Download ENE 429 Antenna and Transmission Lines

ENE 490
Applied Communication Systems
Lecture 1 Backgrounds on Transmission
lines and matching on Smith chart
1
Introduction
 How does information transfer?
mixer
power amp
power amp
mixer
signal
receiving signal
matching
network
Local oscillator
Transmitter
BPF
BPF
matching
network
Local oscillator
Receiver
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High frequency operation
 Microwave frequency range (300 MHz – 300
GHz)
 Microwave components are distributed
components.
 Lumped circuit elements approximations are
invalid.
 Maxwell’s equations are used to explain
circuit behaviors ( Eand )H
3
Applications of high frequency
communications
 Antenna gain
 More bandwidth
 Satellite and terrestrial communication
links
 Radar communication
 Remote sensing, medical diagnostics,
and heating methods
4
Frequency behavior of passive
components
1
XC 
C
X L  L
At 60 Hz
1
9
X C ( 60 Hz) 

2
.
65

10

12
2  60 10
X L ( 60 Hz)  2  60 109  3.77 107   0
An equivalent circuit representation
of high frequency resistor
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Transmission lines (TLs) analysis (1)
 At higher frequencies, voltage and current are not
spatially uniform, must be treated in terms of
propagating waves.
The distributed-parameter model including instantaneous voltage and current
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Transmission lines (TLs) analysis (2)
 Kirchhoff’s circuit laws fail to explain circuit behaviors
at high frequency
 The transmission line must be viewed in terms of
distributed parameters, R, L, C, and G.
 A transmission line theory is applied when
l A   / 10
(lA is the average size of the discrete component)
Example of transmission lines
7
General transmission line equation (1)
 Kirchhoff voltage and current law representations
i ( z , t )
v( z , t )  i ( z , t ) Rz  Lz
 v( z  z , t )
t
v( z  z , t )  v( z , t )
i ( z , t )
lim
  Ri ( z, t )  L
z 0
z
t
dv( z, t )

 ( R  jL)i ( z , t )
dz
i ( z, t )
v( z , t )
 Gv( z, t )  C
z
t
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General transmission line equation (2)
Let VS (t )  Vs cos t  v( z , t )  Re V ( z )e jt 
and
i ( z , t )  Re  I ( z )e jt 
V ( z )  V0 e z  V0 e z
I ( z) 
I 0 e z

I 0 e z
V0 z V0 z

e 
e
Z0
Z0
  ( R  jL)(G  jC )    j
V0 V0
( R  jL)
Z0 
 R0  jX 0     ()
(G  jC )
I0
I0
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Lossless case
 Special case:
Lossless line R = 0  and G = 0
  j LC  j
L
Z0 
C
10
Terminated lossless transmission line
 Convenient representation of source and load ends
for some T.L. problems

V0e
V0e
ZL
Z0

V0e
x=0

-j z
ZL
Z0

+jz
x=l
j d
V0e
d=l
-j d
d=0
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Voltage reflection coefficient
Definition:
  jd
V
Reflected Voltage Wave (d)
0 e
( d ) 
  jd
Incident Voltage Wave (d)
V0 e
V0  j 2d
( d )   e
V0
 Define load reflection coefficient:
V0
 L  (0)  
V0
 Load reflection coefficient in terms of impedance:
Z L  Z0
L 
Z L  Z0
Note: ZL is a load impedance.
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Impedance along the transmission line
V (d ) V0 e jd  V0 e jd
Z (d ) 
  jd
I (d ) V0 e
V0 e jd

Z0
Z0
Z L cos d  jZ 0 sin d
Z (d )  Z 0
Z 0 cos d  jZ L sin d
Z L  jZ 0 tan d
Z (d )  Z 0
Z 0  jZ L tan d
 Impedance anywhere on the transmission line
1  ( d )
Z (d ) 
1  ( d )
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Ex1 Determine the input impedance Zin
when
a)
l = /4
b)
l = /2
c)
ZL = Z0
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Voltage standing wave ratio VSWR (1)
V (d )  V0 (e jd  Le jd )
V (d )  V (d )e jd 1   Le j 2d
V (d ) max  V (d ) 1   L

