Master Degree (LM) in Electronic Engineering MICROWAVES

MICROWAVE RESONATORS

Prof. Luca Perregrini

Università di Pavia, Facoltà di Ingegneria [email protected] http://microwave.unipv.it/perregrini/

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 1 SUMMARY

Chapter 6

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 2 SUMMARY

• Series and Parallel Resonant Circuits • Loaded and Unloaded quality factor Q • Transmission Line Resonators • Rectangular Waveguide Cavity Resonators • Circular Waveguide Cavity Resonators • Dielectric Resonators • Excitation of Resonators • Cavity Perturbations

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 3 MOTIVATION

Microwave resonators are used in a variety of applications, including filters, oscillators, frequency meters, and tuned amplifiers.

The operation of microwave resonators is very similar to that of lumped- element resonators of circuit theory, therefore the basic characteristics of series and parallel RLC resonant circuits will be revised.

Different implementations of resonators at microwave frequencies will be discussed, using distributed elements such as transmission lines, rectangular and circular waveguides, and dielectric cavities.

Finally, the excitation of resonators using apertures and current sheets will be addressed.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 4 SERIES RESONANT CIRCUIT

The input impedance is

and the complex power delivered to the load is

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 5 SERIES RESONANT CIRCUIT

The active power lost in the RLC and the reactive powers of the inductor (magnetic) and of the capacitor (electric) are

Therefore, the complex power and the input impedance can be written as

Resonator losses represented by R may be due to conductor loss, dielectric loss, or radiation loss.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 6 SERIES RESONANT CIRCUIT

Resonance occurs when the average stored magnetic and electric energies are equal ( = ), and the input impedance at resonance become 𝑊𝑊𝑚𝑚 𝑊𝑊𝑒𝑒

(purely real)

From = , the resonant angular frequency = 2 results

𝑊𝑊𝑚𝑚 𝑊𝑊𝑒𝑒 𝜔𝜔0 𝜋𝜋𝑓𝑓0

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 7 SERIES RESONANT CIRCUIT

Another important parameter of a resonant circuit is the quality factor Q, defined as

The quality factor is a measure of the loss of a resonant circuit (lower loss implies higher Q). If there is no interaction with an external network, this is called the unloaded Q.

For the series resonant circuit the unloaded Q can be evaluated as

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 8 SERIES RESONANT CIRCUIT

Since = 1/ , the input impedance can be written as 2 𝜔𝜔0 𝐿𝐿𝐿𝐿

Near the resonant frequency, i.e. for = + with , we have

𝜔𝜔 𝜔𝜔0 Δ𝜔𝜔 Δ𝜔𝜔 ≪ 𝜔𝜔0

and

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 9 SERIES RESONANT CIRCUIT

The last formula for can also be used to infer how to model a lossy high-Q resonator by perturbation of his lossless counterpart. 𝑍𝑍in The input impedance of a lossless series LC circuit is

= = ( )

By substituting 𝑍𝑍in 𝑗𝑗푗𝑗𝑗Δ𝜔𝜔 𝑗𝑗푗𝑗𝑗 𝜔𝜔 − 𝜔𝜔0

1 + 2 0 𝑗𝑗 0 𝜔𝜔 0 → 𝜔𝜔 and remembering = / it results𝑄𝑄

𝑄𝑄0 𝜔𝜔0𝐿𝐿 𝑅𝑅 = = 1 + = + ( ) = + ( ) 2 0 in 0 𝑗𝑗 𝜔𝜔 𝐿𝐿 0 0 𝑍𝑍 𝑗𝑗푗𝑗𝑗Δ𝜔𝜔 𝑗𝑗푗𝑗𝑗 𝜔𝜔 − 𝜔𝜔 0 0 𝑗𝑗푗𝑗𝑗 𝜔𝜔 − 𝜔𝜔 𝑅𝑅 𝑗𝑗푗𝑗𝑗 𝜔𝜔 − 𝜔𝜔 This procedure is general and applies𝑄𝑄 to 𝑄𝑄all the low-losses resonators.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 10 SERIES RESONANT CIRCUIT

