A standing wave ratio of \(2.5\) exists on a lossless \(60 \Omega\) line. Probe measurements locate a voltage minimum on the line whose location is marked by a small scratch on the line. When the load is replaced by a short circuit, the minima are \(25 \mathrm{~cm}\) apart, and one minimum is located at a point \(7 \mathrm{~cm}\) toward the source from the scratch. Find \(Z_{L}\).

In the context of transmission lines, the reflection coefficient, represented as \( \Gamma \), is a measure of how much of an electromagnetic wave is reflected by an impedance discontinuity in the transmission line. The reflection can occur at the interface between the line and the load when there's a mismatch between the load impedance and the characteristic impedance of the line.

The reflection coefficient is calculated using the formula: \[\Gamma = \frac{SWR - 1}{SWR + 1}\] where SWR stands for the Standing Wave Ratio. A reflection coefficient of 1 indicates total reflection (open or short circuit) and a coefficient of 0 indicates no reflection, which means the load is perfectly matched.

In our exercise, an SWR of 2.5 leads to a reflection coefficient of 0.4286, indicating there's a significant mismatch between the load and the characteristic impedance, causing a substantial amount of the wave to be reflected back.

Normalized load impedance (\(z_L\)) is a dimensionless quantity that represents the load impedance relative to the characteristic impedance of the line \(Z_0\). It is used to simplify the analysis of transmission lines by scaling impedances so that the characteristic impedance is unity.

To find the normalized load impedance, we use the reflection coefficient obtained from the SWR with the formula: \[z_L = \frac{1 + \Gamma}{1 - \Gamma}\]. This formula calculates how much the actual load impedance (\(Z_L\)) deviates from the characteristic impedance.

When we plug our \( \Gamma \) into this formula, we get a normalized load impedance of 2.5, implying that the load impedance is 2.5 times greater than the characteristic impedance of the line.

Characteristic impedance (\(Z_0\)) of a transmission line is a key parameter that dictates how electrical signals are conducted by the line. It is a measure of the inherent resistance to the flow of electrical current and is determined by the physical construction of the line, including the dimensions and materials used. \(Z_0\) provides a balance between electric and magnetic fields propagating the wave along the line.

In our problem, the line has a characteristic impedance of 60 ohms. With the normalized load impedance of 2.5 and the actual load impedance formula (\(Z_L = z_L * Z_0\)), the true load impedance comes out to be 150 ohms. This indicates that the load is higher than what the line 'expects', leading to inefficiencies such as reflections and non-optimal power transfer, which are depicted by the standing wave pattern measured on the line.