MWTOOLBOX Reference

The main module used for processing RF network data is rfnetwork. First step is importing the library by:

from mwtoolbox import rfnetwork

SPFILE is the main class in rfnetwork module that is used to define and process multiport RF network data. Its main target is to enable calculations on RF network data in a python script and therefore enable automation of these kind of tasks.

The most usual way to create an RF network object is reading an s-parameter file.

snet = rfnetwork.spfile("amplifier.s2p")

Some parameters of the network can be accessed in following ways:

snet.freqs # frequency points of the network in Hz
snet.S(i,j) # S(i,j)
snet.Y(i,j) # Y(i,j)
snet.Z(i,j) # Z(i,j)
snet.T(i,j) # T(i,j)
snet.ABCD(i,j) # ABCD(i,j)

All these functions return a numpy array and take two optional parameters, data_format and freqs.

snet.S(i,j,data_format="complex",freqs=None)

data_format is the format of the returned data. By default, complex data is returned. Other supported formats are (case insensitive):

• K: Stability factor of 2-port
  
• MU1: Input stability factor of 2-port
  
• MU2: Output stability factor of 2-port
  
• VSWR: VSWR at port i
  
• MAG: Magnitude
  
• DB: Magnitude in dB (20log)
  
• REAL: Real part
  
• IMAG: Imaginary part
  
• PHASE: Phase of in degrees between 0-360
  
• UPHASE: Unwrapped phase in degrees
  
• GDELAY: Group delay in seconds
  

freqs is the frequency points of the returned data. By default, it is the current frequency points of the network.

Frequency points of the network can be changed by interpolation using the command:

snet.set_frequency_points(newfreqs)

where newfreqs is the new frequency points as a list or numpy array.
Or frequencies from the beginning or end of the frequency list can be cropped with:

snet.crop_with_frequency(fstart=fs1, fstop=fs2)

fstart is optional. If given, frequencies below fs1 are cropped out. fstop is also optional and if given, frequencies above fs2 are cropped out.

S-parameter renormalization with different reference impedance can be done with the following command:

snet.change_ref_impedance(impedances)

If impedances parameter is a

• number, it will be the new reference impedance for all ports and frequencies.
  
• list or numpy array, its size should be equal to number of ports and each element is assumed to be reference impedance for each port. If this reference impedance is a:
  
  • number, it is used for all frequencies and reference impedance is frequency independent.
    
  • list or numpy array, its size should be the same as the number of frequency points. Each element of the list is used for corresponding frequency.
    

For complex reference impedances, different S-Parameter definitions are possible. Three of them are supported in the library (3.1.1). By default, power-waves are used.

Reconfiguration of ports can be done through snp2smp method. With this method, ports can be swapped or some ports can be removed from the network.

snet.snp2smp(newports)

where newports is the new ports list. This means port j of new network is newports[j] port of the original network. Unused ports of the original network are terminated with current reference impedances of those ports.

Two-port SPFILE network objects can be cascaded simply using "+" operator. At the following example, port-1 of snet2 is connected to port-1 of snet1 and snet becomes the new two-port network between port-1 of snet1 and port-2 of snet2. snet1 and snet2 are not modified.

snet = snet1+snet2

Similarly, de-embedding of two-port SPFILE network objects can be done using "-" operator. Following the previous example, we can obtain snet1 again by de-embedding snet2 from snet, or obtain snet2 again by de-embedding snet1 from snet.

snet1 = snet-snet2

snet2 = -snet1+snet

All functions which modify the SPFILE object, returns the modified object and takes inplace parameter as the last argument. If this parameter is;

• 0, the object which calls this method is NOT modified but the modified object is returned from the method.
  
• -1 (default value), the inplace attribute of the calling object is used. Default value for new objects is 1.
  
• any other value, the object calling this method is modified.
  

snet_new = snet.snp2smp([2,1], inplace=0) # snet is not modified. snet_new is the new object with new port configuration.
snet.snp2smp([2,1], inplace=1) # snet is modified and returned.

TRL extraction can be done using the function =trl_launcher_extraction= function in rfnetwork module. Inputs are the data for thru, line and reflect measurements. These data can be in the form of s-parameter file name or SPFILE object. 4th parameter is the type of reflect standard ("short" or "open").

