Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 35: Problem 31
Add the quantities \(y_{1}=10 \sin \omega t, y_{2}=15 \sin \left(\omega t+30^{\circ}\right)\), and \(y_{3}=5.0 \sin \left(\omega t-45^{\circ}\right)\) using the phasor method.
Short Answer
Expert verified
The resultant sinusoid is \(y = 26.83 \sin(\omega t + 8.5^{\circ})\).
Step by step solution
01
Convert Sinusoids to Phasors
Phasors are a representation of sinusoids as complex numbers in the form \(A e^{j\theta}\), where \(A\) is the amplitude and \(\theta\) is the phase angle. 1. Convert \(y_1 = 10 \sin \omega t\) to a phasor: \(\mathbf{Y_1} = 10 e^{j0^{\circ}}\).2. Convert \(y_2 = 15 \sin (\omega t + 30^{\circ})\) to a phasor: \(\mathbf{Y_2} = 15 e^{j30^{\circ}}\).3. Convert \(y_3 = 5 \sin (\omega t - 45^{\circ})\) to a phasor: \(\mathbf{Y_3} = 5 e^{-j45^{\circ}}\).
02
Express Phasors in Rectangular Form
Convert each phasor from its polar form \(A e^{j\theta}\) to rectangular form \(A\cos(\theta) + jA\sin(\theta)\).1. \(\mathbf{Y_1} = 10 \cos(0^{\circ}) + j10 \sin(0^{\circ}) = 10 + j0\).2. \(\mathbf{Y_2} = 15 \cos(30^{\circ}) + j15 \sin(30^{\circ}) = 12.99 + j7.5\).3. \(\mathbf{Y_3} = 5 \cos(-45^{\circ}) + j5 \sin(-45^{\circ}) = 3.54 - j3.54\).
03
Add the Phasors
Add the rectangular components separately. First add the real components, then add the imaginary components:Real part: \(10 + 12.99 + 3.54 = 26.53\).Imaginary part: \(0 + 7.5 - 3.54 = 3.96\).Thus, the resultant phasor in rectangular form is: \(26.53 + j3.96\).
04
Convert Resultant Phasor to Polar Form
The magnitude of the phasor is given by: \(\sqrt{(26.53)^2 + (3.96)^2} \approx 26.83\).The angle \(\theta\) is given by \(\theta = \arctan\left(\frac{3.96}{26.53}\right) \approx 8.5^{\circ}\).Thus, the phasor in polar form is \(26.83 e^{j8.5^{\circ}}\).
05
Convert Phasor Back to Sinusoid
The final step is to convert the phasor back into a time domain sinusoid:The resultant sinusoid is:\(y = 26.83 \sin(\omega t + 8.5^{\circ})\). This is the addition of all the given sinusoids.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form of \( a + bj \), where \( a \) is the real component and \( b \) is the imaginary component. In electrical engineering and physics, complex numbers are indispensable tools used to analyze sinusoidal functions and phasors.
- Real Part: The component of the complex number that doesn鈥檛 involve the imaginary unit, \( j \). For example, in the complex number \( 4 + 5j \), 4 is the real part.
- Imaginary Part: The coefficient of the imaginary unit, \( j \), in a complex number. In \( 4 + 5j \), 5 is the imaginary part.
- Imaginary Unit: Represented by \( j \) in electrical engineering (instead of \( i \)) and defined as \( \sqrt{-1} \).
Rectangular Form
Rectangular form is one way to represent complex numbers and phasors. In this form, a complex number is split into its real and imaginary components, expressed as \( a + bj \).
- Usage in Phasors: Phasors in rectangular form allow easy addition and subtraction because you just add or subtract the real parts and the imaginary parts separately. For example, adding \( (x_1 + jy_1) + (x_2 + jy_2) \) results in \( (x_1 + x_2) + j(y_1 + y_2) \).
- Conversion: To convert a phasor from polar to rectangular form, use \( A \cos(\theta) \) for the real part and \( A \sin(\theta) \) for the imaginary part, where \( A \) is the magnitude and \( \theta \) is the phase angle.
Polar Form
Polar form represents complex numbers using a magnitude and a phase angle, written as \( A e^{j\theta} \). It is particularly useful for multiplication and division of complex numbers.
- Magnitude: The distance of the complex number from the origin in the complex plane, calculated using \( \sqrt{a^2 + b^2} \) where \( a \) and \( b \) are the real and imaginary components respectively.
- Phase Angle: The angle made with the positive real axis, found using \( \tan^{-1}\left(\frac{b}{a}\right) \).
- Multiplication & Division: When complex numbers are in polar form, you multiply their magnitudes and add their angles to multiply, or divide their magnitudes and subtract their angles to divide.
Sinusoids
Sinusoids describe smooth periodic oscillations and are foundational in fields like physics, engineering, and signal processing. They can be expressed in the form \( A \sin(\omega t + \phi) \), where each part of this expression has a specific meaning:
- Amplitude (A): The peak value of the sinusoid and a measure of its maximum results.
- Angular Frequency (\( \omega \)): It represents how fast the wave oscillates and is measured in radians per second. It is directly related to the frequency \( f \) by the relation \( \omega = 2\pi f \).
- Phase Angle (\( \phi \)): The starting angle at \( t=0 \), it determines the horizontal shift of the waveform.
