INDIAN INSTITUTE OF TECHNOLOGY ROORKEE
ECN 342 RFM
Network Parameters & Smith chart
Prof. Darshak Bhatt
Two port parameters
2
3
Find the Z parameters?
4
2-Port Network Representation
❑ Reciprocal Network: A network is said to be reciprocal if the voltage
appearing at port 2 due to a current applied at port 1 is the same as the
voltage appearing at port 1 when the same current is applied to port 2.
Exchanging voltage and current results in an equivalent definition of
reciprocity.
❑ Symmetrical networks: A network is symmetrical if its input impedance
is equal to its output impedance.
❑ Lossless: A network containing no lossy elements.
❑ All these properties are easy to link in terms of Z or Y-Parameters, For
rest of the Network parameters it can be derived by using their
conversion from Z or Y parameters matrix.
5
Z: Impedance Parameters
❑
Symmetric: Z11=Z22
❑
Reciprocal:Z12=Z21
❑
Lossless: All components of Z matrix are purely imaginary.
6
Y: Admittance Parameters
❑
Symmetric: Y11=Y22
❑
Reciprocal: Y12=Y21
❑
Lossless: All components of Y matrix are purely imaginary.
7
THE TRANSMISSION (ABCD) MATRIX
8
9
ABCD: Transmission Parameter
❑
Symmetric: A=D
❑
Reciprocal: AD-BC=1
❑
Lossless: A, D are real and B, C are imaginary. (Diag. elements are real and
off-diag. are imaginary.
10
ABCD Parameters of Some Useful TwoPort Circuits
11
ABCD Matrix of Transmission
Lines (1/2)
V (d ) = A1e j d + B1e− j d
A=
V1
  =1
V2 I =0
2
C=
V1   A B  V2 
 I  = C D   I 
 2
 1 
V (d ) = A1 ( e j  d +  0 e − j  d )
I1
  =1
V2 I =0
2
V (d ) = A1 ( e j  d + e − j  d ) = 2 A1 cos (  d ) = V2 cos (  d )
Since V (0) = 2 A1 = V2
Since V (l ) = V1
C=
A=
V1
= cos (  l )
V2 I =0
2
 V1 
I1
V
1
= 1
=
 
V2 I =0 Z inV2 I =0 − jZ 0 cot (  l )  V2  I =0
2
2
2
C=
V1
= jY0 sin (  l )
V2 I =0
2
12
ABCD Matrix of Transmission
Lines (2/2)
I
D= 1
  = −1
I 2 V =0
2
I (d ) =
B=
V1
  = −1
I 2 V =0
2
A1 j d
A
A
e − 0e− j d ) = 1 ( e j d + e− j d ) = 2 1 cos (  d ) = I 2 cos (  d )
(
Z0
Z0
Z0
Since I (0) =
2 A1
= I2
Z0
D=
I1
= cos (  l )
I 2 V =0
2
Since I (l ) = I1
B=
I 
Z I
V1
= in 1
= jZ 0 tan (  l )  1 
I 2 V =0
I 2 V =0
 I 2  V =0
2
2
2
B=
V1
= jZ 0 sin (  l )
I 2 V =0
2
13
Why S Parameters
• No equipment is available to measure total voltage and total
current at these frequencies.
• The open-circuit and short-circuit terminations are difficult.
• Active devices such as transistors may become unstable
under open-circuit or short-circuit conditions.
14
S Matrix
15
VNA
16
Find S parameter of 3 dB attenuator
17
Generalized S Parameters
• S-parameters are defined based on normalized traveling
wave voltages a and b –
18
19
Shifting reference planes for lossless lines
20
• This means that our scattering parameters in terms of the
primed quantities now become
• The relationship between matrices can be written more
conveniently as
21
Properties of S-Parameters
• Reciprocal : Sji = Sij; e.g.S21 = S12
Lossless Network:
• The total power flowing out is equal to the total power flowing
in.
22
Example
• Determine if the network is reciprocal and lossless.
• If port 2 is terminated with a matched load, what is the return
loss seen at port 1?
• If port 2 is terminated with a short circuit, what is the return
loss seen at port 1?
23
24
SIGNAL FLOW GRAPHS
Nodes
Branch
25
Signal Flow Graph Rules
26
Cont..
Series Rule
Parallel Rule
Self Loop Rule
Splitting rule
27
Example
• Use signal flow graphs to derive expressions for Γ𝑖𝑛 and Γ𝑜𝑢𝑡
for the microwave network shown in Figure.
28
Sol
29
Three-Port Circulator
30
4 port Network (Coupler)
31
Parameter conversion
32
Network Parameter Conversion
33
The Smith Chart
• It is polar plot of Γ.
