Complex Numbers in Electrical Engineering

Study Notes

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.

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.

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.

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.

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.

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 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.

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.