2.4 MiB
2.4 MiB
In [48]:
import numpy as np import skrf from skrf.media import DistributedCircuit import skrf.frequency as freq import skrf.network as net import skrf.util import matplotlib.pyplot as plt %matplotlib inline plt.rcParams['figure.figsize'] = [10, 10] f = freq.Frequency(0.4, 2, 1001) tem = DistributedCircuit(f, z0=50) bjt = net.Network('BFU520_Spar_NF_400MHz-2GHz/BFU520_03V3_005mA_NF_SP.s2p').interpolate(f) bjt
Out[48]:
2-Port Network: 'BFU520_03V3_005mA_NF_SP', 0.4-2.0 GHz, 1001 pts, z0=[ 50.+0.j 50.+0.j]
In [49]:
bjt.plot_s_smith()
In [50]:
# calculate the stability circles for the source and load impedances idx_900mhz = skrf.util.find_nearest_index(bjt.f, 915.e+6) sqabs = lambda x: np.square(np.absolute(x)) delta = bjt.s11.s*bjt.s22.s - bjt.s12.s*bjt.s21.s rl = np.absolute((bjt.s12.s * bjt.s21.s)/(sqabs(bjt.s22.s) - sqabs(delta))) cl = np.conj(bjt.s22.s - delta*np.conj(bjt.s11.s))/(sqabs(bjt.s22.s) - sqabs(delta)) rl_900mhz = rl[idx_900mhz][0, 0] cl_900mhz = cl[idx_900mhz][0, 0] rl_900mhz, cl_900mhz
Out[50]:
(2.9580917447631938, (1.2432679422469759+3.3675063614328802j))
In [51]:
def calc_circle(c, r): theta = np.linspace(0, 2*np.pi, 1000) return c + r*np.exp(1.0j*theta) def plot_smith(pts): n = net.Network(s=pts) n.plot_s_smith() cl_points = calc_circle(cl_900mhz, rl_900mhz) plot_smith(cl_points)
In [52]:
rs = np.absolute((bjt.s12.s * bjt.s21.s)/(sqabs(bjt.s11.s) - sqabs(delta))) cs = np.conj(bjt.s11.s - delta*np.conj(bjt.s22.s))/(sqabs(bjt.s11.s) - sqabs(delta)) rs_900mhz = rs[idx_900mhz][0, 0] cs_900mhz = cs[idx_900mhz][0, 0] rs_900mhz, cs_900mhz
Out[52]:
(3.0024614405316727, (-2.914568780972254+2.1685483287786806j))
In [53]:
cs_points = calc_circle(cs_900mhz, rs_900mhz) plot_smith(cs_points)
In [54]:
# let's plot all of them # output stability first for i, f in enumerate(bjt.f): # decimate it a little if i % 100 != 0: continue n = net.Network(name=str(f/1.e+9), s=calc_circle(cl[i][0, 0], rl[i][0, 0])) n.plot_s_smith()
In [55]:
# input stability for i, f in enumerate(bjt.f): if i % 100 != 0: continue n = net.Network(name=str(f/1.e+9), s=calc_circle(cs[i][0, 0], rs[i][0, 0])) n.plot_s_smith()
In [56]:
# let's draw some constant noise circles # first we grab the noise parameters for our target frequency from the network model idx_915mhz = skrf.util.find_nearest_index(bjt.f, 915.e+6) # we need the normalized equivalent noise and optimum source coefficient to calculate the constant noise circles rn = bjt.rn[idx_915mhz]/50 gamma_opt = bjt.g_opt[idx_915mhz] fmin = bjt.nfmin[idx_915mhz] for nf_added in [0, 0.05, 0.1, 0.2, 0.3]: nf = 10**(nf_added/10) * fmin N = (nf - fmin)*abs(1+gamma_opt)**2/(4*rn) c_n = gamma_opt/(1+N) r_n = 1/(1-N)*np.sqrt(N**2 + N*(1-abs(gamma_opt)**2)) n = net.Network(name=str(nf_added), s=calc_circle(c_n, r_n)) n.plot_s_smith() print("the optimum source reflection coefficient is ", gamma_opt)
the optimum source reflection coefficient is (0.087278897277+0.149115719672j)
In [57]:
gamma_s = bjt.g_opt[idx_900mhz] gamma_s
Out[57]:
(0.087278897277004358+0.14911571967162124j)