The half-power fractional bandwidth BW of the resonator is defined as the percentage frequency bandwidth across with the active power delivered not less that one half of the maximum 𝜔𝜔0 Since the active power delivered to the resonator is

= = 2 2 𝑉𝑉 𝑉𝑉 𝑃𝑃a 𝑅𝑅𝑅𝑅 𝑍𝑍in 𝑅𝑅 2 the maximum active power results𝑍𝑍in 𝑍𝑍in = , 2 𝑉𝑉 / = BW𝑃𝑃a /max2 𝑅𝑅 2 Therefore, since , the power𝑅𝑅 is one half when Δ𝜔𝜔 𝜔𝜔0 = + (BW) = 2 2 2 2 from which in 0 𝑍𝑍 𝑅𝑅 𝑗𝑗𝑅𝑅𝑄𝑄 1 𝑅𝑅 BW =

Microwaves, a.a. 2019/20 Prof. Luca𝑄𝑄0 Perregrini Microwave resonators, pag. 11 PARALLEL RESONANT CIRCUIT

In the case of a parallel RLC circuit the results are similar.

In particular, the quality factor is

and the input impedance close to the resonance results

Also in this case the half-power fractional bandwidth BW is 1 BW = See the Pozar’s book for more details and for the derivation.

Microwaves, a.a. 2019/20 𝑄𝑄0 Prof. Luca Perregrini Microwave resonators, pag. 12 LOADED AND UNLOADED Q

An external connecting network may introduce additional loss. Each of these loss mechanisms will have the effect of lowering the .

For a series RLC, the effective resistance 𝑄𝑄 become + .

For a parallel 𝐿𝐿RLC, the effective resistance become 𝑅𝑅 /𝑅𝑅( + ).

If we define𝑅𝑅𝑅𝑅𝐿𝐿an𝑅𝑅external𝑅𝑅𝐿𝐿 quality factor as

𝑄𝑄𝑒𝑒

then the loaded can be expressed as

𝑄𝑄𝐿𝐿

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 13 LOADED AND UNLOADED Q

In summary

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 14 TRANSMISSION LINE RESONATORS

Open/short transmission lines of special length (half wavelength or quarter wavelength) can be used as resonators when lumped element circuits are unfeasible due to the high frequency. To study the quality factor, lossy transmission lines will be considered.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 15 TRANSMISSION LINE RESONATORS

Short-circuited / line At the resonant frequency = , the 𝝀𝝀 𝟐𝟐 length of the line is = /2. 𝜔𝜔 𝜔𝜔0 The input impedanceℓ is 𝜆𝜆

If we are interested in a small frequency band across , we can set = + ( ). 𝜔𝜔0 𝜔𝜔 𝜔𝜔0 Δ𝜔𝜔 Moreover,0 if the mode propagating is TEM ( = = ) and 1, we have Δ𝜔𝜔 ≪ 𝜔𝜔 𝜆𝜆 𝜋𝜋𝑣𝑣𝑝𝑝 + ℓ 2 𝜔𝜔0 αℓ ≪ tan = tan = tan + tanh 𝜔𝜔0 Δ𝜔𝜔 Δ𝜔𝜔 Δ𝜔𝜔 𝛽𝛽ℓ 𝑝𝑝 ℓ 𝜋𝜋 0 𝜋𝜋 ≈ 0 𝜋𝜋 αℓ ≈ αℓ Microwaves, a.a. 2019/20 𝑣𝑣 Prof. Luca Perregrini𝜔𝜔 𝜔𝜔 Microwave resonators, pag. 16 TRANSMISSION LINE RESONATORS

Short-circuited / line By substituting and observing that / 1, the approximated input 𝝀𝝀 𝟐𝟐 impedance results Δ𝜔𝜔𝜔𝜔ℓ 𝜔𝜔0 ≪

It can be cast in the form,

which is the same expression of the series RLC circuit. The lumped element values are

The quality factor is

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 17 TRANSMISSION LINE RESONATORS

Short-circuited / line At the resonant frequency = , the 𝝀𝝀 𝟒𝟒 length of the line is = /4. 𝜔𝜔 𝜔𝜔0 The input impedanceℓ is 𝜆𝜆

We are interested in a small frequency band across : = + ( ).

