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

Рассмотрим синусоиду и соответствующий ей вектор.
Производная синусоиды во временной области равна ее вектору, умноженному на j-омега в фазорной области.
Аналогично, при интегрировании синусоиды во временной области, она преобразуется в свой фазор, разделенный на j-омега в фазорной области.
Эти преобразования дают синусоидное стационарное решение без знания начальных значений.
Теперь рассмотрим два вектора в прямоугольной и полярной формах. Для сложения этих двух векторов используются их прямоугольные формы.
Действительная часть результирующего вектора — это сумма действительных частей двух векторов, а его комплексная часть — это сумма сложных частей отдельных векторов.
Аналогично для вычитания двух векторов используются их прямоугольные формы. Действительная и сложная части результирующего вектора являются различиями действительной и мнимой частей отдельных векторов.
Полярные формы используются для умножения и деления любых двух векторов, а комплексное сопряжение вектора может быть выражено как в прямоугольной, так и в полярной формах.

Фазорная арифметика

Фазоры и соответствующие им синусоиды взаимосвязаны, что дает уникальное представление о поведении цепей переменного тока (AC). Один из способов понять эту взаимосвязь — использовать операции дифференцирования и интегрирования как во временном, так и в фазорном доменах.

Когда производная синусоиды берется во временном домене, она преобразуется в соответствующий ей фазор, умноженный на j-омега (jω) в фазорном домене, где j — мнимая единица, а ω — угловая частота. И наоборот, когда синусоида интегрируется во временном домене, она преобразуется в соответствующий ей фазор, разделенный на j-омега в фазорной области. Эти преобразования позволяют находить стационарные решения для синусоиды, не зная начальных значений переменных.

Далее, рассмотрим два фазора, каждый из которых представлен в прямоугольной и полярной формах. Чтобы сложить или вычесть эти два фазора, используются их прямоугольные формы (которые выражают фазор как комплексное число с действительной и мнимой частями). Действительная часть результирующего фазора — это сумма (для сложения) или разность (для вычитания) действительных частей двух исходных фазоров, а его мнимая часть — это сумма или разность мнимых частей отдельных фазоров.

При умножении или делении любых двух фазоров используются их полярные формы (выражающие фазор как величину и угол). Величина результирующего фазора представляет собой произведение (для умножения) или частное (для деления) величин двух исходных фазоров, а угол результирующего фазора представляет собой сумму или разность углов отдельных фазоров.

Наконец, комплексно-сопряженное фазорное выражение, получаемое путем изменения знака его мнимой части, может быть выражено как в прямоугольной, так и в полярной форме. Эта операция имеет решающее значение во многих приложениях, включая вычисление мощности в цепях переменного тока.

В заключение, фазоры служат мощным математическим инструментом при изучении цепей переменного тока, упрощая анализ и решение проблем, которые были бы значительно более сложными во временной области.

Рассмотрим синусоиду и соответствующий ей вектор.
Производная синусоиды во временной области равна ее фазору, умноженному на j-омега в фазоре ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

AC Circuit Analysis

Анализ цепей переменного тока

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

AC Waveform Basics and Phase

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...

graphical and analytic representation of sinusoids

graphical-and-analytic-representation-of-sinusoids

Measuring Sinusoid Phase on an Oscilloscope

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...

phasor relationships for circuit elements

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...

impedances and admittance

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...

node analysis for ac circuits

AC Circuit Node Method in an Angioplasty Model

AC circuit node analysis is used to study an angioplasty system with a catheter turbine. The system helps remove plaque deposits from coronary arteries. Its electrical model is similar to a dual-node RLC circuit driven by a current-controlled voltage source. The analysis begins by collecting the input source voltage, inductance, and capacitance values. These measurements are used to find the driving voltage for the catheter shaft. The circuit is then viewed in the frequency domain, where...

mesh analysis for ac circuits

Impedance Matching with Mesh Currents

Impedance matching in radio communication uses resistor, capacitor, and inductor circuits to help signals move efficiently between a transmitter and a receiver. The circuit must balance the input and output signals before the current in the capacitor can be found. This makes the problem a good example of AC circuit analysis for high school students. The first step is to identify the input voltage and the output signal. The angular frequency is then taken from the time-domain form of the input...

source transformation for ac circuits

AC Source Conversion in Circuit Analysis

AC source conversion is a circuit analysis method used in the frequency domain. It changes a voltage source in series with an impedance into a current source in parallel with the same impedance, or it converts a current source back into a voltage source. The source values must keep the correct relationship during each change. This method helps solve for an unknown voltage in a circuit that includes a current source and components such as resistors, capacitors, and inductors. Each of these...

th&#233venin equivalent circuits

Finding Voltage and Impedance in Circuit A

The household power distribution system can be simplified with Thevenin's theorem. In this system, distribution lines and transformers form the main network. Electrical appliances can be treated as load impedance, which makes the circuit easier to analyze. An AC circuit can be split into two parts, circuit A and circuit B, with a single pair of terminals between them. Circuit A can then be replaced by its Thevenin equivalent circuit. That equivalent has a voltage source in series with an...

norton equivalent circuits

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...

superposition theorem for ac circuits

Analyzing Mixed-Frequency Linear Circuits

Analyzing mixed-frequency linear circuits starts with the principle of superposition. In a linear circuit, the total output from several inputs is the sum of the outputs from each input acting alone. Here, the inputs are the voltages from independent voltage sources and the currents from independent current sources. When one source is kept active, every other independent source is set to zero. A 0-V voltage source becomes a short circuit, and a 0-A current source becomes an open circuit. This...

op amp ac circuits

Analyzing an Op Amp Audio Filter

An op amp audio filter shapes an amplified sound signal by strongly reducing the lower-frequency parts. In this circuit, the focus is on a basic filter setup used after an amplifier in an audio system. The filter uses an operational amplifier in an inverting arrangement with resistors R1 and R2 and a capacitor C1. The analysis starts with a known input signal and asks for the output signal. To solve it, the capacitor impedance is found first. That requires the angular frequency from the...

design example: capacitance multiplier circuit

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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