Return value is a 3-tuple. First 2 items are SPFILE objects of input and output launchers (1st port for both output is the external port). 3rd item is \(\gamma L\) for the interconnecting line of the line standard. Frequency points for all outputs are the same.

sp1,sp2,gL= rfnetwork.trl_launcher_extraction(thru, line, ref, refstd="short")

In literature there are different types of definitions for \(a\) and \(b\) signals of S-Parameter definition. In this library 3-types are supported; Power Waves, Pseudo-Waves and HFSS Pseudo-Waves. Power Waves is the default definition used. The definition used is set by formulation parameter:

S-Parameter Formulations:

1. Power-Waves
   
2. Pseudo-Waves
   
3. HFSS Pseudo-Waves
   

The following equations are used to renormalize S-Parameters to a different set of reference impedances (Reveyrand 2018) .

Let:

where

• \(Z_1\), \(Z_2\) are the reference impedances for each port.
  
• \(diag{}\) a matrix of which only diagonal elements are non-zero and given by the list in brackets.
  
• \(F\) depends on the S-Parameter formulation used and given as in the related section.
  
• \(A^{*}\) The complex conjugate of matrix A.
  

Then the renormalization formulae are;

T-Matrix definition used in the library is:

For a reciprocal network;

Different definitions of gain for a 2-Port network (Pozar 2005).

Transducer gain;

Available power gain;

Operating Power gain;

Maximum available gain;

For stability, K>1 and \(| \Delta | < 1\) where (Orfanidis 2016);

\(\mu\) parameters (Edwards-Sinsky stability parameter) enable to identify the source of instability as source or load. For unconditional stability, \(\mu_1 > 1\) and \(\mu_2 > 1\).

For simultaneous conjugate match at both ports of a 2-port network, source and load reflection coefficients can be calculated as (Orfanidis 2016):

where (-) should be used when \(B_1 > 0\) or \(B_2 > 0\), (+) otherwise.

Extracting \(\Gamma_L\) at port 2, if \(\Gamma_{{in}}\) of the circuit at port 1 is given.

References used for this section are (Wedge and Rutledge 1993) and (Jahn et al., n.d.) (Section 4.3 Computing with S-Parameters).

Two networks defined by SPFILE objects can be connected together to create a new network and SPFILE object. Assume that we have networks \(S\) and \(T\). We want to create a new network by connecting port \(k\)
of network \(S\) to port \(l\) of circuit \(T\). Then S-Parameters of new network can be calculated as;

with \(i\) and \(j\) being ports of \(S\), and;

with \(m\) being a port of \(T\). With these formulae the new S-Parameter network is built.

If ports \(l\) and \(k\) of the same circuit \(S\) are connected and both ports are removed, the new S-parameters are calculated by:

If number of measurement ports is equal to number of die ports, unique solution is obtained. If there are unmeasured non-die ports, their termination impedance affects the result too. Ports are sorted first. \(b_1\) and \(a_1\) are signals of measurement ports. \(b_2\) and \(a_2\) are signals of chip ports (but \(b\) and \(a\) switched places compared to standard definition). Then extracted chip S-Parameters are derived as:

Everything in this equation can also be matrices.

A good summary of passivity of S-Parameters is given in (S-Parameter Quality Metrics, Yuriy Shlepnev, 2014).

\(P_{net} = \sum (|a_n|^2 - |b_n|^2) = a^H (I - S^H S) a\) should be \(\geqslant 0\) for all possible \(a\). This is a matrix form called Hermitian Quadratic Form. To satisfy this condition, following condition should be met:

where \(S^H\) is conjugate transpose of \(S\).

These conditions should be met from DC to infinite frequency which can only be met approximately by S-Parameter files.

Passivity is enforced with the algorithm given in (Doshi, Sureka, and Pupalaikis 2012). Suppose that \(k\) eigenvalues are larger than 1 and violates passivity criteria.

Procedure:

• Perform SVD (singular value decomposition) of the matrix
  

• Let \(\Delta \Sigma\) be a diagonal matrix of size n, with the diagonal entries defined as
  

• Now form the matrix \(\Delta S\) such that
  

• Thus passivity-enforced s-parameter matrix is given as:
  

Single-Ended Ports: 1,2, Even Port:3, Odd Port: 4

If the single ended S-Parameters to be converted are referenced to 50\(\Omega\), then in the final S-Parameters including common and differential ports, differential port impedance is 100\(\Omega\) and common mode impedance is 25\(\Omega\).

The same equation is used to convert single-ended a and b to mixed-mode counterparts.

where

\(F\) and \(Z_r\) can be interchanged since they are both diagonal. So,

Similarly,

For real characteristic impedances,

Introduction.

