IMTAN

Returns the tangent of a complex number.

Syntax

IMTAN( z )

where z is a complex number

Semantics

IMTAN( z ) is equivalent to IMDIV(IMSIN( z ), IMCOS( z )).

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

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

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

Application:

An application of the IMTAN function can be found in electrical engineering, specifically in the analysis of alternating current (AC) circuits. In AC circuits, components like resistors, capacitors, and inductors have impedances that are often represented by complex numbers. The impedance is a measure of the opposition that a circuit presents to a current when a voltage is applied.

Let's consider an example of a series RLC circuit, which consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series. The total impedance (Z) of this circuit is the sum of the individual impedances of the components:

$Z = R + j X_{L} - j X_{C}$

where:

R is the resistance (real part).
XL=ωL is the inductive reactance.
XC=1/ωC is the capacitive reactance.
ω=2πf is the angular frequency.
j is the imaginary unit (often denoted as i in mathematics).

The phase angle (ϕ) of the impedance is a crucial parameter that describes the phase difference between the voltage and the current in the circuit. The tangent of this phase angle is given by the ratio of the imaginary part to the real part of the impedance:

$tan (ϕ) = \frac{Imaginary part ( Z )}{Real part ( Z )} = \frac{X _{L} - X _{C}}{R}$

However, what if we are working with a more complex scenario where the impedance itself is not a simple real number but a complex number? For instance, in a more advanced analysis or with specific types of components, the impedance might be given as a complex value. Let's say we have a specific circuit component whose impedance is given by the complex number Z=50+j25 Ohms.

We can use the IMTAN function to find the tangent of this complex impedance. While the direct physical meaning of tan(Z) might not be as intuitive as the phase angle, this calculation can be a step in a larger mathematical model used for circuit design, filter analysis, or signal processing. For example, it might be used in the calculation of reflection coefficients or in network analysis where complex functions are applied to complex impedances.

Here's a table demonstrating how to use the IMTAN function with this example:

	Component	Value	Formula	Result
	A	B	C	D
1	Complex Impedance (Z)	50+j25	COMPLEX(50, 25)	50+25i
2	Tangent of Impedance		IMTAN("50+25i")	1.00288−0.00516i

Explanation of the table:

Complex Impedance (Z): This is the input complex number. In a spreadsheet, you would typically create this complex number using a function like COMPLEX(real_part, imaginary_part). In this case, COMPLEX(50, 25) represents the impedance 50+25i.
Tangent of Impedance: This is where the IMTAN function is used. The formula IMTAN("50+25i") calculates the tangent of the complex number 50+25i. The result is another complex number, approximately 1.00288−0.00516i.

The result, a complex number itself, is a mathematical output used in further calculations within the electrical engineering model, such as determining the overall system response, stability, or other frequency-dependent characteristics. The IMTAN function simplifies a complex trigonometric calculation that would otherwise be very cumbersome to perform manually.