Moreover, if the0 mode propagating0 is TEM0 ( = = ) and 1, we have 𝜔𝜔 𝜔𝜔 𝜔𝜔 Δ𝜔𝜔 Δ𝜔𝜔 ≪ 𝜔𝜔 𝜆𝜆 𝜋𝜋𝑣𝑣𝑝𝑝 ℓ 4 2𝜔𝜔0 αℓ ≪ cot = cot + = tan tanh 2 2 2 2 𝜋𝜋 𝜋𝜋Δ𝜔𝜔 𝜋𝜋Δ𝜔𝜔 𝜋𝜋Δ𝜔𝜔 𝛽𝛽ℓ 0 − 0 ≈ − 0 αℓ ≈ αℓ Microwaves, a.a. 2019/20 𝜔𝜔 Prof. Luca𝜔𝜔 Perregrini 𝜔𝜔 Microwave resonators, pag. 18 TRANSMISSION LINE RESONATORS

Short-circuited / line By substituting and observing that /(2 ) 1, the approximated input 𝝀𝝀 𝟒𝟒 impedance results Δ𝜔𝜔𝜔𝜔ℓ 𝜔𝜔0 ≪

It can be cast in the form,

which is the same expression of the parallel RLC circuit. The lumped element values are

The quality factor is

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 19 TRANSMISSION LINE RESONATORS

Open-circuited / line At the resonant frequency = , the 𝝀𝝀 𝟐𝟐 length of the line is = /2. 𝜔𝜔 𝜔𝜔0 The input impedanceℓ is 𝜆𝜆

We are interested in a small frequency band across : = + ( ).

𝜔𝜔0 𝜔𝜔 𝜔𝜔0 Δ𝜔𝜔 Δ𝜔𝜔 ≪ 𝜔𝜔0 Moreover, if the mode propagating is TEM ( = = ) and 1, we have 𝜆𝜆 𝜋𝜋𝑣𝑣𝑝𝑝 ℓ 2 𝜔𝜔0 αℓ ≪ tan = tan + tanh Δ𝜔𝜔 Δ𝜔𝜔 𝛽𝛽ℓ 𝜋𝜋 0 𝜋𝜋 ≈ 0 𝜋𝜋 αℓ ≈ αℓ Microwaves, a.a. 2019/20 𝜔𝜔 𝜔𝜔 Prof. Luca Perregrini Microwave resonators, pag. 20 TRANSMISSION LINE RESONATORS

Open-circuited / line By substituting and observing that / 1, the approximated input 𝝀𝝀 𝟐𝟐 impedance results Δ𝜔𝜔𝜔𝜔ℓ 𝜔𝜔0 ≪

It can be cast in the form,

which is the same expression of the parallel RLC circuit. The lumped element values are

The quality factor is

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 21 TRANSMISSION LINE RESONATORS

Let’s consider a resonator comprising many sections of transmission lines with different characteristic impedances , , …, , loaded at the extremes with arbitrary reactive impedances and : 𝑍𝑍1 𝑍𝑍2 𝑍𝑍𝑁𝑁 𝑗𝑗𝑗𝑗𝑎𝑎 𝑗𝑗𝑗𝑗𝑏𝑏

1 2 𝑁𝑁 𝑗𝑗𝑗𝑗𝑎𝑎 𝑍𝑍 𝑍𝑍 𝑍𝑍 𝑗𝑗𝑗𝑗𝑏𝑏

The characteristic equation𝑍𝑍𝐿𝐿to𝑍𝑍determine𝑅𝑅 the resonant frequencies can be obtained by cutting the structure in any section and imposing the condition = ∗ 𝑍𝑍𝐿𝐿 𝑍𝑍𝑅𝑅 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 22 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

