IMAGINARY

Returns the imaginary part of a complex number.

Syntax:

IMAGINARY(complexnumber)

complexnumber is text representing a complex number, for example as a+bi or a+bj, where a is the real part and b the imaginary part.

IMAGINARY returns the imaginary part as a number.

Example:

IMAGINARY("4+3i")

returns 3.

Application:

AC Circuit Analysis

In alternating current (AC) circuits, electrical engineers use complex numbers to simplify calculations. The impedance (Z) of a circuit component, which is a measure of its opposition to the flow of current, is often represented as a complex number.

The complex number for impedance is in the form of Z=R+jX, where:

R is the resistance (the real part), which dissipates energy.
X is the reactance (the imaginary part), which stores and releases energy (in capacitors and inductors).
j is the imaginary unit, which is used instead of i to avoid confusion with the symbol for current.

The IMAGINARY function, in this context, would return the reactance (X), which is crucial for understanding how the circuit behaves. For example, a positive reactance means the circuit is inductive, while a negative reactance means it's capacitive.

Consider a series of circuit components with different impedances. We can use the IMAGINARY function to determine the reactance of each component.

	Component	Impedance (Z)	Real Part (Resistance, R)	Imaginary Part (Reactance, X)	IMAGINARY(Z)
	A	B	C	D	E
1	Resistor	20+0j Ω	20 Ω	0 Ω	0 Ω
2	Inductor	10+5j Ω	10 Ω	5 Ω	5 Ω
3	Capacitor	15−8j Ω	15 Ω	-8 Ω	-8 Ω
4	Combined	45−3j Ω	45 Ω	-3 Ω	-3 Ω

Explanation:

Resistor: A pure resistor has zero reactance, so its impedance is a purely real number. The imaginary part is 0.
Inductor: An inductor's impedance has a positive imaginary part, indicating it adds inductive reactance to the circuit. The IMAGINARY function returns 5 Ω.
Capacitor: A capacitor's impedance has a negative imaginary part, indicating it adds capacitive reactance. The IMAGINARY function returns -8 Ω.
Combined: When the components are in series, their impedances are added together. The total impedance is (20+10+15)+(0+5−8)j=45−3j. The IMAGINARY function for the total impedance returns -3 Ω.

By using the IMAGINARY function, an engineer can quickly isolate the reactance of the entire circuit. A negative result, like in this example, tells the engineer that the overall circuit is capacitive. This information is vital for designing and troubleshooting the circuit, as the overall reactance determines its frequency response and power factor.