Scattering Parameter Function Library By Joseph L. Cahak Copyright 2013 Sunshine Design Engineering Services

Transcription

1 Scattering Parameter Function Library By Joseph L. Cahak Copyright 2013 Sunshine Design Engineering Services

2 Contents Scattering Parameter Function Library... 1 S-Parameter Library Software DLL LLB Labview Library Theory Scalar Numbers Complex Numbers and Parameters Transmission Lines Reflections Transmission Units of measurement S-Parameters S-Parameter Definitions Network Analysis Network Analyzer measurements Scattering Transfer or T-Parameters Embedding S-parameter Anti Network and De-Embedding Gamma In and Out Gain Equations S-Parameter Re-Normalization Software Data Descriptions Data Clusters for Labview LLB Data Structures in DLL header DLL Header Function Prototypes Complex Function Descriptions CAbs CAdd CDiv CLn... 38

3 1.8.5 CLog CMult CNeg ComplexInput CSqr CSqrt CSub db2lin dbv2lin Deg2Rad Gamma2Z GammaInput ImpedanceInput Lin2dB Lin2dBv Pol2Rect Rad2Deg Rect2Polar Spar2Z Z2Gamma Z2Spar S Parameter Function Descriptions Par_Mat_Multiply Spar2Tpar Spar Input Spar_AntiNetwork Spar_dBDeg2RectLin Spar_DeEmbed Spar_Embed Spar_FixtureDeEmbed Spar_FixtureEmbed Spar_GainAvailable... 43

4 Spar_GainTransducer Spar_GainUnilateralTransducer Spar_GammaInOut Spar_PowerGain Spar_RectLin2dBDeg Spar_ReNormalize SxxInput Tpar2Spar References... 46

5 Figures Figure 1 - Complex Cluster... 8 Figure 2 S11 or Gamma Source Signal Flow... 9 Figure 3 - Power Reflected... 9 Figure 4 - Transmission Network Signal Flow... 9 Figure 5 - Transmitted Power Figure 6-2 port S-Parameter signal flow Figure 11 - Test Device and Fixtures Figure 12 - VNA Response Calibration Figure 13 - VNA w/ Cal Kit Calibration Figure 14 - VNA Fixture measurement using Port Extension correction Figure 15 - VNA Adapter Removal method of correction Figure 16 - VNA Fixture measurement w/adapter Removal Figure 17 - VNA De-Embedding DUT From fixture Figure 7 - Cascaded Network Figure 8-4 Parameter Complex Matrix Multiply Figure 9 - S to T Parameter Figure 10 - T to S Parameters Figure 11 - Cascaded Network S-Parameters Figure 12 - S-parameter Embedding w/s-parameter Labview Library Figure 13 - Anti-Network Computation Figure 14 - S-Parameter Anti-Network Figure 15 De-Embedding S-Par Networks w/s-parameter Labview Library Figure 16 - Fixture De-embedding Process Figure 17 - Fixture De-embedding Figure 18 - Fixture Embedding Figure 19 - Gamma In/Out w/ S-Parameter Labview Library Figure 20 - Power Gain w/s-parameter Library Figure 21 - Transducer Gain w/s-parameter Library Figure 22 - Unilateral Transducer Gain w/s-parameter Library Figure 23 - Available Gain w/s-parameter Library Figure 24 - S-Parameter Re-Normalization to Arbitrary Impedances Figure 25 - Re-Normalize S-Parameters to Arbitrary Impedances with S-Parameter Labview Library Figure 26 Complex Cluster Figure 27 - Impedance Cluster Figure 28 - Gamma Cluster Figure 29 1-Port Spar Cluster Figure 30 2-Port Spar Cluster... 35

6 S-Parameter Library 1.1 Software The software for these functions is available in two forms. It was compiled to a Labview Library Sparameter.LLB and S-parameter.DLL for use with any development language or interface. The software was developed with Labview 2012 and complied as a DLL and a Labview Library (LLB). Both are provided to the user under the license agreement DLL This is a package of all the functions and the associated data structures and files typically packaged with a DLL. This should allow the user to use the library in nearly any control environment that can call DLL s. 01/26/ :51 PM 38 Sparameter.aliases 01/26/ :51 PM 586,240 Sparameter.dll 01/26/ :51 PM 4,698 Sparameter.h 01/26/ :51 PM 297 Sparameter.ini 01/26/ :51 PM 29,416 Sparameter.lib LLB Labview Library I made all the base Complex Data Controls and Methods available along with all the Spar Data Controls and methods. This library make for easy integration into any Labview RF test program and brings the new capabilities to the Vector Network Analyzers of older model or lower order measurement architecture (i.e. Transmit-Receive or TR Test Sets). THE LLB is provided for versions LV8.6, LV2011 SP1 and LV /26/ :51 PM 1,055,733 Sparameter.llb Which consists of the following vi collection: Complex 01/26/ :27 PM 19,488 CAbs.vi 01/25/ :07 PM 17,312 CAdd.vi 01/25/ :08 PM 20,712 CDiv.vi 01/26/ :27 PM 25,695 CLn.vi 01/26/ :27 PM 25,726 CLog10.vi 01/25/ :08 PM 20,673 CMult.vi 01/26/ :25 AM 19,776 CNeg.vi 01/25/ :08 PM 5,156 Complex.ctl 01/25/ :11 PM 11,420 ComplexInput.vi 01/26/ :27 PM 22,784 CSqr.vi 01/26/ :23 AM 21,849 CSqrt.vi 01/25/ :08 PM 20,612 CSub.vi 01/26/ :27 PM 22,310 db2lin.vi 01/25/ :08 PM 25,615 dbv2lin.vi 01/25/ :08 PM 28,631 Deg2Rad.vi 01/26/ :23 AM 5,166 Gamma.ctl 01/26/ :23 AM 25,263 Gamma2Z.vi 01/25/ :12 PM 11,434 GammaInput.vi 01/26/ :23 AM 5,198 Impedance.ctl

