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Basics of Microwave Measurements
Steven Anlage
http://www.cnam.umd.edu/anlage/AnlageMicrowaveMeasurements.htm
1
Electrical Signals at Low and High Frequencies
2
Transmission Lines
Transmission lines carry microwave signals from one point to another
They are important because the wavelength is much smaller than the length of typical T-lines
used in the lab
You have to look at them as distributed circuits, rather than lumped circuits
V
The wave equations
3
Transmission Lines
Take the ratio of the voltage and current waves
at any given point in the transmission line:
Wave Speed
= Z0
The characteristic impedance Z0 of the T-line
Reflections from a terminated transmission line
Z0
ZL
Reflection
coefficient

Vleft
Vright
b Z L  Z0
 
a Z L  Z0
Open Circuit ZL = ∞,  = 1 ei0
Some interesting special cases:
Short Circuit ZL = 0,  = 1 eip
Perfect Load ZL = Z0,  = 0 ei?
These are used in error correction measurements to characterize non-ideal T-lines
4
Transmission Lines and Their Characteristic Impedances
5
Transmission Lines, continued
The power absorbed in a termination is:
Model of a realistic transmission line including loss
Shunt
Conductance
Traveling
Wave
solutions
6
with
How Much Power Reaches the Load?
7
Waveguides
H
Rectangular metallic waveguide
8
Network Analysis
Assumes linearity!
9
N-Port Description of an Arbitrary Enclosure
N Ports
V1
V1
V1 , I1
 Voltages and Currents,
N – Port
 Incoming and Outgoing Waves
System
VN
V N
S matrix
V 1 
V 1 
  
  
V
 2
V 2 
 
 

  [S ]  

 
 
 
 






V N 
V N 
10
VN , IN
Z matrix
V1 
 I1 
V 
I 
2
 
 2
 
 
   [ ]   
 
 
 
 
 
 
VN 
 I N 
S  ( Z  Z 0 ) 1 ( Z  Z 0 )
Z ( ), S ( )
 Complicated
Functions of
frequency
 Detail Specific
(Non-Universal)
Linear vs. Nonlinear Behavior
11
Network vs. Spectrum Analysis
12
Resonator Measurements
Traditional Electrodynamics Measurements
input
~ microwave
wavelength
l
output
Microwave
Resonator
Cavity
Perturbation
B
sample
Hrf
Sample
transmission
T1
df
f0
13
T2
df’
f0’
rf currents
Quality Factor
Q = Estored/Edissip.
Q = f0 / df
inhomogeneities
These measurements
average the properties
over the entire sample
frequency
Df = f0’ – f0  D(Stored Energy)
D(1/2Q)  D(Dissipated Energy)
Electric and Magnetic Perturbations
Electric Field Pert.
E
Magnetic Field Pert.
E
Sample
e1 - i e2
s, r/t
Rs + i Xs
Sample
B
B
m1 + i m2
s, r/t
Rs + i Xs
Varying capacitance (e1) and
inductance (m1) change the stored
energy and resonant frequency Df
Varying sample losses (r/t,
tand  e2/e1, m2) change the quality
factor (Q) of the microscope
Df = f0’ – f0  D(Stored Energy)
D(1/2Q)  D(Dissipated Energy)
14
The Variable-Spacing Parallel Plate Resonator
Vary s
s: contact –
~ 100 mm
in steps of
10 nm to 1 mm

Brf
Principle of Operation: Measure the resonant frequency, f0, and the
quality factor, Q, of the VSPPR versus the continuously variable
thickness of the dielectric spacer (s), and to fit them to theoretical forms
in order to extract the absolute values of l and Rs.
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The measurements are performed at a fixed temperature
In our experiments L, w ~ 1 cm
The VSPPR Experiment
Films held and aligned by two sets
of perpendicular sapphire pins
Dielectric spacer thickness (s)
measured with capacitance meter
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VSPPR: Theory of Operation
Superconducting samples
Resonant Frequency
f 0,SC
Quality Factor
f 0, PC
1

1  2leff / s 1  s
SC Trans.
line resonator
f 0, PC 
fringe
effect
c
2L e r
leff  l coth( d / l )


1 
2



0.423  ln
 sf m e  
pL 
 0 0 
1
1
1
1



QSC Q Qd Qrad
*
Reff
f SC
1

 tan d  s
*
*
QSC pm0 f ( s  2leff ) f
Reff
*  f 0 , SC 
 * 
 Reff
 f 
2
f* is a reference frequency
  1/ L
Assumes: 2 identical and uniform films, local electrodynamics, Rs(f) ~ f2
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V. V. Talanov, et al., Rev. Sci. Instrum. 71, 2136 (2000)
US Patent # 6,366,096
High-Tc Superconducting Thin Films at 77 K
1200
VSPPR, T=77 K
12.4
LN2 dielectric spacer
1000
12.2
800
12.0
600
11.8
Q-factor
Resonant Frequency (GHz)
750nm-YBCO/LAO
400
11.6
200
11.4
0
0
20
40
60
80
100
Dielectric Spacer Thickness ( m m)
l fit: 257 ± 25 nm
Rs fit: 200 ± 20 m @ f* = 10 GHz
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Mutual Inductance Measurements
(l1+l2)/2 = 300 ± 15 nm
L = 9.98 mm, w = 9.01 mm, film thickness d = 760 ± 30 nm, Tc = 92.4 K