Chris Angove, Independent Professional Engineer

Chris Angove is a highly experienced and MSc qualified chartered electronics engineer specialising in electrical and electronics engineering. He manages and owns Faraday Consultancy Limited (FCL).

Scattering and Transmission Parameters

Scattering (S) parameters are widely used in radio frequency (RF) and microwave engineering for the analysis, simulation and design of linear networks. T parameters are also useful, in particular for embedding and de-embedding purposes which are often necessary to achieve precision RF and microwave measurements. Other examples of network parameters are admittance (Y) parameters, impedance (Z) parameters and hybrid (H) parameters [1] [2]. S parameters are useful because the elements of the S parameter matrices relate directly to common RF and microwave measurements including forward gain or transmission, reflection (return loss and VSWR) and reverse isolation. Every S parameter measurement may be expressed in units of magnitude and phase but sometimes only one of these is necessary. For example, to derive VSWR and return loss from parameters of the form Smm does not require phase. S parameters, in general, vary with frequency and other conditions such as temperature, power supply and bias voltages. They are also defined for standard impedance loads, often 50 Ω or 75 Ω. Other parameters require short or open circuit loads for measurements which would not usually be appropriate for RF and microwave equipment. More recent derived versions of S parameters may be used to represent mixed (common and differential) mode excitation and to account for the non-linear properties of networks, especially at higher powers.

  1. An Overview of S parameters in RF and Microwave Engineering?
  2. Transmission Lines and Imperfect Matching
  3. S parameter Matrices
  4. S parameter Definitions
  5. S parameter Definitions for a 2 Port Network
  6. What is a Network?
  7. What is a Port?
  8. How Many Ports May a Network Have?
  9. That 65 Port Thing was Surely Not One Network But Several Smaller Ones Wasn't It?
  10. What Are Balanced and Unbalanced (or Single-Ended) Networks?
  11. How Do You Interpret 'Linear' and 'Log Magnitude' S parameter Measurements?
  12. What is a Vector Network Analyzer?
  13. Why Do You Talk About Spatial Phase? Is it Not Just Phase?
  14. What Are random and systematic errors?
  15. How Would I Characterise, Say, a 4 Port DUT Using a 2 Port VNA?<
  16. What Are T Parameters and How Might They Be Useful?
  17. How Would I Extract Group Delay Data from S parameter Matrices?
  1. An Overview of S parameters in RF and Microwave Engineering?
  2. RF and microwave equipment is typically designed using self contained sub-units, generically called 'networks', connected together to build the equipment for a particular function. Every network must have one or more ports and must, according to the original S parameter definition, behave linearly. A simple network may have just one port, such as a signal generator or a load, but most networks have 2 ports and a few have 3 or more ports. Typical examples of functional equipment usually comprises several networks like these connected together. The transmission lines necessary to connect networks are themselves considered to be 2 port networks. They could be any of many different types: single ended or differential, for example coaxial, microstrip, coplanar strips or stripline, whatever is appropriate for the frequencies and physical hardware in use.

    An example schematic of a piece of RF or microwave equipment comprising interconnected networks, some with transmission lines (cables), is shown in Figure 1-1.

    Figure 1-1 A schematic of a typical RF or microwave module represented by various interconnected networks, each of which may be electrically represented by a S parameter matrix.

    The function of this is not important but it does demonstrate how networks may typically be connected together port to port. The ports of each network have been numbered serially by the circled numbers starting with 1, usually at the input. There is no specific requirement for this numbering convention but the ports must be uniquely identified and this one is often adopted. The port numbers should be allocated for each network at the time of measuring its S parameters stand-alone using, typically, a correctly set up and calibrated, vector network analyzer (VNA). Port numbering like this ensures that the network ports are connected together unambiguously when the equipment is assembled.

    A port is defined as a two conductor interface on the network which may connect to any external components such as other networks, signal sources, loads, amplifiers or test equipment. Historically, these are often called 'terminals' despite the fact that traditional physical terminals may not be appropriate at higher frequencies. At each of the port interfaces in Figure 1-1 two 'wire' connections are therefore shown to avoid ambiguity. For some documents, schematics such as Figure 1-1 might be considered too cluttered and the network port connections may be simplified to single lines.

    The terminal pairs might look like differential connections (differential pairs), but that is not necessarily the case. Despite the trend towards differential transmission lines in recent years, in fact the majority of connections like these are still coaxial. Coaxial connections are single-ended in which the screen or shield connections are all assumed to be, rightly or wrongly, 'at the same potential', so the ground connections have zero impedance between them. That might have been tolerable in the earlier days of RF and microwave engineering using electrically small components, but it receives more criticism today, especially for electromagnetic compatibility (EMC) compliance.

    We have to be cautious when allocating and using port numbers however, because those used will relate directly to the element subscripts within the associated S parameter matrix. For example, 'CABLE 1', a 2 port transmission line, in Figure 1-1 is actually a passive reciprocal network. It was designed to be useable with the signals passing in either direction. The equipment will probably appear to work satisfactorily if it was connected in reverse to that shown, with port 2 on the left and port 1 on the right. However, any precision cascade analysis required would then require modification of its S parameter matrix, to interchange the S parameter indices 1 to 2 and vice versa. Unfortunately, no manufacturer will ever produce a perfectly reciprocal cable and elements of non-reciprocity may become significant, especially at higher frequencies.

  3. Transmission Lines and Imperfect Matching
  4. S parameters call heavily on the theory and equations associated with transmission lines, or more precisely, transmission lines which are imperfectly matched. In fact, the adjective 'scattering' used in 'scattering parameters' is derived from the scattering of propagating plane waves when they pass across media which have differing impedances and are therefore unmatched. On reflection perhaps, this may be a better analogy in the optical domain where phenomina such as diffraction are well known and frequently observed.

