IMEXP

Returns e to the power of a complex number.

Syntax:

IMEXP(complexnumber)

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

IMEXP returns the mathematical constant e raised to the power of complexnumber. The result is a complex number presented as text.

If complexnumber is a+bi, IMEXP(complexnumber) returns $e^{a + bi} = e^{a} (cos b + i s inb)$ .

Example:

IMEXP("1+2i")

returns -1.13120438375681+2.47172667200482i as text.

Application:

Analyzing a RLC Circuit

An RLC circuit is a fundamental circuit in electrical engineering that consists of a resistor (R), an inductor (L), and a capacitor (C). The impedance (Z) of an RLC circuit is a complex number that describes the opposition to alternating current (AC) flow.

Let's consider an AC circuit with the following components:

Resistor (R) = 10Ω
Inductor (L) = 0.05H
Capacitor (C) = 0.0001F
Operating frequency (f) = 60Hz

The angular frequency is ω=2πf=2π(60)≈377rad/s.

The impedance of each component is:

Resistor impedance: ZR=R=10Ω
Inductor impedance: ZL=jωL=j(377)(0.05)=j18.85Ω
Capacitor impedance: $Z_{C} = \frac{1}{jω C} = \frac{- j}{ω C} = \frac{- j}{( 377 ) ( 0.0001 )} \approx - j 26.53Ω$

The total impedance of the series RLC circuit is the sum of these impedances:

$Z_{t o t a l} = Z_{R} + Z_{L} + Z_{C} = 10 + j 18.85 - j 26.53 = 10 - j 7.68Ω$

Now, let's say we want to find the complex exponential of this impedance, which can be a step in a more complex calculation, such as finding the transient response of the circuit or analyzing its behavior in a different mathematical domain.

Using the IMEXP function in a spreadsheet, we can find the exponential of this complex impedance.

Table Example

	Value	Formula	Description
	A	B	C
1	10		Real part of impedance (R)
2	-7.68		Imaginary part of impedance (XL−XC)
3	10-7.68i	COMPLEX(A1, A2)	Complex number representing the impedance
4	0.0007 + 0.0009i	IMEXP(A3)	The exponential of the complex impedance

Explanation of the IMEXP function in the table: The formula IMEXP(A3) takes the complex number in cell A3, which is 10−7.68i, and calculates its exponential using the formula e10−7.68i.

$e^{10 - 7.68 i} = e^{10} (cos (- 7.68) + i sin (- 7.68))$

$e^{10} \approx 22026.47$

$cos (- 7.68) \approx 0.000032$

$sin (- 7.68) \approx - 0.999999$

$e^{10 - 7.68 i} \approx 22026.47 (0.000032 - i \cdot 0.999999)$

$e^{10 - 7.68 i} \approx 0.7048 - i \cdot 22026.45$

The result, a new complex number, could then be used in further calculations, such as finding the complex-valued transfer function of the circuit, which describes how the circuit's output voltage or current changes with respect to its input. In this way, the IMEXP function provides a direct and efficient way to perform a fundamental operation on complex numbers that are central to the analysis of AC circuits.