ECE 5317-6351
Microwave Engineering
Fall 2015
Prof. David R. Jackson
Dept. of ECE
Notes 2
Smith Charts
1
Generalized Reflection Coefficient
I (z)
Z0 , β
+
ZL
V (z)
z
Recall:
z=0
Z − Z0
ΓL = L
Z L + Z0
+2 γ z
+ −γ z
=
V ( z ) V0+ e −γ z (1 + Γ=
e
V
(1 + Γ ( z ) )
)
L
0 e
+
V0+ −γ z
V
+2 γ z
0
=
=
I ( z)
e (1 − Γ
e −γ z (1 − Γ ( z ) )
)
Le
Z0
Z0
 1+ Γ ( z) 
V ( z)
 1 + Γ L e +2 γ z 
=
=
Z=
z
Z
Z
( )

0
0
+2 γ z 
−
Γ
−
Γ
I ( z)
e
z
1
1
( )
L



Generalized reflection Coefficient:
Γ ( z) =
Γ L e +2 γ z
2
Generalized Reflection Coefficient (cont.)
Γ ( z) =
ΓL e
= ΓL e
Z − Z0
ΓL = L
Z L + Z0
+2 γ z
jφL
e
+2 γ z
For
=Γ R ( z ) + jΓ I ( z )
Re {Z L } ≥ 0
⇒ ΓL ≤ 1
Proof:
RL + jX L ) − Z 0
(
ΓL =
( RL + jX L ) + Z 0
RL − Z 0 ) + jX L
(
=
( RL + Z 0 ) + jX L
Lossless transmission line (α = 0)
Γ ( z) =
Γ L e j(φL + 2 β z )
RL − Z 0 ) + X L2
(
=
2
( RL + Z 0 ) + X L2
2
⇒ ΓL
2
3
Complex Γ Plane
Γ = Γ ( z)
=Γ R + jΓ I
= Γ Le
j ( +2 β z )
Im Γ
Decreasing d (toward load)
= Γ L e j(φL + 2 β z )
= Γ L e j(φL −2 β d )
ΓL
ΓL
φL
φL − 2 β d
d
Γ
Increasing d (toward
generator)
Re Γ
z = −d
Lossless line
d = distance from load
4
Impedance (Z) Chart
1+ Γ 
Z ( z ) = Z0 

1
−
Γ


Γ = Γ ( z)
Z ( z)  1+ Γ 
Zn ( z ) ≡
=


1
Z0
−
Γ


Define
Z=
Rn + jX n
n
Hence we have:
 1 + ( Γ R + jΓ I ) 
Rn + jX n =


 1 − ( Γ R + jΓ I ) 
Next, multiply both sides by the RHS denominator term and equate real and imaginary
parts. Then solve the resulting equations for ΓR and ΓI in terms of Rn or Xn. This gives two
equations.
5
Impedance (Z) Chart (cont.)
1) Equation #1:
2

 1 
Rn 
2
 ΓR −
 + Γ I =

+
+
R
R
1
1
n 
n 


2
ΓI
Equation for a circle in the Γ plane
1
ΓR
 R

center =  n , 0 
 1 + Rn 
1
radius =
1 + Rn
6
Impedance (Z) Chart (cont.)
2) Equation #2:
2

1   1 
( Γ R − 1) +  Γ I −  = 
Xn   Xn 

2
2
ΓI
Equation for a circle in the Γ plane
1
ΓR
 1 
center = 1,

X
n 

1
radius =
Xn
7
Impedance (Z) Chart (cont.)
Short-hand version
Xn = 1
Rn = 1
Xn = -1
Γ plane
Γ plane
8
Impedance (Z) Chart (cont.)
Imag. (reactive)
impedance
Inductive (Xn > 0)
plane
ΓΓ plane
Xn = 1
Rn = 1
Short ckt.
(Γ= −1)
Match pt.
(Γ=0)
Open ckt.
(Γ=1)
Real
impedance
Xn = -1
Capacitive (Xn < 0)
9
Admittance (Y) Calculations
1
1  1− Γ ( z) 
Y=
( z) =


Z ( z ) Z0  1 + Γ ( z ) 
Note:
 1 + ( −Γ ( z ) ) 
= Y0 
 1 − ( −Γ ( z ) ) 


Y0 ≡
1
Z0
Y ( z )  1 + ( −Γ ( z ) ) 
⇒ Yn ( z ) =
=
=
 Gn ( z ) + jBn ( z )

Y0
 1 − ( −Γ ( z ) ) 
Define: Γ′ = − Γ
 1 + Γ′ 
Yn ( z ) = 

′
−
Γ
1


Same mathematical form as for Zn:
Conclusion: The same
Smith chart can be used as
an admittance calculator.
1+ Γ 
Zn ( z ) = 

1− Γ 
10
Admittance (Y) Calculations (cont.)
Imag. (reactive)
admittance
Capacitive (Bn > 0)
plane
Γ′Γ′plane
Bn = 1
Gn = 1
Open ckt.
(Γ′ = −1)
Match pt.
(Γ′ =0)
Short ckt.
(Γ′ =1)
Real
admittance
Bn = -1
Inductive (Bn < 0)
11
Impedance or Admittance (Z or Y)
Calculations
Normalized impedance or admittance coordinates
The Smith chart can be used
for either impedance or
admittance calculations, as
long as we are consistent.
The complex plane is either the
Γ plane or the Γ′ plane.
12
As an alternative, we can continue to use the original Γ plane,
and add admittance curves to the chart.
 1 + ( −Γ ( z ) ) 
Yn ( z ) = 
= Gn ( z ) + jBn ( z )
 1 − ( −Γ ( z ) ) 


Admittance (Y) Chart (cont.)
Gn = 0
Inductive (Bn < 0)
plane
ΓΓplane
Bn = -1
Gn = 1
Short ckt.
Match pt.
Bn = 0
Open ckt.
Bn = +1
Capacitive (Bn > 0)
14
Admittance (Y) Chart (cont.)
Short-hand version
Bn = -1
Gn = 1
Bn = 1
Γ plane
15
Impedance and Admittance (ZY) Chart
Short-hand version
Bn = -1
Xn = 1
Gn = 1
Rn = 1
Bn = 1
Xn = -1
Γ plane
16
All Four Possibilities for Smith Charts
Z Chart
Y Chart
Γ′ plane
Γ plane
Y Chart
The top two
are the most
common.
Z Chart
Γ plane
Γ′ plane
17
Standing Wave Ratio
The SWR is given by the
value of Rn on the positive
real axis of the Smith chart.
Proof:
SWR =
ΓL
1 + ΓL
1 − ΓL
1+ Γ ( z)
Zn =
1 − Γ ( z)
Rn = Rnmax
1 + Γ L e +2 j β z
=
1 − Γ L e +2 j β z
Γ plane
⇒ R
max
n
1+ ΓL
=
1− ΓL
1 + Γ L e jφL e +2 jβ z
=
1 − Γ L e jφL e +2 jβ z
18
Electronic Smith Chart
At this link:
http://www.sss-mag.com/topten5.html
Download the following zip file:
smith_v191.zip
Extract the following files:
smith.exe
mith.hlp
smith.pdf
This is the application file
19