Impedance at Points of Voltage Maxima and Minima on a Lossless Line

Introduction:
In the study of transmission lines, the voltage standing wave ratio (VSWR) is an important parameter that characterizes the behavior of the line. The VSWR is the ratio of the maximum voltage to the minimum voltage along the line. In the case of a lossless line with a characteristic impedance R0, we can determine the impedance at the points of voltage maxima and minima.

Explanation:
To understand why the impedance at the points of voltage maxima and minima is given by option 'A' (SR0 and R0/S), let's consider the behavior of the voltage and current along the transmission line.

1. Voltage and Current Distribution:
- When a signal travels along a lossless transmission line, it experiences reflections at the line's ends due to impedance mismatch.
- These reflections result in the formation of standing waves along the line, with voltage and current nodes and antinodes.
- At voltage maxima, the voltage reaches its highest positive peak, while at voltage minima, the voltage reaches its lowest negative peak.
- The voltage and current distribution along the line can be described by the voltage and current standing wave patterns.

2. Relationship between VSWR and Impedance:
- The VSWR is defined as the ratio of the maximum voltage (Vmax) to the minimum voltage (Vmin) along the transmission line.
- VSWR = (Vmax / Vmin)
- The VSWR can also be expressed in terms of impedance as VSWR = (Zmax / Zmin), where Zmax and Zmin are the maximum and minimum impedances along the line, respectively.

3. Impedance at Voltage Maxima and Minima:
- At the points of voltage maxima, the voltage is maximum, and the current is minimum. This implies that the impedance at voltage maxima (Zmax) is equal to the characteristic impedance of the line (R0) multiplied by the VSWR (S).
Zmax = SR0
- Similarly, at the points of voltage minima, the voltage is minimum, and the current is maximum. This implies that the impedance at voltage minima (Zmin) is equal to the characteristic impedance of the line (R0) divided by the VSWR (S).
Zmin = R0/S

Conclusion:
In a lossless transmission line with a characteristic impedance R0 and VSWR = S, the impedance at the points of voltage maxima and minima are given by SR0 and R0/S, respectively (option 'A'). This relationship is derived from the behavior of voltage and current standing waves along the line. Understanding these relationships is crucial in the design and analysis of transmission lines in various electrical and electronic systems.