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Chapter 3: Problem 10

The Complex Plane The complex variable \(z\) is related to the real variable \(u\) according to $$ z=1+e^{j u} $$ \- Sketch the contour of values \(z\) takes on in the complex plane. \- What are the maximum and minimum values attainable by \(|z|\) ? \- Sketch the contour the rational function \(\frac{z-1}{z+1}\) traces in the complex plane.

Short Answer

Expert verified
The contour of \(z\) is a circle centered at (1,0) with radius 1. Max \(|z| = 2\), min \(|z| = 0\). The contour of \(\frac{z-1}{z+1}\) forms a circle through origin.

Step by step solution

01

Expressing Complex Number in Terms of Unit Circle

Given the complex number \( z = 1 + e^{ju} \), where \( e^{ju} \) represents a point on the unit circle in the complex plane, since \( e^{ju} = \cos(u) + j\sin(u) \). Thus, the contour described by \( z \) in the complex plane is the set of all points \( z = 1 + \cos(u) + j\sin(u) \). This describes a circle with its center located at (1,0) in the real axis and a radius of 1.
02

Determine Maximum and Minimum Values of |z|

The magnitude of \( z \) is \( |z| = \sqrt{(1 + \cos(u))^2 + \sin^2(u)} \). Simplifying, \( |z| = \sqrt{1 + 2\cos(u)} \). The maximum value of \( |z| \) occurs when \( \cos(u) = 1 \), giving \( |z| = 2 \). The minimum value occurs when \( \cos(u) = -1 \), giving \( |z| = 0 \).
03

Analysing the Rational Function

Consider the rational function \( \frac{z-1}{z+1} \). Substituting \( z = 1 + e^{ju} \) gives: \( \frac{(1 + e^{ju}) - 1}{(1 + e^{ju}) + 1} = \frac{e^{ju}}{2 + e^{ju}} \). This describes a transformation of the circle centered at the origin with radius 1, but now mapped in a complex transformation.
04

Sketch Contours in the Complex Plane

- The contour of \( z = 1 + e^{ju} \) describes a circle centered at (1,0) with radius 1.- The contour of \( \frac{z-1}{z+1} \) involves a non-trivial M枚bius transformation, resulting in a more complex shape which is symmetric about the real axis and crosses through the origin and length is compressed into the unit disk centered at the origin.

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Key Concepts

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

Complex Variable
In mathematics, a complex variable refers to a variable that can take on the value of a complex number. Complex numbers are composed of two parts: a real part and an imaginary part, represented as \( z = x + yi \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit with \( i^2 = -1 \).
Complex variables allow us to map a plane using these numbers, known as the complex plane, which consists of a horizontal real axis and a vertical imaginary axis.
This plane provides a powerful way to visualize and solve equations involving complex numbers, such as the equation \( z = 1 + e^{ju} \) that describes points on the complex plane.
Unit Circle
The unit circle is a fundamental concept in complex analysis and trigonometry. It is the set of all points in the complex plane that are a distance of exactly one unit from the origin (0,0).
Mathematically, if a complex number \( z = x + yi \) lies on the unit circle, then \( |z| = 1 \). An important representation involves Euler's formula: \( e^{ju} = \cos(u) + j\sin(u) \).
This formula indicates that as the variable \( u \) varies, \( e^{ju} \) traces out a circle of radius 1 centered at the origin.
In the exercise example, this unit circle gets modified by adding 1, such that the new circle's center is at (1,0) and the radius remains 1.
Magnitude of Complex Numbers
The magnitude (or modulus) of a complex number \( z = a + bi \) is calculated using the formula \( |z| = \sqrt{a^2 + b^2} \).
This formula gives the distance of the point \( z \) from the origin in the complex plane.
In our exercise, the magnitude of \( z = 1 + e^{ju} \) is given by \( |z| = \sqrt{(1 + \cos(u))^2 + \sin^2(u)} \). After simplification, this expression becomes \( |z| = \sqrt{1 + 2\cos(u)} \).
Understanding this formula is crucial, as it allows us to find that the minimum magnitude is 0, occurring when \( \cos(u) = -1 \), and the maximum magnitude is 2, when \( \cos(u) = 1 \).
M枚bius Transformation
A M枚bius transformation is a function of the form \( T(z) = \frac{az + b}{cz + d} \), where \( a, b, c, \) and \( d \) are complex numbers and \( ad - bc eq 0 \).
This transformation maps circles and lines in the complex plane to other circles and lines, possibly rearranging them in complex and intricate ways.
In the step involving the rational function \( \frac{z-1}{z+1} \), a M枚bius transformation occurs, creating a notable image in the complex plane.
This transformation compresses the outside of the unit circle into the interior, resulting in a complex shape and illustrating how analytic properties can shift geometric shapes.
The result is particularly elegant and shows the power of M枚bius transformations in mapping the complex plane.

