Which compound has the most complex formula
Complex calculation in electronics
In several subtitles, this chapter deals with important and necessary basic information on complex calculations in the electronics sector. There is a video clip about phasor diagrams in their component and exponential forms. The four basic arithmetic operations for complex numbers are described.
The Cartesian coordinate system
In the plane, the position of a point can be clearly described with the help of suitable coordinate systems. The best known is the right-angled coordinate system, which is also called the Cartesian coordinate system. The plane is spanned by two axes, where the y-axis is perpendicular to the horizontal x-axis. The axis intersection is the origin of the coordinates with the value pair (0; 0). The position of a point in relation to the zero point is determined by two scalar quantities, the coordinates of the point with P (x; y).
The polar coordinate system
The position of a point in relation to the origin of the coordinates can be determined by the scalar path length r, the radius, measured from the axis zero point and an angle of rotation φ around this zero point. The point position is given by P (r; φ). The origin of the coordinates is now called the pole. The x-axis becomes the polar axis to which the angle of rotation φ relates. In this polar coordinate system, the polar axis pointing horizontally to the right is assigned the value φ = 0. The direction of rotation for positive angles is counterclockwise and counterclockwise.
The conversion between the two coordinate systems uses angle functions of the right triangle. The axes of the Cartesian coordinate system form the right angle. The prerequisite is that the zero point and the polar point as well as the x-axis are congruent with the polar axis. The following sketch shows the conversion:
The point P is at a distance of r = 10 units from the pole and lies around φ = 120 ° in the 2nd quadrant. The projections of P onto the axes result in the intercepts x and y and thus a right-angled triangle with r, the hypotenuse. The calculated negative angle of rotation points to the 2nd quadrant and must be supplemented to 180 °.
The pointer representation in the Gaussian plane of numbers
In AC technology, two components are necessary for a clear description of current or voltage. In addition to the amount, the phase position must also be observed. Pointers are therefore suitable for the graphical representation of changing quantities. They can be described mathematically by complex numbers. The pointer positions should begin at the zero point of the Gaussian plane of numbers.
The Gaussian plane of numbers is spanned by a right-angled coordinate system with a horizontal real axis and an imaginary axis perpendicular to it. As in the Cartesian axis system, the point of intersection of both axes is the zero point. A complex number Z can be clearly assigned to each point in the plane.
The imaginary unit is defined as i2= −1. The older representation i = √-1 can also be found. In electrical engineering, the letter j is used because electricity has traditionally been assigned the letter i.
A complex number is the sum of a real number and the square root of a negative number. A negative number is the product of its positive number with −1. The complex representation for Z is obtained with the following transformation.
The component form of a complex number
To distinguish them from conventional numbers, complex numbers are often underlined. They have a real part and the imaginary part as coefficients of j. Both the real and the imaginary part are real numbers. The general representation Z = x + y · j is called algebraic notation or normal form. A complex number can also be written in the trigonometric representation. There is a defined relationship between the two component forms.
In the two-dimensional plane, every complex number can be assigned a pointer. It extends from the pole, the zero point of the coordinate system, to the point coordinates of the complex number. This provides a connection to the polar coordinate system and further to the Cartesian coordinate system. The systems are isomorphic to one another.
The representation of a complex number in the polar diagram is called trigonometric form. The absolute amount and thus the length of the pointer is calculated according to the Pythagorean theorem. It is called the complex number module. The radians of the angle the pointer makes with the positive real axis is called the complex number argument.
The argument φ of a certain complex number has an infinite number of values. They differ by the factor n · 2 · π, where n is an integer. This is justified by the periodic course of the arctan function. In all arithmetic operations, only the main value of the argument with n = 1 is actually used.
A complex number with the imaginary part equal to zero is an element of the real numbers.
A complex number with the real part equal to zero is an element of the imaginary numbers.
Two complex numbers are the same if they match in the real part and in the imaginary part.
In the case of complex numbers, the terms 'greater than' or 'less than' are not defined.
Two complex numbers are the same if they are the same in their real and imaginary parts.
Two complex numbers are conjugate complex if they only differ in the sign of the imaginary part.
The exponential form of a complex number
In addition to the component form or the trigonometric notation, every complex number can be written in another important form of representation, the exponential form. It is derived from the power series that are written instead of the sine and cosine functions.
The exponential form clearly shows that the in section the pointer representation in the Gaussian plane of numbers established condition for the unambiguous description of variable quantities is fulfilled. The two components lead to the pointer length, the module as a scalar value. The phase angle φ is the argument of the complex quantity. All three forms of representation of complex numbers are equal and can be converted into one another.
Vector diagrams in component and exponential form
The following video clip shows the associated calculable values for some settings of the real and imaginary components or the sum pointer (module) and the phase angle (argument). The pointers are displayed graphically according to the input values.
In the second part, two pointers are freely placed in the 1st and 4th quadrant, with the respective exponential form being displayed. The subsequent calculation determines the associated pointer components. The component shapes are displayed and the total pointer is displayed graphically.
Conjugates complex numbers
If the pointer of a complex number is mirrored on the real axis, the pointer of the conjugate complex number is obtained. Only the imaginary component changes sign.
The multiplication of a complex number by its conjugate complex number gives a real result. Due to this fact, complex denominators can be removed from a system of equations. The complex conjugate extension of a fraction makes the expression in the denominator real.
Addition and subtraction of complex numbers
The addition and subtraction of complex numbers is done using the normal form. The rule here is that their real parts and their imaginary parts are added or subtracted. If the trigonometric representation is used for calculations, the corresponding angle theorems must be observed.
Multiplication and division of complex numbers
If the multiplication or division of complex numbers is carried out using the exponential or polar form, the amounts have to be multiplied and the angles added. With division, the amounts are divided and the angles subtracted.
With a few more calculation steps, the multiplication and division of complex numbers can also be done from the component form or Cartesian notation.
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