let us
examine some general features of this aspect.

• Laplace Transform Vs Fourier Transform

– Laplace Transform
∗ Applicable to source and load
∗ Focus is on complex frequency s = σ + jω
∗ Model- Constant Coefficient Differential Equations. Why?
∗ TF model is derived for analysis
∗ Also useful for obtaining Zero State Response, Zero Input Response
– Fourier Transform
∗ All comments apply +
∗ FS ’great’ to analyze periodic functions
∗ Focus is fully on frequency
∗ Helps to explain harmonics
∗ Also helps to explain the frequency spectra
∗ Logarithmic scaling gives the benefit of compression of information
∗ Analysis of magnitude and phase difference can be separated
When sampled data system is involved, the non-linearities are implicit. The conversion
inherent in ZT z = esT linearizes the difference equations representing the system. With
superposition becoming applicable, the analysis is simplified to a large extent. The features of
such a conversion are then comparable to LT and FT. However, Discrete Fourier Transform
becomes applicable.
• Z Transform Vs Discrete Fourier Transform
– Z Transform
∗ Helps to linearize the system model
∗ Focus on complex frequency Ω = ωT
∗ CC Difference E Model
∗ Pulse Transfer Function model,
∗ Zero State Response, Zero Input Response
∗ Mapping between domains?
– Discrete Fourier Transform
∗ All comments on FT applies
∗ FS ’great’ to analyze periodic sequences
∗ Dirichlet’s conditions
∗ Periodic Spectra Why?

 4.3 Signals and System Approach 36

4.3.2 System Models

Algebraic equations are not applicable whenever time dependence comes for the response
of a system. Differential and difference equations represent continuous and discrete systems
respectively. It is the coefficients that are indicative of the entanglement with the dependent
and independent variables of the system. A LTI system (Linear Time Invariant) may be
am impossibility, but is the building block for analysis and prediction of behaviour. Hence
constant coefficient differential or difference equations are taken to model the LTI system for
the limited purpose of analysis about the operating point or for steady state analysis. However
initial conditions are included in the ZIR.

The next is the transfer function or pulse transfer function model. Since initial conditions
are set to zero, the transfer function model is indicative of the functionality of the system it
represents to a particular input. When compared to the impulse response model, it does not
have the design information but includes the frequencies of the input signal in terms of poles
and zeros. Perhaps it is more specific to the input signal, in that case, which is contrary to the
impulse response model which is based on an impulse input ad hence reveals the full nature of
the linearized system.

Once the system is modelled, the analysis of response to an autonomous system or a forced
system or the full transition from transient to steady state response can be made. Convolution
is the finest tool in this context.
Some interesting examples
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
Example 9
To Draw Some Conclusions

5 — CASE STUDIES

5.1 Some interesting examples

Some elementary circuit examples are examined here for applying the concepts discussed so
far discussed. Some circuits appear complex, in which case it can be surmised that the method
depends only on very simple interpretations, based on the ideas covered in this booklet. If the
circuit is simple in appearance, then it is reiterated that some underlying principles need to be
used to be not misled by the simple structure of the network. Let us examine them.

5.1.1 Example 1

Figure 5.1: Example1

Here if VR = 5V and VC = 4 sin 2t the question is to obtain the voltage VL , and is given by
A. 3 − 8 cos 2t B. 32Sin2t
C. 16 sin 2t D. 16 cos 2t
 5.1 Some interesting examples 38

————————————–Space for Rough Work————————————–

Solution: Please note these points carefully.

1. Please note that the legs containing the 2A source and the 5Ω resistor carry dc currents.
(Else, a sin 2t or cos 2t would have shown up.)
2. Both the 2A dc source and the current flowing through the 5Ω resistor cause no drop
di
across the inductor since it is a short to dc ( dt = 0)
3. With no current being supplied to the inductor to cause a voltage drop in it, (applying
superposition in reverse)in the network the above 2 legs are open.
4. The capacitor has a charging current as given by C dvdtC and is equal to 8Cos2t
di
5. This is the ac current supplied by the inductor and hence VL is given by −L dt . This
is equal to 16Cos2t
Very simple!