7 01/25/ :13 PM 11,758 ImpedanceInput.vi 01/26/ :27 PM 28,910 Lin2dB.vi 01/25/ :08 PM 32,215 Lin2dBV.vi 01/25/ :08 PM 22,120 Pol2Rect.vi 01/25/ :08 PM 25,347 Rad2Deg.vi 01/25/ :08 PM 25,410 Rect2Polar.vi 01/26/ :23 AM 28,542 Spar2Z.vi 01/26/ :23 AM 20,679 Z2Gamma.vi 01/26/ :23 AM 24,174 Z2Spar.vi S-Parameter 01/26/ :37 PM 35,089 4Par_Mat_Multiply.vi 01/26/ :33 PM 33,077 Spar2Tpar.vi 01/25/ :08 PM 7,451 SParameter.ctl 01/25/ :10 PM 17,997 SparInput.vi 01/25/ :08 PM 31,188 Spar_AntiNetwork.vi 01/25/ :08 PM 28,778 Spar_dBDeg2RectLin.vi 01/26/ :43 PM 25,980 Spar_DeEmbed.vi 01/25/ :07 PM 37,594 Spar_Embed.vi 01/26/ :22 PM 24,531 Spar_FixtureDeEmbed.vi 01/26/ :22 PM 30,121 Spar_FixtureEmbed.vi 01/26/ :27 PM 32,986 Spar_GainAvailable.vi 01/26/ :27 PM 38,295 Spar_GainTransducer.vi 01/26/ :27 PM 41,753 Spar_GainUnilateralTransducer.vi 01/26/ :23 AM 30,271 Spar_GammaInOut.vi 01/26/ :27 PM 36,246 Spar_PowerGain.vi 01/25/ :08 PM 28,798 Spar_RectLin2dBDeg.vi 01/26/ :23 AM 48,148 Spar_ReNormalize.vi 01/26/ :27 PM 5,176 Spar_xx.ctl 01/25/ :09 PM 11,444 SxxInput.vi 01/26/ :20 PM 32,785 Tpar2Spar.vi 1.2 Theory Scalar Numbers These are values like real resistance, DC voltage, DC Current to name a few. Power is another and this can be measured as a scalar quantity with a power meter and Power in and Power Out measurement can give the gain or Loss of the network as the difference in the measurements. While you can measure the absolute power, there is much about the network (device) you cannot measure at RF frequencies with only scalar measurement values Complex Numbers and Parameters A complex number is represented by 2 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. 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 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 for instance. The 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

8 represented by two equations with 4 parameters. The 4 parameters have different representation depending on what is known and what operations the users desire to use them for. Some network parameters are better than others depending on circuit types and the operations on them. Complex Values I used complex numbers in a variety of ways in calculations. Typical data structures are the Complex, Spar_xx, Impedance, Complex Number and Spar 2 Port. In Labview the data clusters are groups of values and the properties associated with them. Properties that I used are db/linear, Degrees/Radian, Rectilinear/Polar and units. With this data description we know enough about the state of the complex value to operate on it for RF operations. If the value is in db or dbv, I must convert to Linear and Rectilinear for most complex operations. The complex and S-parameter operations use the parameters to determine the state of the value and to convert to the proper data format for the calculations, and then back to the final display format (db-polar). The Impedance cluster would show units of Ohms, with the db/lin = L or Linear, Deg/Rad= D or Degrees and finally the Pol/Rect = R or Rectilinear. The Gamma cluster is dimensionless and is typically in Linear, Radian and Rectilinear formats as shown in the figures. The complex cluster (Figure 1) has no defined properties and can be pre-represented in any way, so I have left that data cluster (structure) blank. See section for more information on the data types used. Figure 1 - Complex Cluster 1.3 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 as a real resistance and a reactance driven by the charge delay or advancement as related to the driving voltage. This Z(ohms) = X & iy, with Z being complex value of the real (X) resistance and Imaginary (Y) reactance components 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 6 and 7. Reflection coefficient represents this value in power measurements and is also known as rho or = = Rho can be converted to the Voltage Standing Wave Ratio or VSWR by the equation =. 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). These

9 matches can also be derived by the Impedances. The load can be represented by Z=X+iY. The measurement system has impedance. This measurement with the device or network sets up an impedance match represented by Gamma or = =. The Gamma or complex reflection coefficient can be transformed by absolute value to rho or ρ=, which can be used to derive the VSWR from ρ for the complex to scalar power measurement transformation. It can also be transformed by the system impedance to the load impedance. 6 7 = 6898 : > 6? = 6898 : 1 +? 1? > 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 = A BC. Figure 4 - Transmission Network Signal Flow

