6.4
Aritmética fasorial
Los fasores y sus correspondientes sinusoides están interrelacionados, lo que ofrece información única sobre el comportamiento de los circuitos de cor…
Consideremos una sinusoide y su correspondiente fasor.
La derivada de la sinusoide en el dominio del tiempo es igual a su fasor multiplicado por j-omega en el dominio del fasor.
De manera similar, al integrar una sinusoide en el dominio del tiempo, se transforma en su fasor dividido por j-omega en el dominio del fasor.
Estas transformaciones producen la solución sinusoide de estado estacionario sin conocer los valores iniciales.
Ahora, consideremos dos fasores en formas rectangulares y polares. Para añadir estos dos fasores, se utilizan sus formas rectangulares.
La parte real del fasor resultante es la suma de las partes reales de los dos fasores, y su parte compleja es la suma de las partes complejas de los fasores individuales.
Del mismo modo, para restar dos fasores, se utilizan sus formas rectangulares. Las partes reales y complejas del fasor resultante son las diferencias entre las partes real e imaginaria de los fasores individuales.
Las formas polares se utilizan para multiplicar y dividir dos fasores cualesquiera, y el conjugado complejo de un fasor se puede expresar tanto en formas rectangulares como polares.
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Q1: How does differentiation of a sinusoid transform between time and phasor domains?
When you differentiate a sinusoid in the time domain, it transforms into its corresponding phasor multiplied by j-omega (jω) in the phasor domain, where j is the imaginary unit and ω is the angular frequency. This transformation allows you to find steady-state solutions without knowing initial values, simplifying AC circuit analysis significantly.
Q2: What happens to a sinusoid when it is integrated in the phasor domain?
Integration of a sinusoid in the time domain translates into its corresponding phasor divided by j-omega in the phasor domain. This transformation provides a direct method to obtain steady-state solutions for sinusoidal signals without requiring initial variable values, making it essential for AC circuit problem-solving.
Q3: Why are rectangular forms used for adding and subtracting phasors?
Rectangular forms express phasors as complex numbers with real and imaginary parts, making component-wise addition and subtraction straightforward. The real part of the resultant phasor equals the sum or difference of the real parts, while the imaginary part equals the sum or difference of the imaginary parts of the individual phasors.
Q4: When should you use polar form for phasor operations?
Polar forms, expressing phasors as magnitude and angle, are used for multiplication and division. The resultant magnitude equals the product or quotient of individual magnitudes, and the resultant angle equals the sum or difference of individual angles. This approach simplifies complex phasor calculations in AC circuit analysis.
Q5: How is the complex conjugate of a phasor expressed and why is it important?
The complex conjugate of a phasor is obtained by changing the sign of its imaginary part and can be expressed in both rectangular and polar forms. This operation is crucial in many AC circuit applications, including the computation of power, making it fundamental to understanding phasor relationships for circuit elements.
Q6: What is the relationship between phasors and their corresponding sinusoids?
Phasors and sinusoids are interrelated mathematical representations of AC signals. Phasors transform sinusoidal time-domain operations into algebraic operations in the phasor domain through differentiation and integration rules. This relationship simplifies AC circuit analysis by allowing steady-state solutions to be found without initial conditions.
Q7: How do phasor arithmetic operations differ based on the form used?
Phasor arithmetic requires selecting the appropriate form for each operation: rectangular forms for addition and subtraction enable direct component combination, while polar forms for multiplication and division enable magnitude and angle operations. Choosing the correct form streamlines calculations and reduces computational complexity in AC circuit problems.