    We often require a source to be 'perfectly matched' to a load for maximum power transfer across our frequency range of interest. That is of course impossible to achieve, but we can usually get much closer to it with the correct application of S and T parameters.

  5. S parameter Matrices
  6. The electrical performance of every network, such as those shown in Figure 1-1, may be expressed as an S parameter matrix of dimensions N×N, where N is the number of ports of the network. As an example, let us take the amplifier (OK, network if you prefer). It has 2 ports numbered 1 and 2 for the input and output respectively. The amplifier would have been designed and then tested using a correctly set up and calibrated VNA, for full 2 port measurements, across the required frequency range, let us assume 1.0 GHz to 1.5 GHz in 10 MHz steps. S parameters are always functions of frequency and very often of other parameters as well such as temperature, DC voltage and bias, so these must all be recorded along with the measurements.

    After the measurement, the VNA will yield a set of data, 2 port S parameter measurements, for a set of discrete CW frequencies, in this case 51. At this point the S parameter indices for the DUT assumed by the VNA, usually 1 and 2 for a 2 port VNA, must be correctly related to the amplifier or device under test. So, for example, if the VNA ports 1 and 2 were connected to the amplifier (or DUT) ports 1 and 2 respectively, the default output of the VNA would be correct. If the DUT had different port numbering, the VNA setup would require modification. The VNA measurement may be displayed in real time on the instrument and, if required, saved electronically to files for archiving and use elsewhere. Examples might include creating plots and tables for reporting purposes or exporting to simulation software.

    For every frequency step, the VNA will internally generate a 2×2 S parameter matrix which will comprise an array of 4 values representing the (linear) magnitude and the (spatial) phase of the parameter concerned. Each of these will be represented in the form Smn, where m is the row and n is the column of the associated S parameter matrix. The subscript indices, m and n, represent the 'response' port and the 'stimulus' port identities respectively. For example, Smn represents the S parameter result for a stimulus applied at port n and the response measured at port m with all other ports terminated in the system impedance.

    An example of an S parameter matrix result at one frequency for a 2 port network in a few different formats is shown by the 2×2 matrix in (3-1).

    S = S 11 S 12 S 21 S 22 = 0.123° 0.0246° 11.2169° 0.215° = 0.092+j0.039 0.014+j0.014 11.0j2.14 0.19+j0.052 = 0.1 e j0.40 0.02 e j0.80 11.2 e j2.95 0.2 e j0.26
    (3-1) An example 2 port (2×2) S parameter matrix assuming port numbering 1 and 2 with the element values shown respectively in: symbolic index notation, (linear) polar, complex rectangular and complex exponential forms. In complex exponential notation, by convention, the real part operator may be omitted for clarity and the angles are in radians. Results such as these also require the measurement conditions to be specified, for example Z0=50 Ω, frequency=1.2 GHz, bias=10 mA.

    Any of the S parameter matrix forms is perfectly valid with perhaps linear polar being the most popular. Notice that the complex exponential form does not include the 'real part' operator which is a common practice in electronics engineering.

    The S parameter values, elements in the S parameter matrices, have been defined to relate directly to common RF parameters: reflection coefficient (from which are derived return loss and VSWR) and transmission from one port to another port (gain and reverse isolation). In fact, reflection coefficient may be considered as transmission from one port (an incident or forward wave) to the same port (a reflected wave). Referring again to the S parameter index notation, Smn: for m=n the result is a reflection coefficient at port m (or n). For m≠n the result represents transmission from port n to port m.

    S parameter measurements also require a defined nominal system impedance. This is most conveniently the same as the nominal system impedance of the equipment under test, commonly 50 Ω. However, it is possible to convert S parameter measurements from one system impedance to another. A common example would be between 50 Ω and 75 Ω, unbalanced coaxial in both cases, as both impedances are used in some communications systems.

    S parameters are used frequently in RF and microwave engineering. T parameters are also popular but less frequently used. Many of the challenges of RF engineering are concerned with the behaviour of components which interface in one or more ways with transmission lines and other components. In most cases these are imperfectly matched, at least across part of the operating frequency range. Definitions for S and T parameters draw heavily from transmission line theory.

    S parameters may be defined or measured for a network comprising an integer number of ports from 1 to n. In general, S parameter values vary with frequency, temperature and possibly other parameters such as bias voltage or current. One set of S parameter measurements comprises an N×N; square matrix in which each element is a value expressed in magnitude and phase. A one port S parameter is a complex number of the form Smm, where m is an integer used to identify the port.

  7. S Parameter Definitions
  8. In (3-1) and all other S parameter matrices, the columns represent the stimuli ports and the rows represent the response ports. So, for example, S21 represents the result for a stimulus applied to port 1 and the response measured at port 2. S22 would represent both the stimulus being applied to and the response being measured from port 2.

    In every measurement, by definition, every port must be terminated exactly in the system impedance Z0, a purely resistive value. This must not change across the whole measurement frequency range. These conditions also apply to VNA source and load impedances as appropriate, during the normal measurement procedure, and to the loads connected to any unused ports. If the equipment under test has more ports than those available on the VNA, the tests must be performed in stages. For each stage, the unused ports must each also be terminated in the system impedance.

    A little caution may be required with port numbering. By convention, these normally correspond to the indices (row and column numbering) of the S parameter matrix so, in (3-1), the port numbers allocated were 1 and 2. However, there may have been some very logical reason why the ports might have been numbered 34 and 92 instead, in which case the corresponding S parameter matrix would be (4-1). This would be fine provided it was still only a 2 port network. If it was actually a 3 port network with ports 34, 92 and 97 for example, then the true S parameter matrix would need to have 3 dimensions (3×3) with the rows and columns named 34, 92 and 97.

    S 34,34 S 34,92 S 92,34 S 92,92
    (4-1) An S parameter matrix for a 2 port network with the ports numbered 34 and 92
  9. S Parameter Definitions for a 2 Port Network
  10. To start with, we will consider the S parameters of a 2 port network which may have either balanced ports, or unbalanced ports such as those applicable to coaxial connectors. These are shown in Figure 5-1.