10 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 1mW. 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 IJ db = 10 log G H K. To convert measurements made in voltages, not true power, and this includes vector network analyzers, vector signal analyzers and spectrum Analyzers use the formula PBQ IJ LMN=20 log G H K. PBQ This can be derived from the fact that RSTUV= PBQW, thus the doubling of the multiplier of the B logarithm of the voltage ratio measurement of power. Convert db to Watts using the inverse formulas: XYY8=10 Z[\]^ _` and XYY8=10 a[bcde] W`. The absolute power can be converted between Watts and dbv with the following equations: XYY8=10 Hfgh _` ik and LMj=10log G kxyy8 1000l 1.4 S-Parameters 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 X1= P _ m Power Incident at Port 2 or X2= P W m Power Emitted at Port 1 or b1= P _ o Power Emitted at Port 2 or b2= P W o p = X + q X q p q = q X + qq X q r p p q s=r q q qq st X X q u

11 Figure 6-2 port S-Parameter signal flow Figure 10 shows the signal flow diagram for the S-Parameters. Note that the dimension of a1,a2 and b1,b2 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. = v or forward reflection coefficient B q = v or reverse transmission coefficient Bq q = vq or forward transmission coefficient B qq = vq or reverse reflection coefficient Bq Note also that the reflection coefficient for the load and source, 7 = vq and Bq = v, respectively are B the same as qq = vq and Bq = v. In other words, if you measure the 1 port S-Parameter of a source B or load, it is the same as the Gamma or. These S-parameters are the basis of many RF and Microwave measurements. 1.5 Network Analysis Network Analyzer measurements In many RF and Microwave measurements the S-Parameters are typically expressed in db Magnitude units and Degrees for polar coordinate system. The units of my data structures allow me to keep track of and manage the conversions necessary to perform the complex calculations for my computations. The Network and Vector Network Analyzers and Spectrum Analyzers all measure with voltage ratio measurements, so to convert to dbv we must use the following equation. SwYXxU yzy LMN=20 log G : SwYXxU { > The Spectrum Analyzer is a frequency discriminating detector that detects the voltage for the signal. It will give the amplitude of signal as a function of frequency. It is scalar in measurement dimension magnitude vs frequency.

12 The Vector Network Analyzer measures the complex amplitude and angle of complex RF signals and also measures in voltage although complex in format magnitude and angle vs frequency. By using reference signals to calibrate the test system response and setting up a reference frame for the measurements, the instrument can measure the amplitude and phase angle of the AC-RF signal for each frequency it is tuned to. The Vector Signal Analyzer is sort of like a cross between a Spectrum and Network Analyzer. In addition it also measures the signal modulation and a number of features about the modulation format. It also measures in voltage and is in complex or scalar format depending on the data being displayed; i.e., the RF signal characteristics or the modulation format and data. Calibration Making measurements of S-Parameters is a process of measurements made with calibrations standards 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 the imaginary plane of reference for the measurements being made. 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 between. There are numerous methods and standards for Coaxial, Waveguide, Planar, probe and other interconnections methods. There are also a number of test fixtures out there 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 that are difficult or nearly impossible to measure with standard equipment (e.g. mixed impedances and more). Some of the most used calibrations methods are: SOLT Short, Open, Load and Thru is the SOLT cal. These are physically and electrically near identical components. The connectors and the calibration component physical parts are constructed to tolerances of 10,000ths of an inch. There are calibration parameters that help define and reference these standards. Some of these are the Open Capacitance frequency response C0, C1, C2, C3 and Short inductance frequency response L0, L1, L2 and L3 that are the polynomial coefficients for the formulas correcting the phase for fringe capacitance and inductance causing ripple in the frequency response that can be modeled with 3 rd order polynomial. There is the delay or electrical length for each of the parts, for the Short, Open and the Thru. The Load has an indeterminate phase due to the small signal level, so of no consequence. The Thru standard loss and phase shift can be calculated and removed from the measurement. What this does is roll the reference plane from the middle of the Thru and ends of the Short, Open and load back to the connector ground ring or plane of reference for phase and loss for the measurement. If no lengths and loss are specified, the calibration is less accurate and the measurement reference 0dB 0Deg position is more difficult to determine. It is usually assumed to be in the middle of the connection Thru and ends of the Open/Short.

13 TRL Thru, Reflect, Line(s) is the TRL cal. This uses these three standards to calibrate. This means that fixture measurements of devices can now be better removed from the device measurement. This allows better device characterization. It require a line of minimum length, reflections of shorts to both ports and finally a Line longer then the Thru. The line must also meet certain requirements for length. Several drawbacks with TRL are the separation of the contacts have to move to accommodate the Line and on some substrate materials like GaAs, for instance, the lines become more reactive at low frequencies due to tangent loss and so system impedance is more difficult to determine. With the LRM calibration(see below), the loads are measured for DC Resistance, which can be entered into the cal standards to offset the VNA measurement System Zo. This accommodates the actual measured load impedance. The Spars are relative to the System Zo, as we will see later when we discuss converting S- Parameters to Arbitrary Impedance. LRM Line, Reflect Match or LRM is actually a variant of TRL. When you have an infinitely long transmission line, this looks to the source just like a perfect load because there is no terminating end point to reflect any energy of the signal and it dissipates thru transmission loss. This load replacement allows the calibration to be performed in a fixture, On-Wafer, coaxial or planar that does not have to be further separated. It can be done in a fixed length fixture. This and the fixed loads standards can determine the System Zo at all frequencies better, verses the issue with high tangent loss with the TRL cal affecting low frequency measurements. LRM does not have this problem. This also is a big advantage for On-Wafer measurements, since the test probes now do not have to have extra motion and alignment to calibrate. Electronic Load/Line For coaxial measurements, ATN Microwave years ago developed an electronic line/load for calibration. Agilent bought the company out. Agilent now offers the Electronic Calibration gear to do the calibration with one connection and simple steps. It uses PIN diodes under automated control to vary the line and reflection parameters of the calibration standard electronically. This is not dependent on physical dimension variation as the mechanical standards are. These can be repeatedy measured and used with little loss of calibration accuracy. Measurements Vector Network Analyzer measurements are made either to coaxial connected devices or to devices with some form of fixture or launch to the device under test. The methods used to measure and calibrate or de-embed with depend on factors such as cost, equipment availability and capability. If the measurement features are not available on your version of Network Analyzer, then software extensions can add to that capability at a low cost related to hardware costs. The type of fixture as well as the capabilities the instrument has may also determine which calibration and Fixturing methods are available to the user.

