**Sunshine Design Engineering Services**

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## Scattered Thoughts on Scattering Parameters

By Joseph L. Cahak

Copyright 2013 Sunshine Design Engineering Services

Scattering parameters or S-parameters (aka Spars) are used by RF and microwave engineers to measure and design
components and systems at those frequency ranges. These S-parameters are typically measured with an instrument
called a vector network analyzer, or VNA.

**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 reactance 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.

**Z Parameters**

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.

**Y Parameters**

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.

**h Parameters**

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.

**g Parameters**

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.

**Network Transforms**

There are matrix transforms that are used to convert from one network description to another. Typically the
S, Z, Y, h and ABCD are direct conversions. The conversion to a few of the others are after the base conversion
and then converting from h to the inverse g for instance, 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.

Figure 1 - Network Description Transformations

**Transmission Lines**

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 element.

**Reflections**

When a RF signal is incident to a load, some of the signal is absorbed and converted to heat and radiated,
what isn’t absorbed is reflected back to the source as shown in Figure 2 and 3. Reflection coefficient represents
this value in power measurements and is also known as rho or
ρ = Γ = |Power Reflected/Power Incident|.
Rho can be converted to the voltage standing wave ratio, or VSWR, by the equation VSWR = (1+ρ)/(1-ρ).
These measurements are still scalar, but dimensionless. Rho and VSWR are measured in units of power (single dimension),
but the ratio cancels the units out (thus dimensionless). Rho and VSWR can also be derived by the system impedances.
The load can be represented by Z = R +jX. The measurement system used to characterize a component or
another system also has impedance. This measurement with the device or network sets up an impedance match represented
by gamma or Γ = (Z_{sys}-Z)/(Z_{sys}+Z) = (Power Reflected)/(Power Incident).
The gamma, or complex reflection coefficient, can be transformed by absolute value to rho, or
ρ=|Γ|
, which can be used to derive the VSWR as given above. It can also be transformed by the system impedance to the
load impedance.

Figure 2 – S11 or Gamma Source Signal Flow

Figure 3 - Power Reflected

**Transmission**

When a signal is incident to a 2-port network, the network looks like a load to the source, but the network
has an output and a match at the output. The device has reflections from the output back to the input in addition
to the transmitted part of the incident signal. As with the Reflection case, there is also some energy absorbed
and converted to heat. The transmitted signal can be represented by Tau or
τ = (Power Transmitted)/(Power Incident).
See Figures 4 and 5.

Figure 4 - Transmission Network Signal Flow

Figure 5 - Transmitted Power

**Units of Measurement**

Resistance is measured in ohms, voltage in volts, current in amps. Power is measured in watts. This can be
referenced to a specific power level. Tradition and convention stipulate 1mW as 0 dBm. With this power can
be represented in logarithmic units or decibels, dB for relative power gain or loss and dBm for absolute power
level referenced to 1 mW. When power is measured the type of measurement makes a difference in how the units
are converted. A Power Meter measures in Volt-Amps or true power as opposed to reactive or apparent power. The
formula to convert real power to decibels is

dB = 10*log_{10} [(Power Out)/(Power In)].

To convert measurements made in voltages, not true power, and this includes vector network analyzers, vector
signal analyzers and spectrum Analyzers use the formula

dB = 20*log_{10} [(Voltage Out)/(Voltage In)].

This can be derived from the fact that Power = Voltage^{2}/Resistance , thus the doubling
of the multiplier of the logarithm of the voltage ratio measurement of power. Convert dB to Watts using the inverse
formulas:

Watts = 10^{Power/10} and Watts=10Voltage20 .

The absolute power can be converted between Watts and dBm with the following equations:

Watts =10 ^{(dBm/10) - 3} and dBm = 10*log_{10} (Watts*1000)

dB units is a relative or ratio measurement. For power measurements adopted convention is to measure the signal
relative to 1 mWatt for dBm and 1 V for dBV.

