Academic Year 2023/2024 - Teacher: Giovanni AIELLO

Expected Learning Outcomes

The course introduces the knowledge of the principles of electrical engineering and aims to provide students with the methods for studying electrical circuits and preparatory knowledge for subsequent courses in electronics, automatic and electrical communications.

After a brief mention of the electric and magnetic fields, useful for the introduction of the model with concentrated parameters, the student engineer learns to analyze simple circuits in the time and sinusoidal regime, methods of systematic analysis and fundamental theorems of analysis of networks.

Finally, the typical use of models and methods of electrical circuit analysis for signal and power applications is highlighted.

Knowledge and understanding.
The knowledge acquired during the course, in particular, the link between the electromagnetic field and the model with concentrated parameters, the solution methods and the theorems of the electrical networks allow the student to fully understand the functioning of the electrical networks, as well as the application and limits of validity of the circuit model.

Applying knowledge and understanding.
At the end of the course, the student acquires the ability to solve linear and time-invariant electric circuits both in stationary and sinusoidal regimes as well as in transient.

Making judgements.
The course also aims to improve critical skills and judgment. In fact, the student is asked to identify the most appropriate solution methods in relation to the complexity of the circuit to be analyzed. Moreover, each time he analyzes an electrical circuit, the student is asked to verify the correctness of the solution obtained both on the basis of the approximate knowledge of the expected solution and the comparison of solutions obtained with different methods (and with IT tools). Finally, he is invited to critically interpret any anomalies found in the solution of a circuit. In this way, he acquires certainty of the result found, an awareness of the functioning of the circuit and is able to judge autonomously the correctness of the obtained solution.

Communication skills.
The student learns the correct use of the circuit and mathematical symbols as well as the technical terms and units used in electrical engineering.

Learning skills.
The study of the subject improves the classification skills of the engineer student. In particular, by solving the electrical circuits with the various systematic methods and theorems studied in the theory of circuits, the student catalogs the circuits in different classes, on the basis of their topology and the bipoles that compose them, in order to identify the most efficient method for analyze the circuit itself. The improvement of the classification capacity and the exercise of the critical spirit contribute to strengthening the student's ability to continue the study autonomously after the course of study.

Course Structure

The knowledge to be acquired during the course is the content of the lectures conducted in the classroom by the teacher and - in order to facilitate personal study - the topics are listed in detail in the course syllabus, with explicit references to the parts in which they are covered in the main set texts.

Practical classes and personal training by solving exercises are the means to acquire the ability to apply knowledge. Examples, with the steps necessary to apply the knowledge acquired to the solution of the circuits, are carried out by the teacher in the classroom during the practical classes that follow the explanation of a new topic. Some of the exercises solved by the teacher are also solved by means of a free software for the numerical solution of the electric networks so as to provide students with an alternative way to independently verify the correctness of the results obtained. In order to guide the student, during the personal training phase, to master the tools to be used for the solution of the circuits, at the end of each practical class, a list of recommended exercises (available on the reference books for the exercises or online) is published. In addition, the student is invited to solve the same circuit with different methods, using all the knowledge acquired and all the tools (including IT ones) at their disposal, thus multiplying the value of the single exercise. Finally, the student is encouraged to deepen the topics covered using materials other than those proposed, especially for what concerns the personal study phase, thus developing the ability to apply the acquired knowledge to contexts different from those presented during the course.

To encourage students to study theory topics and to practice already during the course, as well as to facilitate the passing of the final exam in the sessions immediately following the conclusion of the course, there is an alternative route to the classic exam (typically consisting of a written test and an oral exam) consisting of:

  • a SUITABILITY TEST, to be carried out, approximately, halfway through the period of the lessons;
  • an "IN ITINERE" TEST, to be carried out at the end of the lecture period;
  • a SIMPLIFIED WRITTEN TEST to be held in the exam sessions immediately following the end of the course;
  • an ORAL TEST to be held a few days after the written test.

This route allows students to evaluate if they are up to date with the arguments explained by the teacher and has the advantage of splitting the written exam into two tests to be tackled at different times, ensuring the student has more time available for the solution of the proposed questions.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Required Prerequisites

Typical topics of mathematics and physics courses.
Please see the "Italian version" for more details,

Attendance of Lessons

Class attendance is not compulsory.

