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- 1 - © 1994-99 D. B. Leeson
S-Parameter Matrices
The objective of microwave circuit analysis is to 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 toolbox. However, the tools that are appropriate for lumped circuits must be extended to apply to distributed networks.
A matrix that is of great use in microwave network problems is the "scattering" matrix, so-called by analogy to the scattering or reflection of waves by a free-space reflector. As introduced in the prior notes, S-parameters have become the preferred description of microwave n-ports for the following reasons:
Voltage and current are difficult to define and measure in distributed circuits
The measurement of power in incident and reflected waves is a natural technique for microwave transmission lines. Voltage and current may not be well defined, or even defined at all, in some structures. The specification of voltage and current in a distributed circuit requires a specification of the exact location, and these parameters vary with location in the circuit. The determination of the individual parameters of voltage and current equation sets requires short or open circuit loads, which are sensitive to the precise location; in particular, it is not practical to mount a connector close enough to a microwave lumped device to be measuring its actual port voltages and currents. Also, many active devices cannot be operated with fully reflective terminations (short or open) of arbitrary phase, as they will oscillate, which is a large signal nonlinear condition and may even result in device failure.
Incident and reflected waves are the natural description for microwave structures
The matched condition (Γ = 0) is a unique, repeatable termination. It is insensitive to the length of transmission line to the matched load, so that measurements can be made without requiring the reference planes (the port connectors) to be located directly at the device under measurement (or being described). A matched load is a natural structure that can maintain its character over a very broad frequency range.
Conversion from S-parameters to other parameter sets is a matter of routine algebra
Each of the many equivalent parameter sets is uniquely useful for a given circuit topology. For example, the ABCD and T matrices are adept at cascaded networks, while the Z and Y parameter sets can be directly evaluated for tee and pi networks, respectively. However, the conversion process, while complex appearing, lends itself to repeated routines for hand or computer calculation.
S-parameters (in fact, all the parameter sets) benefit from the matrix toolbox.
The toolbox of established matrix mathematics is directly applicable to the matrices that are the equivalent of the port equations of the parameter sets. For example, the S matrix can be inspected for lossless, reciprocal or unilateral character. If either or both of these conditions is present, many of the individual matrix elements can be determined by inspection.
Equivalence of Matrix and Equation Form
For a single port network, we have the following simple relationships from our study of Γ and Smith chart.
b 1 = Γa 1 = S 11 a 1 b 1 = Γa 1 = S 11 a 1
For a multiport network the reflection coefficient is Γ defined as
bn = Γnan, so Γn =
bn an where n is the port number.
Note that Γn = Snn only if all other ports are terminated, that is, only if all am = 0 for m?n. Otherwise it must be algebraically calculated from all the parameters.
The example of 2-port equations and their equivalent matrix is shown here to emphasize that both forms contain the same information, but the matrix form suggests the use of formal matrix algebra tools as an aid to analysis:
b 1 b 2
^
S 11 S 12
S 21 S 22
^
a 1 a 2
^
^
b 1 = S 11 a 1 + S 12 a 2 b 2 = S 21 a 1 + S 22 a 2
It is not uncommon to consider microwave networks of three and four ports, as in power dividers and directional couplers. The extension to the example of 3-port equations and equivalent matrix should reinforce the concept of equivalence:
b 1 b 2 b 3
S 11 S 12 S 13
S 21 S 22 S 23
S 31 S 32 S 33
a 1 a 2 a 3
b 1 = S 11 a 1 + S 12 a 2 + S 13 a 3 b 2 = S 21 a 1 + S 22 a 2 + S 23 a 3 b 3 = S 31 a 1 + S 32 a 2 + S 33 a 3
Using matrix rules we can inspect an S-parameter matrix and see whether the network is reciprocal and whether it is lossless. We can then use these facts to reduce the number of independent variables in the matrix, so that we can more easily evaluate the matrix elements (the parameters of the equations).
Ya
1 Yb Yc 2
If we consider the admittance at port 1 for shorted port 2, we see that for this network
Y 11 + Y 12 = Ya; similarly
Y 22 + Y 12 = Yc and
Y 12 = -Yb [I] = [Y][V] I 1 = Y 11 V 1 + Y 12 V 2 I 2 = Y 21 V 1 + Y 22 V 2
Now consider the application of the ABCD matrix. The fact that the output voltage and current of the first of two cascaded networks are equal to the input voltage and the negative of the input current, respectively, of the second network makes the ABCD matrix a natural choice because is explicitly deals with the parameters Vn and In.
V 1 = A V 2 + B I 2
I 1 = C V 2 + D I 2
[ABCD] [ABCD]
So by matrix multiplication, we can find the ABCD description of cascaded networks.
A last consideration is the question of a shift in reference planes, which is handled well by the S matrix. If we ask for the S matrix description of the following network
Zo, θ 1 [S] Zo, θ 2
we find that it is simply [S'] = [θ 1 ][S][θ 2 ], where [θn] is defined such that all terms are zero
except the diagonal terms, which are e-j2θn.
