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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