Name: phasor rules for ac circuit analysis
Uploaded: 2024-04-11T12:48:48.000Z
Duration: 1 min 13 s
Channel: JoVE

Considere um sinusóide e seu fasor correspondente.
A derivada do sinusóide no domínio do tempo é igual ao seu fasor multiplicado por j-omega no domínio fasor.
Da mesma forma, ao integrar um sinusóide no domínio do tempo, ele se transforma em seu fasor dividido por j-omega no domínio fasor.
Essas transformações produzem a solução sinusóide em estado estacionário sem conhecer os valores iniciais.
Agora, considere dois fasores em formas retangulares e polares. Para adicionar esses dois fasores, suas formas retangulares são usadas.
A parte real do fasor resultante é a soma das partes reais dos dois fasores, e sua parte complexa é a soma das partes complexas dos fasores individuais.
Da mesma forma, para subtrair dois fasores, suas formas retangulares são usadas. As partes reais e complexas do fasor resultante são as diferenças das partes reais e imaginárias dos fasores individuais.
As formas polares são usadas para multiplicar e dividir quaisquer dois fasores, e o conjugado complexo de um fasorial pode ser expresso em formas retangulares e polares.

aritmética fasorial

Os fasores e suas senoides correspondentes estão inter-relacionados, oferecendo insights exclusivos sobre o comportamento dos circuitos de corrente alternada (AC). Uma maneira de entender essa relação é através das operações de diferenciação e integração nos domínios do tempo e dos fasores.

Quando a derivada de uma senóide é tomada no domínio do tempo, ela se transforma em seu fasor correspondente multiplicado por j-ômega (jω) no domínio fasorial, onde j é a unidade imaginária e ω é a frequência angular. Por outro lado, quando uma senóide é integrada no domínio do tempo, ela se traduz em seu fasor correspondente dividido por j-ômega no domínio fasorial. Essas transformações fornecem um meio de encontrar soluções em estado estacionário para a senóide sem conhecer os valores iniciais das variáveis.

A seguir, considere dois fasores, cada um representado nas formas retangular e polar. Para somar ou subtrair esses dois fasores, são utilizadas suas formas retangulares (que expressam o fasor como um número complexo com partes reais e imaginárias). A parte real do fasor resultante é a soma (para adição) ou diferença (para subtração) das partes reais dos dois fasores originais, e sua parte imaginária é a soma ou diferença das partes imaginárias dos fasores individuais.

Ao multiplicar ou dividir quaisquer dois fasores, suas formas polares são usadas (expressando o fasor como uma magnitude e um ângulo). A magnitude do fasor resultante é o produto (para multiplicação) ou quociente (para divisão) das magnitudes dos dois fasores originais, e o ângulo do fasor resultante é a soma ou diferença dos ângulos dos fasores individuais.

Por último, o conjugado complexo de um fasor - que é obtido alterando o sinal da sua parte imaginária - pode ser expresso tanto na forma retangular quanto na forma polar. Esta operação é crucial em muitas aplicações, incluindo o cálculo de potência em circuitos AC.

Concluindo, os fasores servem como uma poderosa ferramenta matemática no estudo de circuitos AC, simplificando a análise e resolvendo problemas que seriam significativamente mais complexos no domínio do tempo.

Aritmética Fasorial

Considere um sinusóide e seu fasor correspondente.
A derivada do sinusóide no domínio do tempo é igual ao seu fasor multiplicado por j-omega no fasor ...

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

Unidade 1

AC Circuit Analysis

Análise de Circuitos CA

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

Phasor Arithmetics

phasor rules for ac circuit analysis

Phasor rules make AC circuit analysis easier by linking sinusoids to their matching phasors. A phasor is a complex-number form of a sinusoidal signal. This connection helps explain how alternating current changes with time.

Differentiation and integration work in both the time domain and the phasor domain. If you take the derivative of a sinusoid in time, its phasor is multiplied by jω, where j is the imaginary unit and ω is the angular frequency. If you integrate a sinusoid in time, its phasor is divided by jω. These relationships help find steady-state solutions without needing the initial variable values.

Phasor addition and subtraction use rectangular form. In rectangular form, a phasor is written with real and imaginary parts. For addition, the real parts are added and the imaginary parts are added. For subtraction, the real parts are subtracted and the imaginary parts are subtracted.

Phasor multiplication and division use polar form. In polar form, a phasor is written as a magnitude and an angle. For multiplication, the magnitudes are multiplied and the angles are added. For division, the magnitudes are divided and the angles are subtracted.

The complex conjugate of a phasor is found by changing the sign of its imaginary part. It can be written in both rectangular and polar form. This operation is important in many applications, including power calculations in AC circuits. Together, these phasor tools simplify problems that are much harder to solve in the time domain.

