IMSQRT

Returns the square root of a complex number.

Syntax:

IMSQRT(complexnumber)

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

IMSQRT returns the square root of complexnumber as text. That is, if complexnumber is $a_{0} + b_{0} i$ , it returns $a_{1} + b_{1} i$ , such that $(a_{1} + b_{1} i) * (a_{1} + b_{1} i) = a_{0} + b_{0} i$ .

Example:

IMSQRT("16+30i")

returns 5+3i as text.

Application:

An application of the IMSQRT function can be found in electrical engineering, specifically in the analysis of AC (Alternating Current) circuits. In AC circuits, components like resistors, inductors, and capacitors all contribute to a circuit's impedance, which is a measure of the opposition to the flow of current. Unlike resistance in a DC circuit, impedance in an AC circuit is a complex number because it has both a real part (resistance) and an imaginary part (reactance).

The square root of impedance can be used in various calculations, such as determining the characteristics of a filter or a transmission line. For example, the characteristic impedance (Z0) of a transmission line can be calculated using the square root of a product of two complex numbers: series impedance (Z) and shunt admittance (Y).

Let's consider a simplified example where we need to find a value that, when squared, gives a certain complex impedance.

Suppose a component in an AC circuit has a complex impedance of 3+4i ohms, where 3 is the resistance and 4 is the inductive reactance. We need to find a related complex value, which we can call Vsqr, that is the square root of this impedance. This calculation might be a step in a larger analysis to find a specific circuit parameter.

Table: Calculating the Square Root of Complex Impedance

	Component	Complex Impedance (Z)	Formula	Result ( $V_{s q r} = Z$ )
	A	B	C	D
1	Inductive Circuit	3+4i	IMSQRT("3+4i")	2+i
2	Capacitive Circuit	5-12i	IMSQRT("5-12i")	3-2i
3	Purely Inductive Circuit	0+9i	IMSQRT("0+9i")	2.12132+2.12132i
4	Purely Resistive Circuit	16	IMSQRT("16")	4
5	Purely Resistive (Negative)	-9	IMSQRT("-9")	0+3i

Explanation:

Inductive Circuit ("3+4i"): The function returns "2+i". We can verify this by squaring the result: (2+i)2=22+2(2)(i)+i2=4+4i−1=3+4i.
Capacitive Circuit ("5-12i"): The function calculates the square root of a complex number with a negative imaginary part, which is common for capacitive reactance.
Purely Inductive Circuit ("0+9i"): Even without a real part, the function correctly calculates the complex square root.
Purely Resistive Circuit ("16"): The IMSQRT function can also handle real numbers (complex numbers with an imaginary part of zero), returning the expected real square root.
Purely Resistive (Negative) ("-9"): This is a classic example where a standard SQRT function would produce an error. The IMSQRT function correctly handles it, returning the imaginary number "3i" (or "0+3i").