j-operator-and-operation-of-complex-numbers.docx

**Operation of Complex Numbers** - **Complex Numbers and Phasors** - The mathematics used in Electrical Engineering to add together resistances, currents or DC voltages use what are called real numbers. But real numbers are not the only kind of numbers we need to use especially whe...

**Operation of Complex Numbers** - **Complex Numbers and Phasors** - The mathematics used in Electrical Engineering to add together resistances, currents or DC voltages use what are called real numbers. But real numbers are not the only kind of numbers we need to use especially when dealing with frequency dependent sinusoidal sources and vectors. As well as using normal or real numbers, **Complex Numbers** were introduced to allow complex equations to be solved with numbers that are the square roots of negative numbers, √-1. - In electrical engineering this type of number is called an "imaginary number" and to distinguish an imaginary number from a real number the letter " j " known commonly in electrical engineering as the **j-operator**, is used. Thus the letter "j" is placed in front of a real number to signify its imaginary number operation. **Vector Rotation of the j-operator** **Complex Numbers using the Rectangular Form** - In the last tutorial about **Phasors** we saw that a complex number is represented by a real part and an imaginary part that takes the generalised form of:  Where:   Z  --  is the Complex Number representing the Vector   x  --  is the Real part or the Active component   y  --  is the Imaginary part or the Reactive component    j  --  is defined by √-1 **Addition and Subtraction of Complex Numbers** - The addition or subtraction of complex numbers can be done either mathematically or graphically in rectangular form. For addition, the real parts are firstly added together to form the real part of the sum, and then the imaginary parts to form the imaginary part of the sum and this process is as follows using two complex numbers A and B as examples.  - Two vectors are defined as, A = 4 + j1 and B = 2 + j3 respectively. Determine the sum and difference of the two vectors in both rectangular ( a + jb ) form and graphically as an Argand Diagram.  **Multiplication and Division of Complex Numbers** - The multiplication of complex numbers in the rectangular form follows more or less the same rules as for normal algebra along with some additional rules for the successive multiplication of the j-operator where: j^2^ = -1. So for example, multiplying together our two vectors from above of A = 4 + j1 and B = 2 + j3 will give us the following result.  **Polar Form Representation of a Complex Number** - As the polar representation of a point is based around the triangular form, we can use simple geometry of the triangle and especially trigonometry and Pythagoras's Theorem on triangles to find both the magnitude and the angle of the complex number. Remember that, trigonometry deals with the relationship between the sides and the angles of triangles so we can describe the relationships between the sides as:  **Converting between Rectangular Form and Polar Form** - In the rectangular form we can express a vector in terms of its rectangular coordinates, with the horizontal axis being its real axis and the vertical axis being its imaginary axis or j-component. In polar form these real and imaginary axes are simply represented by "A ∠θ". Then using our example above, the relationship between rectangular form and polar form can be defined as.  **Polar Form Multiplication and Division** - Rectangular form is best for adding and subtracting complex numbers as we saw above, but polar form is often better for multiplying and dividing. To multiply together two vectors in polar form, we must first multiply together the two modulus or magnitudes and then add together their angles. - **Division in Polar Form** - Likewise, to divide together two vectors in polar form, we must divide the two modulus and then subtract their angles as shown.  **RLC Parallel Impedance** **Parallel Impedances** - If a single resistance and a single reactance are connected together in parallel, the impedance of each parallel branch must be found. But as there are only two components in parallel, R and X, we can use the standard equation for two resistances in parallel. - It is given as: R~T~ = (R~1~\*R~2~)/(R~1~ + R~2~). **Resistance and Inductance in Parallel**  **Resistance and Capacitance in Parallel** **Resistance, Inductance and Capacitance in Parallel**