Lecture 3:Microwave Network Analysis, Sparameters and impedance matching networks
When do you need to carry out microwave
network analysis?
Objective: move from the requirement to solve for all the fields and
waves of a structure to an equivalent circuit that is amenable to all
the tools of the circuit analysis.
Reasons to use network analysis over Maxwell’s equations:
a) A field analysis using Maxwell’s equations is normally difficult
and provide much more information than we need.
b) We are only interested in the signal flow and the voltage and
current at a set of terminals.
c) Many RF/Microwave components/devices have more than 1 port
and present cumbersome problems for complete field analysis
(multiple interfaces).
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• Circuit dimensions << wavelength
– Lumped passive and active components.
– Negligible phase change throughout the circuit.
– Circuit theory --- Kirchhoff’s laws and Ohm’s law.
• Circuit dimensions ~ wavelength
– Distributed passive and active components.
– Phase depends on position. Components are
characterized by their dimensions, propagation
constants and characteristics impedances.
– Microwave network theory.
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Impedance Matrix
Impedance and Admittance Matrices
• Two-terminal pair --> port
• V and I ---> equivalent V and I.
– Reference planes are defined to provide a phase reference for the
equivalent V and I phasors.
– At the nth reference plane, the total voltage and current are
Vn = Vn+ + Vn−
I n = I n+ − I n−
• The impedance matrix that relates these voltages and
currents:
[V ] = [Z ][I ]
V
Z ij = i
Ij
I k = 0 for k ≠ j
Zii: input impedance
Zij: transfer impedance between ports i and j, (i ≠ j)
An arbitrary N-port network.
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Some Characteristics of Impedance and
Admittance Matrices
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The Scattering Matrix (S-parameter Matrix)
• Reciprocal Networks (no active devices, ferrites, or plasmas -- no electrical or magnetic sources): defined as having identical
transmission characteristics from port one to port two or from
port two to port one --- circuit behavior independent of directions
of waves and currents.
Both Z and Y matrices are symmetric.
Yij = Y ji
Z ij = Z ji
• Lossless Networks
All the Z and Y elements are imaginary.
* However, to determine Z and Y elements, both voltage and
current values need to be measured. This is difficult at microwave
frequencies. Furthermore, open and short circuits can easily result
in oscillations in circuits.
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• The exact value of voltage and current is difficult to define
for non-TEM lines.
• Difficult to deal with voltage and current in high frequency
measurement.
• It is more convenient to deal with the ratio of voltages or
currents reflected or transmitted.
• The scattering matrix of N-port networks with the same
characteristic impedance at all ports is defined as
V1−   S11 S12 L S1N  V1+ 
 − 
 
M  V2+ 
V2  =  S 21

M   M
 M 
 − 
 
S NN  VN+ 
VN   S N 1 L
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Generalized Scattering Matrix
Conversion of impedance to admittance (Z to Y)
Rule: simply rotating the Γ vector of the impedance by 180 degree.
The scattering matrix of N-port networks with characteristic
impedance Z0n at nth port is defined as
 b1   S11
b  S
 2  =  21
M  M
  
bN   S N 1
S12 L S1N   a1 
M   a2 
 
 M 
 
L
S NN  a N 
an =
Vn+
Z0n
Vn−
bn =
Z0n
Z 0 n : Z 0 of the nth port
• Sij is found by driving port j with V j+ , and measuring the reflected
wave amplitude,Vi − , coming out of port i. The incident waves on all
ports except the jth port are set to zero, which means that all ports are
connected to matched loads.
• The matched load has advantages in terms of its insensitivity to the
transmission line length.
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Frequency response of networks in Smith chart
Impedances tend to move
clockwise with frequency for
passive networks
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The Scattering Matrix (S-parameter Matrix)
Tank Circuit: loop
Why do we need this at all?
small loops indicating selfresonances of the inductor and
capacitor.
• It is not practical to measure voltages and currents at the ports at
microwave frequencies.
• It is natural to deal with power in incident and reflected waves for
microwave transmission lines.
• Active devices may not be stable with short or open terminations
due to oscillation.
• The Scattering matrix relates the voltage waves incident on the
ports to those reflected from the ports.
• Most importantly, scattering matrix elements can be measured
without open or short in the load, just matching loads. There is
no reflected wave regardless of the length of the transmission lines
used --- practical to implement.
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Example 4.4 on page 198 of Pozar.
S parameters of the 3 dB attenuator circuit
S-parameters for Reciprocal Networks and
Lossless Networks
S11 S12  0 0.707
S21 S22  = 0.707 0