According to https://en.wikipedia.org/wiki/Skin_effect:

Wheeler's Incremental Inductance Rule is frequently used to analyse conductor loss in transmission lines. The idea is that at high frequencies, for typical conductors, surface resistance is equal to surface reactance. Surface resistance is calculated by calculating surface reactance. Surface reactance is calculated by calculating inductance change of a lossless transmission line after shifting the boundary of conductor by \(\delta / 2\). The formula is:

where \(L_z\) is inductance per unit length, \(x\) is offset from the conductor surface, \(\delta\) is skin depth, \(Z_0\) is characteristic impedance, \(\mu\) is magnetic permeability of conductor. Important notes to keep in mind for derivation:

1- \(\frac{\mu}{\mu_0}\) coeffcieint come from the fact that at the boundary, right inside the conductor, magnetic field is \(\mu_r = \frac{\mu}{\mu_0}\) times stronger than the external magnetic field right outside the conductor.

2- Magnetic field in the conductor is assumed to be uniformly distributed to a thickness of \(\delta / 2\).

In terms of characteristic impedance, following formula can also be used:

where \(c_0\) is speed of light in vacuum and \(\varepsilon_{{eff}}\) is relative effective dielectric permittivity of the transmission line.

In this formula, during calculation of \(\frac{\partial Z}{\partial x}\) the same transmission line without dielectric filling can also be used with \(\varepsilon_{{eff}} = 1\). Mostly, inductance does not depend on the
dielectric filling if it is non-magnetic.

We need to multiply this with 8.68 (\(20 \log (e)\)) to convert the unit from Np/m to dB/m.

The S-Parameters of a block to deembed a 2-port S-Parameter is:

The derivation of formulae for extraction of transition parameters from THRU and LINE measurements taken by connecting the same transition back to back, is being given. The transition is assumed to be reciprocal. Assume that the T-Matrix(definition also used in the mwtoolbox package) of the transition is:

Because of reciprocity, the T-Matrix of reversed transition is:

The T-Matrix of connecting line in LINE measurement is:

T-Matrix for LINE measurement (assuming \(\alpha = e^{- \gamma l}\)):

T-Matrix for THRU measurement (assuming \(\alpha = e^{- \gamma l}\)):

For reciprocal networks,

So,

Let

Since we will have quadratic equations, it is best to target \(b\) variable first, since we know that its magnitude should be greater than 1. If we can solve \(b\), then

With these formulae, the calculation of all cells of T-Matrix can go in this order: \(b \Rightarrow T_{21} \Rightarrow a \Rightarrow T_{11} \Rightarrow T_{22} \Rightarrow T_{11}\). Using these equations and after manipulating matrix equations the quadratic equation for \(b\) is obtained as:

This method is implemented in rfnetwork module with function thru_line_deembedding.

This method is used to measure the performance of a 2-port structure with only 1-port measurements. The block diagram is shown in Figure 3. Reflection coefficient at port-1 of DUT while port-2 is terminated with a load impedance is given as;

where \(\Gamma_{L}\) is the reflection coefficient of load and \(S_{xx}\) are S-parameters of the DUT. This reflection coefficient is measured for 3 different load impedances. By this way, 3 equations are obtained to relate input reflection coefficients (\(\Gamma_{in}^{1,2,3}\)) to load reflection coefficients (\(\Gamma_{L}^{1,2,3}\)). These 3 equations are converted to a single matrix equation to solve for S-parameters.

In this formulation, \(S_{11}\), \(S_{22}\) and \(\Delta=S_{11}S_{22}-S_{12}S_{21}\) are calculated directly. \(S_{21}\) is calculated from \(\Delta\). There are 2 limitations of this technique:

1. Reciprocity is assumed, so \(S_{12}=S_{21}\) which is usually the case.
   
2. \(S_{21}\) can be calculated only with a sign (+/-) ambiguity.
   

This technique can be used to measure microstrip-WG transition. WG side can be terminated with the following loads from WR12 calibration kit, while the measurements are done from PCB side using microstrip probe launchers.

1. Short
   
2. Delay+Short (to implement a case close to open)
   
3. Termination
   

After the extraction, probe launcher should also be deembedded. For this purpose, PCB should have calibration structures to characterize launcher data. Launcher data can be extracted using TRL extraction. The tolerances on launcher data are also a source of error.