    Figure 5-1 Schematics of a 2-port balanced network and a 2-port unbalanced network for the definitions of S parameters.

    We are initially considering 2-port parameters because: (a) they are the most commonly encountered and (b) they are often used in cascades, which is a common architecture. Most vector network analyzers (VNAs) also have 2 ports and therefore, once set up and calibrated, enable a whole set of 2 port S parameter measurements in one operation. Either port of a two port VNA may also be used to measure the S parameters of a one port networks such as a load. The performance of a one port source, such as a signal generator, cannot be described by S parameters because it contains a source and therefore cannot meet the S parameter measurement criteria. The S parameters of a network with 3 or more ports can be measured by a 2 port VNA in stages, processing the results after each stage as described in Section 15.

    In general, at each port there will not be a perfect match so there will be a forward wave component into the port and a reverse wave component out of the port. These are represented respectively as VnF and VnR where 'n' is the port identity and shown by a number enclosed in a circle. So for example the forward wave at port 2 will be V2F and the reverse wave at port 1 will be V1R. For this imperfect match condition at port 'n' a standing wave will occur with the total (resultant) voltage Vn and resultant current In given, using transmission line theory, by the equations in (5-1).

    V n = V nF + V nR Z 0 I n = V nF V nR
    (5-1) Total standing wave voltage (Vn) and current (In).

    Z0 is the system impedance (Ω).

    For each port we may define incident and reflected waves which are sometimes called 'power waves', an and bn, where 'n' is the port identity. These are also complex values, each represented by a magnitude and a phase, similar to S parameters, but in this case normalised to √Z0 as defined in (5-2).

    a n = V nF Z 0 b n = V nR Z 0
    (5-2) Definitions for the incident and reflected power waves, an and bn respectively.

    To re-iterate, Z0 is the defined system impedance, a purely resistive value such as 50 Ω, or to be more explicit, Z0=50+j0 Ω.

    A common question is: "How does the VNA present 'exact' Z0=50+j0 Ω impedances to the DUT during measurements, especially across big frequency ranges?". Of course it cannot, but today's even quite basic VNAs include very powerful and fast processing capability. During the calibration procedure before any measurements, this is used to calculate and correct for systematic errors inherent in the VNA itself. The resulting 'corrected' measurements then significantly account for departures of the source and load impedances from the nominal values.

    Let us consider a 2 port network as an example on which the ports are identified as 1 and 2. As with all network parameters, there are no rules about which port may be an input or an output but whatever is chosen should be related to the DUT in some way, such as with labels. When such measurements are planned, an initial assessment must be made to accommodate any high power levels which may exit any port to avoid possible test equipment damage. This should include the possible consequences of fault conditions which may occur such as instability (oscillation).

    For example, if we were planning to measure the S parameters of a 10 W linear amplifier, it would be unwise to connect its output port directly to a VNA whose maximum incident power was +30 dBm (1 W). Even if we had taken special precautions to reduce the input power to give an expected output power well below +30 dBm, a 10 W output power might have been caused by instability. This would likely cause expensive damage to the VNA and put it out of use for repair.

    By definition, the relationship between the power waves, an and bn, and the 2 port S parameter matrix is given by (5-3).

    b 1 b 2 = S 11 S 12 S 21 S 22 a 1 a 2
    (5-3) The S parameter matrix equation for a 2 port network relating incident and reflected power waves.

    Solving the matrix equation in (5-3) gives the two equations shown in (5-4). These show the relationships between the 2 port S parameter matrix and the incident and reflected power waves, an and bn respectively, for each of the ports.

    b 1 = S 11 a 1 + S 12 a 2 b 2 = S 21 a 1 + S 22 a 2
    (5-4) The S parameter matrix equation from (5-3) as two separate equations.

    Now, in order to get some more useful information, we need to apply the definition requirement that both ports must be loaded with the system impedance Z0.

    We know that all of the definition equations so far have assumed that every port is terminated exactly in the system impedance, Z0, either a source or load as required. Consider port 2 of the 2 port network. If this was connected to a load of exactly Z0, there would be no reflections from the load by our definition and therefore no incident wave at port 2 of the network, so a2 would be zero. Substituting this into the first equation in (5-4) yields the result for the port 1 parameter S11 = b1/a1. Similar expressions may be obtained for all of the 2 port S parameters with appropriate terminations, which are shown in (5-5).

    S 11 = b 1 a 1 S 12 = b 1 a 2 S 21 = b 2 a 1 S 22 = b 2 a 2
    (5-5) The S parameter definitions for a 2 port network.

    The function of the VNA is to accurately measure the power waves, in magnitude and phase, at each of the ports, and across frequency. Then the VNA must perform the necessary calculations from (5-5) to obtain the associated S parameters. As we noted earlier, this is achieved after quite a lengthy setting up procedure and calibration.

  11. What is a Network?
  12. A network may be passive or active but to comply with the S parameter conditions it must behave linearly and this normally means under small signal, steady state conditions. These will normally be continuous wave (CW), either at one frequency but, more commonly, across a range of frequencies. Occasionally S parameters may be specified under non-linear conditions. This is not recommended but, provided the conditions under which they are measured are accurately known and repeatable, they may provide useful information.

  13. What is a Port?
  14. A port is defined as a pair of associated terminals through which the actual currents may flow in either direction. The algebraic definitions of positive and negative current flows are as identified in Figure 5-1. These are not terminals in the traditional sense but simply two conductor connections designed appropriately for the frequencies being used. For single ended ports using the present technology these would usually be coaxial connectors.

  15. How Many Ports May a Network Have?
  16. Any number. The largest I have seen is 65. Fortunately the ports were serially numbered 1 to 65.