**S-Parameter Definitions**

S-parameters a and b components are in units of power represented as the square root of the power or

Power Incident at Port 1 or a_{1} = V_{1}^{+}/√Z_{o}

Power
Incident at Port 2 or a_{2} = V_{2}^{+}/√Z_{o}

Power
Emitted at Port 1 or b1 = V_{1}^{-}/√Z_{o}

Power Emitted at Port 2
or b_{1} = V_{2}^{-}/√Z_{o}

b1 = S_{11}a_{1 }+ S_{12}a_{2}

b2 = S_{21}a_{1 }+ S_{22}a_{2}

Figure 6 - 2-Port S-Parameter Signal Flow

Figure 6 shows the signal flow diagram for the S-Parameters. Note that the dimension of a_{1},a_{2}
and b_{1},b_{2} are complex. These can then be used to derive the Scattering Parameters (Spars)
for the network which are also complex values and dimensionless due to the ratio and in the case below are in
rectangular coordinates and have linear magnitudes, not dB. Note that S-Parameters are dependent on the system
Impedance. S-Parameters at one system impedance are not equal to S-Parameters at another System Impedance.

S_{11} = b_{1}/a_{1} or forward reflection coefficient

S_{12} = b_{1}/a_{2} or
reverse transmission coefficient

S_{21} = b_{2}/a_{1} or forward transmission
coefficient

S_{22} = b_{2}/a_{2} or reverse reflection coefficient

Note also that the reflection coefficient for the load and source,
Γ_{L }= b_{2}/a_{2} and
Γ_{s }= b_{1}/a_{1}
, respectively are the same as S_{22 }= b_{2}/a_{2} and
S_{11 }= b_{1}/a_{1} . In other words, if you measure the 1 port
S-Parameter of a source or load, it is the same as the Gamma or
Γ . These S-parameters are the basis of many
RF and Microwave measurements.

**Network Analyzer Measurements**

In many RF and Microwave measurements the S-Parameters are typically expressed in dB (decibels) Magnitude units
and Degrees in the polar coordinate system. Network and Vector Network Analyzers and Spectrum Analyzers all measure
with voltage ratio measurements, so to convert to dB in terms of volts we must use the following equation.

dB = 20*log_{10} [(Voltage Out)/(Voltage In)]

Making measurements of S-parameters is a process of measurements made with calibrations standards (special
components with values traceable to NIST or other designated agencies) and formulas to compute the correction
factors from the measurement of those standards that determine a reference plane for the measurements. The measurement
reference plane is an imaginary plane of reference for the measurements being made that lies somewhere between
the measurement system’s output and input ports, inclusive. It defines the points to which the network analyzer
is calibrated to have 0 dB magnitude and 0 degrees phase response. It is the input and output planes to which
reflection and transmission measurements are referenced. Another aspect of the relative measurement is that within
the VNA dynamic range, the input power can be varied and the network response will stay the same until the power
level goes outside the dynamic range.

There are numerous methods and standards for coaxial, waveguide, planar, probe and other interconnections methods.
There are also a number of test fixtures available and calibration techniques to help the test engineer make measurement
in fixtures and give them the ability to de-embed fixture components to get at the raw device or subsystem S-parameters,
which are difficult or nearly impossible to measure with standard equipment (e.g. mixed impedances).

**New Stuff**

I recently attended the Agilent “Back to Basics” seminar and as with the first one I attended 20 years
ago, they still cover all the basics of measurements, S-parameters, measurement systems and how to use them. The
new instruments all have the extended measurement capabilities built in with some really exciting capabilities.
Spectrum analyzers can now reach down into the noise floor “Noise Floor Extension” and get to -172 dBm/Hz.
This is a mere 2 dB from the theoretical noise limit at room temperature (recall that 25°C gives a 50 ohm
noise power of -174 dBm in a 1 Hz bandwidth).

The new oscilloscopes have some amazing features across the board and at lower costs. They have very high bit
depth and are being used for the spectrum analyzer and vector signal analyzers (VSA) front ends, as they have
in some cases more than 14 bits of ADC depth, thereby giving much greater dynamic range. Finally the exciting
news on the vector network analyzer (VNA) measurement front is the new X-parameter test sets that can do large
signal multi-harmonic, power and spurious analysis, in addition to the small signal S-parameters. This is important
because now the non-linear characteristics of device and circuit response can be measured in addition to the linear
characteristics. With this new information much more accurate device and circuit models can be formulated, yielding
a more accurate model for designers to use in simulations.