 

• Reciprocal Networks
– [S] is symmetric. For a 2x2 [S], S12=S21.
• Lossless Networks
– [S] is a unitary matrix.
[S ]t [S ]* = [U ]
N
∑S
ki
 N
*
 ∑ S ki S ki = 1
k =1
N
∑ S ki S kj* = 0
 k =1
S = δ ij
*
kj
k =1
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where [U] is the unit matrix.
for i=j
for i≠ j
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Shift in Reference Plane
Features on S-parameters
• The reflection coefficient looking into port n is not equal to Snn,
unless all other ports are connected to matched load.
• The transmission coefficient from port m to port n is not equal to
Snm, unless all other ports are connected to matched load.
• The S parameters are properties of the network itself, and are
defined under the condition that all ports are connected to
matched loads. Changing the terminations or excitations of a
network does not change its S parameters, but may change the
reflection and transmission coefficients.
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Open occurs in practical measurement setup.
S11' = S11e − j 2φ1
'
S 21
= S 21e − j (φ1 +φ2 )
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The Transmission (ABCD) Matrix
• Used for a cascade connection of two or more twoport connection.
• Defined as
V1   A B  V2 
V1 = A V2 + B I2
 I  = C D   I 
I1 = C V2 + D I2
 2 
 1 
A two-port network:
• For the cascade connection of two-port networks, we
have
V1   A1 B1   A2 B2  V3 
 I  = C D  C D   I 
 1  1
1 2
2  3 
ABCD Example: Quarter- and Half-Wave Transmission
I2
Lines
l
V1
Zo
ZL
The ABCD matrix of a length of transmission line l of Zo and ß is
A cascade connection:
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A = cos ßl
B = j Zo sin ßl
C = j Yo sin ßl
D = cos ßl
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• Note that, for cos ßl = 0 (that is, for ßl=π/2, a quarter
wavelength or odd multiple) the ABCD matrix becomes simply
A=0
C = j Yo
B = j Zo
D=0
which implies that I2 = V1/j Zo independent of V2 or I1.
• Similarly, if sin ßl = 0 (that is, for ßl=π, a half wavelength or
multiple) the ABCD matrix becomes simply
A = -1
C=0
B=0
D = -1
which implies that V2 = -V1 and I2 = -I1 independent of the
terminating impedance at end 2.
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Two-port network with certain network parameters can lead to
various equivalent circuit formation.
T equivalent
π equivalent
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Impedance Matching
• Why impedance matching?
– Maximum power is delivered when the load is matched to the line.
– Impedance matching sensitive receiver components (antenna, LNA,
etc.) improves the signal-to-noise ratio of the system.
– Impedance matching in a power distribution network (such as
antenna array feed network) will reduce amplitude and phase
errors.
– Impedance matching uniquely removes the requirement for a
specific reference plane.
– Provide reliable and predictable interconnections between
components in a system.
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Methods of Impedance Matching
•
•
•
•
•
Matching stubs (shunts or series, single or multiple)
Quarter-wavelength transformers (single or multiple)
Lumped elements
Tapered transmission lines
Combination of the above
A lossless network matching an arbitrary load
impedance to a transmission line.
Z0
Load
Matching
network
Two-port matching network
One-port
matching
ZL
Multiple solutions
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Design issues of the matching networks
• Complexity --- Simplest design that satisfies the required
specification is generally the most preferable. Cheaper, more
reliable, less lossy.
• Bandwidth --- Normally, it is desirable to match a load over a
band of frequencies. Increased bandwidth usually comes with
increased complexity, e.g. using multistage matching.
• Frequency --- Matching networks are usually optimized for a
particular frequency.
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L networks consist of two reactive components (inductor
and capacitor), which results in eight different
configurations.
• Implementation --- Choose the right type of matching