  17. That 65 Port Thing Was Surely Not One Network But Several Smaller Ones Wasn't It?
  18. No, it was literally one network as far as I could work out from the circuit schematic. However there may have been unintentional leakage paths between some of ports which would have been included in the results. With some of the latest test equipment and very careful VNA setup and calibrations it is possible to measure isolation values to greater than 100 dB. That would correspond to a log magnitude S parameter transmission measurement of less than -100 dB, since (linear) transmission is the reciprocal of (linear) isolation. So the VNA is capable of measuring, for example, extremely small values of leakage which might have been caused by capacitive coupling between tracks, ground planes and other conductors.

  19. What Are Balanced and Unbalanced (Single-Ended) Networks?
  20. Schematics of these are shown in Figure 5-1: the top diagram shows balanced ports and the bottom diagram shows unbalanced ports. In a balanced network the terminals of every port are assumed to have independent connections, one for the inward current and one for the outward current. There is no assumption of any common connection between any ports, such as that often provided by a ground. Most VNAs however do have single ended (coaxial) ports, of which one terminal is ground, so it is not possible to use one of these directly to make any balanced measurements of a network. There are however some options available for mixed mode S parameter measurements using a 4 port single ended VNA with a 4 port electronic calibration (ecal) device. In these cases the system impedances of the common mode and differential mode measurements are 25 Ω and 100 Ω respectively. For unbalanced networks, generally one of the port terminals, for every port, is connected to ground. This is standard for coaxial connectors, which is still the most common type of port.

  21. How Do You Interpret 'Linear' and 'Log Magnitude' S Parameter Measurements?
  22. So far we have only considered linear S parameters. All S parameters are expressed as unitless complex quantities because they are derived from the ratios of two complex values with the same units, such as shown in (5-5) for the case of a 2 port network. Every S parameter therefore has an amplitude and a (spatial) phase and may be represented in rectangular, polar or exponential form. An example of each of these is shown in (11-1). Linear amplitude (or magnitude) may be obtained directly from the value R in either the polar or the exponential forms. It may be obtained from the rectangular form by squaring the real and imaginary coefficients, adding them and taking the square root of the result as shown in (11-2). The phase angle φ may be obtained directly from either of the polar forms or from the rectangular form using the inverse tangent also shown in (11-2).

    S mn =c+jd S mn =Rϕ° S mn =R e jϕ
    (11-1) Linear modes for describing the S parameter Smn: rectangular, polar and exponential respectively. Coefficients 'c' and 'd' are for the real and imaginary parts respectively. The other parameters are defined in (11-2).
    R= c 2 + d 2 ϕ= tan 1 d c
    (11-2) Definitions of the amplitude (R) and the phase angle (φ) for the S parameter, Smn shown in (11-1).
    e jϕ =cosϕ+jsinϕ
    (11-3) Euler's identity, used for converting complex numbers between exponential and rectangular formats

    Log magnitude is just an abbreviation of logarithmic magnitude (RLM) using the dB definition shown in (11-4). Log magnitude simply means that the magnitude is expressed in the logarithmic units of decibels (dB) and is independent of phase.

    R LM =20 log 10 R =20 log 10 c 2 + d 2
    (11-4) Definitions of log magnitude of an S parameter from (11-1) and expressed in decibels (dB). The logarithm coefficient of 20 is used by definition, because 'R' is the linear ratio of voltage amplitudes.
  23. What is a Vector Network Analyzer?
  24. A vector network analyzer (VNA) is an instrument designed to analyze RF and microwave networks across frequency. VNAs include many high precision mechanical components and substantial signal processing capabilities and therefore represent a significant asset for most companies. The running costs are also high, for example: yearly calibration, calibration kits, precision cables, adapters, low VSWR loads, electronic calibration modules, cables, jigs, adapters and skilled operators. A VNA may not be economic to own unless it needs to be used regularly, otherwise rental of a suitable instrument might be an option.

    Some common VNA features and comments are shown in the following list.

    1. Frequency Range, Number of Ports, Types and Genders.
    2. Most VNAs have coaxial connectors appropriate to the maximum operating frequency, with smaller diameters for higher frequencies to reduce the risk of the wrong transmission modes. Connector genders must be selected carefully to minimise the number of adapters required for its intended main use. Unnecessary adapters, even high quality low VSWR types, can degrade measurement uncertainties significantly. Most VNAs have 2 ports but may be used to measure networks with more ports provided the correct procedures are followed.

    3. Calibration for Correction of Systematic Errors
    4. This must not be confused with the regular equipment calibration required for quality assurance, normally every 1 or 2 years. This is the internal calibration procedure usually done immediately before performing a measurement to correct for systematic (predictable) errors inherent in the test equipment. These include imperfect coupler directivity, source and load match and frequency dependent variations. Provided the equipment and environment are maintained stable, these types of calibrations may be saved and recalled later for similar measurements. Typically it is recommended that this type of calibration is repeated for a minimum frequency of once per day.

    5. Results Reporting and Data Output
    6. With today's display technologies, usually there are many ways that results may be displayed on the instrument for immediate feedback to the operator, for example rectangular, polar, Smith chart and tabular forms. Most measurements will be 'corrected' meaning that a systematic error calibration has been performed. 'Uncorrected' measurements may be appropriate in some situations if the VNA is configured to operate in some dedicated mode.

      Data reporting or exporting is standard on most VNAs including industry standard text file formats such as Touchstone. Routine saving of all measurement results in an electronic format such as this is recommended.

    7. Swept CW and Pulse Capability
    8. Components designed for operation with pulsed waveforms, such as those used with radar equipment, have to meet different criteria from those designed for CW or near-CW. The peak to mean power ratio is effectively 1 for a CW (sine squared mean power) waveform but can be as much as 1000 for a pulsed CW primary radar, depending on the definition used. It would be very wasteful and expensive to design a radar DUT assuming the peak power in the same way that we assume mean CW power only on non-radar equipment. In view of thermal dissipation, it can be designed for the mean power which, in our example would be 1/1000 of the peak power and much easier to do. However, it must still be sufficiently resilient to handle the pulses without suffering any damage due to dielectric breakdown as this is a short=term phenomina.

