1 ene 428 microwave engineering lecture 9 scattering parameters and their properties
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ENE 428Microwave Engineering
Lecture 9 Scattering parameters and their properties.
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Impedance and Admittance Matrices• Consider an arbitrary N-port network
below,
n n n
n n n
V V V
I I I
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The impedance [Z] matrix relates voltages and currents.
So we can write [V] =[Z][I]V1 = Z11I1 + Z12I2V2 = Z21I1 + Z22I2, etc.
1 111 12 1
2 221
1
N
N NNN N
V IZ Z ZV IZ
Z ZV I
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The admittance [Y] matrix relates currents and voltages.
So we can write [I] =[Y][V]I1 = Y11V1 + Y12V2
I2 = Y21V1 + Y22V2, etc.
1 111 12 1
2 221
1
N
N NNN N
I VY Y YI VY
Y YI V
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and
• Zij can be found by driving port j with the current Ij, open-circuiting all other ports and measuring the open-circuit Voltage at port i.
• Yij can be found by driving port j with the voltage Vj, short-circuiting all other ports and measuring the short-circuit current at port i.
Zij or Yij can be found by o/c or s/c at all other ports
0i
ij I for k jkj
VZ
I 0i
ij V for k jkj
IY
V
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• Many practical networks are reciprocal (not containing any nonreciprocal media such as ferrites or plasmas, or active devices)
• Impedance and admittance matrices are symmetric, that is
and
Reciprocal Network
ij jiZ Z
.ij jiY Y
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• If the network is lossless, then the net real power delivered to the network must be zero. Thus, Re{Pav} = 0.
• Then for a reciprocal lossless N-port junction we can show that the elements of the [Z] and [Y] matrices must be pure imaginary
where m, n = port index.
Lossless Network
Re{ } 0mnZ
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Single- and Two-port Networks• The analysis can be done easily through simple input-
output relations. • Input and output port parameters can be determined
without the need to know inner structure of the system. • At low frequencies, the z, y, h, or ABCD parameters are
basic network input-output parameter relations.• At high frequencies (in microwave range), scattering
parameters (S parameters) are defined in terms of traveling waves and completely characterize the behavior of two-port networks.
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Basic definitions
• Assume the port-indexed current flows into the respective port and the associated voltage is recorded as indicated.
Two-portnetwork
Port 1 Port 2
V1
+
-
V2
+
-
I1 I2
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Ex of h and ABCD parameters for two-port network• H parameters
• ABCD parameters
1 11 12 1
2 21 22 2
V H H I
I H H V
1 2
1 2
V VA B
I IC D
These two-port representations (Z, Y, H, and ABCD) are very useful at low frequencies because the parameters arereadily measured using short- and open- circuit tests at the terminals of the two-port network.
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Two-port connected in series
1 1 1 11 11 12 12 1
2 22 2 21 21 22 22
a b a b a b
a b a b a b
v v v z z z z i
v iv v z z z z
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Two-port connected in shunt
1 1 1 11 11 12 12 1
2 22 2 21 21 22 22
a b a b a b
a b a b a b
i i i y y y y v
i vi i y y y y
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Two-port connected in cascade fashion
1 1 2 2
1 1 2 2
a a ba a a a b b
a a ba a a a b b
v v v vA B A B A B
i i i iC D C D C D
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Disadvantages of using these parameters at RF or microwave frequency• Difficult to directly measure V and I• Difficult to achieve open circuit due to stray
capacitance• Active circuits become unstable when terminated
in short- and open- circuits.
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Scattering Matrix (1)
• The scattering matrix relates the voltage waves incident on the ports to those reflected from the ports
• Scattering parameters can be calculated using network analysis techniques or measured directly with a network analyzer.
1 111 12 1
2 221
1
N
N NNN N
V VS S S
V VS
S SV V
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Scattering Matrix (2)
• A specific element of the [S] matrix can be determined as
• Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads.
• Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads.
0.i
ij V for k jkj
VS
V
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Reciprocal networks and lossless networks• [S] matrix for a reciprocal network is symmetric,
[S]=[S]t.
• [S] matrix for a lossless network is unitary that means
1[ ] {[ ] } .tS S
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Ex1 A two-port network has this following scattering matrix
Determine if the network is reciprocal, and lossless
0.15 0 0.85 45[ ]
0.85 45 0.2 0S
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Introduction of generalized scattering parameters (S parameters)1.Measure power and phase2.Use matched loads3.Devices are usually stable with matched loads.
S- parameters are power wave descriptors that permits us to define input-output relations of a network in terms of incident and reflected power waves
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Introduction of the normalized notation (1)
0
0
00
00
( )( )
( ) ( )
( )( ) ( )
( )( ) ( ).