Useful Matrix Operations
Certain simple matrix operations are useful in manipulating and evaluating S-parameter matrices. They include
- Multiplication (cascade ABCD or shift of reference plane
- Test for reciprocity
- Test for losslessness
- Test for unilateral transmission
Multiplication of matrices can be used to determine the ABCD or S parameters of cascaded networks of simpler forms.
The test for reciprocity requires that the matrix be symmetric, that is Smn=Snm. This can
generally be determined by inspection.
The test for losslessness is that the sum of the SmnSmn* of any column must be unity. If the network is reciprocal, the matrix is symmetric and the same can be said of any row.
The test for unilateral transmission is that S 12 = 0.
ABCD Example: Quarter- and Half-Wave Transmission Lines
The usefulness of ABCD parameters can be seen in an example that has been the subject of a homework problem. Consider a transmission line of length l.
Z (^) o^ Z^ L
l
V 1
I 2
The ABCD parameters of this network relate Vi and Ii, such that
V 1 = A V 2 + B I 2 , and
I 1 = C V 2 + D I 2.
- Standing waves resulting from Γ? 0
- Transformation of ZL through line of Zo and length ßl
- Description of Γ and Z on the Smith chart (polar graph of Γ)
Review of Scattering Matrix
- Normalization with respect to Zo of wave amplitudes:
a =
V+
Zo
and b =
V-
Zo
, so power is aa* and bb*
- Relationship of bn and an: bn = Γn an
- Expressions for b 1 and b 2 at reference planes: b 1 = S 11 a 1 + S 12 a 2 b 2 = S 21 a 1 + S 22 a 2
- Definitions of Sii:
S 11 =
b 1 a 1 for a 2 = 0, i.e., input Γ for output terminated in Zo.
S 21 =
b 2 a 1 for a 2 = 0, i.e., forward transmission ratio with Zo load.
S 22 =
b 2 a 2 for a 1 = 0, i.e., output Γ for input terminated in Zo.
S 21 =
b 1 a 2 for a^1 = 0, i.e., reverse transmission ratio with Zo^ source. lS 21 l^2 = Transducer power gain with Zo source and load.
- Definitions of ΓL, Γs, Γin and Γout:
ΓL =
ZL - Zo ZL + Zo , the reflection coefficient of the load
Γs =
Zs - Zo Zs + Zo , the reflection coefficient of the source
Γin =
Zin - Zo Zin + Zo
= S 11 +
S 12 S 21 ΓL
1-S 22 ΓL
, the input reflection coefficient
Γout =
Zout - Zo Zout + Zo = S^22 +
S 12 S 21 Γs 1-S 11 Γs , the output reflection coefficient
- Power Gain G, Available Gain GA, Transducer Gain GT:
G =
PL
Pin
power delivered to the load power input to the network
GA =
Pavout Pavs
power available from the network power available from the source
GT =
PL
Pavs
power delivered to the load power available from the source
Power Gain Equations
The equations for the various power gain definitions are
1) G =
PL
Pin =^
1 - lΓinl^2 lS 21 l^2
1 - lΓLl^2 l1 - S 22 ΓLl^2
2) GA =
Pavout Pavs
1 - lΓsl^2 l1 - S 11 ϖsl^2
lS 21 l^2
1 - lΓoutl^2
3) GT =
PL
Pavs^ =^
1 - lΓsl^2 l1 - ΓinΓsl^2 lS 21 l^2
1 - lΓLl^2 l1 - S 22 ΓLl^2
=
1 - lΓsl^2 l1 - S 11 Γsl^2
lS 21 l^2
1 - lΓLl^2 l1 - ΓoutΓLl^2
The expressions for Γin and Γout are
- Γin = S 11 +
S 12 S 21 ΓL
1-S 22 ΓL
- Γout = S 22 +
S 12 S 21 Γs 1-S 11 Γs
For a unilateral network, S 12 =0 and
- Γin = S 11 if S 12 =0 (unilateral network)
- Γout = S 22 if S 12 =0 (unilateral network)
The transducer gain GT can be expressed as the product of three gain contributions
GT=GsGoGL, where
Go = lS 21 l^2
Gs =
1 - lΓsl^2 l1 - ΓinΓsl^2
and
Review of General Scheme of Solving Microwave Problems
MatrixTools
PhysicalStructure
Fields,Waves Maxwell'sEquations
j(ω
t-
β z)
e Assume TEM,
TE, TMMatch Boundaries
CoordinateSystem
ko,
k
c, β
Equivalent TransmissionLine
Z
o,
α
Losses,Fringing,Lumped Elements
Solve Equationsfor
k
c
,
Smith Chart
Equivalent
Circuit S-parameters
n-port Equations
Free spacePropagation,Antennas
