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

Consider a sinusoid and its corresponding phasor.
The derivative of the sinusoid in the time domain equals its phasor multiplied by j-omega in the phasor domain.
Similarly, when integrating a sinusoid in the time domain, it transforms into its phasor divided by j-omega in the phasor domain.
These transformations yield the sinusoid steady-state solution without knowing the initial values.
Now, consider two phasors in rectangular and polar forms. To add these two phasors, their rectangular forms are used.
The real part of the resultant phasor is the sum of the real parts of the two phasors, and its complex part is the sum of the complex parts of the individual phasors.
Similarly, to subtract two phasors, their rectangular forms are used. The real and complex parts of the resultant phasor are the differences of the real and imaginary parts of the individual phasors.
Polar forms are used to multiply and divide any two phasors, and the complex conjugate of a phasor can be expressed in both rectangular and polar forms.

phasor arithmetics

Phasors and their corresponding sinusoids are interrelated, offering unique insights into the behavior of alternating current (AC) circuits. One way to understand this relationship is through the operations of differentiation and integration in both the time and phasor domains.
When the derivative of a sinusoid is taken in the time domain, it transforms into its corresponding phasor multiplied by j-omega (j&#969;) in the phasor domain, where j is the imaginary unit, and &#969; is the angular frequency. Conversely, when a sinusoid is integrated in the time domain, it translates into its corresponding phasor divided by j-omega in the phasor domain. These transformations provide a means to find steady-state solutions for the sinusoid without knowing the initial variable values.
Next, consider two phasors, each represented in rectangular and polar forms. To add or subtract these two phasors, their rectangular forms are used (which express the phasor as a complex number with real and imaginary parts). The real part of the resultant phasor is the sum (for addition) or difference (for subtraction) of the real parts of the two original phasors, and its imaginary part is the sum or difference of the imaginary parts of the individual phasors.
When multiplying or dividing any two phasors, their polar forms are used (expressing the phasor as a magnitude and an angle). The magnitude of the resultant phasor is the product (for multiplication) or quotient (for division) of the magnitudes of the two original phasors, and the angle of the resultant phasor is the sum or difference of the angles of the individual phasors.
Lastly, the complex conjugate of a phasor - which is obtained by changing the sign of its imaginary part - can be expressed in both rectangular and polar forms. This operation is crucial in many applications, including the computation of power in AC circuits.
In conclusion, phasors serve as a powerful mathematical tool in the study of AC circuits, simplifying analysis and solving problems that would be significantly more complex in the time domain.

Phasor Arithmetics

Consider a sinusoid and its corresponding phasor.
The derivative of the sinusoid in the time domain equals its phasor multiplied by j-omega in the phasor ...

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JoVE Core

Electrical Engineering

Unit 1

AC Circuit Analysis

KEY TERMS AND CONCEPTS

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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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...

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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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