'Real axis' presentation slideshows

Section 6.5 Notes

Section 6.5 Notes. 1 st Day. Today we will learn how to change from rectangular (standard) complex form to trigonometric (polar) complex form and vice versus. A complex number z = a + bi can be represented as a point ( a , b ) in a coordinate plane called the complex plane .

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Class July 8 th ~ 15 th :

Class July 8 th ~ 15 th :. HW #4 & #5 due 7/8 Exam III on 7/10 Exam IV on 7/24 FINAL on 7/29 Lab #3 & #4 ( group of 3~4 ) #3: G244 on 7/10 & 7/15 2:30PM ? report due 7/17 #4: optional : on 7/17 & 7/12 ? report due 7/24. Steady-State Error: e (?). E = R - C

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Root Locus Method

Root Locus Method. Root Locus. Motivation To satisfy transient performance requirements, it may be necessary to know how to choose certain controller parameters so that the resulting closed-loop poles are in the performance regions, which can be solved with Root Locus technique. Definition

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7.3 Complex Numbers: Trig Form Day 1

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Math 112 Elementary Functions

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OFFSET QPSK

s 4. s 1. s 3. s 2. OFFSET QPSK. 90 degree shift in phase. ? 2. (01). (11). ? 1. (10). (00). 180 degree shift in phase. OFFSET QPSK. OFFSET QPSK. Whenever both bits are changed simultaneously, 180 degree phase-shift occurs.

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An introduction to the complex number system

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E E 2415

E E 2415. Lecture 8 - Introduction to Phasors. The Sinusoidal Function. f. Euler Functions. Sinusoid to Phasor. Exponential form:. Separate exponents:. Phasor transform:. Also:. Simple Harmonic Motion - 0 ?. Phasor rotates CCW about origin of complex plane

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Chapter 5: Root Locus Design Method

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Chapter 5: Root Locus

Chapter 5: Root Locus. Nov. 13-15, 2012. Two Conditions for Plotting Root Locus. Given open-loop transfer function G k (s). Characteristic equation. Magnitude Condition and Argument Condition. Rules for Plotting Root Locus. Break-in point. Breakaway point.

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DFT/FFT and Wavelets

DFT/FFT and Wavelets. Additive Synthesis demonstration (wave addition) Standard Definitions Computing the DFT and FFT Sine and cosine wave multiplication and integration Input form Windowing Interpreting the results Thresholds Limitations of the FFT Frequency resolution Artifacts

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10.4 Trigonometric (Polar) Form of Complex Numbers

10.4 Trigonometric (Polar) Form of Complex Numbers. Call the horizontal axis the real axis and the vertical axis the imaginary axis . Now complex numbers can be graphed in this complex plane . The Complex Plane and Vector Representations.

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Review of Complex numbers

Imag. r =|z |. y. f. Real. x. Review of Complex numbers. Rectangular Form:. Exponential Form:. Imag. y = Im (z). Real. x = Re(z). Real & Imaginary Parts of Rectangular Form. The real and imaginary parts of a complex number in rectangular form are real numbers:.

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Warm Up Express each number in terms of i.

Warm Up Express each number in terms of i. 9 i. 2. 1. Find each product. 3. Operations with Complex Numbers. Objective Perform operations with complex numbers .

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Root locus

Root locus. A technique enabling you to see how close-loop poles would vary if a parameter in the system is varied Can be used to design controllers or tuning controller parameters so as to move the dominant poles into the desired region.

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Trigonometric Form of Complex Numbers

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Introduction

Introduction. Today we will look at some complex number questions. These questions will help you with your coursework. If you can not do these questions you must re-read your lecture summaries and the HELM workbooks until you can. .

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Trigonometric Form of Complex Numbers

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Polar Coordinates z= rcis ?

Polar Coordinates z= rcis ?. Write z=2+3i in polar form. . 1: sketch the point. 2. find modulus & argument (angle the line makes with the real axis) Modulus is ?(2 2 +3 2 )=?13 ? =tan -1 (3/2)=0.9828rad (4dp) 2+3i= ?13 cos 0.9828+ ?13 isin 0.983 = ?13( cos 0.9828+ isin 0.983)

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5. Root-Locus Technique

R. (. s. ). Kuo-95 (p.470). 5. Root-Locus Technique . Curves of eigenvalues in the s-plane depending on K. Example 5.1. n1=[1,3];a1=conv([1,5,0],[1,6]); a2=[1,2,2];p=roots(a2) d1=conv(a1,a2) rlocus(n1,d1). Roots: D 1 (s)=0. Zeros: N 1 (s)=0. 5 parts ( D 1 (s) 5. order).

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