Microwave

cavities Physics 401, Spring 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 4/1/2013 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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