Consider the following statements regarding transmission lines : 1. In single-stub matching, the combination of stub and line presents a conductance which is equal to twice of the characteristic conductance of the line. 2. Single-stub matching is useful for all frequencies because the position of the stub can be varied with the variation in frequency. 3. Single-stub matching system is a narrowband system. Which of the above statements are not correct?
Transmission Line Matching Solution
Statement 1 Analysis: Conductance Value
Single-stub matching aims to transform the load impedance ($Z_L$) to the characteristic impedance ($Z_0$) of the transmission line. Using normalized admittances ($y = Y/Y_0$), the goal is to achieve an input admittance of $y_{in} = 1 + j0$.
This implies the input conductance ($g_{in}$) required for matching should be normalized $1$. Statement 1 asserts that the combination of the stub and line presents a conductance equal to twice the characteristic conductance (normalized $g_{in} = 2$). This contradicts the fundamental objective of impedance matching.
Therefore, statement 1 is incorrect.
Statement 2 Analysis: Frequency Dependence
The position ($d$) of the stub along the transmission line is determined by the wavelength ($\lambda$) at the operating frequency ($f$). The electrical length is given by $\beta d$, where $\beta$ is the phase constant. Since $\lambda = c/f$, the required position $d$ is intrinsically dependent on the frequency.
- A change in frequency modifies the wavelength ($\lambda$) and consequently the electrical distance ($\beta d$).
- A single-stub configuration optimized for a specific frequency ($f_0$) will typically not provide a satisfactory match at a different frequency ($f \neq f_0$).
While the stub's position can be physically adjusted, this adjustment is necessary for each frequency change. It does not imply the system is inherently effective across "all frequencies".
Therefore, statement 2 is incorrect.
Statement 3 Analysis: Bandwidth
Single-stub matching networks are designed and optimized for a single, specific frequency. The required line lengths and stub positions are calculated based on the wavelength pertinent to that frequency.
When the operating frequency deviates from the design frequency, the impedance transformations change, and the matching condition is usually no longer met. This inherent sensitivity to frequency variations characterizes single-stub matching as a narrowband technique.
Therefore, statement 3 is correct.
Conclusion
The analysis reveals that statements 1 and 2 are not correct because they misrepresent the goals and characteristics of single-stub matching concerning conductance values and frequency applicability. Statement 3 accurately describes the narrowband nature of this technique.
The question asks to identify the statements that are not correct.
Final Answer Selection
Statements 1 and 2 are identified as incorrect.
This corresponds to Option A.
Important Questions from Transmission Lines
A characteristic impedance does NOT satisfy which of the following statements?
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