14 Test Fixturing Tests Fixtures are used to measure a device that does not connect to a measurement instrument directly. The device might be substrate mounted and wire bonded to microstrip that then connects to a Coaxial connector in the fixture header block. Figure 7 - Test Device and Fixtures Response or first order corrections By measuring a Thru calibration with a response Thru for transmission measurements, the Thru adapter and the phase length and loss is part of the device measurement. You will see the S21 and S11 and converse S12 and S22 measurement to be 0 db Magnitude and 0 Degrees phase. So the thru has become part of the measurement. I always assumed the reference plane for this was basically the center of the Thru adapter. So measuring a coaxial device the reference plane (0dB, 0Deg point) is moved from the ground plane of the connection to the same position inside the device under test. See Figure 12. Figure 8 - VNA Response Calibration Cal Kit definitions By using a fully defined cal kit, the calibration kit cal coefficients are used to accurately model the calibration component magnitude and phase responses. This means they can be de-embedded from the measurement. Let s take the typically used 3.5mm Cal Kit. After calibrating, the Thru response has been removed. What that means is with the thru still connected to the VNA after calibration, you will now not see 0dB magnitude and 0 Deg phase. You will see the loss and phase length of the Thru cal component. The calibration has shifted the reference plane back from the middle of the connection to the ground reference plane of the coaxial connectors on both sides or on the connection planes of the probes on a planar substrate being measured.

15 Figure 9 - VNA w/ Cal Kit Calibration Port Extensions w/simple loss correction The VNA allows the user to set a phase shift for each of the ports or the user can set port extension for ports 1 and/or 2 instead. The port extension method is easier, as each port extension give both the reflection and transmission phase shift, verses the users having to set 4 S-Parameter phase shifts, they only ave to enter the port 1 and 2 port extensions. The S-Parameters phase shift is unity for the transmission S21 and reverse transmission S12. The phase shift for reflections is double the transmission phase shift as the signal goes to the reference point and reflects back, thus double the length or shift. The loss of the fixture is determined thru measurements and is divided in half and applied to the input and output of the device to correct the S21, S11 and S22 magnitudes. Again, the fixture losses are singular for S12 and S21 and double that for S11 and S22. This is a simple first order de-embedding or error correction method shown in Figure 14. Figure 10 - VNA Fixture measurement using Port Extension correction Adapter Removal There is a mixed adapter mode where Port1 is different from Port 2. Each of the ports has its own standards and the calibration from each port calibration and the combined Thru, gives a hybrid calibration for these mixed adapter measurements, say for instance coaxial on one side to On-Wafer on the other, or 3.5Mmm Female on one side to 7mm on the other or 3.5mm (F) to 3.5mm (F). This can also

16 be used to measure a coaxial to probe or pin on a test fixture or a wafer probe set. The last case of a female to female connection is called a Non-Insertable Device. An insertable device is a Male to Female connection or direct connection. The Female to Female or Male to Male is not directly connectable without using an adapter. Figure 11 - VNA Adapter Removal method of correction De-Embedding In this method all components are measured for S-Parameters and the adapters or fixture responses are removed mathematically from the combined response measurement. Figure 12 - VNA Fixture measurement w/adapter Removal Figure 13 - VNA De-Embedding DUT From fixture

17 Old VNA s used to use a step recovery diode driven by a square wave at MHz to generate a harmonic comb that was used for the down conversion in such a wideband measurement system. These old system architectures were susceptible to spurious and harmonic signal lock on. Today they use direct conversion techniques that are less prone to erroneous harmonic and spurious signals. This also gives the new analyzers tuned receiver capability. They can measure IMD or harmonic components and measure active IMD or Harmonic sweeps. There is also the new Agilent PXA. These help with non-linear extension S-Parameters or X-Parameters, which opens up some really new measurement and device modeling capability. The new Vector Network Analyzers have new techniques to allow full real de-embedding of external test fixtures and the ability to measure S-Parameters in mixed impedance. This allows you to measure a device designed to 75 Ohms Zo measured with a VNA at 50 Ohm System Zo. My Labview S-Parameter library will help fill this gap in instrument capability on the older VNA test systems and some new vendor additions with limited measurement capability. There are newer techniques to allow full real de-embedding of external test fixtures as well as measuring S-Parameters in mixed impedance, for when you have to measure a device designed to 75 Ohms Zo and the VNA you measure it with is a 50 Ohm Zo System. My Labview S-Parameter library will help fill this gap in instrument capability on the older VNA test systems 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. This is the T-Parameters 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: p =} X q +} q p q X =} q X q +} qq p q They can be defined another way: r p X s=r } } q } q } qq st X q p q u t X p u=r } } q } q } qq sr p q X q s When I was researching this for the RF calculator and S- Parameters Labview library, I made the discovery that there were 2 different definitions for the 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 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 2 or more 2-port networks by simply multiplying the associated individual T-parameter matrices together. With T-

18 Parameters, we can perform matrix multiplication for the cascading networks operations to get the total network T-Parameters and transform back to S-parameters. } B =} B } v } or r } } q } q } qq s=r } B } Bq } Bq } Bqq sr } v } vq } vq } vqq sr } } q } q } qq s Figure 14 - Cascaded Network T-Parameters are complex just like S-Parameters and there are transform equations between the two Parameters and this must be performed on all elements prior to the complex matrix multiplication. Then the } B must be transformed back to S-Parameters for the final answer in S-parameters. Note also that Matrix Multiplication is not commutative AB does not equal BA. This is very important to the cascading math.

