IMCSCH

Returns the cosecant of a complex number.

Syntax

IMCSCH( z )

where z is a complex number

Semantics

IMCSCH( z ) is equivalent to IMDIV(1, IMSINH(z)).

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

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

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

$c = \pm a^{2} + b^{2}$

Application:

AC Circuit Analysis

Problem: We have a list of complex impedances from different components in an AC circuit. We need to calculate the hyperbolic cosecant of each impedance to use in a larger calculation for a filter design.

The hyperbolic cosecant of a complex number z=x+yi is given by:

$csch (z) = \frac{1}{sinh ( z )} = \frac{1}{s i n h ( x ) c o s ( y ) + i c o s h ( x ) s i n ( y )}$

This is a complex calculation to do manually, which is where the IMCSCH function becomes incredibly useful.

Table:

Let's assume our complex impedances are in column A. We will use the IMCSCH function in column B to find the hyperbolic cosecant.

	Impedance (ohms)	Hyperbolic Cosecant
	A	B
1	3+4i	IMCSCH(A1)
2	1-2i	IMCSCH(A2)
3	5+0.5i	IMCSCH(A3)
4	-2+6i	IMCSCH(A4)

Explanation:

Column A: Contains the complex impedances of the circuit components. The values are entered as text strings in the format "a+bi" so the spreadsheet can recognize them as complex numbers.
Column B: Uses the IMCSCH function. For each cell in column B, the formula IMCSCH(A2) (or A3, A4, etc.) takes the complex number from the corresponding cell in column A and calculates its hyperbolic cosecant.

Results:

The spreadsheet would produce the following results:

	Impedance (ohms)	Hyperbolic Cosecant
	A	B
1	3+4i	-0.016335 - 0.057393i
2	1-2i	0.449774 + 0.556277i
3	5+0.5i	0.013478 - 0.001646i
4	-2+6i	0.002826 - 0.038155i

These results, while not intuitive at a glance, are critical inputs for further, more complex calculations in the circuit design process, such as determining voltage, current, or transfer functions at specific frequencies. The IMCSCH function automates a calculation that would be tedious and prone to error if done by hand.