V xv x
Z
i x Z I x
V xa x Z I x
Z
V xb x Z I x
Z
we can write Let’s define
( ) ( ) ( )
( ) ( ) ( )
v x a x b x
i x a x b x
and( ) ( ) ( ).b x x a x
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Introduction of the normalized notation (2)
We can also show a(x) and b(x) in terms of V(x) and I(x) as
00
1 1( ) [ ( ) ( )] [ ( ) ( )]
2 2 a x v x i x V x Z I x
Z
and
00
1 1( ) [ ( ) ( )] [ ( ) ( )]
2 2 b x v x i x V x Z I x
Z
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Normalized wave generalization• For a two-port network, we can generalize the
relationship between b(x) and a(x) in terms of scattering parameters. Let port 1 has the length of l1 and port 2 has the length of l2, we can show that
1 1 11 1 1 12 2 2
2 2 21 1 1 22 2 2
( ) ( ) ( )
( ) ( ) ( )
b l S a l S a l
b l S a l S a l
or in a matrix form,
1 1 11 12 1 1
2 2 21 22 2 2
( ) ( )
( ) ( )
b l S S a l
b l S S a l
Observe that a1(l1), a2(l2), b1(l1), and b2(l2) are the values of in-cident and reflected waves at the specific locations denoted as port 1 and port 2.
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The measurement of S parameters (1)
• The S parameters are seen to represent reflection and transmission coefficients, the S parameters measured at the specific locations shown as port 1 and port 2 are defined in the following page.
Two-portnetwork
Input port
Output port
Z01
Port 1x1=l1
a1(x)
b1(x)
a1(l1)
b1(l1)
Port 2x2=l2
Z02
a2(x)
b2(x)
a2(l2)
b2(l2)
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The measurement of S parameters (2)
2 2
2 2
1 1
2 1
1 111 ( ) 0
1 1
2 221 ( ) 0
1 1
2 222 ( ) 0
2 2
1 112 ( ) 0
2 2
( )|
( )
( )|
( )
( )|
( )
( )|
( )
a l
a l
a l
a l
b lS
a l
b lS
a l
b lS
a l
b lS
a l
(input reflection coefficient with output properly terminated)
(forward transmission coefficient with output properly terminated)
(output reflection coefficient with input properly terminated)
(reverse transmission coefficient with input properly terminated)
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The advantages of using S parameters• They are measured using a matched termination.
• Using matched resistive terminations to measure the S parameters of a transistor results in no oscillation.
Two-portnetwork
Port 1x1=l1
a1(l1)
b1(l1)
Port 2x2=l2
a2(l2)=0
b2(l2)E1
+
-
Z2=Z02
ZOUT2 2
1 111 ( ) 0
1 1
( )( ) a l
b lS
a l
Z1=Z01
Z01 Z02
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The chain scattering parameters or scattering transfer parameters (T parameters) (1)• The T parameters are useful in the analysis of cascade
connections of two-port networks.
• The relationship between S and T parameters can be developed. Namely,
1 1 11 12 2 2
1 1 21 22 2 2
( ) ( )
( ) ( )
a l T T b l
b l T T a l
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21 2111 12
21 22 11 11 2212
21 21
1
.
S
S ST T
T T S S SS
S S
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The chain scattering parameters or scattering transfer parameters (T parameters) (2)
21 21 1222
11 1111 12
21 22 12
11 11
.1
T T TT
T TS S
S S T
T T
and
We can also write
21 11 12 11 12
1 221 22 21 22
.
x x y yyx
x x y yx y
ba T T T T
b aT T T T
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Review (2)• Normalized notation of the incident a(x) and reflected waves b(x) are defined as
• The relationship between the incident and reflected waves and the scattering matrix of the two-port network,
( )( ) ( )
( )( ) ( )
00
00
V xa x Z I x
Z
V xb x Z I x
Z
( ) ( )
( ) ( )1 1 11 12 1 1
2 2 21 22 2 2
b l S S a l
b l S S a l
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Shifting reference planes
• S parameters are measured using traveling waves, the positions where the measurements are made are needed to be specified. The positions are called reference planes.
Two-portnetwork
Port 1x1=l1
a1(0)
b1(0)
a1(l1)
b1(l1)
Port 2x2=l2
a2(0)
b2(0)
a2(l2)
b2(l2)
Port 1'x1=0
Port 2'x2=0
q1bl1 q2bl2
Reference planes
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Scattering matrix of the shifting planes• At the reference planes at port 1 and port 2, we write the
scattering matrix as
and at port 1’ and port 2’ as
• We can show that
1 1 1 111 12
21 222 2 2 2
( ) ( )
( ) ( )
b l a lS S
S Sb l a l
' '1 111 12
' '2 221 22
(0) (0)
(0) (0)
b aS S
b aS S
1 1 2
1 2 2
2 ( )1 111 12
( ) 22 221 22
(0) (0).
(0) (0)
j j
j j
b aS e S e
b aS e S e
q q q
q q q