# Dispersion relation of a transmission line - questions

• I
• Matej Kurtulik
In summary, the conversation discusses the dispersion relation of a transmission line with the given formula ω=sin(kx). The first question is how to control the value of k inside the line, and the second question is if it is possible to generate a wavelength instead of a frequency. The only way to change k is by varying ω, and this phenomenon is known as aliasing. This happens due to the discrete or sampled nature of the system, where only a finite number of points are defined for voltage. Therefore, the upper degenerate part of the k spectrum is not considered, and the argument of sin is limited to the range of 0 to π/2, called the "reduced zone."

#### Matej Kurtulik

Hi,

I have transmission line with dispersion relation ω=sin(kx), so then means that for one value of ω I have two values of k. I apply voltage with some frequency with is allowed to move in the line. First question is, how can I influence what k will be generated inside the line. The another question is, if there is somehow possible to generate a wave not with generating frequency ω like in the first example but to generate wavelength, basically k.

Thank you

The only way to change k without changing the line construction is to vary ω.

and I see from dispersion formula that each ω has two k, so what key will be inside the line, and how can I switch is for another one.

Are you using a lumped-element model (discrete L's and C's) for the transmission line?

Yes

You are seeing the effects of aliasing, as it is called in digital signal processing. It arises in any discrete or sampled system. In the present case, you can't see a continuous sine wave travel down your transmission line model because voltage is defined (sampled) at only a finite number of points--namely across each capacitor, if it's a low-pass transmission line. For spatial frequencies above a certain maximum k_0, the pattern of sampled voltages looks the same as that for a lower frequency--that is, two values of k appear to have the same ω. As a result, there is no point in considering the upper, degenerate part of the k spectrum. It is conventional to limit the argument of sin to the range 0 to π/2. This is called the "reduced zone."

## 1. What is a dispersion relation?

A dispersion relation is a mathematical relationship that describes how waves propagate through a medium. In the context of a transmission line, it shows the relationship between the frequency of the input signal and the resulting wave's wavelength, phase velocity, and attenuation.

## 2. How is the dispersion relation of a transmission line calculated?

The dispersion relation of a transmission line is calculated using the transmission line equations, which take into account the line's physical properties such as its length, impedance, and capacitance. These equations can be solved using techniques such as the telegrapher's equations or the Smith chart.

## 3. What factors affect the dispersion relation of a transmission line?

The dispersion relation of a transmission line is affected by several factors, including the line's physical properties, such as its length and impedance, as well as the material properties of the line, such as its dielectric constant. Additionally, the type of wave being transmitted, such as a single-mode or multi-mode wave, can also impact the dispersion relation.

## 4. What is the significance of the dispersion relation for signal transmission?

The dispersion relation is crucial for understanding how signals propagate through a transmission line. It helps determine the frequency response of the line, which is essential for maintaining signal integrity and minimizing distortion. By analyzing the dispersion relation, engineers can design transmission lines that are optimized for specific applications and frequencies.

## 5. How does the dispersion relation affect the bandwidth of a transmission line?

The dispersion relation has a direct impact on the bandwidth of a transmission line. A wider bandwidth is achieved when the dispersion relation is flatter, meaning that the phase velocity of the transmitted signal remains relatively constant over a range of frequencies. A narrow bandwidth, on the other hand, is the result of a steep dispersion relation, which causes the phase velocity to vary drastically with frequency.