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Microwave ResonatorsWhat is resonance?1. The natural modes of a system.
a. Metallic cavity.b. A long beam.c. Musical instrument.d. LC circuit.
2. Self-sustained if lossless.3. Energy grows to infinity if fed by a source which has a
spectrum containing the resonant frequency if lossless.
Quality Factor
Series Resonant Circuit
Power Loss:
Average Stored Magnetic Energy:
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Average Stored Electric Energy:
Resonant Frequency:
Quality Factor:
At ,
Near resonance, let
Let complex resonant frequency be
Treat the circuit as lossless, and use the complex resonantfrequency to account for the loss
Half-power bandwidth
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Parallel Resonant Circuit
Similarly,
Loaded and Unloaded Q
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Define the Q of an external load as , then
The loaded Q can be expressed as
Short-Circuited Transmission Line
Assume small loss, Assume a TEM line,
Since at ,
and
Thus
Similar to a series RLC circuit,
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Short-Circuited Line
Similar to a parallel RLC circuit,
Open-Circuited LineSimilar to a parallel RLC circuit,
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Open-Circuited LineSimilar to a series RLC circuit,
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Other Types of Resonators
Rectangular Waveguide CavitiesQ of modes:
Circular Waveguide Cavities
Q of modes:
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Dielectric Resonators
• High Q ( ).• Smaller size by using high K (dielectric constant) material.• Lower cost comparing to metallic cavities.• Difficult to design. No close form formula. Full-wave
simulation is needed to achieve accuracy.
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Simple design formula (error 10%):
Excitation of ResonatorsCritical Coupling: theresonator is matched tothe feed at the resonantfrequency to obtainmaximum power transferbetween the resonator andthe feedline.
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A Gap-Coupled Microstrip Resonator
where .
Condition of resonance:
which is a function of .
Note: assume ideal transmission line such that
Characteristic near resonanceBy Taylor’s expansion near resonant frequency
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First,
So,
Compare to series RLC circuit ,
Using complex frequency to include the effect of loss,
we have
where is approximated by the of the open-circuit
transmission line since the gap capacitance is very small. For criticalcoupling