      S parameter measurements have traditionally been performed by frequency swept continuous wave (CW), with usually linear or perhaps logarithmic frequency scaling for larger frequency ranges. The mean power level of the swept CW waveform is normally levelled accurately to a particular value appropriate to keeping the DUT operating comfortably within its linear region. In many types of communications equipment this represents a reliable and representative test method. However, some careful analysis must be directed at the choice of power level incident at the DUT input port, appropriate for the DUT power supply, linear operation, heat sinking and avoiding any risk of dielectric breakdown. This should be representative of the mean power level the DUT would expect to handle in normal service. Swept CW is essentially narrow band with perhaps a little frequency spread caused by the sweeping function itself, which is effectively a type of modulation applied by the test equipment.

      Some more recent VNAs have pulse capability as well as swept CW. Correctly set up pulse tests are appropriate for DUTs which are designed to handle pulse waveforms like those used in radar systems. S parameters may be measured similarly to those swept CW but the full pulse parameters must of course be recorded with the S parameter results.

    9. Instrument Control and Data Acquisition
    10. Nearly all modern test equipment is equipped with Ethernet (IEEE 802.3) derived LAN capability with the 'Gigabit' version currently being the most popular standard. Because test equipment is such a significant investment, the assets are often retained for many years and the data acquisition technology available is sometimes rather mature such as earlier versions of Ethernet or even the early interface bus, GP-IB (IEEE-488-2). Fortunately Ethernet Gigabit is designed for backward compatibility with several earlier Ethernet versions. Also, GP-IB options are frequently available, even on modern test equipment.

    11. Mixed Mode Capability
    12. For more information on this subject, I can recommend a very good reference by Hall and Heck (5).

      Test equipment with coaxial ports are likely to remain dominant for many years. Coaxial connectors are single ended (SE) and the transmission lines which connect to them operate with common mode excitation as the 'return' path is via the ground connection, shield or screen. There are several EMC advantages however in using ports with differential mode excitation (DE) but this does complicate connector interfaces between the associated DUT and other DUTs or test equipment which has single ended ports. Differential mode transmission lines are common on PCBs and many high speed ICs are designed to interface with these because of the EMC advantages. There are various ways in which DUTs with one or more differential mode ports may be tested using test equipment with coaxial ports. So, by mixed mode, we mean ports operating in both common mode (the traditional coaxial interface) and differential mode (using a pair of balanced conductors).

      Good EMC compliant design of RF and microwave equipment includes both screening and grounding features. Screens are normally connected to ground. The ground connections require to be as close to zero impedance as possible at the highest operating frequencies. Therefore, even if a DUT port requires to be balanced (differentially excited), we can substitute two single ended (coaxial) connections for this, provided that they are excited at exactly 180° 'out of phase'. This will be described with some help from Figure 12-1.

      Figure 12-1 Thevenin equivalent circuits based on two 50 Ω source impedance which show how source impedances of 25 Ω and 100 Ω are derived for CM and DM respectively.

      In Figure 12-1 (1), each of the source vector arrows represents the phase of the source at the same instant. (1) shows the Thevenin equivalent circuit for a simple single ended 50 Ω one port source, such as used on a typical VNA, correctly terminated with a 50 Ω load. (2) shows how two sources, each like (1), may be combined both exactly in phase which results in the common mode (CM) equivalent circuit, like (1) but with a 25 Ω source and load impedance. In (3) two sources like (1) are combined exactly 180° out of phase to give the differential mode (DM) equivalent circuit with source and load impedances of 100 Ω. In CM, by definition, the signal return path is via the ground connection, i in (2). For perfect DM, there should not be any signal current via the ground path.

      Starting with a 4 port VNA with installed mixed mode software, a 4 port electronic calibration (ECAL) unit and the necessary cables and adaptors, we may perform a variety of mixed mode S parameter measurements. Some examples are shown in Figure 12-2.

      Figure 12-2 A selection of mixed networks N1 to N7 to consider with several port interfaces, single ended (coaxial), differential (coaxial) and differential (open terminals). Also shown are 4 port VNA configurations for calibration and one measurement. The impedance of every coaxial connection, considered in isolation, is 50 Ω.

      Referring to Figure 12-2, most network DUTs will include a 'ground plane' and some form of screened box as part of good EMC compliant design. Despite this, it is still possible to create differential ports using two (unbalanced) coaxial cables, each with a nominal impedance of 50 Ω. However, each of these must be of electrically identical lengths with perfect differential (180°) excitation. The resulting differential mode impedance would be 100 Ω. If this is not achieved, there will be a finite and unwanted common mode content. Now we will look at the examples in Figure 12-2 in more detail.

      • N1. Port 1: differential, 100 Ω, Port 2: differential, 100 Ω.
      • Each port must be interfaced with a pair of single ended connections with perfect differential excitation. If a VNA is used to provide this, part of the calibration procedure is designed to achieve the necessary excitation at the connectors of the DUT. In most differential DUTs the quality of the differential signal must be accurately maintained through the network to achieve good EMC compliance. Any deviations from this will be shown in the 2 port multimode S parameter matrix, SMM, the format of which is shown in (12-1).

        S MM = S DD11 S DD12 S DC11 S DC12 S DD21 S DD22 S DC21 S DC22 S CD11 S CD12 S CC11 S CC12 S CD21 S CD22 S CC21 S CC22
        (12-1) The mixed mode S parameter matrix for a 2 port network.

        For each element of the SMM matrix, the first 2 letters of the subscript identify the applicable mode: differential (D) or common (C) and the numbers identify the ports, 1 or 2 in this case. These 4 characters represent: response, stimulus, response, stimulus in that order. For example, SCD12 represents a differential mode stimulus applied at port 2 and the common mode response measured at port 1.