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Most popular questions from this chapter

Cartesian to Polar Conversion Convert the following expressions into polar form. Plot their location in the complex plane \(^{36}\). a) \((1+\sqrt{-3})^{2}\) b) \(3+j^{4}\) c) \(\frac{2-j \frac{6}{\sqrt{3}}}{2+j \frac{6}{\sqrt{3}}}\) d) \(\left(4-j^{3}\right)\left(1+j \frac{1}{2}\right)\) e) \(3 e^{j \pi}+4 e^{j \frac{\pi}{2}}\) f) \((\sqrt{3}+j) 2 \times \sqrt{2} e^{-\left(j \frac{\pi}{4}\right)}\) g) \(\frac{3}{1+j 3 \pi}\)

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Big is Beautiful Sammy wants to choose speakers that produce very loud music. He has an amplifier and notices that the speaker terminals are labeled " \(8 \Omega\) source." a) What does this mean in terms of the amplifier's equivalent circuit? b) Any speaker Sammy attaches to the terminals can be well-modeled as a resistor. Choosing a speaker amounts to choosing the values for the resistor. What choice would maximize the voltage across the speakers? c) Sammy decides that maximizing the power delivered to the speaker might be a better choice. What values for the speaker resistor should be chosen to maximize the power delivered to the speaker?

Linear, Time-Invariant Systems For a system to be completely characterized by a transfer function, it needs not only be linear, but also to be time- invariant. A system is said to be time-invariant if delaying the input delays the output by the same amount. Mathematically, if \(S(x(t))=y(t),\) meaning \(y(t)\) is the output of a system \(S(\bullet)\) when \(x(t)\) is the input, \(S(\bullet)\) is the time-invariant if \(S(x(t-\tau))=y(t-\tau)\) for all delays \(\tau\) and all inputs \(x(t) .\) Note that both linear and nonlinear systems have this property. For example, a system that squares its input is time- invariant. a) Show that if a circuit has fixed circuit elements (their values don't change over time), its input-output relationship is time-invariant. Hint: Consider the differential equation that describes a circuit's inputoutput relationship. What is its general form? Examine the derivative(s) of delayed signals. b) Show that impedances cannot characterize time-varying circuit elements \((\mathrm{R}, \mathrm{L},\) and \(\mathrm{C}) .\) Consequently, show that linear, time-varying systems do not have a transfer function. c) Determine the linearity and time-invariance of the following. Find the transfer function of the linear, time-invariant (LTI) one(s). i) diode ii) \(y(t)=x(t) \sin \left(2 \pi f_{0} t\right)\) iii) \(y(t)=x\left(t-\tau_{0}\right)\) iv) \(y(t)=x(t)+N(t)\)

Fun in the Lab You are given an unopenable box that has two terminals sticking out. You assume the box contains a circuit. You measure the voltage \(\sin \left(t+\frac{\pi}{4}\right)\) across the terminals when nothing is connected to them and the current \(\sqrt{2} \cos (t)\) when you place a wire across the terminals. a) Find a circuit that has these characteristics. b) You attach a \(1 \mathrm{H}\) inductor across the terminals. What voltage do you measure?
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