Find the characteristic impedance of a loss-less coaxial cable in which the diameter of the inner conductor is one-third of that of the inner surface of the outer conductor. If the voltage standing wave ratio in the cable is \(2.5\), what is the resistance of the termination?

When we talk about transmission line theory, we refer to the analysis and understanding of how electrical signals travel through cables such as coaxial cables. A transmission line is defined by parameters like resistance, inductance, capacitance, and conductance per unit length that characterize how the signal degrades as it travels. In a perfect world, we'd want our signals to travel without any loss, meaning the signal at the end of the cable would be identical to the signal at the start.

But, reality introduces reflections, standing waves, attenuation, and distortion. To handle these, we use the characteristic impedance, denoted as \(Z_0\), a complex number that represents the ratio of the voltage to the current of a signal traveling along the line. When the line is 'loss-less', it means the signal can travel without experiencing resistive losses, attributing any impedance purely to the line's inherent inductance and capacitance.

Understanding transmission line theory allows us to design systems that better mitigate signal losses and reflections, ensuring more efficient and reliable communication.

The voltage standing wave ratio (VSWR) is a measure of how efficiently RF power is transmitted from the cable to the load, or in this case, the antenna or device at the end of the cable. A VSWR of 1:1 is ideal, indicating that no power is reflected and all is absorbed by the load. A higher VSWR indicates reflections are occurring due to an impedance mismatch, meaning some of the power is being reflected back towards the source.

As the VSWR increases, it implies a greater mismatch and increased reflections, which can lead to reduced transmission efficiency and potentially damage sensitive equipment. Thus, maintaining a low VSWR is important for system performance. Coaxial cables in practical use often have a VSWR greater than 1:1 but strive to keep it as low as possible to ensure maximum power delivery.

The reflection coefficient, commonly described using the Greek letter \(\rho\), quantifies the amount of signal that is reflected back down a transmission line when it encounters a termination or load with an impedance different from the line's characteristic impedance. If the termination impedance, \(Z_L\), matches the characteristic impedance \(Z_0\), \(\rho\) equals zero, indicating no reflections and that signal power is optimally transferred.

If \(\rho\) is nonzero, power is reflected, causing standing waves. The magnitude of \(\rho\) ranges between -1 and 1, where a value of 1 represents total reflection, and -1 would mean a total reversal of the wave. By calculating the reflection coefficient using \(Z_0\) and \(Z_L\), we can assess the efficiency of power transfer and design better systems to reduce power loss and potential interference caused by reflections.

Inductance (\(L_0\)) and capacitance (\(C_0\)) per unit length are fundamental properties of transmission lines that influence their characteristic impedance (\(Z_0\)). Inductance arises from the magnetic field that forms around a conductor carrying a current, while capacitance is related to the ability of the line to store charge.

Both quantities depend on the physical characteristics of the line. For example, in a coaxial cable, the inductance per unit length increases with the natural log of the ratio of the outer to inner conductor diameters, while the capacitance per unit length decreases as this logarithmic ratio increases.

Having a precise understanding of these properties is essential for designing cables with the desired impedance and frequency characteristics, ensuring that signals are carried efficiently with minimal reflection or loss.