Microwave
cavities Physics 401, Fall 2013
Eugene V. Colla
Agenda
• Waves in waveguides
• Standing waves and resonance
• Setup
• Experiment with microwave cavity
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Maxwell’s
Equations uniform plane wave traveling
in z-direction H ┴ E
wave equation
general form of solution
propagation speed
E vs H
( )
0
i t kz
xE E e
Y
X
z Ex
Hy
0D
0B
BE
t
DH
t
2 2
2 2 2
1x x
z v
E
t
E
( , )z
z zE z t f t g t
v v
Z
xyH E
yx
E ZH
1v
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Y
X
Z
( )
0sin
i t kz
y xE E k x e
X
Y
Zi
Ey=Ey(x) at Zi
Z
Y
Xi
Ey=Ey(z) at xi
v
a
b
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Ey
Z
Z
X
Y
X
H-field
Z
X
Y Ey=Ey(z)
Ey=Ey(x or z)
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2 2 2
2 2
0mnp
m n pv
a b c
2
0v -phase velocity
TE101 mode: m=1, n=0, p=1
2 2
2 2 2
101 0
1 1v
a c
a
b c
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coupling loop
coaxial wave guide
cavity
inner conductor
outer
conductor
Y
Z
X
M R
C L L0 line
R
C
L
Z0
Impedance of
wave guide
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Z0
Impedance of
wave guide
0
0
0
0
0
, 1
1
: coupling coefficient
Maximum power transfer:
Z 1
1 ,
2
quality factor without external load
L
L
L
LQ
R Z
QLQ
ZR
R
R
Q Q
coupling loop
coaxial wave guide cavity
inner conductor
outer conductor
Y
Z
X
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Gunn diode
MW oscillator 11/4/2013 9
Resonance Cavity
A
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Use detector to find distance between minimums in the slotted
line (wave guide)
Slotted line detector
Tuner Open
end
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Use detector to find distance between minimums in the slotted line
(wave guide). Distance between consequent minima correspond l/2
0 2 4 6 8 10 12 14 16 180
10
20
30
40
50E
(mV
)
x (cm)
l/2
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Movable plunger (c direction)
Use plunger to change the dimension of the
cavity in z-direction and search for maxima in
power stored using the cavity detector. Identify
TE101 and TE102.
cavity
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2 2
2 2 2
102 0
1 2v
a c
2 2
0
102
1 2
2
vf
a c
Df
f0
0 ~ 450f
Qf
D
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By moving the plunger we
changing the resonance
frequency of the cavity
1st position of the
plunger
2nd position of
the plunger
Frequency of the oscillator
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1. Oscilloscope should run in X-Y mode
2. To plot the I(f) dependence you have to download both Ch1 and Ch2 data
3. Use triangular waveform as a voltage applied to modulation input of the
oscillator
4. Use a proper time scale setting on the scope which could estimated from
scanning frequency
5. Apply the calibration equation to calculate the frequency of the oscillator
from the modulation voltage
mod0.03706 2.9349f V
G
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Voltage tunable
oscillator ZX95-3250a-
S+ from
FM Calibration for microwave oscillator
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By changing of the coupling between oscillator and cavity we can control
the quality factor of the cavity resonance but in the same time we changing
the power delivered to the cavity
While in resonance: turn orientation of the
input loop from the vertical direction in 10o
steps to 360o.
Read cavity detector.
Detector B field
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0 50 100 150 200 250 300 350
0
2
4
6
8
10
12
I (m
A)
grad)
Experimental result.
Fitted to
𝑨 (cos(+ ))𝒏 + 𝑨𝟎
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Presence of dielectric reduces length of cavity at a given resonance
frequency ω0.
This effect grows with the electric field strength Ey.
(0) Without dielectric the cavity length at resonace is c0.
(1) Place dielectric into cavity and move in 0.5cm steps, li.
(2) At each place tune plunger to resonance and record ci.
(3) Plot ∆ci=|c0-ci| versus li : this measures now Ey vs l!
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Z
X
Y
TE102
Courtesy of P. Debevec
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Z
X
Y
TE102
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4. Part IV: (a) Plot cD versus . (b) On the same graph, plot sin c versus . (c)
A small writeup correction:
This works for TE101 mode but more general it should be: where p is
mode index (TEmnp); p=1 for TE101 and p=2 for TE102 modes
sinp l
c
2 2
0 3 3 2 22
abc a cQ
b a c ac a c
Quality factor (TE101 mode) of unloaded cavity can be calculated as:
a
b c
is the skin depth at frequency 0
2 /
– resistivity of the cavity material
r0≈04x107
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a
b c
2 / For red brass =6x10-8m ≈4x107
2.25x10-6m a=7.22cm, b=3.42 cm, c=6.91cm (TE101)
2 2
0 3 3 2 22
abc a cQ
b a c ac a c
Q0~7700
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