Complex Numbers in Electrical Engineering

Complex Numbers and Phasors

• Real numbers are not sufficient to add together resistances, currents, or DC voltages in Electrical Engineering, especially when dealing with frequency-dependent sinusoidal sources and vectors.
• Complex numbers were introduced to allow complex equations to be solved with numbers that are the square roots of negative numbers, √-1, called "imaginary numbers" in Electrical Engineering.
• The letter "j" (j-operator) is used to distinguish an imaginary number from a real number.

Complex Numbers using the Rectangular Form

• A complex number is represented by a real part and an imaginary part, taking the general form of Z = x + jy, where Z is the complex number, x is the real part, y is the imaginary part, and j is defined by √-1.

Addition and Subtraction of Complex Numbers

• Complex numbers can be added or subtracted mathematically or graphically in rectangular form.
• To add or subtract, the real parts are added/subtracted together to form the real part of the sum, and then the imaginary parts are added/subtracted to form the imaginary part of the sum.

Multiplication and Division of Complex Numbers

• The multiplication of complex numbers in rectangular form follows the same rules as normal algebra, along with additional rules for the successive multiplication of the j-operator, where j^2 = -1.
• The multiplication of two complex numbers A and B is performed by multiplying the real and imaginary parts separately and combining them.

Polar Form Representation of a Complex Number

• The polar representation of a point is based on the triangular form, using simple geometry and trigonometry to find both the magnitude and angle of the complex number.
• The relationships between the sides of the triangle can be described using trigonometry and Pythagoras' Theorem.

Converting between Rectangular Form and Polar Form

• The rectangular form expresses a vector in terms of its rectangular coordinates, with the horizontal axis being the real axis and the vertical axis being the imaginary axis or j-component.
• The polar form represents the vector as "A ∠θ", where A is the magnitude and θ is the angle.
• The relationship between rectangular form and polar form can be defined using trigonometric functions.

Polar Form Multiplication and Division

• Polar form is often better for multiplying and dividing complex numbers.
• To multiply two vectors in polar form, multiply the two modulus (magnitudes) and add their angles.
• To divide two vectors in polar form, divide the two modulus and subtract their angles.

RLC Parallel Impedance

• When a single resistance and a single reactance are connected in parallel, the impedance of each parallel branch must be found.
• The impedance of each branch can be found using the standard equation for two resistances in parallel.
• The impedance of a parallel RLC circuit can be found using the combined impedance of the resistance, inductance, and capacitance.