Han Q Le© ECE 3336 Introduction to Circuits & Electronics Lecture Set #9 Linear Circuit-Fourier-Digital Dr. Han Le ECE Dept.

Han Q Le© Introduction to Circuits & Electronics Input Output It does something useful: -information -power

Han Q Le©

Concept of System (Contraption)

Han Q Le©

Concept of System in Electronics R C input v in [t] output v out [t] i(t) R C output v out [t] i(t)input v in [t] Low pass High pass System (circuits) Input Output Control x k (t)y j (t) k=1,..., m j=1,..., p

Han Q Le© Linear (Time-Invariant) System System (circuits) Input Output Control x k (t)y j (t) k=1,..., m j=1,..., p Input Output This is called “linear” Time-invariant DOES NOT mean time-constant. It means that it does not matter when you input the signals (today or 1,000,000 yr BC), you’ll get the same output

Han Q Le© The most significant application of electronic systems is signal processing: generating: signal synthesis: converting information (logic) into electrical signal (voltage or current), encoding conditioning (transforming): signal shaping, filtering, amplifying transmitting: sending signal to other parts of the circuit, or over transmission media: conductor, air (EM waves), fiber optics (for long distance) receiving: transforming the signal, converting into information

Han Q Le© Fourier transform Harmonic function Complex numbers &analysis Phasors

Han Q Le© Applications of mathematical techniques Fourier transform Harmonic function Complex number &analysis Phasors Signal and AC circuit problems RLC or any time-varying linear circuits. Applicable to linear portion of circuits that include nonlinear elements Signal processing signal analysis (spectral decomposition) filtering, conditioning (inc amplification) synthesizing

Han Q Le© Outline Time-varying circuits and signals Introduction to mathematical techniques: – Harmonic functions – Complex analysis – Phasors – Fourier analysis and transform Applications of mathematical techniques to physical problems and circuits

Han Q Le© Link to Mathematica file

Han Q Le© the digital electronic trend

Han Q Le© Automatic Fan Control: The Analog Way We have +-15 V and 5 V DC power input. +15 V -15 V Temperature sensor input (low=0.5 V; high=4 V) User preset (controllable) voltage input. Voltage comparator

Han Q Le© Automatic Fan Control: The Digital Way We have +-15 V and 5 V DC power input. Temperature sensor User preset digital: Micro- processor However, the physical world is analog; the digital core is still based on the analog circuit principle. And outside the digital core, it is still an analog periphery. The digital core

Han Q Le© Digital Electronics Signal processing is fundamentally mathematical operation In the past, electronic circuits were used to do signal processing mathematics directly on the signal voltage or current (analog). With digital electronic, voltage and current no longer represents the signal directly, they represent the numerical (digital) value of the signal Electronic circuits (ICs) are designed to perform numerical (digital) mathematical operations: microcontrollers, digital signal processors, FPGA, microprocessors…

Han Q Le© Electronics become “mathematical operation”- oriented at the functional design level However, fundamental electrical circuit behavior is still essential at the physical implementation level Logical layer (Digital design) Logical layer (Digital design) Physical layer (Analog implementation)

Han Q Le© A summary of phasor, Fourier, signal processing and all that

Han Q Le© The basics Voltage or current signal is time-dependent (otherwise, there is no information transmitted) Solving time-dependent differential equations for circuits involving R, L, C is difficult & tedious But if the signal is a pure harmonic of single frequency, we can use phasor transform, which is easy As it turns out, even if the signal is not a harmonic, Fourier theorem tells us that it can be break down as a sum (or integral) of harmonics Hence, we can use phasor method, along with Fourier theorem to solve for any signal

Han Q Le© Linear Circuit - Frequency domain Circuit Input Output Harmonic function Harmonic transfer function Phasor Fourier Arbitrary input