A complex number is a point on the complex plane shown in Figure 1.1.1.įigure 1.1.1 Complex plane represents complex numbers as points When a real number and an imaginary number are added or subtracted, as in 12 + j5 and -4 + j3, the result is a complex number in rectangular form. The rationalizing step was when the numerator and denominator were each multiplied by j.ġ.1.3.1 Rectangular Form of a Complex Number This view of j should be used as a memory aid only, because j is not a number itself it just designates a number as being imaginary.ĭivision involving a real and an imaginary number involves a special step, called rationalizing, whenever the imaginary number is in the denominator. For instance, the examples given above can be viewed in the following ways: These rules for multiplying and dividing imaginary numbers may be easier to remember if you view the j operator at being equal to Ö It results with in a real number, but in this case the sign is the same as you would get if they were real numbers. Dividing two imaginary numbers also uses a different rule.It results with a real number whose sign is opposite of what you would get if they were real numbers. Multiplying two imaginary numbers, however, uses a different rule than that for real numbers.Adding and subtracting use the same rules as those for real numbers.Imaginary numbers sometimes require special rules when combining them: In the electrical field, however, it is standard practice to use the letter j, as in j2, j0.01, and -j5.6. There is no one universally accepted way of writing them. They have an unfortunate name, because they exist more than in the imagination. Imaginary numbers were invented to provide solutions to square roots of negative numbers. Then ordinary algebra can be used to manipulate the equations into forms more suited for factoring and/or plotting the roots in the complex plane. When using the technique, the differential equations (describing the system to be controlled) are converted into algebraic equations which are functions of a complex variable ( s = s Some of the complex arithmetic is avoided, however, when using the Laplace transformation technique as part of the control system analysis. This is because complex numbers have their own special rules for adding, subtracting, multiplying, and dividing. These complex variables - phasors and impedance - are complex numbers.Ĭomplex numbers must be manipulated using complex algebra, which is an extension of the algebra of real numbers. For example, sinusoidal signals are often transformed into phasors, and resistance, capacitance, and inductance are often transformed into impedance. The analysis of ac circuits can also use complex numbers. And complex variables in turn are functions of complex numbers. Why start this course work with a review of complex numbers? Because classical control system analysis is based on the application of complex variables and functions of complex variables. Review of complex numbers and complex algebra This chapter covers fundamentals of Industrial Control System Analysis, Control System Stability, and Laplace Transforms. Reading through this first section is a great way to brush up on Laplace transforms, using them to solve differential equations, and applying these methods to control system design, using Bode Plot methods. This course has been taught to engineers at Rockwell Automation, Ford Motor Co., and others. The following is the first section of the 1st Chapter of the "Advanced Dynamic Simulation Course".
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