**Conclusion**

We showed that S-parameters can be used for a number of network computations that 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.

**References**

www.AstroCalculator.com See RF Calculator
and soon Labview S-Parameter Library

http://en.wikipedia.org/wiki/Two-port_network

http://en.wikipedia.org/wiki/S-parameters

http://en.wikipedia.org/wiki/ABCD-parameters#ABCD-parameters

http://en.wikipedia.org/wiki/Transmission_line

http://en.wikipedia.org/wiki/Reflection_coefficient

http://en.wikipedia.org/wiki/Standing_wave_ratio

http://en.wikipedia.org/wiki/Electrical_impedance

http://en.wikipedia.org/wiki/Admittance

http://en.wikipedia.org/wiki/Two-port_network#Hybrid_parameters_.28h-parameters.29

http://en.wikipedia.org/wiki/X-parameters

http://www.AstroCalculator.com

Making S-Parameter measurements in Mixed Impedances, Jim Hillstrom, Microwaves & RF, January 1992

Agilent PN 8720-2, In-fixture Microstrip Device Measurements

Using TRL* Calibration, Product Note

Agilent PN 8510-5A, Specifying Calibration Standards for the Agilent 8510

Agilent 8510-13, Measuring Noninsertable Devices, Product Note

Agilent AN 1287-1, Understanding the Fundamental Principles of Vector Network Analysis, Application Note

Agilent AN 1287-3, Applying Error Correction to Network Analyzer Measurements, Application Note

Agilent Application Note 1287-7, Improving Network Analyzer Measurements of Frequency-translating Devices

Agilent AN 1287-6, Using a Network Analyzer to Characterize High-Power Components, Application Note

Agilent AN 1287-9, In-Fixture Measurements Using Vector Network Analyzers, Application Note

Agilent AN 1364-1, De-embedding and Embedding S-Parameter Networks Using a Vector Network Analyzer

Agilent DE-EMBEDDED MEASUREMENTS USING THE HP 8510 MICROWAVE NETWORK ANALYZER

Agilent Network Analyzer Basics

Agilent 5989-5765EN Agilent Signal Integrity Analysis Series Part 3: The ABCs of De-Embedding Application
Note

Agilent AN 154, S-Parameter Design, Application Note

Agilent AN 95-1 S-Parameter Techniques for Faster more accurate Network Design

Microwave Engineering, David M. Pozar, Wiley, ISBN 9971-51-263-7

Networks and Systems, D Roy Choudhury, Wiley, ISBN 0-470-20867-8

Fields and Waves in Communications Electronics, Ramo, Whinnery & Van Duzer, Wiley, ISBN Lib Congress
Catalog 65-19477

High-Frequency Circuit Design and measurements, Peter Yip, Chapman and hall, 0-442-31185-0

**Sunshine
Design Engineering Services** is located in the sunny San Vicente Valley near San Diego, CA, gateway to the
mountains and skies. Are you looking for new things to design, program or create and need assistance? I offer design
services with specialties in electronic hardware, CAD and software engineering, and 25 years of experience with Test
Engineering services in RF/microwave, transceiver and semiconductor parametric test, test application program development,
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**See also:** |
- RF Connectors and Cables -
Searching for the Q - Hybrid
Heaven - Noise and Noise Measurements -
Solace in Solar -
Measuring Semiconductor Device Input Parameters
with Vector Analysis -
Computing with Scattering Parameters -
Measurements with Scattering Parameters -
Ponderings on Power Measurements -
Scattered Thoughts on Scattering Parameters |

**Sunshine Design Engineering Services** 23517 Carmena Rd Ramona, CA 92065 760-685-1126
Featuring: Test Automation Services, RF Calculator and S-Parameter Library (DLL & LLB)
www.AstroCalculator.com
SunshineDesign@cox.net |

Posted August 1, 2013