Detailed Course Content

1. Introductory knowledge.

Scientific notation and order of magnitude. International System of Units (SI); electrical
quantities and their SI units. Review of vector calculus.

Introduction to the discipline.

2. Circuit model.

Assumptions, deduction and validity limits of the Circuit model. Electrical elements and
electrical networks. Electrical voltages and currents of an element; systems of descriptive quantities.
Topological classification of elements: two-pole, n-poles, two-port networks and n-port networks;
physical classification of elements: characteristic variables, form and naming of characteristic

relationships. Nodes, sides and network variables. Kirchhoff's laws and equations. Electrical regimes.

3. General properties and theorems of electrical networks.

Main properties of an electrical network: zero-dimensionality, solvability, linearity, time
invariance, dynamicity. Analysis and solution of a network.
Power, work and electrical energy; energy classification of elements; passivity, Tellegen
General network theorems: Substitution Theorem, Superposition Theorem, Thevenin-Norton

Equivalent Network Theorem, Reciprocity Theorem.

4. Bipolar elements.

Adynamic bipoles. Characteristic variables, form and naming of characteristic relationships;
absorbed power, condition of passivity.
Linear time invariant resistor: equivalent resistance of resistors connected in series and/or
parallel, voltage divider, current divider, Millman networks. Thevenin-side-Norton-side
transformation and vice versa. Star-delta transformation and vice versa.
Non-linear resistors: independent voltage generators and independent current generators, typical
waveforms of generators; ideal diode and junction diode.
Dynamic bipoles. Characteristic variables, form and naming of characteristic relationships; time
invariant linear capacitors and inductors: equivalent circuits, continuity of energy variables, stored
energy, passivity condition; dynamic characteristic; series and/or parallel connections; equivalent

capacitance and equivalent inductance.

5. Two-port elements.

Adynamic two-port: characteristic variables, form and naming of characteristic relationships;
absorbed power, passivity condition. Ideal transformer, gyrator, time invariant linear dependent generators.
Dynamic two-port: characteristic variables, form and naming of characteristic relationships;
absorbed power, passivity condition. Linear time invariant coupled inductors. Inductance and
reluctance representation; coupling coefficients; stored energy, passivity condition. Series
connection and parallel connection of coupled inductors. Partial leakage coefficients, leakage

inductances, equivalent circuit.

6. Graphs and methods of network analysis.

Definition of graph, connected graph, planar graph, mesh (or cycle), cut set (or cocycle), ring, tree and cotree; tree
theorem; fundamental matrices of meshes and of cut sets, orthogonality relation; current space and
voltage space of a network.
Reformulation of Kirchhoff's laws in terms of the graph of a network; generalisation of
Tellegen's theorem.
Methods for analysing nodes, modified nodes, meshes, modified meshes, rings and rules for

systematically writing the relevant equations.

7. Time-invariant linear networks.

Transient regime and time domain analysis. Response with zero input: natural frequencies,
exponential stability and order of a network. Response with zero state: impulse response, response
to canonical input and sine wave response; convolution integral. Complete response: minimum
order differential equation. Transient response and permanent response. Sinusoidal regime and
constant regime; fundamental theorem of the isomorphic regime. Study, classification and  response calculation of first and second
order networks.
Inputs, outputs, input-output relations, state of an electrical network; method of state variables.
Application of Laplace's operator method to the study of electrical networks. Analysis in the

frequency domain; topological and characteristic equations; definition and classification of network functions; poles and zeros.

8. Networks in sinusoidal regime.

Periodic quantities and their parameters: mean value, rms value, peak factor, form factor. Ways of representing
sinusoidal quantities; definition, operations and properties of phasors. Topological and
characteristic equations in sinusoidal regime; network functions in sinusoidal regime and their
Energy behaviour of a network in sinusoidal regime: instantaneous, fluctuating, active, reactive,
apparent and complex power. Maximum active power transfer theorem, losses and efficiencies.
Boucherot's theorem; single-phase compensation of inductive loads.
RC, RL and RLC elementary networks: frequency response, resonance, quality factor. Vector
Three-phase systems: motivations for their use and most frequently used types. Three-phase
generators and loads. Pure, balanced, symmetrical voltages. Interphase connections, phase, line and
barycentric quantities; dissymmetrical systems, unbalanced systems. Equipotentiality of star centres
of balanced loads. Forteschue's theorem and its applications. Symmetrical and balanced networks; equivalent singlephase
networks. Energy behaviour and powers in three-phase systems. Voltmeters, ammeters,
wattmeters, varmeters. Aron theorem and insertion. Power in symmetrical and balanced systems.
9. External characterisation of two-port networks.
Representations of linear time invariant double bipoles: immittance matrices, hybrid matrices,
transmission matrices. Reciprocity and symmetry of double bipoles. Intrinsic, extrinsic, balanced
and unbalanced two-port networks. Interconnection of two-port networks in series and/or parallel,

in cascade.