In [58]:
# so I need to calculate the load reflection coefficient to get a conjugate match when the input sees 50 ohms gamma_l = np.conj(bjt.s22.s - bjt.s21.s*gamma_s*bjt.s12.s/(1-bjt.s11.s*gamma_s)) is_gamma_l_stable = np.absolute(gamma_l[idx_900mhz, 0, 0] - cl_900mhz) > rl_900mhz gamma_l = gamma_l[idx_900mhz, 0, 0] gamma_l, is_gamma_l_stable
Out[58]:
((0.40368792779274587+0.4094872772482992j), True)
In [59]:
def calc_matching_network_vals(z1, z2): flipped = ((abs(np.imag(z2)) < 1e-6 and np.real(z1) < np.real(z2)) or (abs(np.imag(z2)) > 1e-6 and np.real(z1) < np.real(1/(1/z2-1/(1.j*np.imag(z2)))))) if flipped: z2, z1 = z1, z2 # cancel out the imaginary parts of both input and output impedances z1_par = 1e+10 if abs(np.imag(z1)) > 1e-6: # parallel something to cancel out the imaginary part of # z1's impedance z1_par = 1/(-1j*np.imag(1/z1)) z1 = 1/(1./z1 + 1/z1_par) z2_ser = 0.0 if abs(np.imag(z2)) > 1e-6: z2_ser = -1j*np.imag(z2) z2 = z2 + z2_ser Q = np.sqrt((np.real(z1) - np.real(z2))/np.real(z2)) x1 = -1.j * np.real(z1)/Q x2 = 1.j * np.real(z2)*Q x1_tot = 1/(1/z1_par + 1/x1) x2_tot = z2_ser + x2 if flipped: return x2_tot, x1_tot else: return x1_tot, x2_tot z_l = net.s2z(np.array([[[gamma_l]]]))[0,0,0] # note that we're matching against the conjugate; # this is because we want to see z_l from the BJT side # if we plugged in z the matching network would make # the 50 ohms look like np.conj(z) to match against it, so # we use np.conj(z_l) so that it'll look like z_l from the BJT's side z_par, z_ser = calc_matching_network_vals(np.conj(z_l), 50) z_l, z_par, z_ser
Out[59]:
((63.95922663128475+78.255758361282744j), -619.6484445486951j, 74.06320245900419j)
In [60]:
# let's calculate what the component values are c_par = np.real(1/(2j*np.pi*915e+6*z_par)) l_ser = np.real(z_ser/(2j*np.pi*915e+6)) c_par, l_ser
Out[60]:
(2.8070727853180874e-13, 1.2882540735045174e-08)
In [93]:
# the capacitance is kind of low but the inductance seems reasonable # let's test it out output_network = tem.inductor(l_ser) ** tem.shunt_capacitor(c_par) amplifier = tem.inductor(1.3e-9) ** bjt ** output_network amplifier.plot_s_smith()
In [94]:
amplifier.s11.plot_s_db() amplifier.s22.plot_s_db() 10*np.log10(amplifier.nf(50.)[idx_900mhz])
Out[94]:
0.77997948391045657
In [95]:
amplifier.s21.plot_s_db()
In [91]:
# that's probably good enough for the first stage # for the second stage let's stabilize the transistor by adding some parallel resistance # and then simultaneous conjugate matching so that both the input and output are matched to 50 ohms # let's calculate the stability factor to see where it's unstable sqabs = lambda x: np.square(np.absolute(x)) delta = bjt.s11.s*bjt.s22.s - bjt.s12.s*bjt.s21.s K = ((1 - sqabs(bjt.s11.s) - sqabs(bjt.s22.s) + sqabs(delta))/(2*np.absolute(bjt.s12.s*bjt.s21.s)))[:,0,0] plt.plot(bjt.f, K)
Out[91]:
[<matplotlib.lines.Line2D at 0x7f305c0c94e0>]
In [64]:
#so it's basically always unstable #let's add a 250 ohm shunt to the output and see how much that improves it bjt_comp = bjt ** tem.shunt(tem.resistor(250) ** tem.short()) bjt_comp.plot_s_smith()
