ACOTH

Application:

Designing a Quarter-Wave Transformer for Impedance Matching

In microwave engineering, a quarter-wave transformer is a common device used to match a load impedance to a transmission line. However, sometimes we need more flexible impedance matching, and a short-circuited stub is used.

Scenario: We have a 50Ω transmission line (Z0​=50Ω). We need to design a short-circuited stub to provide an input impedance of j100Ω at a frequency of 1 GHz. The phase constant β at this frequency is 2π radians/meter.

We can use the formula for the input impedance of a short-circuited stub:

$Z_{in}=j{Z}_0\tan (\beta L)$

This is the tangent function, not the hyperbolic one. However, the hyperbolic tangent and cotangent are often used in transmission line equations, especially for lossy lines. For example, the input impedance of an open-circuited stub is:

$Z_{in}=-j{Z}_0\cot (\beta L)$

The relationship between the hyperbolic and trigonometric functions is: tanh(jx)=jtan(x) and coth(jx)=−jcot(x).

So, a more general expression for the input impedance of an open-circuited stub is:

$Z_{in}=Z_0\coth (\gamma L)$

For a lossless line, γ=jβ, so:

$Z_{in}=Z_0\coth (j\beta L)=Z_0[-j\cot (\beta L)]$

Now let's use the acoth function directly in a less common but valid scenario. The hyperbolic cotangent is related to the impedance of an infinite transmission line. The acoth function is useful for determining the length of a finite transmission line that behaves like an infinitely long one, or to find the length required for a specific impedance transformation.

Let's construct a scenario where the formula $L=\frac{1}{\gamma }\text{acoth}\left(\frac{Z_{in}}{Z_0}\right)$ is directly applicable.

Imagine a lossy transmission line with a propagation constant γ=α+jβ. We want to find the length of a line segment that, when open-circuited, has a specific input impedance Zin​.

The input impedance of an open-circuited line is $Z_{in}=Z_0\coth (\gamma L)$.

To find the length L, we rearrange the equation:

$\frac{Z_{in}}{Z_0}=\coth (\gamma L)$

Taking the inverse hyperbolic cotangent of both sides:

$\text{acoth}\left(\frac{Z_{in}}{Z_0}\right)=\gamma L$

$L=\frac{1}{\gamma }\text{acoth}\left(\frac{Z_{in}}{Z_0}\right)$

Let's use some example values.

Scenario: A lossy transmission line has a characteristic impedance Z0​=75Ω and a propagation constant γ=0.1+j(2π). We want to find the length L of an open-circuited segment that gives an input impedance Zin​=80−j20Ω.

Calculations:

1. Calculate the ratio Zin/Z0:

$\frac{Z_{in}}{Z_0}=\frac{80-j20}{75}\approx 1.0667-j0.2667$

2. Calculate acoth(1.0667−j0.2667):

The acoth function for a complex number z is:

$\text{acoth}(z)=\frac{1}{2}\ln \left(\frac{z+1}{z-1}\right)$

Let z=1.0667−j0.2667.

z+1=2.0667−j0.2667

z−1=0.0667−j0.2667

$\frac{z+1}{z-1}=\frac{2.0667-j0.2667}{0.0667-j0.2667}\approx 1.157+j7.375$

Now, take the natural logarithm of this complex number. Let the number be w=x+jy.

$\ln (w)=\ln (|w|)+j\arg (w)$

$|w|=\sqrt{1.157^2+7.375^2}\approx 7.465$

$\arg (w)=\arctan \left(\frac{7.375}{1.157}\right)\approx 1.414\text{ radians}$

$\ln (w)\approx \ln (7.465)+j(1.414)\approx 2.01+j(1.414)$

3. Finally, calculate acoth(z):

$\text{acoth}(z)=\frac{1}{2}\ln \left(\frac{z+1}{z-1}\right)\approx \frac{1}{2}(2.01+j1.414)=1.005+j0.707$

4. Calculate the length L:

$L=\frac{1}{\gamma }\text{acoth}(z)=\frac{1}{0.1+j(2\pi )}(1.005+j0.707)$

$L=\frac{1.005+j0.707}{0.1+j6.283}\approx 0.113+j0.149$

The length of the transmission line is the real part of this result, so L≈0.113 meters, or 11.3 cm. The imaginary part indicates a phase shift, which is naturally accounted for in the propagation constant.

This example demonstrates how the acoth function is a powerful tool for inverse problems in transmission line theory, allowing engineers to determine physical dimensions (like length) from electrical properties (like impedance).