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Computing with Scattering Parameters
By Joseph L. Cahak
Copyright 2013 Sunshine Design Engineering Services
Complex Numbers and Parameters
A complex number is represented by two values X and Y, as in X + iY = Z. X is
the real component and iY is the imaginary component with Z being the complex representation.
The letter 'i' is used to designate the complex operator (√-1) by scientists and
mathematicians, but in engineering the letter 'j' is most often used. This is used
to represent AC or RF signals. X may be the resistance and Y the reactance in ohms
of the component to an AC signal. A standard RF load for instance is represented
by 50 ohms real resistance and some reactance with good loads having reactances
of close to 0 ohms. X + iY can also represent a voltage and current as a function
of time. Current is a function of the load and the voltage applied, so the resistances,
voltages and currents are all important in describing a network for DC and AC (or
RF) signals. A 2-port network can be represented by two equations with four parameters.
The four parameters have different representations depending on what is known and
what operations the users desire to use them for. Some network parameters (ABDC,
S-, Z-, Y-, h-, etc.) are better than others depending on circuit types and the
operations on them.
Complex numbers are used in a variety of ways in calculations. Typical data structures
are data clusters and groups of values and the properties associated with them.
Properties most often used are units of dB/linear, degrees/radians, rectilinear/polar.
With this data description we know enough about the state of the complex value to
operate on it for RF operations. Values in dB or dBv must first be converted to
Linear and Rectilinear for most complex operations other than simple addition or
subtraction. S-parameter and any other complex number must first be to converted
to the proper non-decibel data format for the calculations, and then back to the
final display format (dB-Polar, for instance).
Open circuit impedance parameters are used to represent the impedances of the
network. The values are complex and represent real resistance (R) and reactance
(jX) of the elements of the network.
Short circuit parameters or admittance parameters are used to represent the inverse
of impedance. The real and imaginary parts are conductance (G) and susceptance (jB).
The units are mhos or Siemens.
Another model commonly used to analyze BJT (bipolar junction transistor) circuits
is the h-parameter model, closely related to the hybrid-pi model and the y-parameter
two-port model, but the h-parameter model uses input current and output voltage
as independent variables, rather than input and output voltages. In this case it
is a hybrid of an open circuit on the input and a short circuit on the output.
Often this circuit is selected when a voltage amplifier is wanted at the output.
The off-diagonal g-parameters are dimensionless, while diagonal members have dimensions
the reciprocal of one another.
ABCD or T Parameters
ABCD-parameters are known variously as chain, cascade, or transmission line parameters.
This is useful for cascading 2-port network responses.
ABCD' or T' Parameters
These are the inverse of the ABCD or T-parameters, respectively. They are useful
for de-embedding 2-port network responses when multiplied with the ABCD matrix for
the overall network.
Figure 1. Network Description Transformations
These are the matrix transform that are used to convert from one network description
to another. Typically the –parameter, Z, Y, h and ABCD are direct conversions. The
conversion to a few of the others are after the base conversion then the conversion
from h to the inverse g, ABCD with the inverse ABCD-1 and S-parameter
to the T (one of the two types) or the anti-S parameter (S-1) . Note
that the ABCD matrix is also known as a “T” matrix; this is not to be confused with
the other two descriptions of a “T” matrix connected with the S-parameters calculations
and will be presented in another paper. See Figure 1 for a graphical representation
of the transformations available.
Some of the transmission line functions and parameters are complex in value.
They are excited by RF signals with most having complex modulation applied to them.
Some basic measurement parameters need to be defined. A load can be represented
at RF frequencies as a real resistance and an imaginary reactance driven by the
charge delay or advancement as related to the driving voltage. This complex impedance
is written as Z(ohms) = X + iY, with Z being complex value of the real (X) and imaginary
(Y) components. Y is positive for an inductive element and negative for a capacitive
Scattering Transfer or T-Parameters
While S-parameters are useful for measurements and describing network response,
it is not as useful for network embedding and de-embedding. To perform these operations
another network parameter better suited: either the T-parameter or ABCD matrix.
The scattering transfer parameters or T-parameters of a 2-port network are expressed
by the T-parameter matrix and are closely related to the corresponding S-parameter
matrix. The T-parameter matrix is related to the incident and reflected normalized
waves at each of the ports as follows:
They can be defined another way:
Figure 2. Cascaded Network
When I was researching this for the RFCalculator™ and S- parameters library,
I made the discovery that there were two different definitions for these T-parameters.
The RF Toolbox add-on from MATLAB and several other references use the last definition
and the operations do not mix. The "From S to T" and "From T to S" in this article
are both based on the first definition. Adaptation to the second definition is trivial
(interchanging T11 for T22, and T12 for T21). The advantage of T-parameters compared
to S-parameters is that they may be used to readily determine the effect of cascading
two or more 2-port networks by simply multiplying the associated individual T-parameter
matrices together. With T-Parameters, we can perform matrix multiplication for the
cascading networks operations to get the total network T-Parameters and transform
back to S-parameters. See Figure 2.
