ECE 451 – Jose Schutt-Aine 1 ECE 451 Advanced Microwave Measurements Transmission Lines Jose E....
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Transcript of ECE 451 – Jose Schutt-Aine 1 ECE 451 Advanced Microwave Measurements Transmission Lines Jose E....
ECE 451 – Jose Schutt-Aine 1
ECE 451Advanced Microwave Measurements
Transmission Lines
Jose E. Schutt-AineElectrical & Computer Engineering
University of [email protected]
ECE 451 – Jose Schutt-Aine 2
Faraday’s Law of Induction
Ampère’s Law
Gauss’ Law for electric field
Gauss’ Law for magnetic field
Constitutive Relations
BE
t
H J D
t
D
0B
B H D E
Maxwell’s Equations
ECE 451 – Jose Schutt-Aine 3
l
l
z
Wavelength : l
l = propagation velocity
frequency
Why Transmission Line?
ECE 451 – Jose Schutt-Aine 4
In Free Space
At 10 KHz : l = 30 km
At 10 GHz : l = 3 cm
Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength.
Why Transmission Line?
ECE 451 – Jose Schutt-Aine 5
Let d be the largest dimension of a circuit
If d << l, a lumped model for the circuit can be used
circuit
z
l
Justification for Transmission Line
ECE 451 – Jose Schutt-Aine 6
circuit
z
l
If d ≈ l, or d > l then use transmission line model
Justification for Transmission Line
ECE 451 – Jose Schutt-Aine 7
L: Inductance per unit length.
C: Capacitance per unit
length.
L
z
C
I
V
+
-
V IL
z t
I VC
z t
Vj LI
z
Ij CV
z
Assumetime-harmonicdependence
, ~ j tV I e
Telegraphers’ Equations
ECE 451 – Jose Schutt-Aine 8
L
z
C
I
V
+
-
22
2
VLCV
z
22
2
ICLI
z
2V V Ij L j Lj CV
z z z z
2 I I V
j C j Lj CIz z z z
TL Solutions
ECE 451 – Jose Schutt-Aine 9
( )
Forward Wave Backward Wave
j z j zV z V e V e
z
Zo
forward wave
backward wave
LC
o
LZ
C
(Frequency Domain)
( ) j z j z
o o
Forward Wave Backward Wave
V VI z e e
Z Z
TL Solutions
ECE 451 – Jose Schutt-Aine 10
( , ) cos cos
Forward Wave Backward Wave
V z t V t z V t z
LC
o
LZ
C
( , ) cos coso o
Forward Wave Backward Wave
V VI z t t z t z
Z Z
1v
LC
v
f
Propagation constant
Characteristic impedanceWavelength
Propagation velocity
TL Solutions
ECE 451 – Jose Schutt-Aine 11
At z=0, we have V(0)=ZRI(0)
But from the TL equations:
(0)V V V
(0) o o
V VI
Z Z
Which gives
R
o
ZV V V V
Z
RV V
R oR
R o
Z Z
Z Z
is the load reflection coefficientwhere
Reflection Coefficient
ECE 451 – Jose Schutt-Aine 12
- If ZR = Zo, GR=0, no reflection, the line is matched
- If ZR = 0, short circuit at the load, GR=-1
- If ZR inf, open circuit at the load, GR=+1
( ) j z j zRV z V e e
( ) j z j zR
o
VI z e e
Z
2( ) 1j z j zRV z V e e
2( ) 1j z
j zR
o
V eI z e
Z
Reflection Coefficient
V and I can be written in terms of R
ECE 451 – Jose Schutt-Aine 13
( )( )
( )
j z j zR
o j z j zR
e eV zZ z Z
I z e e
tan
tanR o
oo R
Z jZ lZ l Z
Z jZ l
Generalized Impedance
Impedance transformatio
n equation
ECE 451 – Jose Schutt-Aine 14