19 Complex Matrix Multiply Note that Complex Matrix Multiply, like its scalar matrix counterpart, is non-commutative. This means that AXB=AB is not the same as BXA=BA. This is very important to the order of the operations to insure the calculations for the Embedding and De-embedding process work properly. qm qq r ~ ~ q sr M M q s=r ~ M +~ q M q ~ M q +~ q s ~ q ~ qq M q M qq ~ q M +~ qq M q ~ q M q +~ qq qm qq Figure 15-4 Parameter Complex Matrix Multiply

20 S to T: To convert from S-Parameter to T-Parameter the following formulas apply: } = =k qq = q q l q } q = q } q = = qq q } qq = 1 q Figure 16 - S to T Parameter

21 T to S To convert from T-Parameter to S-Parameter the following formulas apply: = } q } qq q = } } qq =} q } q } qq q = 1 } qq qq = =} q } qq Note that this screen shots shows the T Parameters from the S>T conversion and this displays the T>S converting back to the original S-Parameters, thus proving the inverse transform of the two functions. Figure 17 - T to S Parameters

22 1.5.3 Embedding These are the formulas that can be used directly with S-Parameters to embed network response of two network components N & N# for the response of the two as a singular network S. The network embedding equation will take both network S-Parameter responses and compute the S-Parameters if the combined response of the two as a whole network. Network Embedding Equations = + k q q # l k1= qq # l q = k q# q l k1= qq # l q = k q# q l k1= qq # l qq =# qq + k# q# q # qq l k1= qq # l Figure 18 - Cascaded Network S-Parameters In the example shown, SparA is a 6dB attenuator with 14dB S11/S22 and SparB an amplifier with -20 db S11/S22, 20dB S21 gain and -20dB S12 isolation. Note these values for the de-embed function. Figure 19 - S-parameter Embedding w/s-parameter Labview Library

23 1.5.4 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 use the Network to be inversed as one branch and the sum network to be ideal S11=S22=0 and S12=S21=1. Now solve for the unknown sub-network and that is 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. = qq =0 and q = q =1 0= + k q q # l k1= qq # l 1= k q# q l k1= qq # l 1= k q# q l k1= qq # l 0=# qq + k# q# q # qq l k1= qq # l Figure 20 - Anti-Network Computation

24 Anti-Network Equation Solving for # gives; # = k qq = q q l # q = k1= qq# l q # q = k1= qq# l q # qq = k qq# q # q l k qq # =1l This display shows the Anti-network computation of a 6dB attenuator used for the de-embedding. Figure 21 - S-Parameter Anti-Network

25 De-Embedding The process 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-Net you are removing and then Cascading with the network to remove it from the response. This leaves remaining the desired sub-network. The Labview screen displays the 6dB attenuator as the Spar A component and the Spar B component is the combined attenuator and amplifier response. When run it computes the original amplifier s-parameter response. Figure 22 De-Embedding S-Par Networks w/s-parameter Labview Library In this example, I remove the 6dB attenuator from the combined attenuator and amplifier from the embed example previously. Note that I get the original amplified values of -20dB S11/S22, 20dB S21 gain and -20dB S12 isolation all within dB rounding error.

26 Fixture De Embedding Process This section describes the process of fixture embedding and de-embedding. I included the Embedding function to make simulated networks to test the Fixture De-Embedding process. This case is an extension of the de-embed AB network. Instead of just computing ~M ~ =M,, we double up the math to compute AB and the (AB)C to get the total. The operations must be performed in the order shown to be correct. Figure 23 - Fixture De-embedding Process This screen shows an A and C fixture set and a network response of the fixture and device under test, which is the B component of the ABC matrix. In the next section Fixture Embedding, I created a combine network response of an amplifier with 20db gain -20dB isolation and Return losss of -14dB. I made the fixture components to -20dB s111 A input, A & B S21 & S12 of -0.2dB and the A S22 of -14dB and B S11 of -14dB. This fixture can simulate a balun of 50 to 75 ohms transformation and low loss. In the Dethe computation within Embed below you can see the original amplifier response being recreated from rounding error. Figure 24 - Fixture De-embedding

27 Fixture Embedding Process This screen shows an A and C fixture set and a network response of the fixture and device under test, which is the B component of the ABC matrix. In the next section Fixture Embedding, I created a combine network response of an amplifier with 20db gain -20dB isolation and Return losss of -14dB. I made the fixture components to -20dB s111 A input, A & B S21 & S12 of -0.2dB and the A S22 of -14dB and B S11 of -14dB. This fixture can simulate a balun of 50 to 75 ohms transformation and low loss. We used to to simulate a device and fixture measurement response to use for the De-Embed process previously. Figure 25 - Fixture Embedding

28 1.5.5 Gamma In and Out This value of Gamma represents the source and load impedance compared to the measurement system impedance. It gives a measure of power reflected. The Gamma In and Out 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. q = B = k6 q=6 l k6 q +6 l = J = k6 =6 l k6 +6 l q q q = = + k1= qq q l q q J = qq = qq + k1= l Figure 26 - Gamma In/Out w/ S-Parameter Labview Library

