Han Q Le©

ECE 3336 Introduction to Circuits & Electronics

Lecture Set #9Linear Circuit-Fourier-Digital

Dr. Han LeECE Dept.

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Introduction to Circuits & Electronics

Input Output

It does something useful:- information- power

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Han Q Le©

Concept of System (Contraption)

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Han Q Le©

Han Q Le©

Concept of System in Electronics

R

Cinput vin[t] output vout[t]i(t)

R

C

output vout[t]i(t)input vin[t]

Low pass

High pass

System(circuits)

Input Output

Control

xk(t) yj(t)

k=1,..., mj=1,..., p

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Linear (Time-Invariant) System

System(circuits)

Input Output

Control

xk(t) yj(t)

k=1,..., mj=1,..., p

tbxtax 21 tbytay 21 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

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Fourier transform

Harmonic function

Complex numbers &analysis

Phasors

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Applications of mathematical techniques

Fourier transform

Harmonic function

Complex number

&analysisPhasors

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

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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

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Link to Mathematica file

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the digital electronic trend

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refv

sv

Automatic Fan Control: The Analog WayWe 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

refout vvgv s

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Automatic Fan Control: The Digital WayWe have +-15 V and 5 V DC power input.

Temperature sensor

User preset digital:

10101101

refout vvgv s

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

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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…

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• 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)

Physical layer (Analog

implementation)

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A summary of phasor, Fourier, signal processing and all that

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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

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Linear Circuit - Frequency domain

CircuitInput Output

Harmonic function𝑒𝑖𝜔𝑡

Harmonic transfer function

𝐻 (𝜔 )𝑒𝑖 𝜔𝑡

𝑠𝑜𝑢𝑡 (𝑡 )=∫ 𝐴𝑠 (𝜔 )𝐻 (𝜔 )𝑒𝑖 𝜔𝑡 𝑑𝜔𝑠𝑖𝑛 (𝑡 )=∫ 𝐴𝑠 (𝜔 )𝑒𝑖 𝜔𝑡 𝑑𝜔

Phasor

Four

ier

Four

ier

Arbitrary input