      • N2. Port 1: single ended, 50 Ω, Port 2: differential, 100 Ω.
      • This is also known as a balanced to unbalanced transformer (balun). In this case the balanced port comprises open, unscreened terminals such as might be used to feed a twisted pair transmission line or an antenna. The differential impedance at port 2 is not necessarily 100 Ω as with both ports of N1, but would typically be 120 Ω or 300 Ω. The S parameters cannot be measured directly due to the difficulty in interfacing the VNA to port 2.

      • N3. Port 1: differential, 100 Ω, Port 2: single ended, 50 Ω.
      • Apart from the reversed port numbering, this is electrically identical to N2. Again, the S parameters cannot be measured directly, but the balanced ports of N2 and N3 may be connected assuming they are the same impedance to form an equivalent 2 port device with 50 Ω impedances. A crude approximation of the insertion loss of one device would be would be half the log mag S21 value for both devices.

      • N4. Ports 1 to 4: single ended, 50 Ω.
      • This is a 4 port network with every port being single ended coaxial, 50 Ω, so would be represented by a traditional 4×4 S parameter matrix.

      • N5. Ports 1: single ended, 50 Ω, port 2: differential via single ended legs.
      • This is like each of the baluns (N2 and N3) but differs in having connections to each of the balanced legs via a single ended coaxial 50 Ω connector. This may be easier to interface with other similar networks by using 2 electrically identical cables, one for each leg of the differential pair. This may be difficult to achieve high frequencies without any differential delay skew.

      • N6. Port 1 and port 2: single ended 50 Ω.
      • This is a classic 2 port single ended (coaxial unbalanced) network, therefore represented by a 2×2 S parameter matrix.

      • N7. Port 1, port 2 and port 3: single ended 50 Ω.
      • This is a 3 port single ended (coaxial unbalanced) network, therefore represented by a 3×3 S parameter matrix.

    13. External adjustments for High and Low Power Modes
    14. Traditional S parameters are defined for linear conditions, so this usually implies using relatively low powers. However, the input and/or output power levels appropriate for the DUT may require extra amplification and/or attenuation between the VNA and the DUT. Normally it is straightforward to reduce the VNA output power levels, but to increase them would need external amplification. For receiving signals, the VNA includes internal amplification and features such as IF bandwidth reduction to cater for low signal levels, but there is a maximum input power level at each port to avoid internal damage. Therefore external attenuation may be required.

      Most 2 port VNAs in the default full 2 port operating mode, after calibration will alternately send and receive test signals at each of its test ports. When port 1 is sending, port 2 will be receiving and vice versa. Say, for example, there was an external amplifier with its input connected to the VNA port 1, to increase the output power from the VNA port 1, required for a DUT. When the VNA port 1 is sending this might work fine in the forward path and present an appropriate signal level to the DUT. However, when the VNA port 1 is receiving and VNA port 2 is sending, the signal path would be in reverse through, not only the DUT, but the external amplifier. Amplifiers are usually designed to operate with good reverse isolation so it may not be possible to get reliable results for the S12 parameter of the DUT if this requires measurements in reverse through the external amplifier as well.

      To summarise, it is certainly possible to add external components, typically amplifiers or attenuators, in order to measure a particular DUT. However, the effects of these must be de-embedded in some way which may not be straightforward.

    15. Options
    16. Not all options are added expense. Some reduce the cost of the equipment by removing capabilities which may not have been required. However, some options may provide extra facilities relatively cheaply, possibly cheaper than buying another dedicated piece of test equipment.

  25. Why Do You Talk About Spatial Phase? Is It Not Just Phase?
  26. This subject is being prepared, please check back soon.

  27. What Are Random And Systematic Errors?
  28. This subject is being prepared, please check back soon.

  29. How Would I Characterise, Say, a 4 Port DUT Using a 2 Port VNA?
  30. We know that vector network analyzers (VNAs) are expensive investments, especially if they are specified to high frequencies. That is because of the mechanical and electrical precision necessary for the connectors and other components operating at those higher frequencies. Even small imperfections in these cause degraded VSWRs and increase systematic errors which again are worse at the higher frequencies. Systematic errors can, to some extent, be corrected during the calibration procedure but far better overall measurement uncertainty is achieved if these are minimised 'at source'.

    In theory VNAs may be built with many ports but 2 port types are common, 4 port versions are less common and I have not seen one with more than 4 ports. However, we will see shortly that 4 port VNAs are useful for measuring mixed mode S parameters. For many companies, a 4 port VNA would be such a big investment that it would not be cost effective unless they were perhaps implementing a significant contract with many deliverable 4 port devices that required testing.

    If we wish to measure the S parameters of a DUT which has more ports than the VNA we have available, we may choose either of the following methods.

    1. Perform multiple measurements, each with unused ports terminated in low VSWR loads and then process the results to allocate the correct S parameter indices.
    2. Use various switching arrangements to perform the measurements, de-embed the effects of the switches and extra cables and then process the results to allocate the correct S parameter indices.

    Each of these methods is a compromise and requires an analysis of the proposed test method to determine whether it will produce final results with acceptable uncertainties.

    As an example, consider that we wish to measure the S parameters of a 4 port DUT using a 2 port VNA. The test equipment configurations to achieve this are shown in Figure 15-1.

    Figure 15-1. The calibration test equipment configurations necessary to acquire data to determine the S parameters of a 4 port DUT using a 2 port VNA.

    We could achieve this with the following steps.

    1. Design the test configuration using the best possible precision test cables of minimum lengths (A and B in Figure 15-1) and the fewest number of adapters possible.
    2. Perform an uncertainty analysis, including the stand-alone Z0 loads, to determine if the method will yield sufficient accuracy in the final result.
    3. This must account for the effects of the measurement cables used (A and B) in Figure 15-1 and any essential adaptors.

