Math Problem Statement
Calculate what L network components will match a 42 internal resistance in series with an internal inductance of 9 nH to a 72-2 load impedance in parallel with a stray capacitance of 24 pF at a frequency of 46 MHz.
Solution
To design an L-network that matches a source impedance of (where ) to a load impedance of (where ), we can follow these steps:
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Calculate the Impedances of the Source and Load at 46 MHz:
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Source Impedance:
- Here, .
- The inductive reactance, .
- So, .
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Load Impedance:
- The capacitive reactance, .
- Thus, .
-
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Choosing L-Network Configuration: Given that the source has a real part lower than the load's real part, a "high-pass" L-network configuration (series inductance and parallel capacitance) is suitable.
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Calculate the Required Components:
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Calculate the Q-factor (Q):
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Determine the Series Inductor (L):
-
Determine the Parallel Capacitor (C):
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Summary of Component Values:
- Series Inductor (L):
- Parallel Capacitor (C):
This L-network will match the given source and load impedances at the operating frequency of 46 MHz.
Would you like a detailed explanation of each step or any clarifications?
- How do L-networks differ from Pi-networks in impedance matching?
- What happens if the Q-factor calculated is very low or very high?
- How can you determine if a high-pass or low-pass L-network configuration is needed?
- Why does the operating frequency affect the L and C values in the network?
- How do stray capacitances affect the matching network at high frequencies?
Tip: When designing matching networks, always verify the operating frequency and ensure that component tolerances align with design requirements for accuracy.
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Math Problem Analysis
Mathematical Concepts
Impedance Matching
Complex Impedance
Reactance Calculations
L-Network Design
Formulas
Impedance of inductor: Z = jωL
Impedance of capacitor: Z = -j / ωC
Q-factor: Q = √(R_L/R_s - 1)
Series inductance: X_s = Q * R_s
Parallel capacitance: X_p = R_L / Q
Theorems
Impedance Matching for L-Networks
Reactive Power Compensation
Suitable Grade Level
Undergraduate Engineering
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