Introduction / Context:
In transmission line theory, when the load is mismatched with the line impedance, standing waves are produced. Matching networks, including reactive stubs, are used to cancel out the reactive component and achieve maximum power transfer. This question deals with stub matching for a coaxial line terminated with a complex load.
Given Data / Assumptions:
- Characteristic impedance of line Z0 = 75 Ω.
- Load impedance ZL = 75 − j40 Ω.
- Operating frequency = 6 MHz (for context, but not essential in the calculation).
- Goal: Eliminate reflections by matching ZL to Z0.
Concept / Approach:
The load has the correct real part (75 Ω) but contains an excess capacitive reactance (−j40). To cancel this, a reactive element must be added. Direct addition of inductance at the load could fix it, but practical transmission line methods prefer stub matching. A short-circuited stub, placed at a specific distance from the load, provides adjustable susceptance that can cancel the load’s reactive part, achieving conjugate matching.
Step-by-Step Solution:
Step 1: Express load as ZL = 75 − j40 Ω.Step 2: Compare with Z0 = 75 Ω. The resistive part is already matched.Step 3: The reactive part (−j40) must be neutralized.Step 4: A stub line provides adjustable susceptance. By carefully choosing the stub length and distance from load, the reactance is canceled and matching achieved.Step 5: Hence the correct method is a short-circuit stub at a specific distance.
Verification / Alternative check:
Matching charts like the Smith Chart show that ZL can be rotated toward the real axis using a short-circuited stub, confirming the approach.
Why Other Options Are Wrong:
A: Stub at the load could only cancel reactance, not adjust position optimally.B: Adding inductance directly at load is possible but less practical in RF transmission-line matching.D: Capacitance at distance does not cancel the −j40 reactance; it would worsen mismatch.E: Quarter-wave transformer requires real load impedance transformation, not reactive cancellation.
Common Pitfalls:
Confusing stub placement (direct vs at a distance); forgetting that the load already has matched resistance and only needs reactive compensation.
Final Answer:
a short-circuit stub placed at a specific distance from the load
Discussion & Comments