Microwave cavity

1. Microwave Cavity
By:-
Mr. Himanshu Diwakar
Assistant Professor
JETGI
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2. Microwave Cavity
• In general, a cavity resonator is a metallic enclosure that confines the
electromagnetic energy. The stored electric and magnetic energies
inside the cavity determine its equivalent inductance and capacitance.
• The energy dissipated by the finite conductivity of the cavity walls
determines its equivalent resistance.
• In practice,
1. The rectangular-cavity resonator,
2. Circular-cavity resonator,
3. And reentrant-cavity resonator
Are commonly used in many microwave applications.
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3. • Theoretically a given resonator has an infinite number of resonant
modes, and each mode corresponds to a definite resonant frequency.
• The mode having the lowest resonant frequency is known as the
dominant mode.
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4. Rectangular cavity
• A microwave cavity acts similarly to a resonant circuit with extremely low loss at
its frequency of operation, resulting in quality factors (Q factors) up to the order of
106, compared to 102 for circuits made with separate inductors and capacitors at
the same frequency.
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Rectangular cavity
MR. HIMANSHU DIWAKAR

5. Where
𝑚 = 0,1,2,3, … represents the number of the half-
wave periodicity in the x direction
n = 0,1,2,3, … represents the number of the half-
wave periodicity in the y direction
p = 1,2,3, … represents the number of the half-
wave periodicity in the z direction
• The wave equations in the rectangular resonator should satisfy the
boundary condition of the zero tangential E at four of the walls.
• These functions can be found as
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6. And
where m = 1, 2, 3, 4, …
n = I, 2, 3, 4, ...
p = 0, I, 2, 3, …
The separation equation for both TE and TM modes is given by
𝑘2
=
𝑚𝜋
𝑎
2
+
𝑛𝜋
𝑏
2
+
𝑝𝜋
𝑑
2
For a lossless dielectric, 𝑘 𝑐 = 𝜔2
𝜇𝜖 therefore, the resonant frequency is expressed by
𝑓𝑟 =
1
2 𝜇𝜖
𝑚
𝑎
2
+
𝑛
𝑏
2
+
𝑝
𝑑
2
(𝑇𝐸 𝑚𝑛𝑝, 𝑇𝑀 𝑚𝑛𝑝)
for 𝑎 > 𝑏 < 𝑑 the dominant mode is the 𝑇𝐸101 mode
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7. Figure shows the methods of excitation for the rectangular resonator.
Methods of exciting wave modes in a resonator.
The maximum amplitude of the standing wave occurs when the
frequency of the impressed signal is equal to the resonant frequency.
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8. Cylindrical cavity
• A circular-cavity resonator is a circular waveguide with two ends closed by a
metal wall.
• The field solutions of a cylindrical cavity of length L and radius R follow from the
solutions of a cylindrical waveguide with additional electric boundary conditions
at the position of the enclosing plates.
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Cylindrical cavity
MR. HIMANSHU DIWAKAR

9. The wave function in the circular resonator should satisfy Maxwell's
equations, subject to the same boundary conditions described for a
rectangular-cavity resonator.
• Substitution of 𝑘 𝑐 = 𝜔2
𝜇𝜖 yields the resonant frequencies for TE and TM modes,
respectively, as
𝑓𝑟 =
1
2𝜋 𝜇𝜖
𝜒′ 𝑛𝑝
𝑎
2
+
𝑞𝜋
𝑏
2
• It is interesting to note that the 𝑇𝑀110 mode is dominant where 2a > d, and that
the 𝑇𝐸111 mode is dominant when 𝑑 ≥ 2𝑎.
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10. Microwave resonant cavities can be represented and thought of as simple LC
circuits.For a microwave cavity, the stored electric energy is equal to the stored
magnetic energy at resonance as is the case for a resonant LC circuit
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LC circuit equivalent for microwave resonant cavity
MR. HIMANSHU DIWAKAR

11. Q Factor of a Cavity Resonator
• The quality factor Q is a measure of the frequency selectivity of a
resonant or antiresonant circuit, and it is defined as
𝑄 = 2𝜋
maximum energy stored
energy dissipated per cycle
=
𝜔𝑊
𝑃
• At resonant frequency, the electric and magnetic energies are equal and in time
quadrature. The total energy stored in the resonator is obtained by integrating the
energy density over the volume of the resonator:
• Where E and H are the peak values of the field intensities.
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12. Cont’d
• The average power loss in the resonator can be evaluated by
integrating the power density as given
• So
• Since the peak value of the magnetic intensity is related to its
tangential and normal components by
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13. Cont’d
• where 𝐻 𝑛 is the peak value of the normal magnetic intensity, the value
of 𝐻 𝑛
2 at the resonator walls is approximately twice the value of
𝐻 2 averaged over the volume.
• So the Q of a cavity resonator
• An unloaded resonator can be represented by either a series or a
parallel resonant circuit. The resonant frequency and the unloaded 𝐻 𝑛
of a cavity resonator are
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14. Cont’d
• If the cavity is coupled by means of an ideal N: 1 transformer and a series
inductance 𝐿 𝑠 to a generator having internal impedance 𝑍 𝑔, then the
coupling circuit and its equivalent are as shown
Coupling circuit. Equivalent circuit.
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15. Cont’d
• The loaded 𝑄 𝑒 of the system is given by
• The coupling coefficient of the system is defined as
• And the loaded 𝑄 𝑒 would become
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16. Cont’d
• There are three types of coupling coefficients:
1. Critical coupling:
If the resonator is matched to the generator, then
𝐾 = 1
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17. 2. Overcoupling: If K > 1
• The cavity terminals are at a voltage maximum in the input line at
resonance. The normalized impedance at the voltage maximum is the
standing-wave ratio 𝜌. That is
𝐾 = 𝜌
The loaded 𝑄𝑙 is given by
3. Undercoupling: If K < 1
The cavity terminals are at a voltage minimum and the input terminal impedance
is equal to the reciprocal of the standing-wave ratio. That is
𝐾 =
1
𝜌
The loaded 𝑄𝑙 is given by
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18. The relationship of the coupling coefficient K and the standing-wave
ratio is shown in Fig.
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Thank you
MR. HIMANSHU DIWAKAR