ece 350 fields and waves ii the smith chartjsa.ece.illinois.edu/bach_khoa/ece350/notes/smith.pdf ·...
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ECE 350 – Jose Schutt‐Aine 1
ECE 350Fields and Waves II
The Smith Chart
Jose E. Schutt-AineElectrical & Computer Engineering
University of [email protected]
ECE 350 – Jose Schutt‐Aine 2
TL Equations
tantan
R oo
o R
Z jZ lZ l ZZ jZ l
2( ) j lRl e
1 ( )( )1 ( )o
zZ z Zz
( )( )( )
o
o
Z z ZzZ 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 eZ
2( ) 1j z j zRV z V e e
Voltage Current
ECE 350 – Jose Schutt‐Aine 3
Derivation of the Smith Chart
o1 ( z )Z( z ) Z1 ( z )
n1 ( z ) 1Z ( z )1 ( z ) 1
The relationship between impedance and reflection coefficient is given by:
where Zo is the characteristic impedance of the system. The normalized impedance is
r ij nZ r jx
The reflection coefficient and the normalized impedance have the form:
and
ECE 350 – Jose Schutt‐Aine 4
( ( =
(r i r i
2 2r i
1 ) j 1 ) j1 )
( (=(
2 2r i r i r i
2 2r i
1 j 1 ) j 1 )r jx1 )
Therefore
Separating real and imaginary components,
r i
r i
1 jr jx1 j
Derivation of the Smith Chart
(
2 2r i
2 2r i
1r1 )
Isolating the real part from both sides
ECE 350 – Jose Schutt‐Aine 5
2 2 2r r i r ir 1 2 1
2 2r i r( r 1 ) ( r 1 ) 2r 1 r
2 2 rr i
2r 1 r1 r 1 r
2 22 2 rr i 2 2
2r r 1 r r1 r (1 r ) 1 r (1 r )
22
r i 2
r 11 r (1 r )
Multiplying through by the denominator,
Completing the square
Derivation of the Smith Chart
or
ECE 350 – Jose Schutt‐Aine 6
r ,01 r
11 r
This is the equation of a circle centered at
and of radius
22
r i 2
r 11 r (1 r )
Derivation of the Smith Chart
Equating the imaginary parts gives
i2 2
r i
2x(1 )
2 2r r i ix 1 2 2
2 2r r i ix 2x x 2 x or
ECE 350 – Jose Schutt‐Aine 7
11,x
1x
This is the equation of a circle centered at
of radius
2 2 ir r i 2 2
2 1 12 1 1 1x x x
2
2r i 2
1 11x x
Derivation of the Smith Chart
ECE 350 – Jose Schutt‐Aine 8
n
n
Z 1 r 1 jxZ 1 r 1 jx
1/ 22 2
2 2
( r 1 ) x 1( r 1) x
n1 ( z )Z1 ( z )
n
1 1 ( z )yZ 1 ( z )
The reflection coefficient is given by
We also have
The Smith Chart
Thus, going from normalized impedance to normalized admittance corresponds to a 180 degree shift.
ECE 350 – Jose Schutt‐Aine 9
The Smith Chart
3 ways to move on the Smith chartConstant SWR circle displacement along TLConstant resistance (conductance) circleaddition of reactance (susceptance)Constant reactance (susceptance) arc addition of resistance (conductance)
ECE 350 – Jose Schutt‐Aine 10
The Smith Chart
ECE 350 – Jose Schutt‐Aine 11
2270 3 0 4 0 6 jR Rz . j . . e
0 5 0 435 0 065mind . . .
2. SWR on the lineSWR=4.0
1. Reflection coefficient at load
3. dmin
Smith Chart Example
FIND:
ECE 350 – Jose Schutt‐Aine 12
50 0 26 0 09 13 4 5. j . j .
3 5 1 2 50 0 068 0 025. j . / . j .
0 14 0 325 0 185 0 14. . j . .
4. Line impedance at 0.05 to the left
6. Location nearest to load where Real[y]=1
5. Line admittance at 0.05l
Smith Chart Example
ECE 350 – Jose Schutt‐Aine 13
Smith Chart Example