IMCOSH

Application:

Analyzing an Electrical Transmission Line

Consider a long electrical transmission line. The voltage and current at any point along the line can be described by a set of complex numbers that depend on the distance from the source. The relationship between the input voltage and current (VS​, IS​) and the output voltage and current (VR​, IR​) is often described by a set of equations that use hyperbolic functions.

The hyperbolic cosine of a complex number can be used to model the voltage distribution along the line. Let's say we have a complex number, z=γl, where:

• γ is the complex propagation constant of the transmission line.
• l is the length of the transmission line.

The propagation constant γ is a complex number that accounts for the line's properties, such as resistance, inductance, capacitance, and conductance. It's often expressed as γ=α+iβ, where:

• α is the attenuation constant, which represents the signal loss.
• β is the phase constant, which represents the change in phase of the signal.

The voltage at the sending end of the transmission line can be related to the voltage at the receiving end by an equation that includes cosh(γl). In a simplified case, if the line is terminated with a specific impedance, the voltage at the sending end (VS​) is given by:

$V_S=V_R\cosh (\gamma l)+I_R{Z}_C\sinh (\gamma l)$

where VR​ and IR​ are the voltage and current at the receiving end, and ZC​ is the characteristic impedance of the line.

The IMCOSH function would be used to calculate the value of cosh(γl) when the propagation constant γ and the line length l result in a complex argument.

Example Calculation

Let's assume a transmission line with the following parameters:

• Propagation constant γ=0.001+0.02i
• Line length l=100 km

The complex number for which we need to find the hyperbolic cosine is γl: z=γl=(0.001+0.02i)∗100=0.1+2i

Now, we can use the IMCOSH function to find the hyperbolic cosine of 0.1+2i. The formula for the hyperbolic cosine of a complex number x+yi is:

cosh(x+yi)=cosh(x)cos(y)+isinh(x)sin(y)

Let's apply this to our example, where x=0.1 and y=2 (in radians):

• cosh(0.1)≈1.005004
• cos(2)≈−0.416147
• sinh(0.1)≈0.100167
• sin(2)≈0.909297

cosh(0.1+2i)=(1.005004)(−0.416147)+i(0.100167)(0.909297) cosh(0.1+2i)≈−0.41804+0.09109i

Table: IMCOSH Function for a Transmission Line Model