Waveguide cavities can be made without dielectrics, and therefore they are preferred when high-Q is required. A cavity based on a rectangular waveguide is usually obtained by short circuiting the waveguide at both ends. An infinite number of modes resonate in the cavity, and they can be derived from propagating TE and TM modes in the waveguide. The resonant frequencies and field expressions will be derived under the hypothesis of a lossless resonator. The Q factor will be derived by using a perturbation technique.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 23 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

It is sufficient to impose the perfect electric wall condition at = 0, to the field of the TE or TM propagating modes 𝑧𝑧 𝑑𝑑

to determine the resonant frequency and the resonant modal field. Enforcing , , 0 = , , = 0 leads to

𝐸𝐸�𝑡𝑡 𝑥𝑥 𝑦𝑦 𝐸𝐸�𝑡𝑡 𝑥𝑥 𝑦𝑦 𝑑𝑑 and, therefore

where

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 24 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

The resonance wavenumber is therefore

and the resonance frequency results

If < < the dominant resonant mode (lowest resonant frequency) will

be the TE101 mode, corresponding to the TE10 dominant waveguide mode 𝑏𝑏 𝑎𝑎 𝑑𝑑 in a shorted guide of length /2. The dominant TM resonant mode is the TM mode. 110 𝜆𝜆𝑔𝑔

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 25 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

To evaluate the Q, we10needℓ to determine the electric and magnetic stored energies, as well as the dissipated power.

Taking into account the expression of the field of the propagating TE10 mode, and using the conditions deriving from the enforcement of the resonance, the resonant field results

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 26 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

The stored energies are10ℓderived with some simple calculations:

Since and , we have

and, therefore,

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 27 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

The power dissipated10ℓ in the metallic boundary (assuming a good conductor) is obtained using the perturbation technique, i.e., using the field calculated in the lossless case and the formulas for the skin effect:

where is the surface resistance.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 28 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

The resulting unloaded10Qℓ -factor due to only conductor losses ( ) is

𝑄𝑄𝑐𝑐

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 29 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

The power dissipated10inℓ the dielectric (if any) with a dielectric constant = (1 tan ) is

𝜖𝜖 𝜖𝜖𝑟𝑟𝜖𝜖0 − 𝑗𝑗 𝛿𝛿

Therefore, the unloaded Q-factor due to only dielectric losses ( ) is

𝑄𝑄𝑑𝑑

It is interesting to notice that does not depend on the shape of the

cavity, but only on the intrinsic losses𝑑𝑑 of the dielectric. Therefore this formula is very𝑄𝑄 general and applies to all the cavities fully filled with a dielectric, independently on their shape. 𝑄𝑄𝑑𝑑 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 30 RECTANGULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

Considering together 10theℓ conductor and dielectric dissipated power (i.e., = + ), it can easily be demonstrated that the overall unloaded Q- factor ( ) is a combination of the quality factors: 𝑃𝑃𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑃𝑃𝑐𝑐 𝑃𝑃𝑑𝑑 𝑄𝑄0

In general, the dielectric losses are dominating, and < . This is the reason to avoid dielectrics in waveguide resonators. 𝑄𝑄𝑑𝑑 𝑄𝑄𝑐𝑐

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 31 CIRCULAR WAVEGUIDE CAVITY RESONATORS

A cavity based on a circular waveguide can be obtained by short circuiting the waveguide at both ends. The dominant cylindrical cavity mode is

the TE111, derived from the TE11 of the waveguide. An example of application is the frequency meter in the photo: the cylindrical cavity is in between the two waveguides, and its resonance frequency is tuned until there is a transmission from the input to the output waveguide. Due to the high frequency selectivity (high-Q), this allows to determine the frequency with an high accuracy.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 32 CIRCULAR WAVEGUIDE CAVITY RESONATORS

The field of the TE or TM propagating modes is

Imposing the perfect electric wall condition at = 0, the resonant frequency and the resonant modal field can𝑧𝑧 be 𝑑𝑑obtained The resonant frequencies are

TE modes

TM modes

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 33 CIRCULAR WAVEGUIDE CAVITY RESONATORS