29 1.5.6 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. q q = q q q = 1= qq q = k1+ qq q l+ q q q q Figure 27 - Power Gain w/s-parameter Library

30 Transducer Gain Transducer gain is the ratio of the power delivered to the load to the power available from the source. A = q q k1= q lk1= q q l k1= lk1= qq q l= q q q q Figure 28 - Transducer Gain w/s-parameter Library 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). AJ = q q k1= q lk1= q q l 1= q 1= q qq q Figure 29 - Unilateral Transducer Gain w/s-parameter Library

31 Available Gain 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 (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 device noise figure measurement q q k1= q l = k1= qq q l+ q k q = q l=2uk ˆl with = qq = q q ˆ= = qq = qq = Figure 30 - Available Gain w/s-parameter Library

32 1.5.7 S-Parameter Re-Normalization When using a Vector Network Analyzer, a Spectrum Analyzer or Vector Signal Analyzer the measurement impedance is defined by the system hardware which is designed to a specified system impedance. When one designs and has to measure a part that does not have a match at the typical 50 Ohm Measurement System Impedance, what can one do? 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 formulas 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 impedance measurement to any device Impedance like 75 or 90 Ohms. The newer model VNA s have these capabilities built in. In those cases where you don t have the feature built in, these formulas let you use a 50 Ohm VNA to make any arbitrary impedance or a 75 Ohm amplifier measurements, for instance. Figure 31 - S-Parameter Re-Normalization to Arbitrary Impedances Forward Parameters q =6 [k1+ lk1= qq q l+ q q q ] q =6 [k1= lk1= qq q l= q q q ] q = k6 q=6 l k6 q +6 l = k q= q l k q + q l q =6 Œ: 6 2 > 6 q k1+ q lž: q k q + q l > Reverse Parameters q =6 [k1+ qq lk1= l+ q q ] q =6 q [k1= qq lk1= l= q q ] = k6 =6 l k6 +6 l

33 qq = k q= q l k q + q l q =6 Œ: 6 q 2 > 6 q k1+ lž: k q + q l > Example This example is a 75 Ohm Amplifier that has S11 and S22 = -14 db and S21=6dB and S12=--10dB in 50 Ohm Test System. Note that after the transformation the S11=-32dB and S22=-30dB showing the better match for the 75 Ohm simulated measurement from the 50 Ohm measurement and arbitrary impedance transformation. Figure 32 - Re-Normalize S-Parameters to Arbitrary Impedances with S-Parameter Labview Library

34 1.6 Software Data Descriptions This section will define the data structures or Labview controls and the functions available Data Clusters for Labview LLB Complex Cluster The complex cluster (Figure 26) has no defined properties and can be pre-represented in any way, so I have left that data cluster (structure) blank. Figure 33 Complex Cluster Impedance The impedance cluster (Figure 27) is used to represent terminating loads and the real (resistive) and imaginary (reactive) parts. The units are Ohms and the units dblin= L Linear, Pol/Rect= R Rectilinear and Rad/Deg= R Radian. These make sure the calculations are done correctly in complex format. Figure 34 - Impedance Cluster Gamma The Gamma cluster (Figure 28) is unitless as the dimension is Ohms divided by Ohms. So it is a ratio value. The units dblin= L Linear, Pol/Rect= R Rectilinear and Rad/Deg= R Radian. These make sure the calculations are done correctly in complex format. Figure 35 - Gamma Cluster

35 1 Port S-Parameter The 1 Port S-Parameter is a power ratio complex value with real and imaginary components. The units can be dblin= L Linear, Pol/Rect= R Rectilinear and Rad/Deg= R Radian like the Gamma and they are the same ratio as the Gamma raion matches the S-parameter reflection ratio and has equivalency that can be demonstrated. Most S-Parameter measurements are with units dblin= D db, Pol/Rect= P Polar and Rad/Deg= R Degrees as I show in the cluster below. Figure 36 1-Port Spar Cluster 2-Port S-Parameters The 2-Port network has four set of complex S-parameters S11, S12, S21, S22. Figure 37 2-Port Spar Cluster

36 1.6.2 Data Structures in DLL header Labview DLL header data definition #ifdef cplusplus extern "C" { #endif typedef struct { double Mag; double Ang; LStrHandle units; LStrHandle dblin; LStrHandle RadDeg; LStrHandle PolRect; } TD2; typedef struct { TD2 S11; TD2 S21; TD2 S12; TD2 S22; } TD1; typedef struct { double Mag; double Ang; LStrHandle units; LStrHandle dblin; LStrHandle RadDeg; LStrHandle PolRect; } TD3; typedef struct { double Mag; double Ang; LStrHandle units; LStrHandle dblin; LStrHandle RadDeg; LStrHandle PolRect; } TD4; 1.7 DLL Header Function Prototypes void cdecl Spar_AntiNetwork(TD1 *SParA, TD1 *SParameterOut); void cdecl Spar_dBDeg2RectLin(TD1 *SParameter, TD1 *SParameterOut); void cdecl Spar_DeEmbed(TD1 *SParA, TD1 *SParB, LVBoolean *NetworkToRemoveFrontTrue, TD1 *SParameterOut); void cdecl Spar_Embed(TD1 *SParA, TD1 *SParB, TD1 *SParameterOut); void cdecl Spar_GainAvailable(TD3 *ZIn, TD3 *ZOut, TD1 *SParameter, TD3 *ZSystem, TD2 *GainAvailable); void cdecl Spar_GainTransducer(TD3 *ZIn, TD3 *ZOut, TD1 *SParameter, TD3 *ZSystem, TD2 *GainTransducer); void cdecl Spar_GainUnilateralTransducer(TD3 *ZIn, TD3 *ZOut,