• Bilateral Transform connecting the impedance Z and the
Reflection coefficient Γ . The smith chart maps the x-plane
on the Γ plane.
34
•
-→
35
Resistance & Reactance circles
• Γ𝑖 = 0,
36
• For negative resistance r<0 we have |Γ|>1
37
Active and Passive Load
38
Inductance and Capacitance on Smith
Chart
• Locus of the reflection coefficient for an inductor and a
capacitor In a Z Smith chart
39
Reflection Coefficient
• The reflection coefficient is the ratio of the reflected to the
incident wave. For a loss less line we have :
40
Reflection Coefficient along a line
For a loss less line the reflection coefficient can be written :
• As we move along the line, Γ(d) moves along a circle of
radius |Γ𝐿 |.
• The reflection coefficient Γ(d) rotates clock wise as d
increases, and we moves towards generator
41
Measurement Point Definition
42
Voltage Standing Wave Ratio (VSWR) on
the smith chart
• A perfect matching (|Γ|=0) corresponds to VSWR of 1.
• VSWR should be less than 2.
43
Y Smith Chart
• The Y - smith chart can be obtained by expressing Γ in terms
of Y
• The Y Smith Chart is obtained by inverting the Z smith chart.
• In the rotated Y-Smith Chart the short and open are
exchanged.
44
Impedance and Admittance Smith Charts
45
Addition of an Inductor in Series with a
Load
• Given the load z = 0.5 - j1.0 by adding 𝑍𝐿 = 𝑗1.5
• 𝑧𝑖𝑛 = 𝑧 + 𝑧𝐿 = 0.5 + 𝑗0.5.
46
Addition of a Capacitor in Series with a
Load
• Given the load z = 0.5 – j0.5 by adding 𝑧𝐶 = −𝑗0.5
• 𝑧𝑖𝑛 = 𝑧 + 𝑧𝑐 = 0.5 − 𝑗1.
47
Addition of an Inductor in Shunt with a
Load
• Given the load y = 0.5 + j0.5 by adding yL = −𝑗1.5.
• 𝑦𝑖𝑛 = 𝑦 + 𝑦𝑙 = 0.5 − 𝑗1.
48
Addition of an Capacitor in Shunt with a
Load
• Given the load y = 0.5 – j1 by adding yc = 𝑗0.5.
• 𝑦𝑖𝑛 = 𝑦 + 𝑦𝑐 = 0.5 − 𝑗0.5.
49
How to match a load with a series L and
Shunt C
• We want to realize a transformer with a series inductor and
shunt capacitor to match the complex impedance
z = 0.5 – j1 to 50 ohm load.
50
• Given the load z=0.5- j1.0, by moving up on the constant
resistance r=0.5 circle on the Z Smith chart we obtain
• 𝑧𝑖𝑛 = 𝑧 + 𝑧𝑙 = 0.5 + 𝑗0.5. using 𝑧𝐿 = 𝑗1.5
• Since 𝑦𝑖𝑛 = 1 − 𝑗1, using yc = 𝑗1. we obtain 𝑦𝑖𝑛 + 𝑦𝑐 = 1 by
moving on the constant conductance g=1 circle on the Y
Smith chart.
51
Alternate Design: Matching a Load with a
series L and shunt L
• Given the load z=0.5- j1.0, by moving up on the constant
resistance r=0.5 circle on the Z Smith chart we obtain
• 𝑧𝑖𝑛 = 𝑧 + 𝑧𝑙 = 0.5 − 𝑗0.5. using 𝑧𝐿 = 𝑗0.5
• Since 𝑦𝑖𝑛 = 1 + 𝑗1, using yc = −𝑗1. we obtain 𝑦𝑖𝑛 + 𝑦𝑐 = 1 by
moving on the constant conductance g=1 circle on the Y
Smith chart.
52
Matching a Load with a series C and
Shunt C
• Given the load z=0.5+j1.0, by moving down on the constant
resistance r=0.5 circle on the Z Smith chart we obtain
• 𝑧𝑖𝑛 = 𝑧 + 𝑧𝑙 = 0.5 + 𝑗0.5. using 𝑧𝑐 = −𝑗0.5
• Since 𝑦𝑖𝑛 = 1 − 𝑗1, using yc = 𝑗1. we obtain 𝑦𝑖𝑛 + 𝑦𝑐 = 1 by
moving down on the constant conductance g=1 circle on the
Y Smith chart.
53
Stub Matching Example
• A 50 Ω transmission line is connected to a cellular phone
antenna with load impedance ZL =25-j50 Ω. Find the
position and the length of a shunt short-circuit stub required
to match the 50 Ω line.
54
55