A talent to drop parts of networks, from one’s purview, if they are irrelevant in solving a
specific problem is something to be developed. This is what is expressed as a feel for a circuit
as is useful in the next problem.

5.1.2 Example 2

Figure 5.2: Example 2

————————————–Space for Rough Work————————————–

 5.1 Some interesting examples 39

Solution: Please note that Thevenin’s Impedance is a steady state impedance and the
sources are dc. Also do note that the inductor is a short (offers no impedance) and the
capacitor is an open circuit (offering infinite impedance). Thirdly do note that for Thevenin’s
impedance, the sources are killed and represented by their internal impedance, making the
voltage source a short circuit and the current source an open circuit. Hence, finally, when
looking at the path for current to flow from P to Q, there is only a 1Ω resistor, offering
impedance. 

If now, terminal P is placed to the right of and after the capacitor, it becomes tricky. This
is because the inductor and the capacitor are in series and start the shuttling of energy and the
Thevenin’s impedance cannot be estimated using intuition as employed before. The answer
then is D which ensures that the the frequency of this shuttling is incorporated via s!!! Having
come this far, let us move into full-fledged AC circuits.

5.1.3 Example 3

Figure 5.3: Example 3

————————————–Space for Rough Work————————————–

As complex as the circuit looks, so simple is going to be the solution.

 5.1 Some interesting examples 40

Solution:
1. Thevenin’s impedance, from the terminals to which RL is connected, absolves the
current I2 of any responsibility as it becomes 0.
2. This results in the welcome situation of shorting the dependent voltage source in the
left loop or part of the circuit. As it is, this loop is independent of the right loop.
3. Now the calculation of the current in the left loop is very simple since the impedance
is seen and not calculated to be equal to 5∠53.130 . Hence the current flowing through
j4 inductor, I1 is equal to 20∠0A.
4. Then VL1 is 80∠900 , the ∠900 contributed by the inductor, and so Thevenin’s voltage
is 10VL1 = 800∠900


So far, no problems with dependent sources have been done. Students are generally wary of
dependent sources as such. It could be because the nature of dependence is mostly abstract in
nature and no physical situations are visualized. The two port networks that explain transistors
giving rise to dependent sources, have actually made the understanding more complex, than
given an insight. The outcome is that students do not see the reduction of choices in analysis
pattern as an opportunity. The authors suggest to make the best result of the situation when
dependent sources are there. Only Kirchoff’s laws are relevant. One cannot kill sources since
dependent sources do not reveal their internal impedances. Hence both Thevenin’s voltage and
Norton’s current equivalents are required to compute Thevenin’s impedance. Superposition
theorem is also non-applicable.

5.1.4 Example 4

Figure 5.4: Example 4

————————————–Space for Rough Work————————————–

 5.1 Some interesting examples 41

Solution: Here we have a circuit with a closed loop on the RHS which has a dependent
source. Dependent sources are generally feared but are not only harmless, but are useful
creatures. The independence of both loops with respect to each other gives us the current
i as equal to Vab . This current creates a voltage drop of Vab across the 1Ω resistor in this
loop ( point ’b’ with respect to reference), taking the voltage at point ’a’ with respect to
reference as equal to 2Vab . However the potential divider in the LHS loop fixes the voltage
at ’a’ as equal to 2.5V , Solving, i = Vab = 1.25A. 

One important lesson learnt here is that the reference is of paramount importance when
voltages are spoken of. In this example, it may be observed that the line joining the bottom
parts of both the 1Ω resistors has only one purpose since it does not purport to carry any
current. It keeps the reference of both the loops as the same or at the same potential. Hence
we speak of voltage of point a and of point or node b.
Perhaps we have been conscientiously working on the source angle so far. But whatever
happens in a circuit , rather current flow, is ostentatiously decided by the topology of the
circuit and the impedance contents. That is the relevance of Graph Theory in circuit theory.
In the circuit given below, essentially an RC circuit, the transient will be decided by the time
constant as perceived from the terminals of the single capacitor, the network will work out to
be.