Plotting the resonant frequencies as a function of the dimension, this design chart can be obtained.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 34 CIRCULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

From it we obtain the stored𝑛𝑛𝑛𝑛ℓ energy

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 35 CIRCULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

and the dissipated power𝑛𝑛𝑛𝑛ℓ

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 36 CIRCULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

The resulting unloaded𝑛𝑛𝑛𝑛Qℓ-factor due to only conductor losses ( ) is

𝑄𝑄𝑐𝑐

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 37 CIRCULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

Normalized unloaded 𝑛𝑛Q𝑛𝑛 forℓ various cylindrical cavity modes (air filled)

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 38 CIRCULAR WAVEGUIDE CAVITY RESONATORS

Unloaded Q of the TE mode

The power dissipated 𝑛𝑛in𝑛𝑛ℓthe dielectric = (1 tan ) is

𝜖𝜖 𝜖𝜖𝑟𝑟𝜖𝜖0 − 𝑗𝑗 𝛿𝛿

Therefore, the unloaded Q-factor due to only dielectric losses ( ) is

𝑄𝑄𝑑𝑑

As expected, does not depend on the shape of the cavity, ad is the same as in the rectangular case. 𝑄𝑄𝑑𝑑 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 39 DIELECTRIC RESONATORS

A small disc or cube (or other shape) of (not metalized) dielectric material can be used as a microwave resonator. Low loss and a high dielectric constant ( = 10 ÷ 100) materials (e.g.,

BaO9Ti4, TiO2), the field is confined within𝑟𝑟 the dielectric (some field leakage leads to a small radiation loss lowering𝜖𝜖 Q). Smaller in size, cost, and weight than metallic cavity, easy to incorporate in microwave integrated circuits and to couple to planar transmission lines. No conductor losses, but dielectric loss usually increases with dielectric constant. Q of up to several thousand can sometimes be achieved. The resonant frequency can be mechanically tuned using an adjustable metal plate above the resonator. Because of these desirable features, dielectric resonators have become key components for integrated microwave filters and oscillators.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 40 DIELECTRIC RESONATORS

Resonant frequencies of TE mode

Due to the high-dielectric 01permittivity,𝛿𝛿 the boundary condition on the surface of the dielectric can be approximated as a perfect magnetic wall (dual of the electric wall). In fact, as a first order approximation of the reflection on the boundary is

1 = 1 when + 1 𝜖𝜖𝑟𝑟 − Γ → 𝜖𝜖𝑟𝑟 → ∞ Which is the opposite𝜖𝜖𝑟𝑟 of the electric wall ( 1) Considering a cylindrical waveguide madeΓ →of−not metalized dielectric, the solution of the Helmoltz equation is dual with respect ot the metalized counterpart (TE TM, TM TE, , ).

Microwaves, a.a. 2019/20 → → 𝐸𝐸� →Prof.𝐻𝐻� Luca𝐻𝐻� Perregrini→ 𝐸𝐸� Microwave resonators, pag. 41 DIELECTRIC RESONATORS

Resonant frequencies of TE mode

The electric and magnetic01field𝛿𝛿 of the TE01 mode of a cylindrical waveguide is

The propagation constant in the dielectric is

while in air the mode is attenuated with

and the characteristic impedance is

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 42 DIELECTRIC RESONATORS

Resonant frequencies of TE mode

In the dielectric ( < /2) 01In𝛿𝛿 air ( < /2)

𝑧𝑧 𝐿𝐿 𝑧𝑧 𝐿𝐿

Matching the fields at the interfaces we have

which leads to

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 43 DIELECTRIC RESONATORS

Resonant frequencies of TE mode

This solution is approximate01𝛿𝛿 (ignores fringing fields at the sides), with error on the order of 10% (not accurate enough for practical purposes). It illustrates the basic behavior of dielectric resonators, and more accurate solutions are available in the literature. The unloaded Q of the resonator can be calculated by determining the stored energy (inside and outside the dielectric cylinder), and the power dissipated in the dielectric and possibly lost to radiation. If the radiation is small, the unloaded Q can be approximated as 1 tan 𝑄𝑄0 ≈ as in the case of the metallic cavity resonators𝛿𝛿 .

