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Chapter 5: Problem 5

Sketch the imaginary part of \(A e^{j \varphi} e^{\text {jut }}\) to show that this is \(A \sin (\omega t+\varphi)\). What do we mean when we say that the real and imaginary parts of \(A e^{j \varphi} e^{j \omega t}\) are "90 \(^{\circ}\) out of phase"?

Short Answer

Expert verified
The imaginary part of \(A e^{j \theta} e^{j \text{ut}}\) is \(A \text{sin}(\theta + \text{ut})\). The real and imaginary parts are 90° out of phase because sine and cosine functions are phase-shifted by 90 degrees.

Step by step solution

01

Express the Given Equation

Start with the given expression: \(A e^{j \theta} e^{j \text {ut}}\). We will reformat it to a form that separates the real and imaginary parts. Recall that \(e^{j \theta} = \text{cos}(\theta) + j\text{sin}(\theta)\) (Euler's formula).
02

Combine Exponential Terms

Combine the exponential exponentials into one: \(A e^{j (\theta + \text{ut})}\). Again apply Euler's formula to rewrite: \(A e^{j (\theta + \text{ut})} = A [\text{cos}(\theta + \text{ut}) + j\text{sin}(\theta + \text{ut})]\).
03

Isolate the Imaginary Part

Focus on the imaginary part of the expanded form: \(A j\text{sin}(\theta + \text{ut})\). Since the imaginary part is what is needed, we only concern ourselves with \(A \text{sin}(\theta + \text{ut})\).
04

Interpretation of Phase Relationship

When we say the real and imaginary parts of \(A e^{j \theta} e^{j \text{ut}}\) are 90° out of phase, we mean that the sine and cosine components are shifted by 90 degrees relative to each other. Cosine and sine functions reach their peaks at different times: when cosine is at a maximum, sine is at zero, and vice versa. This phase difference is a quarter of a period, which corresponds to 90 degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Euler's Formula
Euler's formula is a key concept in electrical engineering. It states that for any real number \(\theta\), the following identity holds: \[ e^{j\theta} = \text{cos}(\theta) + j\text{sin}(\theta) \]
This formula connects complex exponentials with trigonometric functions and is crucial in analyzing sinusoidal functions.
By using Euler's formula, complex exponentials can be expressed in terms of their real and imaginary parts, which are the cosine and sine components, respectively.
This ability to separate the real and imaginary parts is particularly important when dealing with alternating currents and signal analysis in electrical engineering.
Phase Shift
A phase shift refers to the displacement of a wave in time. In the context of sinusoidal functions, it indicates how much a wave is shifted horizontally.
The general form of a sinusoidal function is \(A \text{sin}(\text{ω} t + \text{ϕ})\), where \(\text{ϕ}\) represents the phase shift.
For instance, in the expression \(A e^{j (\theta + \text{ωt})} \), the term \(\theta + \text{ωt}\) includes both the angular frequency \(\text{ω}\) and the phase shift \(\theta)\).
This term dictates the wave's position over time.
A positive phase shift means the wave is delayed, while a negative phase shift means it is advanced.
Imaginary Part of a Complex Number
The imaginary part of a complex number is the component that is multiplied by the imaginary unit 'j'.
For example, in the expression \( A [ \text{cos}( \theta + \text{ωt}) + j \text{sin}( \theta + \text{ωt}) ] \), the imaginary part is \( A \text{sin}( \theta + \text{ωt}) \).
This part of the complex number denotes the sine component of the sinusoidal function.
Imaginary parts are essential in representing oscillatory behavior in electrical systems, such as in alternating current (AC) signals.
These components allow engineers to analyze and manipulate the waveforms effectively.
Sinusoidal Functions
Sinusoidal functions are fundamental to describing periodic phenomena in electrical engineering.
They are represented by sine and cosine functions, which model the oscillatory nature of AC signals.
The general form of a sinusoidal function can be given as: \(\text{A sin} ( \text {ωt} + \text{ϕ})\), where \(\text{A}\) is the amplitude, \( \text{ωt} \) is the angular frequency times time, and \( \text{ϕ} \) is the phase shift.
These functions help in expressing complex exponentials in terms of real and imaginary parts using Euler's formula.
Understanding sinusoidal functions is vital for analyzing circuit behavior, signal processing, and communications.

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Most popular questions from this chapter

Show how to compute \(A\) and \(\varphi\) in the equation \(x(t)=A \cos (\omega t+\varphi)\) from the initial conditions \(x(O)\) and \(\left.\frac{d}{d t} x(t)\right|_{t=0}\).

Show that the Lissajous figure \(z(t)=A_{z} \cos \left(\omega t+\varphi_{x}\right)+j A_{y} \cos \left(\omega t+\varphi_{y}\right) \quad\) is just the rotating phasor \(A e^{j(\omega t+\varphi)}\) when \(A_{z}=A_{y}=A, \varphi_{z}=\varphi\), and \(\varphi_{y}=\varphi+\frac{\pi}{2}\)

Fix \(r_{n}\) in \(E\left(r_{n}, t\right)\) and show that \(E\left(r_{n}, t\right)\) varies cosinusoidally with time \(t .\) Sketch the function and interpret it. What is its period?

Write the phasor \(A_{1} e^{j \varphi_{1}}+A_{2} e^{j \varphi_{2}}\) as \(A_{3} e^{j \varphi_{3}} ;\) determine \(A_{3}\) and \(\varphi_{3}\) in terms of \(A_{1}, A_{2}, \varphi_{1}\), and \(\varphi_{2} .\) What is the corresponding real signal?

Sketch the imaginary part of \(A e^{j \varphi} e^{j \omega t}\) to show that this is \(A \sin (\omega t+\varphi)\). What do we mean when we say that the real and imaginary parts of \(A e^{j \varphi} e^{j \omega t}\) are " \(90^{\circ}\) out of phase"?
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