37 TD1 *SParameter, TD3 *ZSystem, TD2 *GainUnilateralTransducer); void cdecl Spar_GammaInOut(TD3 *ZIn, TD3 *ZOut, TD1 *SParameter, TD3 *ImpedanceSystem, TD4 *GammaIn, TD4 *GammaOut); void cdecl Spar_PowerGain(TD3 *ZIn, TD3 *ZOut, TD1 *SParameter, TD3 *ZSystem, TD2 *GainAvailable); void cdecl Spar_RectLin2dBDeg(TD1 *SParameter, TD1 *SParameterOut); void cdecl Spar_ReNormalize(TD1 *SParameter, TD3 *ZIn, TD3 *ZOut, TD3 *ZSystem, TD1 *SParameterOut); void cdecl SparInput(TD1 *SParameterIn, TD1 *SParameterOut); void cdecl SxxInput(TD2 *ComplexIn, TD2 *ComplexOut); void cdecl CAbs(TD2 *Input, TD2 *Output); void cdecl CAdd(TD2 *Input1, TD2 *Input2, TD2 *Output); void cdecl CDiv(TD2 *Input1, TD2 *Input2, TD2 *Output); void cdecl CLn(TD2 *Input, TD2 *Output); void cdecl CLog10(TD2 *Input, TD2 *Output); void cdecl CMult(TD2 *Input1, TD2 *Input2, TD2 *Output); void cdecl CSqr(TD2 *Input, TD2 *Output); void cdecl CSqrt(TD2 *Input1, TD2 *Output); void cdecl CSub(TD2 *Input1, TD2 *Input2, TD2 *Output); void cdecl DB2Lin(TD2 *Input, TD2 *Output); void cdecl DBv2Lin(TD2 *Input, TD2 *Output); void cdecl Deg2Rad(TD2 *Input1, TD2 *Output); void cdecl Gamma2Z(TD4 *Gamma, TD3 *ZSystem, TD3 *Z); void cdecl GammaInput(TD2 *ComplexIn, TD2 *ComplexOut); void cdecl Lin2dB(TD2 *Input, TD2 *Output); void cdecl Lin2dBV(TD2 *Input, TD2 *Output); void cdecl Pol2Rect(TD2 *Input, TD2 *Output); void cdecl Rad2Deg(TD2 *Input, TD2 *Output); void cdecl Rect2Polar(TD2 *Input, TD2 *Output); void cdecl Spar2Z(TD4 *Spar, TD3 *ZSystem, TD3 *Z); void cdecl Z2Gamma(TD3 *Z, TD3 *ZSystem, TD4 *Gamma); void cdecl Z2Spar(TD2 *Z, TD3 *ZSystem, TD4 *Spar); void cdecl CNeg(TD2 *Input1, TD2 *Output); void cdecl ComplexInput(TD2 *ComplexIn, TD2 *ComplexOut); void cdecl ImpedanceInput(TD2 *ComplexIn, TD2 *ComplexOut); void cdecl _4Par_Mat_Multiply(TD1 *A, TD1 *B, TD1 *AB); void cdecl Spar2Tpar(TD1 *SParametersIn, TD1 *TParameterOut); void cdecl Spar_FixtureDeEmbed(TD1 *SParA, TD1 *SParB, TD1 *SParC, TD1 *SParameterOut); void cdecl Spar_FixtureEmbed(TD1 *SParA, TD1 *SParB, TD1 *SParC, TD1 *SParameterOut); void cdecl Tpar2Spar(TD1 *TParametersIn, TD1 *SParametersOut); long cdecl LVDLLStatus(char *errstr, int errstrlen, void *module); #ifdef cplusplus } // extern "C" #endif #pragma pack(pop)

38 1.8 Complex Function Descriptions This section describes the complex functions of this library CAbs Complex Absolute Value Function finds the Absolute value of the complex value input. Outputs a complex value to aid the complex calculations, but the Absolute Magnitude value is the only value. The imaginary component is zero. Input: Complex Value Output: Complex Value CAdd Complex Addition Value Function finds the additive value of the complex values input. Outputs a complex value computed from two input values. A+B=C Input: Complex Value A Complex Value B Output: Complex Value C CDiv Complex Division Value Function finds the division value of the complex values input. Outputs a complex value computed form two input values. A/B=C Input: Complex Value A Complex Value B Output: Complex Value C CLn Complex Natural Log Value Function finds the Natural Log value of the complex value input. Ln(A)=C Input: Complex Value Output: Complex Value CLog10 Complex Log base 10 Value Function finds the Sx G value of the complex value input. Sx G (A)=C Input: Complex Value Output: Complex Value CMult Complex Multiplication Value Function finds the multiplicative value of the complex values input. Outputs a complex value computed from two input values. A*B=C Input: Complex Value A Complex Value B Output: Complex Value C CNeg Complex Log base 10 Value Function finds the Sx G value of the complex value input. -A=C