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 44 DIELECTRIC RESONATORS

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 45 EXCITATION OF RESONATORS

Resonators are not useful unless they are coupled to external circuitry. The way in which resonators can be coupled to transmission lines and waveguides depends on the type of resonator under consideration. Some common coupling techniques (i.e., gap coupling and aperture coupling) will be discussed. The coupling coefficient for a resonator connected to a feed line will be defined and discussed. The determination of the Q of a resonator from two-port measurement will be addressed.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 46 EXCITATION OF RESONATORS

Example of couplings:

Microstrip resonator gap coupled Rectangular cavity resonator to a microstrip feedline. fed by a coaxial probe.

Circular cavity resonator aperture Dielectric resonator coupled coupled to a rectangular waveguide. to a microstrip line

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 47 EXCITATION OF RESONATORS

The level of coupling depends on the application. If the high-Q of a waveguide cavity resonator is to be preserved (e.g., in a frequency meter), is usually loosely coupled to its feed guide. On the contrary, resonators used in oscillators or in tuned amplifiers, may be tightly coupled in order to achieve maximum power transfer. A measure of the level of coupling between a resonator and a feed is given by the coupling coefficient. To obtain maximum power transfer between a resonator and a feed line, the resonator should be matched to the line at the resonant frequency; the resonator is then said to be critically coupled to the feed.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 48 COUPLING COEFFICIENT

To define the coupling coefficient, let’s consider a series resonant circuit connected to a transmission line.

The input impedance near resonance of the series RLC is

where

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 49 COUPLING COEFFICIENT

At resonance ( = , = 0) the input impedance is = , therefore 𝜔𝜔 𝜔𝜔0 Δ𝜔𝜔 𝑍𝑍in 𝑅𝑅

Moreover, assuming an infinite transmission line (or, equivalently, a matched generator), the external resistance seen from the resonator is = , and the external Q-factor become

𝑅𝑅external 𝑍𝑍0

Therefore, the external and unloaded Q are identical at the critical coupling. In this case, the loaded Q is one half of the unloaded Q.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 50 COUPLING COEFFICIENT

We can define the coupling coefficient

which can be applied to both series ( = / ) and parallel ( = / )

resonant0 circuits connected to 0a transmission𝑔𝑔 𝑍𝑍 𝑅𝑅 line of characteristic𝑔𝑔 𝑅𝑅 𝑍𝑍 impedance .

0 Three cases𝑍𝑍can be distinguished: < 1: resonator undercoupled to the feedline = 1: resonator critically coupled to the feedline 𝑔𝑔 > 1: resonator overcoupled to the feedline The𝑔𝑔Smith chart sketch of the impedance loci for the series resonant circuit exemplifies𝑔𝑔 the three cases.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 51 GAP-COUPLED MICROSTRIP OPEN RESONATOR

Consider a /2 lossless open-circuited microstrip resonator proximity coupled to the open end of a microstrip transmission line 𝜆𝜆

The gap between the resonator and the microstrip line can be modeled as a series capacitor.

The normalized input impedance seen by the feedline is

where = is the normalized susceptance of

𝑐𝑐 0 Microwaves, a.a. 2019/20𝑏𝑏 𝑍𝑍 𝜔𝜔𝐶𝐶 Prof. Luca Perregrini Microwave resonators, pag. 52 GAP-COUPLED MICROSTRIP OPEN RESONATOR

Imposing the resonance condition = 0, we have

𝑧𝑧

A graphical solution is shown in the plot. Since usually 1, the first resonance is just 𝑏𝑏𝑐𝑐 ≪ below the resonance1 of the transmission𝜔𝜔 line 𝜔𝜔0 (corresponding to = ). = /

Therefore, the effect𝛽𝛽ℓ of 𝜋𝜋the 𝛽𝛽ℓ 𝜔𝜔ℓ 𝑣𝑣𝑝𝑝 coupling is to lower the resonance frequency.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 53 GAP-COUPLED MICROSTRIP OPEN RESONATOR