networks, either tuning stub or transmission line.
• Adjustability --- This maybe required for applications
where a variable load impedance occurs.
Matching with Lumped Elements (L Networks)
Network for zL inside the
1+jx circle (smith chart).
Network for zL outside the
1+jx circle (smith chart).
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Analytic solution for the matching network elements
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and
Case 1: load impedance inside the 1+jx circle ---> RL>Z0
For a match looking into the matching network, we have
Z 0 = jX +
Z
1 X L Z0
+
− 0
B
RL
BRL
Both solutions are applicable for impedance matching at a
single frequency. But one solution may be preferable over
the other one when other performance, e.g. bandwidth, is
considered.
1
jB + 1 /( RL + jX L )
Solving for X and B from the two equations for real and
imaginary parts,
Case 2: load impedance outside the 1+jx circle ---> RL<Z0
Solutions are:
X ± RL / Z 0 RL2 + X L2 − Z 0 RL
B= L
RL2 + X L2
X = ± RL ( Z 0 − RL ) − X L
B=±
*Note: B is always real (RL>Z0) and has two solutions. One solution
is capacitive (positive) and the other one is inductive (negative).
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X=
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27
( Z 0 − RL ) / RL
Z0
*Analytic solution is computing intensive and lack of
intuition.
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Smith Chart Solutions of Matching Networks
Impedance effect of series and shunt connections of L and
C to a complex load in the Smith Chart
The effect of connecting a single reactive component
(either capacitor or inductor) to a complex load
zL
• The addition of a reactance connected in series with a complex
impedance results in motion along a constant-resistance circle
in the combined Smith Chart.
• A shunt connection produces motion along a constantconductance circle.
A general rule of thumb for rotation in the Smith Chart
• When an inductor is involved, we rotate in the direction that
moves the impedance into the upper half of the Smith Chart.
• In contrast, a capacitance results in the move toward the
lower half.
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Solution:
Example:
Step 1: Compute normalized transmitter and antenna
impedances. Since no characteristic impedance Z0 is given,
we arbitrarily select Z0 = 75 Ω for simplicity.
1
We have
zT = ZT /Z0 = 2 + j 1
zA = ZA /Z0 = 1 + j 0.2
Step 2: Taking into account the first element (the shunt
capacitor) connected to the transmitter.
Move down on the circle of the constant conductance.
Step 3: Taking into account the next element (the series
inductor) connected to the transmitter.
Figure 1: Transmitter to antenna matching circuit design.
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Move up on the circle of the constant resistance.
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Step 4: Draws the complex conjugate of the antenna
impedance in the Smith Chart for maximum power
transfer. This should be the output impedance of the
matching network.
Design of the
matching network
using ZY Smith
Chart
zM = zA* = 1 - j 0.2
Step 5: Find the normalized impedance of the intersection
of two circles. zTC = 1 - j 1.22 and the corresponding
admittance of yTC = 0.4 + j 0.49.
There is
another path
connecting zM
and zT.
The normalized susceptance of the shunt capacitor is
jbC = yTC - yT = j 0.69
C = bC /(ωZ 0 ) = 0.73 pF
What does this
mean?
and the normalized reactance of the inductor is
jxL = zA - zTC = j 1.02
L = ( xL Z 0 ) / ω = 6.09nH
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Procedures of designing impedance matching networks
using Smith Chart
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Design procedures cont’
1 Find the normalized source and load impedances.
2 In the Smith Chart, plot ircuits of constant resistance and
conductance that pass through the point denoting the source
impedance.
3 Plot circles of constant resistance and conductance that pass
through the point of the complex conjugate of the load
impedance.
4 Identify the intersection points between the circles in steps 2
