Probe Set Investigation

Import Measurement and Simulation data

# Define function to extract measurements data
import csv
import numpy as np
def read_s1p(filename):
    """
    Arguments:
    filename - filename including directory to measurement file.
    
    Returns
    freq - Measured frequency in Hz
    s11 - Measured scattering parameters in linear complex form 
    """
    freq = []
    s11 = []
    with open(filename, 'r') as csvfile:
        reader = csv.reader(csvfile, delimiter=' ', quotechar='|')
        for row in reader:
            row = list(filter(None, row))
            if len(row) == 3:
                freq.append(float(row[0]))
                s11.append(float(row[1]) + 1j*float(row[2]))
        freq = np.array(freq)
        s11 = np.array(s11)
        return freq, s11

Lets import all of our measurements and simulations using the read_s1p function

freq, s11_open = read_s1p("measurements/open.s1p")
freq, s11_short = read_s1p("measurements/short.s1p")
freq_cst, s11_open_cst = read_s1p("measurements/cst_open.s1p")
freq_cst, s11_short_cst = read_s1p("measurements/cst_short.s1p")
freq_cst, s11_open_cst_shifted = read_s1p("measurements/cst_open_shifted.s1p")
# Convert the CST frequency from GHz to Hz
freq_cst = freq_cst*1e9

Lets compare our probe-set measurements to their ideal simulation using CST Microwave Studio

import matplotlib.pyplot as plt

# Define S11 plotting function
def plt_s11(freq, s11):    
    plt.subplot(1, 2, 1)
    plt.plot(freq/1e6, 20*np.log10(np.abs(s11)))
    plt.plot(freq/1e6, 20*np.log10(np.abs(s11)))
    plt.xlim(freq[0]/1e6, freq[-1]/1e6)
    plt.ylim(-1,0.1)
    plt.xlabel("Freq [MHz]")
    plt.ylabel("S11 [dB]")
    plt.grid(True)
    plt.subplot(1, 2, 2)
    plt.plot(freq/1e6, np.angle(s11)*180/np.pi)
    plt.plot(freq/1e6, np.angle(s11)*180/np.pi)
    plt.xlim(freq[0]/1e6, freq[-1]/1e6)
    plt.ylim(-180,180)
    plt.xlabel("Freq [MHz]")
    plt.ylabel("S11 [Deg]")
    plt.grid(True)
    
# Plot Open
fig = plt.figure(figsize=(20, 4))
fig.suptitle("Open", fontsize=16)
plt_s11(freq, s11_open)
plt_s11(freq_cst, s11_open_cst)

# Plot Short
fig = plt.figure(figsize=(20, 4))
fig.suptitle("Short", fontsize=16)
plt_s11(freq, s11_short)
plt_s11(freq_cst, s11_short_cst)

plt.show()

The phase of both simulations match up well with their measurements. However, the short does have a substantial amount of loss across the entire band. This is most likely due to the copper tape that was used to create the short.

Shifting the Reference Plane to the Probe Interface

Using the open probe, which matched well with the simulation, we will add a phase offset to shift the reference plane from the connector to the probe interface.

l = 7.49e-3 # Distance from connector to probe interface
eps_r_teflon = 2.1 # Permittivity of teflon in coax
wavelength = 3e8/(freq)
vf = 1/np.sqrt(eps_r_teflon)
phase_offset = 2*np.pi*((2*l)/wavelength)/vf

# Apply the phase offset to all frequencies
s11_open_ref = s11_open*np.exp(1j*phase_offset)

# Plot the result compared to the simulated phase shifted result
fig = plt.figure(figsize=(20, 4))
fig.suptitle("Open Phase Shifted", fontsize=16)
plt_s11(freq, s11_open_ref)
plt_s11(freq_cst, s11_open_cst_shifted)
plt.ylim(-5,5)
plt.show()

Compared with the simulation the phase is about 1 degree off at the highest frequencies. This translates to geometrical errors in the submilimeter range. Polishing the end of the probe might improve this result.

Probe Capacitance and Radiating Conductance

Using the phase shifted measurement, we can calculate the probe calibration parameters.

e_c = 1 - 1j*0 # Permittfivity of a vacuum

# Define some parameters
w = 2*np.pi*freq
Z0 = 50
Y0 = 1/Z0

# Calculate the admittance of the air
Y = Y0*(1-s11_open_ref)/(1+s11_open_ref)

# Calculate the probe parameters
# Calculate A and B terms
A = 1j*w*e_c
B = np.power(e_c,5/2)

# Calculate the probe parameters
G_0 = (np.real(A)*np.imag(Y) - np.real(Y)*np.imag(A))/(np.real(A)*np.imag(B) - np.real(B)*np.imag(A))
C_0 = (np.imag(Y) - np.imag(B)*G_0)/np.imag(A)

# Plot the probe parameters 
fig = plt.figure(figsize=(20, 4))
plt.subplot(1, 2, 1)
plt.plot(freq/1e6, G_0)
plt.xlim(freq[0]/1e6, freq[-1]/1e6)
plt.xlabel("Freq [MHz]")
plt.ylabel("$G_0$")
plt.title("Radiating Conductance")
plt.grid(True)
plt.subplot(1, 2, 2)
plt.plot(freq/1e6, C_0)
plt.xlim(freq[0]/1e6, freq[-1]/1e6)
plt.xlabel("Freq [MHz]")
plt.ylabel("$C_0$")
plt.title("Fringing Capacitance")
plt.grid(True)
plt.show()
print("Fringing capacitance: "+str(np.average(C_0[1:])))
print("Radiating Conductance: "+str(np.average(G_0[1:])))

Fringing capacitance: 1.64635151707e-14
Radiating Conductance: 5.35396130766e-06

Measurement:
Fringing capacitance: 1.64773857969e-14
Radiating Conductance: 5.34207558123e-06

Simulation:
Fringing capacitance: 3.27689464957e-14
Radiating Conductance: 8.31421262357e-07

We find that only 1 degree in phase gives us double the fringing capacitance and almost 10 times less conductance. We are also ignoring losses in the short piece of coax transmission line in the probe. As a result, we really do need a good reference material to calibrate the probe. Just using a simulation and callipers will not be enough.