The goal is to represent the resonator around the first resonance as a RLC equivalent circuit. 𝜔𝜔1 Expanding in a Taylor series around we have

𝑧𝑧 𝜔𝜔1 0 where

If the transmission line support a TEM mode, then 1

( ) ( / ) 𝑐𝑐 = = 𝑏𝑏 ≪ 𝑑𝑑 𝛽𝛽� 𝑑𝑑 𝜔𝜔ℓ 𝑣𝑣𝑝𝑝 ℓ and, finally 𝑑𝑑𝜔𝜔 𝑑𝑑𝜔𝜔 𝑣𝑣𝑝𝑝

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 54 GAP-COUPLED MICROSTRIP OPEN RESONATOR

In conclusion, for a lossless transmission line the input impedance near the resonance is

Since the impedance vanishes at the resonance, the resonator is equivalent to a RLC series.

To take into account losses for high-Q resonators, the perturbation technique can be used, replacing the resonant frequency :

1 1 + 𝜔𝜔 2 𝑗𝑗 leading to 𝜔𝜔1 → 𝜔𝜔1 𝑄𝑄0

= Remember that, for a transmission𝑅𝑅 line, 𝛽𝛽 Microwaves, a.a. 2019/20 Prof. Luca Perregrini𝑄𝑄0 2𝛼𝛼 Microwave resonators, pag. 55 GAP-COUPLED MICROSTRIP OPEN RESONATOR

Since = 2 𝜋𝜋𝑍𝑍0 𝑅𝑅 2 The coupling coefficient results: 𝑄𝑄0𝑏𝑏𝑐𝑐

Therefore = gives the critical coupling ( = ) 2 𝜋𝜋 𝑏𝑏𝑐𝑐 0 𝑄𝑄0 𝑅𝑅 𝑍𝑍 < gives the undercoupling 2 𝜋𝜋 𝑏𝑏𝑐𝑐 𝑄𝑄0 > gives the overcoupling 2 𝜋𝜋 𝑏𝑏𝑐𝑐 𝑄𝑄0 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 56 APERTURE-COUPLED RECTANGULAR CAVITY

Consider a /2 rectangular waveguide short circuited at the two ends, and coupled with another waveguide (maybe identical) through a small aperture 𝜆𝜆𝑔𝑔 at one end.

The small aperture acts as a shunt inductance, and the structure can be modeled by the following equivalent circuit

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 57 APERTURE-COUPLED RECTANGULAR CAVITY

Assuming = 1 , the 𝜔𝜔𝐿𝐿 normalized input admittance 𝑥𝑥𝐿𝐿 𝑍𝑍0 ≪ seen by the feedline is

Imposing the resonance condition = 0, we have

𝑦𝑦

whose solutions provides the resonances

It is noted that the structure is lossless. Losses can be intruduced later on by the perturbation technique.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 58 APERTURE-COUPLED RECTANGULAR CAVITY

The goal is again to represent the resonator around the first resonance as a resonant RLC circuit. 𝜔𝜔1 Expanding in a Taylor series around we have

𝑦𝑦 𝜔𝜔1 0 where

For the TE or TM mode of a waveguide it results 1

𝑏𝑏𝑐𝑐 ≪

and, finally

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 59 APERTURE-COUPLED RECTANGULAR CAVITY

In conclusion, for a lossless resonator the input admittance near the resonance is

Since the admittance vanishes at the resonance, the resonator is equivalent to a RLC parallel.

