# CIRCUIT THEORY

OVERVIEW

Module of Electric Circuit Analysis (Resistive-only DC, Transient, Steady State AC and Three-phase analyses).

## AIMS AND CONTENT

LEARNING OUTCOMES

The module is designed to provide to Students the knowledge and operational skills indispensable to properly describe, analyze and solve Electric Circuits, both in time and frequency domain.

AIMS AND LEARNING OUTCOMES

The aim of the module is to enable Students to master, both theorically and practically, the basic knowledge necessary to study and solve simple Circuital Models, with particular attention to power aspects.

The basic methods for Linear, Time Independent and Lumped Parameter Circuit Analysis are taught, and the techniques for their application to the problems is developed. The analysis is made within time-domain (Resistive-Only DC, Transient Analysis), and within frequency domain (Steady State AC).

At the end of the module, the Student shall have correctly understood the basic concepts of Electric Circuit Analysis, shall be able to correctly classify the different types of Circuit problems, and to correctly formulate the solution, arriving, when possible, to determine their analytical solution.

Teaching methods

Online theory lectures and related exercises (6 credits) in the first semester. The module will continue in the second semester, with further exercises (3 credits), aimed at deepening the knowledge about Circuits of particular industrial importance in the area of Power Electrical Engineering (Three-phase Circuits).

SYLLABUS/CONTENT

*The Circuit Model.*

Current and Voltage. Potential Difference. The Electric Circuit: Model inherent Hypotheses and Limits. Circuit components: Terminals and Connectors, Bipoles and Multipoles, Limit Surface. Lumped Parameter Circuits. Reference directions for Voltage and Current. Kirchhoff’s Voltage and Current Laws. Linearly independent relations among Kirchhoff’s Laws, and elementary selection techniques.

Component’s Equations expressed on voltage-current plane. Elementary bipoles: Resistor, Open Circuit, Short Circuit, Voltage and Current Ideal Generators. Representation of Components on voltage-current plane.

Instantaneous Electric Power. Power of a Bipole. Power Conventions for Generators and Loads. Power dissipated by a Resistor. Joule effect. Tellegen’s Theorem. Conservation of Power. Graphs, oriented Graphs, and their application to Circuit Analysis.

*Resistive-only Circuits.*

Definitions and inherent Hypotheses. Resistor: Linear, Time-Independent Resistor, its constitutive equation (Ohm’s Law). Definition of Resistance and Conductance. Concept of equivalent network, formulae of equivalent network of Resistors in series and in parallel, Voltage Divider and Current Shunt. Network reduction techniques. The star-delta transform. Theorems for resistive networks: Thevenin’s, Norton’s and determination of equivalent networks. Maximum power transfer theorem. Real generators, Millmann’s Theorem. Superposition Theorem and its application. Description of general techniques for Circuit Analysis (Nodal and Loop Analysis).

*Capacitors, inductors, coupled inductors.*

Ideal capacitor and inductor, elementary properties. Characteristic equations, stored energy, initial conditions, state variables. Formulae of equivalent network of Capacitors and Inductors in series and in parallel. Real components. Two port components: description in terms of impedance, admittance, hybrid parameter, transmission parameter matrices. Ideal transformer.

*Typical waveforms for Circuit Problems.*

Unitary step functions. Finite time impulse. Unitary slope function. Dirac impulse function, and distributions. Integral and differential relationships among elementary functions. Construction of stepwise continuous functions as combination of elementary functions. Sinusoidal waveforms. Periodic waveforms. Alternate, odd, even, and with half-waveform symmetry. Fourier series: harmonic functions, their definition and main properties.

*Dynamic Circuit Equations, and their solution.*

The solution of Linear Ordinary Differential Equations with constant coefficients. General and Particular Integral, initial conditions, characteristic polynomials. Solution of simple first-order R-C or L-R Circuits. Behaviour of inductors and capacitors during sudden variations of state variables. Circuit of second order with Inductors and Capacitors. Taxonomy of the roots of characteristic polynomial, and their relationship with output waveforms. Natural response and forced response. Example of solution of simple dynamic circuits. Introduction to circuit simulators.

*Equations of circuits in Steady State Alternate Current, and their solution.*

Representation of sinusoidal waveforms through the use of complex numbers: the phasors. Definition of impedance and admittance for all types of linear components. Extension of already defined network theorems to the networks in SSAC. Voltage drop across a line. Power factor correction. Example of solution of simple linear circuits of applicative significance.

Power in SSAC: instantaneous power, active power, reactive power, complex power. Tellegen’s Theorem for SSAC networks. Balance of active and reactive power.

Resonance and antiresonance conditions. Concept of filter. Elementary R-C and L-R filters.

Techniques for the practical solution of SSAC networks: power balance method, methods based on impedance computations.

Example of solution of simple linear circuits for high and low power applications.

*Three-phase circuits.*

Definitions and reasons leading to introduce three-phase circuits. Three-phase circuits with balanced (3 conductors) or unbalanced (4 conductors) load. Phase and line voltages. Symmetric/unsymmetrical systems. Balanced/unbalanced loads. Positive, negative, zero sequences. Phase and line currents. Power in three-phase circuits. Solution of simple circuits.

Harmonic content of voltage and current in three-phase circuit. Unbalanced circuits.