Ttotal = TaTbTc
T-Parameters are complex just like S-parameters and there are transform equations
between the two parameter types and this must be performed on all elements prior
to the complex matrix multiplication. Then the Ttotal must be transformed
back to S-parameters for the final answer in S-parameters. Note also that matrix
multiplication is not commutative; that is, AB does not equal BA. This is very important
to the cascading math.
S to T:
To convert from S-parameters to T-parameters the following formulas apply:
T to S:
To convert from T-parameters to S-parameters the following formulas apply:
There are the formulas that can be used directly with S-parameters to embed the
network response of two network components, N and N#, for the response as a singular
network, S. The network embedding equation will take both network S-parameter responses
and compute the S-parameters as the combined response of the two as a whole network.
See Figure 3.
Network Embedding Equations
Figure 3. Embedding Cascaded Network S-Parameters
S-Parameter Anti Network and De-Embedding
To perform de-embedding, we must first find the inverse network or anti-network.
This is done by using the embedding formula and the network to be de-embedded as
one branch and the sum network to be ideal S11=S22=0 and S12=S21=1.
Then, solve for the unknown sub-network to obtain the anti-network. This can then
be used to embed to the network to be de-embedded to remove that network element
from the total response for the sub-net response of the remaining component. See
S11 = S22=0 and S21
Figure 4. Anti-Network Computation
Solving for N#xx gives:
The process used to de-embed a sub-network from the network to get the remaining
sub-network result is to calculate the anti-network of the sub-network you are removing
and then cascading the result with the network to remove it from the response. Doing
so produces the desired sub-network.
Fixture Embedding and De-Embedding Process
Figure 5. Fixture De-embedding Process
This section describes the process of fixture embedding and de-embedding. The
embedding function is included to facilitate making simulated networks to test the
fixture de-embedding process. This case is an extension of the de-embedded AB network.
Instead of just computing AB x A-1 = B , double up the math to compute
AB and then (AB)C to get the total. The operations must be performed in the order
shown in Figure 5 to be correct.
1.1.1 Gamma In and Out
This value of gamma (reflection coefficient) represents the source and load impedance
compared to the measurement system impedance. It gives a measure of power reflected.
The gamma in and out values are both representative of the source or load looking
through a network. So the gamma in is the combined response of the source and an
intervening network and similarly for gamma out on the load side.
1.1.2 Gain Equations
The network S-parameters can be used to calculate the scalar gain of the network.
The gains are the operating power gain, transducer gain, unilateral transducer gain
and the available gain.
Operating Power Gain
The operating power gain is the ratio of the power delivered to the load and
the power input to the network.
Transducer gain is the ratio of the power delivered to the load to the power
available from the source.
Unilateral Transducer Gain
The unilateral transducer gain is the ratio of the power delivered to the load
to the power available from the source for a device with little to no S12
reverse transmission (high isolation).
Available gain is the ratio of the power available from the 2-port network to
the power available from the source. This gain is useful to calculate the network
gain (or loss) of an input network to a device being tested for noise figure. This
loss is the noise figure in dBF of the input network for the cascaded gain equation
when making a device noise figure measurement.
D = S11S22 - S21S12
M = S11 - DS'22
N = S22 - DS'11
When using a vector network analyzer (VNA), a spectrum analyzer (SA), or vector
signal analyzer (VSA), the measurement impedance is defined by the system hardware
which is designed to a specified system impedance. When a component must be designed
and measured with an impedance not at the typical 50 ohm measurement system
impedance, special accommodations must be made. In a scalar measurement system,
this can be accommodated with a simple voltage impedance conversion and gain adjustment.
In a complex signal environment this has to be done through the use of complex mathematical
operations to account for all the complex impedances, matches and gains. I found
the following formulas (see references) for the conversion of 50 ohm or any
other impedance measurements to any device impedance like 75 ohms commonly
found in commercial broadcast equipment or the less common video 90 ohms. The
newer model VNAs have these capabilities built in. In those cases where you do not
have the feature built in, these normalization formulas let you use a 50 ohm
VNA to make measure an arbitrary impedance a 75 ohm amplifier measurement,
for instance. See Figure 6.
Figure 6. S-Parameter Re-Normalization to Arbitrary Impedances
N21 = Zo[(1+S11)(1-S22Γ2)+S12S21Γ2]
D21 = Z1[(1-S11)(1-S22Γ2)-S12S21Γ2]
N12 = Zo[(1+S22)(1-S11Γ1)+S12S21Γ1]
D12 = Z2[(1-S22)(1-S11Γ1)-S12S21Γ1]
Shown was that S-parameters can be used for a variety of network computations
and can add value to measurements where the equipment is limited in features. The
reader can find these equations and more in my S-Parameter Library (DLL & LLB)
and my RF Calculator products.
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Posted August 11, 2013