5.1.5 Example 5

Figure 5.5: Example 5

————————————–Space for Rough Work————————————–

This is a simple example to show that the capacitor assembly resolves itself to have an
effective capacitance of 23 farads and with an open circuit for the current source yields a 6Ω
resistor, as a series assembly from the capacitors’ perspective giving a τ = 4s.
 5.1 Some interesting examples 42

The next three problems are taken from GATE 14. The difference in these problems is
that they are not objective questions requiring exact solutions or answers. Hence approximate
methods are not sought. However an expectation of nice, full rounded answers are there!

5.1.6 Example 6

Figure 5.6: Example 6

————————————–Space for Rough Work————————————–

This circuit, with a ’Wheatstone structure’, is in truth, quite tricky. There is the earlier
made statement that impedance is the path seen by the current. We also have a tendency to add
XL and XC from earlier semesters. Perhaps our solution method should cover all such routes
the mind can take.
Solution:
1. Each leg with an inductor and a capacitor of equal ohmic values provide a shorted
path on either side of the voltmeter.
2. If this is so, the current drawn is in phase with voltage and equal to 200 sin ωt.
3. Now this current divides equally into 2 at the top junction giving equal voltage drops
across the inductor and capacitor of equal ohmic value. However, there is a phase
difference of 180o .
4. Then the voltage across the voltmeter is = 0

 5.1 Some interesting examples 43

Next we go for another ac circuit. In this problem too, the data given is in excess of what
is required. At the same time, all numerical values given are useful.

5.1.7 Example 7

Figure 5.7: Example 7

————————————–Space for Rough Work————————————–

As mentioned before, more the elements in a circuit, more the focus required to pick only
the required information.
Solution:
1. To compute XL , we need only the voltage across it and the current through it.
2. The 10A that leaves the voltage source also enters the series RL circuit, no doubt.
Hence current through XL is 10A. This is also true about R.
3. The voltage across the the inductor is the vectorial difference between the total
voltage as measured by the voltmeter and the drop across R.
4. Since current through R is 10A, voltage can be computed based on power loss in the
resistor.
5. Power expended in the resistor R is 1 kW minus 4W, dissipated in the 1Ω resistor.
6. Then the voltage across the resistor thus is 99.6V and across the inductor 173.4V .
The impedance or the reactance is thus XL = 17.34Ω.

 5.1 Some interesting examples 44

The next problem in the booklet is again a DC circuit problem with many shades. This implies
that the problem can be done in several ways. As usual, the simplest , yet intuitive method is
chosen.

5.1.8 Example 8

Figure 5.8: Example 8

————————————–Space for Rough Work————————————–

When the circuit has too many sources, it is useful, to zero in on the interior part of the
circuit from the outside. Here the requirement being, the power delivered by the innermost
source, let us try this methodology.
Solution:
1. Examining the outermost loop made of two voltage sources and a iΩ resistor, it
becomes clear that there is no voltage drop across the resistor. This implies no current
flow through it and hence an open circuit in it.
2. Now, the circuit is simplified because of dropping of a resistor. Let us apply
Thevenin’s method and superposition theorem now with the 2A source open cir-
cuited as a ’curtain caller’ problem.
3. Thevenin’s voltage is easily computed. The horizon voltage source gives a 1V drop
across the open circuited terminals with upper terminal positive. The horizontal
voltage source in this case parallels the two remaining 1Ω resistors in its short
circuited state. When roles are exchanged with the horizontal voltage source active,
it is seen that only 12 V appears across the open circuited terminals but that the upper
 5.1 Some interesting examples 45
terminal is negative. (This is a case of the 2 resistors functioning as a potential
divider)
4. Thus the Thevenin’s voltage is 12 V ( the upper terminal is positive).
5. Then the Thevenin’s impedance is a parallel combination of the two resistors and so
RT hevenin = 12 Ω.
6. Now please imagine the Thevenin’s circuit It is a simple series combination of 12 V a
1
2 Ω and a 2A source with upper terminal positive, The current source creates a drop
of 1V across the Thevenin’s impedance resulting in 1.5V across the current source
making the power delivered ( upper terminal positive) equal to 3W
7. A very satisfying problem!!!


The concluding problem in the booklet is conventionally done with an eye on the transfer
impedance as against driving point impedance. Here vector diagram approach is used in order
to reap the benefits of the diagrammatic approach. This approach is used because a pictorial
representation appeals to many students and is less vulnerable to making mistakes while doing
problems within limited time, especially in the on-line mode.

