Aim of this course is to present the fundamental methods used for analyzing the linear circuits, in their steady state and in time domain, using the Steinmetz and Laplace transform. Then, Bode diagrams for representing network functions are introduced. D1 - Knowledge and comprehension ability The student, at the end of the course, will know the basis principles of electric linear circuits working. D2 - Ability of applying knowledge and comprehension The student will be able to analyze completely a power supply circuit, as well as to verify on the whole the dimensioning of an electric installation. For what regards the information linear circuits, the student will be able to analyze the main characteristics and properties, as well as the fequency behaviour. D3 - Autonomy of assessment The studente will be able to assess, deciding among more cases, how to model correctly an electric circuit. D4 - Communication ability The student will be able to describe the way an electric circuit works very correctly.

Notions of mathematics analysis and physics of electricity.

TOPOLOGY AND CIRCUITS
Lumped circuits. Electric quantities and reference directions. Theory of graphs associated to two-terminal, n-terminal elements and circuits. Reduced incidence matrix and Kirchhoff’s laws. Tellegen’s theorem. Concept of cut-sets and I law of Kirchhoff.
CONSTITUTIVE RELATIONS AND TABLEAU
Classification of elements and circuits by models. Tableau analysis for resistive, linear and time-invariant circuits. Superposition theorem.
TWO-TERMINAL ELEMENTS AND TRANSFORMS
Implicit and explicit representations of two-terminal resistors, power analysis. Real voltage and current sources. Sinusoids and phasors. Linear combination of phasors, first derivative and integral. Laplace transform overview. Capacitors and inductors: constitutive equations in time and transforms. Circuit equations in sinusoidal steady-state and Laplace transform. Concept of impedance and admittance, polar and Cartesian representations. Resistive-capacitive, resistive-inductive and resonant two-terminal elements. Series and parallel connections of two-terminal elements, voltage and current dividers. Thevenin and Norton theorems. Millman theorems. Power and energy. Power classification of elements. Istantaneous power in AC steady state, real power. RMS values. Complex, real, reactive and apparent power, power triangle and power factor. Complex power balance and Boucherot theorem. Rephasing capacitor of an inductive load.
LINEAR TWO-PORTS
Implicit and explicit representations of two-ports. Properties of reciprocity, simmetry, and unidiretionality. Wye-delta (Tee-Pi) and delta-wye (Pi-Tee) conversions. Ideal and real controlled sources, ideal transformer and linear ideal operational amplifier. Interconnected two-ports. Connections of operational amplifiers. Miller theorems.
GENERAL CIRCUIT ANALYSIS
Modified Node analysis. Mesh analysis.
THREE-PHASE CIRCUITS
Balanced three-phase voltage sources in wye and delta connections. Phase and line voltages (abc and acb), common point O. Three-phase wires and line currents. Balanced and unbalanced three-phase loads in wye and delta connections. Three-phase power systems having a neutral wire. Complex, real and reactive power in a balanced load. Notes of rephasing a Three-phase load. Expression of an asymmetrical set of three phasors as a linear combination of three symmetrical sets of phasors. Complex, real and reactive power in an unbalanced three-phase power system. Comparison between a mono-phase and a three-phase power system. Equivalence of star-connected loads. Aron power measuring system.
FIRST-ORDER TRANSIENT ANALYSIS
Transient analysis of first-order RC e RL circuits, with application of Thevenin and Norton theorems. Homogeneous equations, particular and general solutions. List of fundamental particular solutions. Superposition principle of particular solutions. Stability of a linear circuit. Series RLC Resonant Circuit. Notes of the n-order circuits in time domain. Output variables and associated resistive circuit. Ideal switch and continuity property of state variables. Degenerate circuits, parallel and series connection of capacitors and inductors. Solution with the Laplace traform of transient responses. Definition and analytic espression of network functions.
MAGNETIC CIRCUITS
Mutual inductance in time and phasors. Series real resonant circuits: resonant frequency, selectivity, network function, amplitude and phase plot. Reference to Ampere and Faraday laws. Magnetic hysteresis. Magnetic reluctance, Hopkinson’s law e magnetic circuits in air and iron. Magnetic circuit formed by a torus in air and an iron torus with an air gap. Kirchhoff magnetic laws and correspondence to electric circuits.

Teacher notes;
"Circuiti elettrici", Renzo Perfetti, Zanichelli;
"Elettrotecnica", Daniele et alii, Monduzzi.