( ) tanoZ l jZ l
( )tan
oZZ lj l
- Short circuit ZR=0, line appears inductive for 0 < l < l/2
- Open circuit ZR inf, line appears capacitive for 0 < l < l/2
2
( ) o
R
ZZ l
Z
- If l = l/4, the line is a quarter-wave transformer
Generalized Impedance
ECE 451 – Jose Schutt-Aine 15
( )Backward traveling wave at( )
Forward traveling wave at ( ) b
f
V zzz
z V z
2 2( )
j z
j z j zRj z
V e Vz e e
V e V
2( ) j lRl e Reflection coefficient
transformation equation
1 ( )( )
1 ( )
o
zZ z Z
z( )
( )( )
o
o
Z z Zz
Z z Z
Generalized Reflection Coefficient
ECE 451 – Jose Schutt-Aine 16
2( ) 1j z j zRV z V e e
2 2( ) 1 1 j z j z j z
R RV z V e e V e
max 1 RV
min 1 RV
Maximum and minimummagnitudes given by
Voltage Standing Wave Ratio (VSWR)
We follow the magnitude of thevoltage along the TL
ECE 451 – Jose Schutt-Aine 17
max
min
1
1
R
R
VVSWR
V
Voltage Standing Wave Ratio (VSWR)
Define Voltage Standing Wave Ratio as:
It is a measure of the interaction between forward and backward waves
ECE 451 – Jose Schutt-Aine 22
Application: Slotted-Line Measurement
- Measure VSWR = Vmax/Vmin
- Measure location of first minimum
ECE 451 – Jose Schutt-Aine 23
Application: Slotted-Line Measurement
At minimum, ( ) pure real Rz
Therefore, min2min( ) j d
R Rd e
min2 j dR R e So,
1
1
R
VSWR
VSWRSince min21
1
j dR
VSWRe
VSWRthen
1
1
R
R oR
Z Zand
ECE 451 – Jose Schutt-Aine 24
Summary of TL Equations
tan
tanR o
oo R
Z jZ lZ l Z
Z jZ l
2( ) j lRl e
1 ( )( )
1 ( )
o
zZ z Z
z
( )( )
( )
o
o
Z z Zz
Z z Z
Impedance Transformation
Reflection Coefficient Transformation
Reflection Coefficient – to Impedance
Impedance to Reflection Coefficient
2( ) 1 j z j z
Ro
VI z e e
Z 2( ) 1j z j z
RV z V e e
Voltage Current
ECE 451 – Jose Schutt-Aine 25
+- Vg =100 V
Zo = 50
Zg =(10+j10)
d=l d=0
ZR=(30+j40) Zin
Ig
Consider the transmission line system shown in the figure above.(a) Find the input impedance Zin.
(30 40) 500.5 90
(30 40) 50R o
RR o
Z Z j
Z Z j
4.0.7252( ) 0.5 90 0.5 72
jj lRd l e e
Example 1
ECE 451 – Jose Schutt-Aine 26
1 ( ) 1 0.5 7250
1 ( ) 1 0.5 72in o
lZ Z
l
64.361 51.738 39.86 50.54inZ j
(b) Find the current drawn from the generator
100 01.552 39.11
(10 10)(39.86 50.54)g
gg in
VI A
Z Z j j
1.207 0.981gI j A
Example 1 – Cont’
ECE 451 – Jose Schutt-Aine 27
c) Find the time-average power delivered to the load
1( ) 64.361 51.73 1.5562 39.11 100.159 12.62
2in gV l Z I
*1 1Real ( ) Real 100.159 12.62 1.5562 39.11
2 2gP V l I
*1Real ( ) 48.26
2 gP V l I W
Example 1 – Cont’
ECE 451 – Jose Schutt-Aine 28
+- Vg
Zo = 50
d=l d=0
L=1 H
C=100 pF
R=50
In the system shown above, a line of characteristic impedance 50 W is terminated by a series R, L, C circuit having the values R = 50 W, L = 1 mH, and C = 100 pF.
(d) Find the source frequency fo for which there are no standing waves on the line.
Example 1 – Cont’