Phasor Rules for AC Circuit Analysis

sinusoidal sources

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Alternating current, or AC, changes direction and magnitude over time, unlike direct current, or DC, which flows in one direction with constant polarity. AC is the main form of electricity used for modern power transmission and distribution because it can travel long distances efficiently. In homes, power supplies use sinusoidal sources to deliver electricity. A sinusoidal source creates a voltage that rises and falls in a smooth wave-like pattern. This waveform is called a sinusoid, which is...

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graphical-and-analytic-representation-of-sinusoids

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Measuring sinusoid phase on an oscilloscope helps compare two sinusoidal voltages with the same amplitude and period. An oscilloscope is an instrument used to display and analyze waveforms. The process starts by reading the peak-to-peak value, which is twice the amplitude of the sinusoid. This shows the maximum voltage swing of the waveform. Next, the period and angular frequency are measured. The period is the time for one complete cycle. Angular frequency, often written as omega, is the rate...

phasors

Phasors for AC Circuit Analysis

Phasors are a mathematical tool for analyzing alternating current, or AC, circuits. They represent sinusoids with complex numbers. The phasor magnitude matches the sinusoid amplitude, and the phasor angle shows the phase measured from the positive x-axis. Phasors make AC circuit analysis simpler by removing time from the current and voltage expressions. That lets an AC circuit be treated like its equivalent direct current, or DC, form for calculations. Phasors can also be written in...

phasor arithmetics

Phasor rules make AC circuit analysis easier by linking sinusoids to their matching phasors. A phasor is a complex-number form of a sinusoidal signal. This connection helps explain how alternating current changes with time. Differentiation and integration work in both the time domain and the phasor domain. If you take the derivative of a sinusoid in time, its phasor is multiplied by jω, where j is the imaginary unit and ω is the angular frequency. If you integrate a sinusoid in time, its...

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phasor-relationships-for-circuit-elements

AC Circuit Phasors: Resistor, Inductor, Capacitor

AC circuit phasors help show how voltage and current behave for resistors, inductors, and capacitors. Phasor representation moves these relationships from the time domain to the frequency domain, which makes AC circuit analysis easier. For a resistor, Ohm's law still applies. The voltage across the resistor and the current through it stay in phase, so their phasors point in the same direction. An inductor works differently because its voltage depends on the rate of change of current. The...

kirchoff's laws using phasors

AC Circuit Rules with Phasor Diagrams

AC circuit analysis uses Kirchhoff's voltage law and Kirchhoff's current law with phasors. In household wiring, AC power from a distribution panel is sent to appliances connected in parallel. These rules help describe how voltage and current behave in sinusoidal steady state. Kirchhoff's voltage law, or KVL, says that the sum of phasor voltages around a closed loop in an AC circuit is zero. The voltages can first be written in the time domain and then changed into phasor form. In phasor...

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AC Circuit Response: Impedance and Admittance

AC circuit response changes with frequency, and impedance describes how resistors, inductors, and capacitors affect sinusoidal current flow. In phasor form, voltage and current are used to show this behavior in a clear way. Impedance is a key idea in AC circuit analysis because it measures opposition to current in an alternating circuit. These components behave differently in AC than in DC. In DC circuits, inductors act like short circuits and capacitors act like open circuits. In AC circuits,...

impedance combination

Series and Parallel Impedance Rules

Impedance in series and parallel circuits follows the same rules used for resistors. A string of Christmas lights is a simple example. Each bulb represents one impedance element. In a series connection, the current stays the same through every component. This series behavior matches Kirchhoff’s Voltage Law, or KVL. The total impedance in the circuit is the sum of the individual impedances. The source voltage is then divided among the components based on their values. This is the voltage...

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Norton Current in AC Circuit Analysis

Norton’s theorem helps simplify complex AC circuits by replacing a two-terminal linear network with a smaller equivalent circuit. That equivalent circuit uses an impedance in parallel with a constant current source. The method lets students break one complicated circuit into two parts and analyze them more easily. To build the Norton equivalent, the source is replaced by its internal impedance. The result is the Norton impedance, which is the same as the Thévenin impedance. This parallel...

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design-example-capacitance-multiplier-circuit

Capacitance Multiplier Circuit for Larger Capacitance

A capacitance multiplier circuit creates a larger effective capacitance from a small physical capacitor. In integrated circuit technology, this approach is useful when a small capacitor is not enough. The circuit can multiply capacitance by as much as 1000, so a 10-pF capacitor can act like a 100-nF capacitor. The design uses two op-amps. The first op-amp works as a voltage follower, which helps isolate the generated capacitance from the loading caused by the second stage. The second op-amp...

key terms and concepts

Learn how phasors simplify AC circuit analysis through differentiation, integration, rectangular and polar forms, and complex conjugates in steady-state signals.

Phasor Rules for AC Circuit Analysis | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Phasor Rules for AC Circuit Analysis

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