V (d ) min  V (d ) (1   L )
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Voltage standing wave ratio VSWR (2)
I (d ) max
V (d )

1 L

Z0

I (d ) min
V (d )

1  L

Z0

V (d ) max
V (d ) min

I (d ) max
I (d ) min
1  L

 VSWR
1  L
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Ex2 Determine VSWR and the locations
of maximum and minimum voltages for
a)
matched load
b)
ZL = 100 , Z0 = 50 
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c) short circuit
d) open circuit
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Source and loaded transmission lines
L
S
ZS
Z 0±
ZL
+
VS
IN
Z in  Z 0
in  (d  l ) 
  0e 2 jl
Z in  Z 0
OUT
Z S  Z0
S 
Z S  Z0
out  S e j 2l
2Z (d )
Transmission coefficient: T (d )  1  ( d ) 
Z (d )  Z 0
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Power transmission of a transmission
line
 
1
*
 Average power: Pav  Re VI
2
2
2
1  S
1 VS
2
 For lossless line: Pin 
1  in
2
8 Z0 1   

S

in
 For lossless and matched condition: we have Pavs,
maximum available power provided by the source.
2
1 VS
Pin 
 Pavs
8 ZS
P W 
 Power in decibel: P  dBm   10 log
1 mW
20
Ex3 For the circuit shown above, assume a lossless line with Z0 = 50
, ZS = 75, and ZL = 100. Determine the input power and
power delivered to the load. Assume the length of the line to be /2
with a source voltage of VS = 10 V.
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Input impedance matching
+
ZS
+
VS
-
V
Zin
-
 Optimal power transfer requires conjugate complex
matching of the T.L. to the source impedance:
Zin = ZS*
 Similarly for output matching:
Zout = ZL*
22
The Smith Chart
 a graphical tool to analyze circuit impedance
 design of matching networks
 computations of noise figures, gain, and stability
circles.
23
Using of the Smith Chart
 Impedance transformation
Step 1 – Normalize the load impedance ZL with respect to
the line impedance Z0 to determine zL.
Step 2 – Locate zL in the Smith Chart
Step 3 – Identify the corresponding load reflection
coefficient 0 in the Smith Chart both in terms of
its magnitude and phase.
Step 4 – Rotate 0 by the length in terms of wavelength 
or twice its electrical length d to obtain in(d).
Step 5 – Record the normalized input impedance zin at
this spatial location d.
Step 6 – Convert zin into actual impedance Zin.
24
Ex4 Given the load impedance ZL to be 30+j60,
Determine the input impedance if the T.L. is 2 cm
long and is operated at 2 GHz.
25
Standing wave ratio in Smith chart
 The numerical value of SWR can be found from the
Smith chart by finding the intersection of the circle of
radius with the right hand side of the real axis.
Ex5 Three different load impedances:
a) ZL = 50 ,
b) b) ZL = 25+j75 , and
c) c) ZL = 40 + j20 ,
are sequentially connected to a 50  transmission line.
Find the reflection coefficients and the SWR circles.
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Admittance transformation
1
Yin 
yin  Y0 yin
Z0
 Rotations by 180 degrees convert the impedance to
the admittance representation.
 Y-Smith chart
 ZY-Smith chart
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Parallel and Series Connections (1)
 Parallel connection of R and L elements
Yin
R
L
ZS
AC
Z0
yin (L )  g  j
L
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Parallel and Series Connections (2)
 Parallel connection of R and C elements
Yin
R
C
ZS
AC
yin (L )  g  jZ0LC
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Parallel and Series Connections (3)
 Series connection of R and L elements
Z in
L
R
ZS
AC
L L
zin (L )  r  j
Z0
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Parallel and Series Connections (4)
 Series connection of R and C elements
Z in
C
R
ZS
AC
1
zin (L )  r  j
LCZ0
31
Ex6 of a T-network (operated at 2 GHz)
ZL
C=1.91 pF
L=4.38 nH
C= 2.39 pF
L=3.98 nH
R=31.25 
Z in
T-type network
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