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

Betrachten Sie eine Sinuskurve und den entsprechenden Phasor.
Die Ableitung der Sinuskurve im Zeitbereich entspricht ihrem Phasor multipliziert mit j-Omega im Phasorbereich.
Ähnlich verhält es sich bei der Integration einer Sinuskurve in den Zeitbereich, die sich in ihren Phasor verwandelt, der durch j-Omega im Phasorbereich geteilt wird.
Diese Transformationen ergeben die sinusförmige stationäre Lösung, ohne die Anfangswerte zu kennen.
Betrachten wir nun zwei Phasoren in rechteckiger und polarer Form. Um diese beiden Phasoren hinzuzufügen, werden ihre rechteckigen Formen verwendet.
Der reale Teil des resultierenden Phasors ist die Summe der realen Teile der beiden Phasoren, und sein komplexer Teil ist die Summe der komplexen Teile der einzelnen Phasoren.
In ähnlicher Weise werden zum Subtrahieren von zwei Phasoren ihre rechteckigen Formen verwendet. Die realen und komplexen Teile des resultierenden Phasors sind die Differenzen der realen und imaginären Teile der einzelnen Phasoren.
Polare Formen werden verwendet, um zwei beliebige Phasoren zu multiplizieren und zu dividieren, und das komplexe Konjugat eines Phasors kann sowohl in rechteckiger als auch in polarer Form ausgedrückt werden.

zeigerarithmetik

Zeiger und ihre entsprechenden Sinuskurven sind miteinander verbunden und bieten einzigartige Einblicke in das Verhalten von Wechselstromkreisen. Eine Möglichkeit, diese Beziehung zu verstehen, sind Differenzierungs- und Integrationsoperationen sowohl im Zeit- als auch im Zeigerbereich. Wenn die Ableitung einer Sinuskurve im Zeitbereich genommen wird, wandelt sie sich in den entsprechenden Zeiger multipliziert mit j-Omega (jω) im Zeigerbereich um, wobei j die imaginäre Einheit und ω die Winkelfrequenz ist. Wenn umgekehrt eine Sinuskurve in den Zeitbereich integriert wird, wird sie in den entsprechenden Zeiger geteilt durch j-Omega im Zeigerbereich übersetzt. Diese Transformationen bieten eine Möglichkeit, stationäre Lösungen für die Sinuskurve zu finden, ohne die anfänglichen Variablenwerte zu kennen. Betrachten Sie als Nächstes zwei Zeiger, die jeweils in rechteckiger und polarer Form dargestellt sind. Um diese beiden Zeiger zu addieren oder zu subtrahieren, werden ihre rechteckigen Formen verwendet (die den Zeiger als komplexe Zahl mit Real- und Imaginärteil ausdrücken). Der Realteil des resultierenden Zeigers ist die Summe (bei Addition) oder Differenz (bei Subtraktion) der Realteile der beiden ursprünglichen Zeiger, und sein Imaginärteil ist die Summe oder Differenz der Imaginärteile der einzelnen Zeiger. Beim Multiplizieren oder Dividieren zweier beliebiger Zeiger werden deren Polarformen verwendet (wobei der Zeiger als Betrag und Winkel ausgedrückt wird). Die Größe des resultierenden Zeigers ist das Produkt (für die Multiplikation) oder der Quotient (für die Division) der Größen der beiden ursprünglichen Zeiger, und der Winkel des resultierenden Zeigers ist die Summe oder Differenz der Winkel der einzelnen Zeiger. Schließlich kann das komplexe Konjugierte eines Zeigers – das durch Ändern des Vorzeichens seines Imaginärteils erhalten wird – sowohl in rechteckiger als auch in polarer Form ausgedrückt werden. Dieser Vorgang ist in vielen Anwendungen von entscheidender Bedeutung, einschließlich der Berechnung der Leistung in Wechselstromkreisen. Zusammenfassend lässt sich sagen, dass Zeiger ein leistungsstarkes mathematisches Werkzeug bei der Untersuchung von Wechselstromkreisen sind, das die Analyse vereinfacht und Probleme löst, die im Zeitbereich deutlich komplexer wären.

Zeigerarithmetik

Betrachten Sie eine Sinuskurve und den entsprechenden Phasor.
Die Ableitung der Sinuskurve im Zeitbereich ist gleich ihrem Phasor multipliziert mit j-Omega im Phasor ...

Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

AC Circuit Analysis

Analyse von Wechselstromschaltungen

KEY TERMS AND CONCEPTS

ZENTRALE BEGRIFFE UND KONZEPTE

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

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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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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

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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...

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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,...

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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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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...

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Impedance Matching with Mesh Currents

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AC Source Conversion in Circuit Analysis

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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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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...

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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...

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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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