    4. Specify high quality, low VSWR, loads of sufficient quantity suitable for terminating the unused ports of the DUT, as denoted by Z0 in Figure 15-1.
    5. Expressed as return loss values, these should be at least 10 dB better than the return loss values specified for the DUT.

    6. Power up and allow the DUT and VNA to stabilise. Set up and calibrate the VNA, using sufficiently long cables to reach all of the DUT ports and the necessary adapters, preferably using an electronic calibration unit (ECAL), as shown in the bottom left diagram in Figure 15-1.
    7. Choose parameters such as frequency range, incident power levels, IF bandwidth (which sets the noise floor), number of points and smoothing. Perform a 'full 2 port' calibration and save the calibration data, traceable to the specific test cables used. A mechanical calibration kit may be used but this will be a more manual procedure and slow down the calibration process.

    8. Connect the VNA and DUT and loads as shown in diagram (1) in Figure 15-1 and perform the first S parameter measurement.
    9. Save the results electronically to a file, typically a 'Touchstone' 2 port (.s2p) file, with a filename which is traceable to this measurement.

    10. Similarly repeat measurements for each of the other configurations (2) to (6) shown in Figure 15-1, saving each set of results in a unique file for later processing. Perform these measurements as quickly as possible under stable conditions. If the tests are interrupted in any way, return to the start and repeat the full set of tests.

    The output of this procedure will be a set of 6 'Touchstone .s2p' files exported from the VNA. These are industry standard text files which may be edited with a text editor or imported into applications such as Excel® or Matlab® and processed as required. The final S parameter output data for the 4 port DUT in Touchstone format will be a .s4p file. This test method inevitably causes some duplication of measurements. For example, measurements (1), (2) and (4) in Figure 15-1 will each provide theoretically identical data for the DUT reflection at port 1 (S11). There will be some small differences due to imperfect repeatability between measurements. For these, a consistent policy should be adopted, for example keeping the results from (1) and ignoring the others. Similar arguments will apply to the other ports.

    Figure 15-2 represents the transmission signal flow paths internal to the 4 port DUT, via each of its ports 1 to 4, which are represented by its S parameter matrix.

    Figure 15-2. Each node represents one of the ports of a 4 port network (DUT). Each line represents a transmission path from one port to another in both directions which will be defined by one of the S parameters. For example, the line between port 3 and 4 represents S34 and S43.
  31. What Are T Parameters and How Might They Be Useful?
  32. The S parameter matrix for a 2 port network may be symbolically defined of the form shown in (16-1).

    S= S 11 S 12 S 21 S 22
    (16-1) The 2 port S parameter matrix with ports designated 1 and 2.

    Alternatively, for the same network we could have defined the transmission (T) parameter matrix as shown in (16-2) [4].

    T= T 11 T 12 T 21 T 22
    (16-2) The 2 port T parameter matrix with ports designated 1 and 2

    T parameters are often useful for circuit architectures which include cascaded networks. Fortunately, it is straightforward to convert between S and T parameters in either direction as required using simple equations. The S to T and the T to S conversion formulas for 2 port networks are shown in (16-3) and (16-4) respectively [4].

    As we noted in the introduction, one of the reasons why S parameters are so useful is that the S parameter matrix includes elements which may be easily converted to properties often used in RF and microwave engineering such as reflection coefficient, VSWR, return loss, gain and reverse isolation.

    Often however we require to analyse two or more 2 port networks cascaded together, such as those shown schematically in Figure 16-1.

    Figure 16-1. Two cascaded 2 port networks represented by S parameters (1) and T parameters (2).

    Suppose the networks A and B, in the direction of the signal flow, are cascaded together as shown in (1) where (SA) and (SB) are their respective S parameter matrices. This cascade is equivalent to a single 2 port network with S parameters (SAB).

    This type of configuration often occurs where the VNA does not have immediate access with short cables to all of the networks that require measuring. We may know the S parameters of the whole cascade (SAB) and one of the networks, (SA) or (SB), and require the S parameters of the other. Alternatively we may know (SA) and (SB) and require (SAB). These situations are where T parameters are useful.

    S and other parameter matrices including T parameters obey the rules of linear (matrix) algebra and representing the matrix elements as complex numbers to describe magnitude and phase is compliant with these. Note that these include keeping the correct matrix multiplication order. We can use these to help us analyse this configuration.

    Performing a matrix multiplication of (SA) and (SB) will give us an apparently plausible result but unfortunately not what we are looking for, (SAB). However, this procedure does work if we had used T parameters instead as shown in Figure 16-1 (2). As we can easily convert between S and T parameters, we may use the following procedure to find (SAB).

    1. Using a VNA, measure (SA) and (SB) separately.
    2. Convert (SA) and (SB) to their respective T parameter matrices (TA) and (TB) using (16-3).
    3. Multiply (TA) by (TB) in that order by applying the rules of matrix multiplication to obtain the result (TAB).
    4. Convert (TAB) to the S parameter equivalent,(SAB) using (16-4).
    T 11 = 1 S 21 T 12 = S 22 S 21 T 21 = S 11 S 21 T 22 = S 12 S 11 S 22 S 21
    (16-3) The equations required to convert from S parameters to T parameters.
    S 11 = T 21 T 11 S 12 = T 22 T 21 T 12 T 11 S 21 = 1 T 11 S 22 = T 12 T 11
    (16-4) The equations required to convert from T parameters to S parameters.

    For most networks, especially active devices, there will be one S parameter matrix for each set of fixed conditions such as frequency, temperature and DC voltage. So, for a typical measurement session, many such conversion calculations like those we have discussed will be required. These may provide an opportunity to enlarge the matrix dimensions starting with frequency.