In [65]:
# let's calculate K again delta2 = bjt_comp.s11.s*bjt_comp.s22.s - bjt_comp.s12.s*bjt_comp.s21.s K2 = ((1 - sqabs(bjt_comp.s11.s) - sqabs(bjt_comp.s22.s) + sqabs(delta2))/(2*np.absolute(bjt_comp.s12.s*bjt_comp.s21.s)))[:,0,0] plt.plot(bjt_comp.f, K2)
Out[65]:
[<matplotlib.lines.Line2D at 0x7f30736c25c0>]
In [66]:
# that's a pretty nice improvement! let's check the delta to be sure, and # find the new source stability circles to see if we need to add some series resistance there # K > 1 and |delta| < 1 for stability plt.plot(bjt_comp.f, np.absolute(delta2[:,0,0]))
Out[66]:
[<matplotlib.lines.Line2D at 0x7f30739fd668>]
In [67]:
rs2 = np.absolute((bjt_comp.s12.s * bjt_comp.s21.s)/(sqabs(bjt_comp.s11.s) - sqabs(delta2))) cs2 = np.conj(bjt_comp.s11.s - delta2*np.conj(bjt_comp.s22.s))/(sqabs(bjt_comp.s11.s) - sqabs(delta2)) # input stability for i, f in enumerate(bjt_comp.f): if i % 100 != 0: continue n = net.Network(name=str(f/1.e+9), s=calc_circle(cs2[i][0, 0], rs2[i][0, 0])) n.plot_s_smith()
In [68]:
# that doesn't look too bad, so let's move forward and try to conjugate match delta2 = bjt_comp.s11.s*bjt_comp.s22.s - bjt_comp.s12.s*bjt_comp.s21.s B1 = 1 + sqabs(bjt_comp.s11.s) - sqabs(bjt_comp.s22.s) - sqabs(delta2) B2 = 1 + sqabs(bjt_comp.s22.s) - sqabs(bjt_comp.s11.s) - sqabs(delta2) C1 = bjt_comp.s11.s - delta2*np.conj(bjt_comp.s22.s) C2 = bjt_comp.s22.s - delta2*np.conj(bjt_comp.s11.s) gamma_s_all = (B1 - np.sqrt(np.square(B1) - 4*sqabs(C1) + 0j))/(2*C1) gamma_l_all = (B2 - np.sqrt(np.square(B2) - 4*sqabs(C2) + 0j))/(2*C2) gamma_s = gamma_s_all[idx_900mhz, 0, 0] gamma_l = gamma_l_all[idx_900mhz, 0, 0] z_s = net.s2z(np.array([[[gamma_s]]]))[0,0,0] z_l = net.s2z(np.array([[[gamma_l]]]))[0,0,0] # this section broke when I switched to a lower bias current so just ignore this z_s, z_l
Out[68]:
((9.4205547521026507e-15+38.020469196561926j), (-1.2560739669470201e-14+123.33562619268822j))
In [69]:
x_s_1, x_s_2 = calc_matching_network_vals(np.conj(z_s), 50) x_l_1, x_l_2 = calc_matching_network_vals(np.conj(z_l), 50) x_s_1, x_s_2, x_l_1, x_l_2
/usr/lib/python3/dist-packages/ipykernel_launcher.py:19: RuntimeWarning: invalid value encountered in sqrt
Out[69]:
(38.02046988287653j, (4.710277376051326e-23-6.863146054144066e-07j), (nan+nanj), (nan+nanj))
In [70]:
c_s_shunt = np.real(1/(2j*np.pi*915e+6*x_s_1)) c_s_ser = np.real(1/(2j*np.pi*915e+6*x_s_2)) l_s_ser = np.real(x_s_2/(2j*np.pi*915e+6)) c_l_shunt = np.real(1/(2j*np.pi*915e+6*x_l_1)) l_l_ser = np.real(x_l_2/(2j*np.pi*915e+6)) c_s_shunt2 = np.real(1/(2j*np.pi*915e+6*x_s_2)) l_s_ser2 = np.real(x_s_1/(2j*np.pi*915e+6)) c_l_shunt2 = np.real(1/(2j*np.pi*915e+6*x_l_2)) l_l_ser2 = np.real(x_l_1/(2j*np.pi*915e+6)) #c_s_shunt, c_s_ser, c_l_shunt, l_l_ser c_s_shunt, l_s_ser, c_l_shunt, l_l_ser
Out[70]:
(-4.574899496286098e-12, -1.1937744477362458e-16, nan, nan)
In [71]:
