Goal of this course
The Advanced Circuit Theory course is designed to complement the basic students' knowledge on electrical circuits obtained in the course of the Junior Electrical Engineering degree, with more "in depth" or complementary topics that are typical of a rigourous and scientifical approach to the study of circuits.  In particular, the analysis of linear and non-linear circuits, purely resistive or dynamic is formalised using general methods such as modified nodal analysis  and state equations. The basic framework of circuit simulators is presented, along with explanations related to possible simulation failures. Finally, network functions of linear dynamic circuits are introduced in order to present some simple examples of one-port synthesis and filter design. In all the topics covered, specific attention is given to  the application of general methods.

Knowledge and understanding (Dublin Descriptor 1)

At the end of the course and after successfully passing the exam, the student:

- knows the fundamental principles of graph theory applied to electrical circuits

- has knowledge of the state space representation of nonlinear dynamic circuits, of the linearisation methods and of energy based stability properties of circuits

- knows how to apply the standard modified nodal analysis method for the solution of general circuits (nonlinear, dynamic etc.) including integration methods and nonlinear solvers.

- Understands the basic properties of linear passive one-ports and knows the standard methods for their realization

- Moreover, as a more general and methodological goal, a rigorous, math based approach to circuits.

Applying knowledge and understanding (Dublin Descriptor 2)

At the end of the course and after successfully passing the exam, the student:

- is able to write solving equations for general circuits and analyzing their solutions both using numerical methods and using closed form solutions

- is able to write, using standard programming languages, the basic structure of a circuit simulator

- knows how to synthesize a passive linear one port using Cauer or Foster methods

1 Fundamental laws of electrical networks

Topological properties. Implicit and explicit Kirchhoff's laws. Graph matrices and their relations. Power and energy. Implicit and explicit Constitutive formulations of one-port and two-port components. Unidirectionality, 'reciprocity', symmetry and other properties of linear two-ports. Special two-ports: transformer, gyrator and operational amplifiers. Connection of two-ports.

2 Analysis of linear and nonlinear resistive networks

General methods of analysis. Nodal analysis (NA) and Modified Nodal Analysis (MNA). Numerical methods for solving linear algebraic systems. Sparsity. Characterization of nonlinear elements and their linearization. Examples: diode, BJT and MOS transistors, OP-AMP. Small-signal analysis. Graphical and numerical methods for solving systems of nonlinear algebraic circuit and their interpretation. Properties of circuits composed of  strictly passive non-linear and strictly monotonous non-linear bipoles.

3 Dynamic circuit analysis

Formulation of state equations in the time domain. Solution of the state equation of linear dynamic circuits in the time domain: free and forced components of the solution. Natural frequencies and stability. Second-order linear circuits. Qualitative study of simple dynamic nonlinear circuits: oscillators and bistable. MNA formulation for general circuits. Simple numerical methods for the solution of the MNA problem.

4 Network Functions

Analysis of linear dynamic circuits in sinusoidal steady state. Network functions in the phasor domain. Frequency response. Analysis of linear dynamic circuits using the Laplace transform. State equations and their solution in the Laplace domain. Immittances and general network functions H(s), poles and zeros.

5 Synthesis of one-ports and passive filter design

Properties of immittance functions Z(s) and Y(s) of RLC passive one-ports. Properties and synthesis of one-ports with only two elements (LC, RC, RL) according to the canonical forms of Foster and Cauer. Design of a low pass filter. Sensitivity analysis with respect to a parameter of the circuit and method for that network.

Prerequisites
All topics that are part of the basic circuit theory course (Elettrotecnica)
Laplace transforms
Electronic components (diodes, transistors, operational amplifiers, ...)
Linear algebraic equations
Eigenvalues ​​and eigenvectors of matrices
Linear time-invariant ordinary differential equations

The exam is composed of two parts, a first "practical" written test, followed by a second written test related to the theory. The goal of the practical part is assessing the student's capability of application of the main topics of the course with a direct approach to simple problems (Dublin descriptors 1 and 2), the theory part to test the knowledge and understanding of the main methods used in the course aiming at the self-sufficiency in the solution of problems not considered in this course (Dublin descriptor 1).

The date of the practical part is published online on the POLIMI webpage, students can access the theory part either immediately after the practical part (and will be corrected only if the practical part is sufficient), or (in this case only if the practical part is sufficient) after about 7 to 10 days or in any other following exam session (in this case the student will be temporarily marked as non-passed and will have to enroll to the chosen session).

The practical part lasts 2h30' and consists in the solution of three problems covering the main areas of the course, the students must work autonomously but can have access to a single self produced A4 sheet containing formulas or anything else the student believes can be useful to pass this part of the exam. Clarity in the description of the methods used and in the results obtained and terseness of language will be evaluated positively. The practical part is marked from 0/30 to 30/30; access to the theory part is reserved to students with a mark greater or equal to 16/30. The theory part lasts 45' and consists in three questions that must be answered in written form each question taking no more than two A4 pages and where terseness of language and a quantitative approach where possible are encouraged. The theory part, if not sufficient, invalidates the whole exam, if sufficient it can increment or decrement the practical part mark usually in the range -4/+4.

Communication of results, times and places for the theory part will be via mail, using the official POLIMI student mail address.

Lecture Notes available on the POLIMI beep platform.