In this notebook, we try to deduce the equivalent circuit of a WEST ICRH capacitor which has been simulated under HFSS.
In [12]:
%matplotlib inline
import skrf as rf
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
rf.stylely()
First we import all the scattering parameters of the capacitor simulated at different positions. S-parameters are imported as skrf networks in a NetworkSet object. This NetworkSet will allow interpolating the capacitors are intermediate values of the $D_{cylinders}$ values, by interpolating the scattering parameters between individual Networks.
In [14]:
capas = rf.read_all('S_Matrices/', f_unit='MHz')
capas_set = rf.NetworkSet(capas)
f_band = '35-65MHz'
f = capas_set[0].f
omega = 2*np.pi*f
D_cylinders = [110, 100, 90, 80, 77, 76]
print(f'{len(capas)} s-parameter files representing {len(D_cylinders)} values of D_cyclinders')
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# check the data
capas_set[f_band].plot_s_re(m=1, n=0)
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In [5]:
idx = (f > 40e6) & (f < 60e6)
In [41]:
# dummy transmission line to match port 3
coax = rf.media.Coaxial(frequency=capas_set[0].frequency)
In [158]:
def Z1(omega, R, C, L):
'''
Resistor R in series with a capacitance C and an inductance L
R in Ohm, C in pF and L in nH
'''
return (R +1j*(- 1/(C*1e-12*omega) + L*1e-9*omega))
def Z1_re(omega, R, C, L):
return np.real(Z1(omega, R, C, L))
def Z1_im(omega, R, C, L):
return np.imag(Z1(omega, R, C, L))
def Z3(omega, Cshunt):
'shung capacitance. C_shunt in pF'
return -1j/(Cshunt*1e-12*omega)
def Z3_im(omega, Cshunt):
return np.imag(Z3(omega, Cshunt))
In [159]:
Z1s, Z2s, Z3s = {}, {}, {}
Cs, Rs, Ls, C_shunts = [], [], [], []
# For each network in the network set, extract the equivalent network parameters
for (ntw_name, ntw) in capas.items():
# connect the port 3 (voltage probe) to match load to make a 2-port network
# (since equivalent T network is only for a 2-ports network)
ntw = rf.connect(ntw, 2, coax.match(), 0)
# extract impedances
Z3s[ntw_name] = 1/ntw.a[:,1,0]
Z1s[ntw_name] = ntw.a[:,0,1]
Z2s[ntw_name] = 0
# fit the equivalent impedances values to the equivalent circuit elements R,l,Cs
(C_shunt,), cov = curve_fit(Z3_im, omega[idx], np.imag(Z3s[ntw_name][idx]))
(_, C, L), cov = curve_fit(Z1_im, omega[idx], np.imag(Z1s[ntw_name][idx]))
(R, _, _), cov = curve_fit(Z1_re, omega[idx], np.real(Z1s[ntw_name][idx]))
print(f'{ntw_name}: C={C:.1f} pF, L={L:.1f} nH, C_shunt={C_shunt:.1f} pF, R={R:0.1e} Ohm')
Cs.append(C)
Rs.append(R)
Ls.append(L)
C_shunts.append(C_shunt)
In [160]:
fig, ax = plt.subplots()
ax.plot(D_cylinders, Cs, '.', ms=10)
ax.set_xlabel('D_cylinders')
ax.set_ylabel('Capacitance [pF]')
Out[160]:
The equivalent circuit parameters R,L can be set as constant:
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R_eq = np.mean(Rs)
L_eq = np.mean(Ls)
C_shunt_eq = np.mean(C_shunts)
Now to test the performance of the equivalent model, we create a Network from the equivalent circuit and we compare S-parameter values to the ones of the NetworkSet :
In [169]:
def equivalent_capa_ntw(omega, C, R=R_eq, L=L_eq, C_shunt=C_shunt_eq):
_Z1 = Z1(omega, R, C, L)
_Z3 = Z3(omega, Cshunt=C_shunt)
A11 = 1 + _Z1/_Z3
A12 = _Z1
A21 = 1/_Z3
A22 = np.ones_like(omega)
ntw = rf.Network()
ntw.f_unit = 'Hz'
ntw.f = omega/(2*np.pi)
ntw.s = rf.a2s(np.array([[A11, A21], [A12, A22]]).T)
return ntw
In [176]:
eq_capa = equivalent_capa_ntw(omega, C=100, C_shunt=50)
fig, ax = plt.subplots()
capas_set[-2].plot_s_db(m=0, n=0, ls='--', ax=ax)
ax.plot(omega/(2*np.pi), eq_capa.s_db[:,0,0])
Out[176]:
Reminding that the ABCD parameters of a serie impedance is (D. Pozar, Microwave Engineering Fourth Edition (Wiley, 2011), p.754, "The ABCD Parameters of Some Useful Two-Port Circuits."):
Assuming the impedance consist in a resistor, a self and a capacitor, then one can extract the element values of the network set from the ABCD parameter $Z=\textrm{ABCD}_{12}$: $$ Z \doteq R + j(L\omega - 1/C\omega) $$
In [48]:
# L expressed in nH and C in pF to make the fit to work!
def ZZ(omega, R, C, L):
return R + 1j*(L*1e-9*omega - 1/(C*1e-12*omega))
def re_Z(omega, R, C, L):
return np.real( ZZ(omega, R, C, L) )
def im_Z(omega, R, C, L):
return np.imag( ZZ(omega, R, C, L) )
In [50]:
fix, (ax1, ax2) = plt.subplots(2,1, sharex=True)
# dummy transmission line to match port 3
coax = rf.media.Coaxial(frequency=capas_set[0].frequency)
for (ntw_name, ntw) in capas.items():
ntw = rf.connect(ntw, 2, coax.match(), 0)
Z = ntw.a[idx,0,1]
(R,_,_), pcov = curve_fit(re_Z, omega[idx], np.real(Z))
(_,C,L), pcov = curve_fit(im_Z, omega[idx], np.imag(Z))
print(f'C={C:0.1f} pF, L={L:0.1f} nH, R={R:0.1e} Ohm')
ax1.plot(f[idx]/1e6, np.real(Z))
media = rf.media.DefinedGammaZ0(frequency=ntw.frequency, z0=50)
ntw2 = media.capacitor(C*1e-12) ** media.inductor(L*1e-9) ** media.resistor(1e-1)
ax1.plot(f[idx]/1e6, ntw2.a_re[idx,0,1], ls='--')
ax2.plot(f[idx]/1e6, np.imag(Z))
ax2.plot(f[idx]/1e6, ntw2.a_im[idx,0,1], ls='--')