IMARGUMENT

Returns the argument of a complex number.

Syntax:

IMARGUMENT(complexnumber)

complexnumber is text representing a complex number, for example as a+bi or a+bj.

IMARGUMENT returns the argument of complexnumber in radians - that is, in a polar representation, the angle relative to the horizontal axis. For a complex number $a + bi = r (cos φ + i s in φ)$ the argument is $φ$ .

Example:

IMARGUMENT("3+3i")

returns 0.785398163397 - that is $\frac{2 π}{8}$ radians.

Application:

Calculating the Phase Angle of Circuit Impedance

Let's consider a simple AC circuit with a resistor and an inductor in series. The impedance of this circuit, which is the total opposition to the flow of current, is a complex number.

The resistance, R, is the real part of the impedance.
The inductive reactance, XL, is the imaginary part of the impedance.

The total impedance, Z, is given by Z=R+jXL. Here, j is used to denote the imaginary unit in electrical engineering (instead of i) to avoid confusion with current.

The phase angle, θ, of the impedance tells us the phase difference between the voltage across the circuit and the current flowing through it. We can find this angle using the IMARGUMENT function.

Let's say we have a circuit with the following values:

Resistance (R): 30 ohms
Inductive Reactance (XL): 40 ohms

The complex impedance, Z, is 30+40j.

The formula would be IMARGUMENT("30+40i").

Here's how a table would look:

	Circuit Component	Value (ohms)	Complex Number Representation	IMARGUMENT Formula	Result (radians)	Result (degrees)
	A	B	C	D	E	F
1	Resistance (R)	30	30	IMARGUMENT("30")	0.0000	0.00°
2	Inductive Reactance (XL)	40	40i	IMARGUMENT("40i")	1.5708	90.00°
3	Total Impedance (Z)		30+40i	IMARGUMENT("30+40i")	0.9273	53.13°

Analysis of the Results

Resistance: The impedance is purely real (30 + 0i), so the phase angle is 0. This makes sense because voltage and current are in phase for a resistor.
Inductive Reactance: The impedance is purely imaginary (0 + 40i), so the phase angle is approximately 1.5708 radians or 90°. This is a fundamental property of inductors: the voltage leads the current by 90°.
Total Impedance: The total impedance is a combination of both. The IMARGUMENT function calculates the phase angle of 0.9273 radians (or 53.13°). This tells us that in this specific series R-L circuit, the voltage leads the current by a phase angle of 53.13°. This information is vital for power calculations and for designing the circuit to operate correctly at a given frequency.