10. Transmission lines.

Assumptions, deduction and limits of validity of the transmission line model; primary
parameters and telegrapher's equations of a uniform two-conductor line. Lines in sinusoidal regime;
characteristic impedance and propagation parameter; reflection and transmission coefficient. Ideal


11. Electromagnetism.

Static electric field and its properties. Capacitance coefficients, potential coefficients and energy of a conductor
system. Capacitor bipole. Examples of capacitance calculations.
Stationary current field and its properties. Ohm's and Joule's laws for field quantities. Electromotive field. Bipole
resistor. Examples for calculation of resistances.
Stationary magnetic field and its properties. Magnetic vector potential. Inductance coefficients, reluctance
coefficients and energy of a current system. Inductor bipole. Calculation examples of self and
mutual inductance.
Ferromagnetic media: first magnetisation curve, normal and differential permeability, hysteresis
loop, permanent magnets.
Magnetic circuits: magnetic voltage, law of magnetic voltages and magnetic fluxes, Hopkinson's
law, reluctance and permeance; equivalent electric circuit; calculation of self and mutual induction
Quasi-stationary electromagnetic field and its properties. Equation of magnetic field diffusion in conductors.
Induced currents, skin and proximity effect. Sinusoidal electromagnetic fields; penetration depth in
a conducting half-space.
Non-stationary electromagnetic field and its properties. Faraday-Neumann-Lenz law, dynamic and motional
induced electromotive force; moving coil in a uniform magnetic field. Ampere-Maxwell's law,
displacement current.
Computational electromagnetism; illustrative examples of the use of numerical methods for the
calculation of electromagnetic fields.

