COMPLEX

Returns a complex number, given real and imaginary parts.

Syntax:

COMPLEX(realpart, imaginarypart, suffix)

returns a complex number as text, in the form a+bi or a+bj.

realpart and imaginarypart are numbers. suffix is optional text i or j (in lowercase) to indicate the imaginary part of the complex number; it defaults to i.

Example:

COMPLEX(3, 4, "i")

returns the text 3+4i.

COMPLEX(2, -3, "j")

returns the text 2-3j.

Application:

A real-world example of the COMPLEX function is in electrical engineering, particularly in the analysis of alternating current (AC) circuits. In AC circuits, voltage and current are not simply scalar values but have both magnitude and phase. Complex numbers provide a convenient way to represent these quantities.

The COMPLEX function, in this context, could be used to create a complex number representing the impedance of a component. Impedance (Z) is the total opposition a circuit presents to alternating current. It has a real part, resistance (R), and an imaginary part, reactance (X). Reactance is further divided into inductive reactance (XL) and capacitive reactance (XC).

Let's consider a series RLC circuit (a circuit with a resistor, an inductor, and a capacitor in series). The total impedance (Z) is given by:

$Z = R + j (X_{L} - X_{C})$

where j is the imaginary unit ( $- 1$ ), which is used in electrical engineering instead of i to avoid confusion with current.

The COMPLEX function can be used to calculate this impedance. The real part of the complex number would be the resistance (R), and the imaginary part would be the net reactance (XL−XC).

Here is a table demonstrating this with some example values:

	Component	Resistance (R) (Ω)	Inductive Reactance (XL) (Ω)	Capacitive (XC) (Ω)	Real Part (for COMPLEX function)	Imaginary Part (for COMPLEX function)	COMPLEX Function Output (Impedance, Z)
	A	B	C	D	E	F	G
1	Circuit 1	50	30	10	50	20	50+20j
2	Circuit 2	100	50	80	100	-30	100-30j
3	Circuit 3	25	20	40	25	-20	25-20j

In this table:

The Real Part is the resistance (R).
The Imaginary Part is the net reactance (XL−XC).
The COMPLEX Function Output is the impedance (Z), which is a complex number.

This complex number representation of impedance is crucial for calculating other quantities in the circuit, such as the total current, which is found by dividing the complex voltage by the complex impedance.