Motional Inductance Calculation from Reactance Slope

      Wes Hayward, w7zoi,  1 August 2009  (converted to html on 26dec09)

The motional inductance of a quartz crystal can be calculated from a Smith Chart sweep of the crystal that is observed with a VNA (vector network analyzer.)     If the VNA is the popular N2PK design, the output file from the program Refl.exe can be read with as a comma delimited text file.   The data can then be extracted for calculation, or for plotting.   In my case, I have built a routine in a Windows version of MicroSmith that will read the files directly.  

This version of MicroSmith is not available at this time.  However, other linear simulation programs can be used.    Motional L can be calculated from the raw data even if no Smith Chart program is used.    The data from the N2PK program Refl.exe is extensive; I used the columns showing the reflection coefficient, which is given as |rho| and the related angle.    This can be solved for impedance, Z:

    
This conversion is of course done directly in the Smith Chart programs.  A calculation is shown below.

 
Fig 1.  A 5 MHz crystal with Lm=0.1 H and C0=4 pF is analyzed in MicroSmith08. The sweep is over a 1 kHz range while the analysis frequency (little blue square) data is presented in the upper left.

Note the reactance at 5 MHz is 0.46185 Ohms.   If we back step by 5 Hz, we see the next figure:
 
Fig 2.  The same analysis at a frequency 5 Hz below 5 MHz.

The reactance in the lower figure is -5.817 Ohms.   The difference between the two is 6.27885 Ohms.   So, the reactance slope is 1.25577 Ohms per Hz.

It can be shown that         where Lm is the motional inductance and the reactance slope is approximated by  ΔX/ΔF    .   For this case, Lm is 0.09993, which rounds to the original value used in the simulation.

Shown below is the same calculation done in the original DOS version of MicroSmith.
 
Fig 3.   The same calculation can be done with the original version of MicroSmith.  VNA data can be imported into this program, although it will take some work on the files from the N2PK VNA.

The next figures will show some of the data we saw.  Fig 4 shows the 5 MHz crystal when operated on the fundamental at 5 MHz.  
 
Fig 4.   VNA data displayed on the SC.   The data in the calculation box (upper left) is at a frequency of 4.99858 MHz.    Reactance slope calculations showed Lm=0.1017 H.  The program that came with the VNA (xtal2.exe) extracted .09836 H.  

The next figure shows the 5 MHz crystal operated at the third overtone near 15 MHz.  

  Fig 5.  Operation of the 5 MHz crystal at the 3rd overtone.   Calculations showed Lm=0.7677 H.  The circle near the right side of the chart indicates a larger series resistance of 123 Ω.

A 4 MHz crystal was also investigated.   The following shows that crystal at the 3rd overtone.  The fundamental operation was much like Fig 4.

  Fig 6.  Third overtone operation of a 4 MHz crystal.   Data is at 11.96983 MHz.   The series resistance is very high at 592 Ohms.    Although Lm was high at 0.628 H, the Q was 85K, which was less than the fundamental mode Q of 155K.