and 3. The number of intersection points determins the
number of possible L-section matching networks. (cont’)
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5 Find the values of the normalized reactances and susptances
of the inductors and capacitors by tracing a path along the
circles from the source impedance to the intersection point
and then to the complect conjugate of the load impedance.
--- there are usually multiple paths (multiple solutions).
6 Determine the actual values of inductors and capacitors for a
given frequency.
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Smith Chart Solution 2 (not using combined ZY Smith Chart)
Example 5.1 on Page 254 of Pozar
Design an L section matching network to match a series RC load
with an impedance ZL = 200 - j 100 Ω, to a 100 Ω line, at a
frequency of 500 MHz.
Step 3: Convert back to
impedance.
Step
Solution:
Step
2
1
Therefore we have b = 0.3, x = 1.2 (check this result with the
analytic solution). Then for a frequency at f = 500 MHz,
we have
C=
b
= 0.92 pF
2πfZ 0
L=
xZ 0
= 38.8nH
2πf
3
37
ep
St
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Step 2: Move the load impedance to the impedance circle of 1+ jx
(done in admittance Smith Chart) -- add j 0.3 in susceptance
ep
St
Step 1: Convert the load impedance to admittance by drawing the
SWR circle through the load, and a straight line from the load
through the center of the Smith Chart.
Step 4: Move to the center
of the Smith Chart by
adding an series inductor
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There are two solutions for the matching networks. In this
case, there is no substantial difference in bandwidth
between the two solutions.
Is there another solution?
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Impedance Matching Using a Single Shunt Stub
Microstrip Matching Networks
• A microstrip line can be used
as a series transmission line, as
an open-circuited stub, or as a
short-circuited stub.
Single-stub tuning
• A series microstrip line
together with a short- or opencircuited shunt stub can
transform a 50-Ω resistor into
any value of impedance.
Shunt stub
Tuning Procedures:
• Find the proper d so that Y = Y0 + jB
• Choose the stub susceptance (decided by l) to be -jB
Example 5.2 on Page 259 of Pozar
Tuning parameters: d and stub
reactance or susceptance
Series stub
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Design a single-stub shunt tuning network to match a load
impedance ZL = 15 + j 10 Ω to a 50 Ω line.
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Solution:
y2
zL
Working with the
Smith Chart!
yL
d2
y1
Solution 1 has a
significantly better
bandwidth than
solution 2.
d1
Shorter stub produces
wider bandwidth.
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Microstrip Discontinuities
90O bend or corner:
Microstrip lines and Propagation Velocity
Fields exist partly in air
and partly in dielectric.
c
εr
Propagation velocity is
between the velocity in air
c and the velocity in the
dielectric c .
d
w
< v PCB < c
εr
Define an “effective dielectric constant” εre
v PCB =
c
ε re
ε re =
εr +1 εr −1
2
+
2
(1) capacitance arises through additional charge accumulation at the
corners --- particularly around the outer point of the bend where
electric fields concentrate.
(2) inductances arise because of current flow interruption.
(3) the impedances of these additional components can be comparable
to the line characteristic impedance at microwave frequencies.
1
1 + 12d / W
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Matched microstrip bends: compensation techniques
Using curved and mitered bends to reduce the effect of the additional
capacitance --- mitered bends are as good as, or better than curved
bends at frequencies up to 10 GHz.
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Step changes in width
(impedance steps)
T-junctions
Effect: the length of the w2
section is lengthened.
Typical value for the mitering fraction is
1 − b / 2 w = 0 .6
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Effect: length correction is
needed.
Mitering fraction 1− b / 2 w
b = 0.57 w
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