Examples of solution of three-phase circuits, both balanced and unbalanced.

RECOMMENDED READING/BIBLIOGRAPHY

Besides the books in Bibliography, available in Department Library, on AulaWeb are available copies of written examination problems, with their solutions.

**Reference (for OPTIONAL study deepening on specific subjects)**

- M. Repetto, S. Leva: “Elettrotecnica – Elementi di Teoria ed Esercizi”, 2^ edn, Città Studi Edizioni, Torino, 2018
- L. Verolino: “Elementi di Reti Elettriche”, 1^ edn, EdiSES, Napoli, 2019

More oriented towards circuit analysis for signal applications:

- C. K. Alexander, M.N.O. Sadiku: “Circuiti elettrici” III edn., McGraw Hill Italia, 2008.
- G. Rizzoni: “Elettrotecnica – Principi e applicazioni”, II edn., McGraw Hill Libri Italia, 2008.

For further theoretical study:

- C.A. Desoer, E. S. Kuh: “Fondamenti di teoria dei circuiti”, Franco Angeli, Milano, 1977.

## TEACHERS AND EXAM BOARD

**Ricevimento:** Students in need of further clarifications are received via Teams, on appointment (tel. 010-353-2044, e-mail: mario.nervi@unige.it)

**Ricevimento:** Students wishing further clarifications are received on appointment (e-mail: massimo.brignone@unige.it), both using remote meetings (via Microsotf Teams) as well as frontal meeting. Obviously the second option is strectly dependent on the "nCOVID-19" emergency.

Exam Board

MARIO NERVI (President)

MANSUETO ROSSI

GIORGIO MOLINARI

PAOLO MOLFINO

PAOLA GIRDINIO

MASSIMO BRIGNONE (President Substitute)

## LESSONS

Teaching methods

Online theory lectures and related exercises (6 credits) in the first semester. The module will continue in the second semester, with further exercises (3 credits), aimed at deepening the knowledge about Circuits of particular industrial importance in the area of Power Electrical Engineering (Three-phase Circuits).

LESSONS START

According to the official teaching calendar

## EXAMS

Exam description

Written (or through 4 partial written tests during the year), and oral.

Assessment methods

**Warning: depending on the health emergency "nCOVID-19", the examination structure reported below may be subject to significant variations.**

The examination of Electric Circuit is based on a set of four partial written tests (three during the first semester, one at the end of the second) that the Student must solve (all four, unless there are serious issues to be evaluated case-by-case), and one oral discussion, lasting about 30 minutes, after the end of the second semester. In case the Student was absent to one or more partial tests without valid reasons, or the result was not sufficient, the examination will be based on a written test comprising the complete program of the module. The marking will be organized as follows: max. 14 points for the written examination (both for the set of partial tests, and for the complete written examination), max. 17 points for the oral examination. To be admitted to the oral examination, the marking of written test (partial or complete) must be at least 8/14. The final marking of Electric Circuit is the sum of the marks of written and oral examinations.

The final marking of Elettrotecnica is the average of the marks (rounded to the higher integer figure) of its modules (Electric Circuit and Electric and Magnetic Fields). The partial examinations must be passed in the following order: at first Electric Circuit, and afterwards Electric and Magnetic Fields.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

11/06/2021 | 09:30 | GENOVA | Scritto | Prova scritta di Circuiti Elettrici (60336) Lo scritto di Circuiti -se sarà effettuato in presenza- si tiene in Aula B4. |

14/06/2021 | 09:30 | GENOVA | Orale | L'orale di Campi -se sarà effettuato in presenza- si tiene in aula 3^ piano, edificio Via Opera Pia 11, Genova ("ex-CNR") |

17/06/2021 | 09:30 | GENOVA | Orale | L'orale di Circuiti -se sarà effettuato in presenza- si tiene in aula 3^ piano, edificio Via Opera Pia 11, Genova ("ex-CNR") |

02/07/2021 | 09:30 | GENOVA | Scritto | Prova scritta di Circuiti Elettrici (60336) Lo scritto di Circuiti -se sarà effettuato in presenza- si tiene in Aula B4. |

05/07/2021 | 09:30 | GENOVA | Orale | L'orale di Campi -se sarà effettuato in presenza- si tiene in aula 3^ piano, edificio Via Opera Pia 11, Genova ("ex-CNR") |

08/07/2021 | 09:30 | GENOVA | Orale | L'orale di Circuiti -se sarà effettuato in presenza- si tiene in aula 3^ piano, edificio Via Opera Pia 11, Genova ("ex-CNR") |

19/07/2021 | 09:30 | GENOVA | Orale | |

22/07/2021 | 09:30 | GENOVA | Orale | L'orale di Circuiti -se sarà effettuato in presenza- si tiene in aula 3^ piano, edificio Via Opera Pia 11, Genova ("ex-CNR") |

10/09/2021 | 09:30 | GENOVA | Scritto | Prova scritta di Circuiti Elettrici (60336) Lo scritto di Circuiti -se sarà effettuato in presenza- si tiene in Aula B4. |

13/09/2021 | 09:30 | GENOVA | Orale | L'orale di Campi -se sarà effettuato in presenza- si tiene in aula 3^ piano, edificio Via Opera Pia 11, Genova ("ex-CNR") |

16/09/2021 | 09:30 | GENOVA | Orale |