39 Input: Complex Value Output: Complex Value ComplexInput This function allows easy and type safe inputs to the Complex Data Structure or Labview Cluster CSqr Complex Square Value Function finds the Square value of the complex value input. ~ q =C Input: Complex Value Output: Complex Value CSqrt Complex Square Root Value Function finds the Square Root value of the complex value input. ~=C Input: Complex Value Output: Complex Value CSub Complex Subtraction Value Function finds the subtractive value of the complex values input. Outputs a complex value computed from two input values. A-B=C Input: Complex Value A Complex Value B Output: Complex Value C db2lin Complex db to Linear Value Function finds the db to Linear value of the complex value input. Input: Complex Value Output: Complex Value dbv2lin Complex dbv to Linear Value Function finds the dbv to Linear value of the complex value input. Input: Complex Value Output: Complex Value Deg2Rad Complex Deg to Radian Value Function finds the Deg to Radian value of the complex value input. Input: Complex Value Output: Complex Value Gamma2Z Complex Gamma and System Impedance are used to compute Impedance Z. Input: Complex Gamma Value Input: Complex System Impedance Value Output: Complex Impedance Value

40 GammaInput This function allows easy and types safe inputs to the Gamma Data Structure or Cluster ImpedanceInput This function allows easy and types safe inputs to the Impedance Data Structure or Cluster Lin2dB Complex Linear to db Value Function finds the Linear to db value of the complex value input. Input: Complex Value Output: Complex Value Lin2dBv Complex Linear to dbv Value Function finds the Linear to dbv value of the complex value input. Input: Complex Value Output: Complex Value Pol2Rect Complex Polar to Rectilinear Function finds the Polar to Rectilinear value of the complex value input. Input: Complex Value Output: Complex Value

41 Rad2Deg Complex Radian to Degree Function finds the Radian to Degree value of the complex value input. Input: Complex Value Output: Complex Value Rect2Polar Complex Rectilinear to Polar Function finds the Rectilinear to Polar value of the complex value input. Input: Complex Value Output: Complex Value Spar2Z Complex S-Parameters and System Impedance are used to compute Impedance Z. Input: Complex S-parameter Value Input: Complex System Impedance Value Output: Complex Impedance Value Z2Gamma Complex Impedance and System Impedance are used to compute Gamma or Γ. Input: Complex Impedance Value Input: Complex System Impedance Value Output: Complex Gamma Value Z2Spar Complex Impedance and System Impedance are used to compute S-Parameters ( or qq ). gamma value and the system impedance inputs. Input: Complex Impedance Value Input: Complex System Impedance Value Output: Complex S-Parameter Value

42 1.9 S Parameter Function Descriptions Par_Mat_Multiply Complex Impedance and System Impedance are used to compute S-Parameters ( or qq ). gamma value and the system impedance inputs. Input: Complex Impedance Value Input: Complex System Impedance Value Output: Complex S-Parameter Value Spar2Tpar This function converts S-Parameter input to T-Parameter Output. Input: S-Parameter values Input: T-Parameter values Spar Input This function allows easy and types safe inputs to the S-Parameter Data Structure or Cluster Spar_AntiNetwork This function converts S-Parameter input to Anti-Network or -Parameter Output. Input: S-Parameter values Output: -Parameter values Spar_dBDeg2RectLin This function converts S-Parameter input to S-Parameter db/deg to Output in Rectilinear/Linear. Input: S-Parameter values in db/deg units Output: S-Parameter values in Rect/Lin units Spar_DeEmbed Complex S-Parameter for Network A and B are used to compute S-Parameters the Network being De- Embedded from the A and B Network System input. The user must select to remove the front or rear network, so the proper anti-network is computed and the embed is calculated in the correct order as the operation is non-commutative or M~. Input: Complex S-Parameter A Value Input: Complex S-Parameter B Value Input: Boolean - Network to Remove (Front=True) Output: Complex S-Parameter De-Embedded Value

43 1.9.7 Spar_Embed Complex S-Parameter for Network A and B are used to compute S-Parameters the Network being Embedded from the A and B Network System input. The user must input the front and rear network in the proper order, so that embed is calculated in the correct order as the operation is non-commutative or ~M M~ Input: Complex S-Parameter A Value Input: Complex S-Parameter B Value Output: Complex S-Parameter De-Embedded Value Spar_FixtureDeEmbed Complex S-Parameter for Network A, B and C are used to compute S-Parameters the Network being De- Embedded from the A, B and C Network System input. The proper anti-networks for A and B are computed and the de-embed is calculated in the correct order as the operation is non-commutative or M~. B Matrix is the Embedded Network response and A and C matrices are the Fixture Input and Output networks being removed from the combined network B response. Input: Complex S-Parameter A Value Input: Complex S-Parameter B Value Input: Complex S-Parameter C Value Output: Complex S-Parameter De-Embedded Value Spar_FixtureEmbed Complex S-Parameter for Network A, B and C are used to compute S-Parameters the Network being Embedded from the A, B and C Network System input. The proper matrices A, B and C are computed and the embed is calculated in the correct order as the operation is non-commutative or ~M M~. B Matrix is the Embedded Network response. You can use this one to create fixture measurements to test the de-embed with or simulate fixture device responses. Input: Complex S-Parameter A Value Input: Complex S-Parameter B Value Input: Complex S-Parameter C Value Output: Complex S-Parameter Embedded Value Spar_GainAvailable This function takes the Network S-Parameters and the Source and Load Impedances to compute the Available Gain. This can be used for gain/loss of noise or power measurements. The loss would equal the input network available gain (loss in this case). Input: S-Parameter Network Input: Complex Impedance of Source (default to 50Ohms) Input: Complex Impedance of Load (default to 50Ohms) Input: Complex Impedance of the System (default to 50Ohms) Output: Complex Available Gain in db