To take into account losses for high-Q resonators, the perturbation technique can be used, replacing the resonant frequency :

1 1 + 𝜔𝜔 2 𝑗𝑗 leading to 𝜔𝜔1 → 𝜔𝜔1 𝑄𝑄0

1/

Microwaves, a.a. 2019/20 𝑅𝑅 Prof. Luca Perregrini Microwave resonators, pag. 60 APERTURE-COUPLED RECTANGULAR CAVITY

The critical coupling (maximum power transfer) is achieved when ( = ), which corresponds to 𝑅𝑅 𝑍𝑍0

For a TE mode = / , and the coupling coefficient results:

𝑍𝑍0 𝑘𝑘0𝜂𝜂0 𝛽𝛽

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 61 UNLOADED Q FROM TWO-PORT MEASUREMENTS

Direct measurement of the unloaded Q of a resonator is generally not possible because of the loading effect of the measurement system. However, it is possible to determine unloaded Q from measurements of the frequency response of the loaded resonator when it is connected to a transmission line. Both one-port (reflection) and two-port (transmission) measurement techniques are possible.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 62 UNLOADED Q FROM TWO-PORT MEASUREMENTS

Let us consider the (equivalent) RLC series circuit embedded in a two-port measurement.

The unloaded Q is related to the loaded Q and to the coupling factor:

We need from measurements the loaded Q and the coupling factor.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 63 UNLOADED Q FROM TWO-PORT MEASUREMENTS

For the RLC series the unloaded Q is

= 𝜔𝜔0𝐿𝐿 𝑄𝑄0 Moreover, the𝑅𝑅external Q is twice the one of the single line connection: 2 = = = 2 0 𝑄𝑄0 𝑍𝑍0 𝑒𝑒 𝜔𝜔 𝐿𝐿 𝑄𝑄 𝑔𝑔 𝑒𝑒 Since for the RLC at𝑍𝑍0 the resonance we have =𝑄𝑄 , it𝑅𝑅can be easily demonstrated that 𝑍𝑍 𝑅𝑅

Where is the measured transmission parameter, which is purely real at the resonance, and provides the coupling factor. 𝑆𝑆21 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 64 UNLOADED Q FROM TWO-PORT MEASUREMENTS

The determination of the loaded Q is done from the transmission measurement around the resonance:

3 dB BW = BW0 𝐿𝐿 𝑓𝑓 3 dB 𝑄𝑄

BW

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 65 UNLOADED Q FROM TWO-PORT MEASUREMENTS

The same reasoning can be repeated for an RLC parallel resonator. The only difference is that

1 ( ) = ( ) − 𝑆𝑆21 𝜔𝜔0 𝑔𝑔 21 0 Nothing changes for the determination𝑆𝑆 of𝜔𝜔the loaded Q.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 66 CAVITY PERTURBATIONS

Cavity resonators are often modified by making small changes in their shape, or by introducing small pieces of dielectric or metallic materials. For example, the resonant frequency of a cavity resonator can be easily tuned with a small screw (dielectric or metallic) that enters the cavity volume, or by changing the size of the cavity with a movable wall. Another application involves the determination of dielectric constant of materials by measuring the shift in resonant frequency when a small dielectric sample is introduced into the cavity. In some cases, the effect of such perturbations on the cavity performance can be calculated exactly, but often approximations must be made. One useful technique for doing this is the perturbational method, which assumes that the actual fields of a cavity with a small shape or material perturbation are not greatly different from those of the unperturbed cavity.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 67 CAVITY PERTURBATIONS

In the case of material perturbation, starting from the Maxwell’s equations it can be demonstrated that the shift of the resonant frequency is given by Original cavity Perturbed cavity

(exact expression)

If we assume and , and approximate with in the denominator, we obtain 𝐸𝐸� ≈ 𝐸𝐸� 0 𝐻𝐻� ≈ 𝐻𝐻� 0 𝜔𝜔 𝜔𝜔0

(approximate expression)

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 68 CAVITY PERTURBATIONS

In the case of shape perturbation, starting from the Maxwell’s equations it can be demonstrated that the Δ𝑉𝑉 Δ𝑆𝑆 shift of the resonant frequency is given by Original cavity Perturbed cavity

(exact expression)

If we assume and , we obtain

𝐸𝐸� ≈ 𝐸𝐸� 0 𝐻𝐻� ≈ 𝐻𝐻� 0 (approximate expression)

The frequency shift is related to the change in the stored energy.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Microwave resonators, pag. 69