Name: phasors for ac circuit analysis
Uploaded: 2024-04-11T12:48:48.000Z
Duration: 1 min 12 s
Channel: JoVE

Фазоры — это комплексные представления синусоид, где величина вектора равна амплитуде, а угол равен фазе, измеренной по положительной оси x.
Они упрощают анализ цепей переменного тока, устраняя зависимость тока и напряжения от времени, преобразуя цепь в эквивалентную форму постоянного тока.
Векторы могут быть представлены в прямоугольной, полярной или экспоненциальной форме с использованием тождества Эйлера.
Чтобы получить вектор синусоиды в синусоиде, сначала преобразуйте его в косинусную форму, а затем выразите как действительную часть комплексного числа.
Вектор этой синусоиды равен независимой от времени части комплексного числа.
И наоборот, синусоида данного вектора получается путем умножения фазора на фактор времени и взятия его действительной части.
Векторы могут быть представлены в виде вращающихся векторов или синоров против часовой стрелки на сложной плоскости с постоянной угловой частотой, а диаграммы, изображающие их, известны как векторные диаграммы.
Проекция вращающихся синоров на действительную ось представляет синусоиды.

Фазоры

Фазоры — это мощный математический инструмент, используемый для анализа цепей переменного тока (AC). Они обеспечивают представление синусоиды в терминах комплексных чисел, при этом величина фазора равна амплитуде синусоиды, а угол фазора представляет фазу, измеренную от положительной оси x.

Одним из существенных преимуществ использования фазоров является то, что они упрощают анализ цепей переменного тока, устраняя временную зависимость тока и напряжения. Это преобразование позволяет анализировать цепь переменного тока, как если бы она была эквивалентной формой постоянного тока (DC), что делает расчеты более простыми.

Фазоры можно представить в различных формах — прямоугольной, полярной или экспоненциальной — с помощью тождества Эйлера — фундаментальной формулы комплексного анализа, которая устанавливает глубокую связь между тригонометрическими и экспоненциальными функциями.

Чтобы получить фазор синусоиды в форме синуса, необходимо сначала преобразовать его в форму косинуса, а затем выразить как действительную часть комплексного числа. Фазор этой синусоиды равен независимой от времени части этого комплексного числа. И наоборот, синусоида данного фазора может быть получена путем умножения фазора на временной коэффициент и взятия его действительной части.

В графическом контексте фазоры можно визуализировать как вращающиеся векторы или «синоры», вращающиеся в направлении против часовой стрелки на комплексной плоскости с постоянной угловой частотой. Диаграммы, изображающие эти вращающиеся синоры, известны как фазорные диаграммы.

Ключевая концепция в понимании фазорной диаграммы заключается в том, что проекция вращающихся синоров на действительную ось представляет собой синусоиды. Это означает, что горизонтальное положение синора в любой момент времени соответствует мгновенному значению синусоидальной функции, которую он представляет.

Фазоры — это комплексные представления синусоид, где величина вектора равна амплитуде, а угол равен фазе, измеренной от ...

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

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 rectangular, polar, or exponential form.

Euler’s identity links trigonometric functions and exponential functions, and it helps convert between these phasor forms. To find the phasor of a sinusoid in sine form, the signal is first written in cosine form. It is then expressed as the real part of a complex number, and the phasor is the time-independent part of that number.

A sinusoid can also be recovered from a phasor. The phasor is multiplied by a time factor, and then the real part is taken. In a graph, phasors appear as rotating vectors, or sinors, on the complex plane.

These rotating sinors move counterclockwise with constant angular frequency. The diagrams that show them are called phasor diagrams. The projection of a sinor onto the real axis gives the sinusoid, so the horizontal position at any moment shows the instantaneous value of the wave.

Phasors 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 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 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 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 represent AC sinusoids, simplify circuit analysis, and appear as rotating vectors on the complex plane in phasor diagrams.

Phasors 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

Phasors for AC Circuit Analysis

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