5.1.9 Example 9

Figure 5.9: Example 9

————————————–Space for Rough Work————————————–

5.1 Some interesting examples 46

In vector diagram approach, phasors are to be represented with their phase angle. Hence
1
the sinusoidal input is examined to get ω first. This works out to be RC making the impedance
1
of the capacitors C equal to ωC = R. This gives an interesting chain of impedances in the
given network. We have a series and then a parallel combination of an ohmic resistance and
a capacitive impedance of RΩ. Thus the input voltage is now applied across R − jR and R
parallel − jR. Hence a vector diagram better represents such a network The following steps
lead to the vector diagram given in Fig. 5.10.
Solution:
1. As the solution required is the transfer voltage, let us take that as reference and equal
to v. This is the voltage across both resistance R and capacitive reactance R in the
parallel combination.
2. Hence both the elements draw identical currents i as shown in the Fig. 5.10. This
current is in phase with the voltage v in the resistor, and leads the voltage by 90◦
while flowing through the capacitor..
√
3. Hence the total current flowing through the series impedance R − jR is equal to 2i
as shown in Fig. 5.10 with a leading angle of 45◦ .
4. This current through resistor R is in phase with the voltage across it, this voltage
√
is 2v, at an angle of lead 45◦ with respect to reference. This very same current ,
√
while flowing through the capacitor creates a voltage drop equal to 2v as shown
in Fig. 5.10 with a lagging angle of 90◦ , i.e.at an angle of lag of 45◦ with respect to
reference, as shown in Fig. 5.10.
5. Hence the total voltage across the series combination R − jR is 2v, which is in phase
with v across the parallel combination. Thus the total input voltage is equal to 3v.
v t
6. Then the voltage across the resistor -capacitor parallel combination is 3p cos RC
without any phase shift.


Figure 5.10: Vector diagram representation

 5.2 To Draw Some Conclusions 47

5.2 To Draw Some Conclusions

Only a few examples and some aspects have been given here. GATE examination gives a
repertoire of such excellent problems. Though DC operation has been given the focus, it may
be remembered that, application of a DC source is sudden and hence a harbinger of transients.
It may have been surmised from the first example that, the linearity concept was used to split
the response of the inductor first to DC and then to AC and the results were added up.

Another aspect to be borne in mind is that we must be conscious of the path of current
flow and understand that this alone signifies the impedance. It is only if our mind travels with
the current as it flows, we will accept as natural that when more paths are there as it happens
when parallel paths exist, then the impedance is lowered. The second problem talks about the
path of possible current flow in the Thevenin’s impedance concept.

The dependent sources are mysterious about their internal impedances. Hence they cannot
be replaced by their internal impedances when a Thevenin’s or Norton’s equivalent is sought.
Only way to obtain the Thevenin’s impedances in such cases is to get the Thevenin’s voltage
and Norton’s current and take this ratio. Superposition Theorem also becomes irrelevant while
dealing with dependent sources. The next two problems had dealt with different facets of
having dependent sources.

The fifth problem implies that the topology in which the energy storage device finds itself
is indicative of the time constant, applicable to circuits with only one energy storage devices.
Even if the 3Ω resistor was nearer to the parallel assembly of capacitors, the capacitors (else
inductors) would need to be assembled together first in such problems. Another point worth
remembering is to remove the sources while picking up both the capacitor(inductor) and
resistor assemblies separately.

The next three problems are more or less a revision of the concepts covered. The ’curtain
caller’ is a problem where the lure of the phasor diagram could not be controlled!
 6 — Closure

What is given above is no more than an introduction to fundamentals. It is hoped that it

is more user-friendly, since the language is more or less informal. The methods used are
non-conventional and hence seen to be different from methods used in GATE coaching classes
elsewhere.

Though, advanced courses in Machines, Power Systems and Electronics Controls, etc. are
very important, without the foundation laid in circuit theory, these super-structures stand the
risk of being wobbly and wavery and shall tumble or crumble when practical problems arise.
 Department of Electrical & Electronics Engineering
Vidya Academy of Science & Technology
Thalakkottukara, Thrissur - 680 501
(http://www.vidyaacademy.ac.in)

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