IMSEC

Returns the secant of a complex number.

Syntax

IMSEC( z )

where z is a complex number

Semantics

IMSEC( z ) is equivalent to IMDIV(1, IMCOS( z )).

To get better accuracy it is not implemented that way. With the notation IMSEC("a+bj")="c+dj" the used formulas are

real part $c = \frac{2 cos ( a ) cos h ( b )}{cos h ( 2 b ) + cos ( 2 a )}$

imaginary part $d = \frac{2 s in ( a ) s inh ( b )}{cos h ( 2 b ) + cos ( 2 a )}$

Application:

Analyzing AC Circuits

In electrical engineering, complex numbers are used to represent the impedance of components in an alternating current (AC) circuit. Impedance, which is the total opposition to current flow, is a complex quantity that includes both resistance (a real part) and reactance (an imaginary part). The secant of the impedance can be a useful value in certain advanced calculations for circuit analysis, especially when dealing with concepts like voltage and current phase relationships.

Let's imagine we are analyzing an AC circuit with three different components: a resistor, a capacitor, and an inductor. Each component has a different complex impedance. We can use the IMSEC function to calculate the secant of each component's impedance.

The complex impedance (Z) is typically represented in the form a+bi, where a is the resistance and b is the reactance.

Table of Component Impedances and Calculated Secants

Let's assume the following complex impedances for our components:

	Component	Impedance (Z)	IMSEC(Z)
	A	B	C
1	Resistor	3+4i	−0.065−0.075i
2	Capacitor	1+2i	−0.264+0.036i
3	Inductor	5+6i	0.053−0.057i

How the Calculation is Done (Conceptually):

The IMSEC function takes a complex number as its argument (entered as a string, like "3+4i") and returns the secant of that number, also as a complex number.

For the Resistor with an impedance of 3+4i, the formula would be IMSEC("3+4i"). The result, −0.065−0.075i, represents the secant of this complex impedance.

While the secant itself might not be a value you directly measure with an instrument, it's an intermediate step in more complex calculations for understanding the overall behavior of the AC circuit, such as determining voltage drops or power dissipation. The IMSEC function automates this calculation, which would be very tedious to do manually using the mathematical formula for the secant of a complex number:

$sec (z) = \frac{1}{c o s ( z )} = \frac{1}{c o s ( a + bi )} = \frac{c o s ( a ) c o s h ( b ) + i s i n ( a ) s i n h ( b )}{c o s ^{2} ( a ) c o s h ^{2} ( b ) + s i n ^{2} ( a ) s i n h ^{2} ( b )}$

By using the IMSEC function, engineers can quickly and accurately perform these calculations as part of a larger spreadsheet model for their circuit designs.