TOPOLOGY AND CIRCUITS
Lumped circuits. Electric quantities and reference directions. Theory of graphs associated to two-terminal, n-terminal elements and circuits. Reduced incidence matrix and Kirchhoff’s laws. Tellegen’s theorem. Concept of cut-sets and I law of Kirchhoff.
CONSTITUTIVE RELATIONS AND TABLEAU
Classification of elements and circuits by models. Tableau analysis for resistive, linear and time-invariant circuits. Superposition theorem.
TWO-TERMINAL ELEMENTS AND TRANSFORMS
Implicit and explicit representations of two-terminal resistors, power analysis. Real voltage and current sources. Sinusoids and phasors. Linear combination of phasors, first derivative and integral. Laplace transform overview. Capacitors and inductors: constitutive equations in time and transforms. Circuit equations in sinusoidal steady-state and Laplace transform. Concept of impedance and admittance, polar and Cartesian representations. Resistive-capacitive, resistive-inductive and resonant two-terminal elements. Series and parallel connections of two-terminal elements, voltage and current dividers. Thevenin and Norton theorems. Millman theorems. Power and energy. Power classification of elements. Istantaneous power in AC steady state, real power. RMS values. Complex, real, reactive and apparent power, power triangle and power factor. Complex power balance and Boucherot theorem. Rephasing capacitor of an inductive load.
LINEAR TWO-PORTS
Implicit and explicit representations of two-ports. Properties of reciprocity, simmetry, and unidiretionality. Wye-delta (Tee-Pi) and delta-wye (Pi-Tee) conversions. Ideal and real controlled sources, ideal transformer and linear ideal operational amplifier. Interconnected two-ports. Connections of operational amplifiers. Miller theorems.
GENERAL CIRCUIT ANALYSIS
Modified Node analysis. Mesh analysis.
THREE-PHASE CIRCUITS
Balanced three-phase voltage sources in wye and delta connections. Phase and line voltages (abc and acb), common point O. Three-phase wires and line currents. Balanced and unbalanced three-phase loads in wye and delta connections. Three-phase power systems having a neutral wire. Complex, real and reactive power in a balanced load. Notes of rephasing a Three-phase load. Expression of an asymmetrical set of three phasors as a linear combination of three symmetrical sets of phasors. Complex, real and reactive power in an unbalanced three-phase power system. Comparison between a mono-phase and a three-phase power system. Equivalence of star-connected loads. Aron power measuring system.
FIRST-ORDER TRANSIENT ANALYSIS
Transient analysis of first-order RC e RL circuits, with application of Thevenin and Norton theorems. Homogeneous equations, particular and general solutions. List of fundamental particular solutions. Superposition principle of particular solutions. Stability of a linear circuit. Series RLC Resonant Circuit. Notes of the n-order circuits in time domain. Output variables and associated resistive circuit. Ideal switch and continuity property of state variables. Degenerate circuits, parallel and series connection of capacitors and inductors. Solution with the Laplace traform of transient responses. Definition and analytic espression of network functions.
MAGNETIC CIRCUITS
Mutual inductance in time and phasors. Series real resonant circuits: resonant frequency, selectivity, network function, amplitude and phase plot. Reference to Ampere and Faraday laws. Magnetic hysteresis. Magnetic reluctance, Hopkinson’s law e magnetic circuits in air and iron. Magnetic circuit formed by a torus in air and an iron torus with an air gap. Kirchhoff magnetic laws and correspondence to electric circuits.

Lessons and exercises

Slides of lessons are uploaded on Moodle.

The examination is composed of both a written test and an oral test.
The written test consists of two parts:
1) The first test consists of solving some simple numerical exercises using Moodle. The result is a positive or negative tag.
2) If the result of the first test is positive, then the student is admitted to the second written test. The student has to solve two or more exercises on a paper sheet.
A pass mark of 18/30 has to be achieved to be admitted to the oral test. the passed written test is valid for a year, that is until the corresponding session of the next
accademic year; afterwards, the student will have to take an oral test to top the examination, composed of at least three questions regarding the arguments of the course.

Otherwise:
During the course, there are three written intermediate tests in succession; one of these three tests is performed on Moodle;
to take the second and third tests, it is necessary to achieved a pass mark (18/30) in each previous test; if all the three tests are
posistive, a mark, obtained by the arithmetic mean of the three marks, is proposed; the student is free
either to accept the mark (that is registered without any further test), or to ask an oral test to integrate
the examination, or reject the intermediate tests and take the ordinary examination.

Important note: Possible changes to the normal examination procedure, due to the safety protocols for COVID19, will be communicated on the DIA web sites.