Adding dielectric materials to electromagnetic simulations can be a pain if the correct dielectric properties are not known. To solve this I decided to build a poor-mans dielectric probe. The probe consists of a coaxial interface that is placed flush with a material under test. From a single reflection measurement of your material and a reference material, the complex permittivity can be calculated. As a first step, I simulated a probe with some dielectric materials and used the data to write and test my permittivity extraction code shown below. The procedure is common and can be found in multiple scientific papers.

The hardest part of the extraction is solving the 5th order polynomial. Because it was late in the evening, I opted for the brute force route and wrote a simple gradient decent algorithm to solve the complex permittivity.

The next step is to create the probe with some calibration standards and see if it works in practice.

Set measured values and ref material parameters

# Measured Distilled Water
S11 = -0.097844 - 1j*0.99137
e_c = 78.4 + 1j*9.9762e-5

# Measured Teflon
S11_mut = 0.9983-1j*0.05831

Calulate admittance using S11

f = 1e9
w = 2*np.pi*f
Z0 = 50

Y0 = 1/Z0
Y = Y0*(1-S11)/(1+S11)
print("Admittance of distilled water: "+str(Y))

Admittance of distilled water: (8.473395759728993e-05+0.022070908693777157j)

Calculate probe capacitance parameters using distilled water measurement, $\epsilon_c$ is known.

\begin{equation*}
Y = j \omega \epsilon_c C_0 + \sqrt{\epsilon^5_c} G_0
\end{equation*}

To keep things clean, lets define:
\begin{equation*}
A = j \omega \epsilon_c \\\\
B = \sqrt{\epsilon^5_c}
\end{equation*}

Then we can calculate the probe parameters using:
\begin{equation*}
G_0 = \frac{Re(A)Im(Y) – Re(Y)Im(A)}{Re(A)Im(B) – Re(B)Im(A)}
\end{equation*}
\begin{equation*}
C_0 = \frac{Im(Y) – Im(B) G_0}{Im(A)}
\end{equation*}

import numpy as np

# Calculate A and B terms
A = 1j*w*e_c
B = np.power(e_c,5/2)

# Calculate the probe parameters
G_0 = (np.real(A)*np.imag(Y) - np.real(Y)*np.imag(A))/(np.real(A)*np.imag(B) - np.real(B)*np.imag(A))
C_0 = (np.imag(Y) - np.imag(B)*G_0)/np.imag(A)

print("C0: "+str(C0))
print("G0: "+str(G0))

C0: 4.48047731779e-14
G0: 1.55692371696e-09

Calculate admittance of MUT

Y_mut = Y0*(1-S11_mut)/(1+S11_mut)
print("Admittance of MUT: "+str(Y_mut))

Admittance of MUT: (-1.474302070865214e-08+0.0005835956264497135j)

Calculate the permittivity of the MUT using the probe parameters calculated earlier.

The equation is a 5th order polynomial which does not have any easy solutions. In order to save time, a simple gradient decent algorithm has been implemented to find the complex permittivity.

D = 1j*w*C_0

# Use gradient decent on this 2D space to calculate the complex permittivity
# Initialise permittivity (this should be your best guess)
e_mut = 1 + 0*1j
# Start loop
loss = [np.abs((D*e_mut + np.power(e_mut, 5/2)*G_0 - Y_mut))]
a = loss[-1]
count = 0
while loss[-1] > 1e-12:
    # Calculate derivatives
    rp = np.abs((D*(e_mut + a) + np.power(e_mut + a, 5/2)*G_0 - Y_mut))
    rn = np.abs((D*(e_mut - a) + np.power(e_mut - a, 5/2)*G_0 - Y_mut))
    ip = np.abs((D*(e_mut + a*1j) + np.power(e_mut + a*1j, 5/2)*G_0 - Y_mut))
    ine = np.abs((D*(e_mut - a*1j) + np.power(e_mut - a*1j, 5/2)*G_0 - Y_mut))

    # Calculate next point
    directions = np.array([rp, rn, ip, ine])
    steps = np.array([a, -a, a*1j, - a*1j])
    step = steps[directions - np.min(directions) == 0]
    e_mut = e_mut + step
    e_mut = e_mut[0]
    
    loss.append(np.abs((D*e_mut + np.power(e_mut, 5/2)*G_0 - Y_mut)))  
    a = 1000*np.abs(loss[-1])
    count += 1
print("Loss: "+str(loss[-1]))  
print("Iterations:"+str(count)) 
print("")    
print("E' : "+str(np.real(e_mut)))  
print("E'': "+str(np.imag(e_mut))) 
print("Loss tangent: "+str(np.imag(e_mut)/np.real(e_mut)))
print("Conductivity: "+str(np.imag(e_mut)*w)+str("S/m")) 

plt.figure()
plt.plot(loss[1:])
plt.show()

Loss: 9.08241678757e-13
Iterations:76

E' : 2.0730409503586182
E'': 8.659979628088184e-05
Loss tangent: 4.17742815287e-05
Conductivity: 544122.567597S/m