Textbook Information

Course Planning

 SubjectsText References
1Introductory knowledge. Introduction to the discipline. Material provided by the lecturer. 1) XIII-XVII; 3) 1.1-1.6.
2Circuit model. Electrical elements and electrical networks. Kirchhoff's laws and equations.1) XIII-XVII; 3) 1.1-1.6. Material provided by the lecturer. 
3Elements of graph theory and their application to electrical networks. Tellegen's theorem. 1) 3.1-3.4, 3.7; 2) *3) 4.1-4.5. *Material provided by the lecturer. 
4Systematic methods of network analysis.1) 3.5-3.6; 2) 10.1-10.5, 11.1-11.2, 11.4-11.5.
5Power, work and electrical energy; energy classification of elements; passivity.1) 1.4; 3) 3.8-3.9. *Teaching material provided by the lecturer.
6Adynamic bipoles. Resistor. Equivalence transformations of networks. Independent generators. 1) 1.6, 2.1, 4; 2) 2.1-2.2, 2.6, 3.1-3.3; *3) 6.1-6.4.
7 Adynamic two-port networks. Ideal transformer, dependent generators. 1) 6.1-6.4; 2) 8.2-8.3, 17.5-17.6; *3) 7.1-7.6.
8Representations of two-port networks. Reciprocity and symmetry. Interconnection of two-port networks.1) 6.1-6.4; 2) 8.2-8.3, 17.5-17.6; *3) 7.1-7.6.
9Use of systematic methods of analysis of electrical networks . 1) 3.5-3.6; 2) 10.1-10.5, 11.1-11.2, 11.4-11.5.
10Dynamic bipoles.  Capacitors and inductors. 1) 1.7; 2) 2.3-2.4, 2.6; *3) 9.1-9.9.
11Dynamic two-port networks. Coupled inductors. Leakage coefficients. Equivalent circuits.1) 6.4; 2) 8.1; *3) 10
12Ferromagnetic media. Magnetic circuits; calculation of self and mutual induction coefficients.3) 24.11, 24.13;*24.14. *Teaching material provided by the lecturer.
13Periodic quantities.  Sinusoidal steady-state, phasors, network functions.  Frequency response of RC, RL and RLC networks.1) 5.1-5.2, 5.4-5.6, 5.8; 2) 7.1-7.6; *3) 2, 14.1-14.4, 14.7-12.
14Energetics of a network in sinusoidal steady-state. Maximum Active Power Transfer Theorem, Boucherot's Theorem. Single-phase rephasing.1) 5.3, 5.9; 2) 7.7; *3) 14.5, 14.13.
15Three-phase systems and quantities. Equipotentiality of star centres. Symmetrical and balanced networks. Powers in three-phase systems. Aron's theorem. Forteschue's theorem.1) 5.10;  3) 15.1-15.5, 15.7, 15,9-15.10.
16Properties, analysis and solution of an electrical network. Analysis in the time domain.1) 7.3; 2) 6.1-6.4.
17Responses in the time domain: properties and calculation. 1) 7.3; 2) 6.1-6.4.
18Minimum order differential equation. Natural fequencies and stability of a network. Isomorphic regime theorem.1) 7.3; 2) 6.1-6.4; 14.1, 14.3-14.4.
19Detailed study of first-order networks.1) 2.3; 2) 4.1-4.3.
20Detailed study of second-order networks.1) 7.2; 2) 12.
21State of an electricity grid and method of analysing state variables. .1) 7.2; 2) 12.
22Laplace's operator method and electrical networks. Analysis in the complex pulsation domain. Network functions; poles and zeros.1) 7.4; 2) 13, 15.
23Substitution Theorem, Superposition Theorem, Thevenin-Norton Theorem, Reciprocity Theorem. 2) 16.1.1, 16.2.1, 16.3.1, 16.4.1. 1) 4.2.3, 4.3, 6.3.1.
24Static electric field.  Capacitor bipole. Examples of capacitance calculations.Teaching material provided by the lecturer.  *3) 23.1-32.12.
25Stationary current field. Bipole resistor. Examples of resistor calculations.Teaching material by the lecturer. *3) 21.1- 21.5; 21.8-21.9;21.11-21.12.
26Stationary magnetic field. Inductor bipole. Calculation examples of self and mutual inductance.Teaching material by the lecturer. *3) 24.1, 24.3-24.8.
27Quasi-stationary electromagnetic field. Induced currents and skin effect .Teaching material provided by the lecturer. *3) 24.15-25.22.
28Non-stationary electromagnetic field. Electromotive force, displacement current.Teaching material provided by the lecturer. *3) 25.1-25.4; 24.23.
29Computational electromagnetism.Teaching material provided by the lecturer.
30Transmission Line Model. Telegraphers' equations. Lines in sinusoidal steady-state.Teaching material provided by the lecturer. *3) 27.1-27.2; 27.7-27.9;27.12.
31Teaching material marked with * is intended as in-depth material on the subject.
32list of exercisesSuggested exercises divided by subject available on the Studium platform. Suggested exercise texts.
33 Analysis of elementary resistive networks; calculation of network functions. 
34Application of the method of node potentials to the analysis of resistive networks.
35Application of the mesh current method to the analysis of resistive networks.
36Determination of representations of resistive two-port networks.
37Magnetic circuits and calculation of self and mutual inductances.
38Analysis of elementary sinusoidal steady-state networks; calculation of network functions.
39Analysis of sinusoidal steady-state networks using systematic methods; calculation of network functions.
40Analysis of sinusoidal steady-state networks using systematic methods; calculation of network functions.
41Calculation of powers and application of theorems for sinusoidal steady-state networks.
42Calculation of powers and application of theorems for sinusoidal steady-state networks.
43Analysis of three-phase networks.
44Analysis of three-phase networks.
45Time domain analysis of transient networks by means of node potentials and mesh currents methods.
46Time domain analysis of transient networks using the state variable method.
47Complex pulsation domain analysis with systematic methods of transient regime networks.
48Complex pulsation domain analysis with systematic methods of transient regime networks.
49End-of-course exercise.
50End-of-course exercise.
51End-of-course exercise.
52End-of-course exercise.

Learning Assessment

Learning Assessment Procedures

Both written and oral exams.
Please see the "Italian version" for more details,

Examples of frequently asked questions and / or exercises

Available on Studium e on the course website.