    Taking this a stage further, a very common requirement is to measure the S parameters of a device (DUT) at a range of ambient temperatures which deviate from normal room temperature. The DUT needs to be installed in a thermal chamber, powered up and then its S parameter performance measured across a range of temperatures as required. The VNA would normally be positioned external to the chamber but close to it and the connections to the DUT made via relatively long cables passing through suitable insulated chamber apertures. Other cables necessary for the DUT operation would be similarly routed via apertures. Such a configuration would effectively comprise 3 cascaded 2 port devices as a whole connected to the VNA: the input cables, the DUT and the output cables. We will name the associated S parameters (SA), (SB) and (SC) respectively. These are shown schematically with associated matrix equations in Figure 16-2.

    Figure 16-2 3 cascaded 2 port networks represented by S parameters and T parameters.

    As with the simpler 2 network cascade, we cannot readily remove or de-embed the effects of the input and output cables using their S parameters directly. Initially, we need to measure the S parameters of the input and output cables separately and then convert each set of these measurements to T parameters using the S/T conversion equations (16-3) and (16-4). Referring to Figure 16-2 this procedure would provide T parameters (TA) and (TC) for the input and output cables respectively. Then the S parameters of the whole cascade (SABC) would be measured and also converted to T parameters to give (TABC). Using the rules of linear algebra for matrix equations the T parameters of the DUT (TB) may then be calculated using the expression for (TB) which is included in (16-5). This may be converted back to the corresponding S parameters using the conversion equation (16-4).

    ( T A )=( T ABC ) T C 1 T B 1 T B = T A 1 T ABC T C 1 T C = T B 1 T A 1 T ABC
    (16-5) The matrix equations for extracting the various T parameters shown in Figure 16-2

    (16-5) also shows matrix formulas for the other T parameters which may be used in a similar way.

  33. How Would I Extract Group Delay Data from S parameter Matrices?
  34. For a 2 port network, the group delay (τ) is defined as the rate of change of phase with respect to frequency with a coefficient of -1 (17-1).

    τ= dϕ dω = 1 360 dθ df
    (17-1) Group delay τ (s)

    Where:

    Taking the example of a 2 port network with input and output ports named 1 and 2 respectively, S21 is the forward transmission through the network in magnitude and phase. Therefore, if S21 phase only is measured across frequency with appropriate frequency steps, group delay may be calculated from the slope using either equation in (17-1) appropriate to the phase and frequency units used.

    A plot of the type shown in Figure 17-1 is a good starting point for analysing the various forms of group delay.

    Figure 17-1. S21 transmission phase (unwrapped) against frequency for a 1 m transmission line (about λ/2) with fully enclosed dielectric (eg. coaxial) of relative permittivity (εr =2.1).

    This represents an ideal (dispersion and loss free) transmission line, close to what we should see from a high quality coaxial cable at these frequencies. The length is 1 m and the relative permittivity (dielectric constant) is 2.1 (PTFE). The frequency range and cable characteristics were chosen to demonstrate the interesting group delay properties under investigation and is therefore is 'unwrapped' for clarity. Most VNAs default to display wrapped phase but this can be converted to unwrapped form if it is required. However, the accuracy with which this can be used for extrapolation degrades with higher frequencies and longer cables which therefore contain 'many wavelengths'. In this case the cable length is only about λ/2 at 100 MHz and the spatial phase constant (β=2π/λ) is about π rad/m.

    At these low frequencies the phase frequency slope of this cable should remain constant down to zero frequency or DC. This may be verified by extrapolating Figure 17-1 with some simple geometry. By examining the plot, the slope is approximately 1.73 °/MHz. Substituting this value into (17-1) and doing the necessary unit conversions yields a group delay of τ =4.8 ns. In this 'close to ideal' case the group delay is actually constant and equivalent to the absolute delay of the transmission line. To verify this, the propagation velocity through the cable would be c/√(εr)=2.07×108 m/s. The length is 1 m so the absolute delay is 1/2.07×108 =4.8 ns.

    To take a more of a real world example, Figure 17-2 is a plot of log magnitude transmission against frequency for a LC bandpass filter which is not yet properly tuned.

    Figure 17-2. Log magnitude S21 transmission against frequency for a (poorly tuned) LC bandpass filter for phase frequency and group delay.

    This filter was designed aiming at an Elliptic architecture so the poles and zeros were formed using series and parallel resonant circuits. The reactive components used for resonant circuits cause all sorts of interesting phenomena when we examine the transmission phase against frequency as we may see in Figure 17-3.

    Figure 17-3. Unwrapped S21 transmission phase against frequency for the bandpass filter featured in Figure 17-2.

    In the passband range of approximately 60 MHz to 80 MHz the phase frequency slope is fairly constant at about 13.1 °/MHz. By again substituting this into (17-1) and doing the necessary unit conversions the group delay around these frequencies comes out to about 36.4 ns. Figure 17-4 shows a group delay against frequency plot of the same filter.

    This plot was obtained from the same S parameter data used for the other plots in this section, the group delay being calculated numerically using Equation 17-1.

    Figure 17-4. Group delay against frequency for the bandpass filter featured in Figure 17-2.

    This is only effective if there are sufficient points to get a reasonable approximation to dθ/df, in this 101.

References

  1. Scattering Parameters (Wikipedia article).
  2. Transmission Lines for Communications with CAD Programs, Second Edition
  3. Davidson, C. W.; Macmillan Education Ltd.; pp. 85-88. (ISBN 0-333-47398-1)

  4. Microwave Engineering, Third Edition
  5. Pozar, David M.; John Wiley & Sons, Inc.; pp. 174-183. (ISBN 0-471-44878-8)

  6. Microwave Transistor Amplifiers, Analysis and Design, 2nd. Ed.
  7. Guillermo Gonzalez; Prentice Hall, New Jersey; p 25. (ISBN 0-130254335-4)

  8. Advanced Signal Integrity for High-Speed Digital Designs
  9. Hall, Stephen H., Heck, Howard L.; John Wiley & Sons Inc., New Jersey, USA; pp. 400-406 (ISBN 978-0-470-19235-1).