#input_network2 = tem.capacitor(c_s_ser) ** tem.shunt_capacitor(c_s_shunt) #input_network2 = tem.inductor(l_s_ser) ** tem.shunt_capacitor(c_s_shunt) input_network2 = tem.shunt_capacitor(c_s_shunt2) ** tem.inductor(l_s_ser2) #output_network2 = tem.shunt_capacitor(c_l_shunt) ** tem.inductor(l_l_ser) output_network2 = tem.inductor(l_l_ser2) ** tem.shunt_capacitor(c_l_shunt2) amplifier2 = input_network2 ** bjt_comp ** output_network2 #amplifier2 = input_network2 ** bjt #** output_network2 amplifier2.plot_s_smith()
/home/kelvin/.local/lib/python3.6/site-packages/scikit_rf-0.14.8-py3.6.egg/skrf/network.py:4713: RuntimeWarning: invalid value encountered in true_divide ((z01.conj() + s[:,0,0]*z01)*(1 - s[:,1,1]) + s[:,0,1]*s[:,1,0]*z01) / denom, /home/kelvin/.local/lib/python3.6/site-packages/scikit_rf-0.14.8-py3.6.egg/skrf/network.py:4714: RuntimeWarning: divide by zero encountered in true_divide ((1 - s[:,0,0])*(1 - s[:,1,1]) - s[:,0,1]*s[:,1,0]) / denom, /home/kelvin/.local/lib/python3.6/site-packages/scikit_rf-0.14.8-py3.6.egg/skrf/network.py:4717: RuntimeWarning: invalid value encountered in true_divide ((z01.conj() + s[:,0,0]*z01)*(z02.conj() + s[:,1,1]*z02) - s[:,0,1]*s[:,1,0]*z01*z02) / denom, /home/kelvin/.local/lib/python3.6/site-packages/scikit_rf-0.14.8-py3.6.egg/skrf/network.py:4718: RuntimeWarning: divide by zero encountered in true_divide ((1 - s[:,0,0])*(z02.conj() + s[:,1,1]*z02) + s[:,0,1]*s[:,1,0]*z02) / denom, /home/kelvin/.local/lib/python3.6/site-packages/scipy-1.3.0-py3.6-linux-x86_64.egg/scipy/interpolate/interpolate.py:609: RuntimeWarning: invalid value encountered in subtract slope = (y_hi - y_lo) / (x_hi - x_lo)[:, None] /home/kelvin/.local/lib/python3.6/site-packages/scipy-1.3.0-py3.6-linux-x86_64.egg/scipy/interpolate/interpolate.py:612: RuntimeWarning: invalid value encountered in add y_new = slope*(x_new - x_lo)[:, None] + y_lo /home/kelvin/.local/lib/python3.6/site-packages/scikit_rf-0.14.8-py3.6.egg/skrf/network.py:3282: RuntimeWarning: invalid value encountered in multiply a = a_real(noise_freq.f) + 1.j * a_imag(noise_freq.f)
In [72]:
amplifier2.s21.plot_s_db()
In [73]:
amplifier2.s11.plot_s_db()
In [74]:
amplifier2.s22.plot_s_db()
In [75]:
10*np.log10(amplifier2.nf(50.)[idx_900mhz])
Out[75]:
153.2299538314351
In [76]:
# check stability again delta2 = amplifier2.s11.s*amplifier2.s22.s - amplifier2.s12.s*amplifier2.s21.s K2 = ((1 - sqabs(amplifier2.s11.s) - sqabs(amplifier2.s22.s) + sqabs(delta2))/(2*np.absolute(amplifier2.s12.s*amplifier2.s21.s)))[:,0,0] plt.plot(amplifier2.f, K2)
Out[76]:
[<matplotlib.lines.Line2D at 0x7f305c20bda0>]
In [77]:
# of course, it matches the other one since all we did was transform the impedances at the input and output # let's try again with 10 ohms of resistance in series with the base to stabilize that side bjt_comp2 = tem.resistor(10) ** bjt_comp delta3 = bjt_comp2.s11.s*bjt_comp2.s22.s - bjt_comp2.s12.s*bjt_comp2.s21.s B1 = 1 + sqabs(bjt_comp2.s11.s) - sqabs(bjt_comp2.s22.s) - sqabs(delta3) B2 = 1 + sqabs(bjt_comp2.s22.s) - sqabs(bjt_comp2.s11.s) - sqabs(delta3) C1 = bjt_comp2.s11.s - delta3*np.conj(bjt_comp2.s22.s) C2 = bjt_comp2.s22.s - delta3*np.conj(bjt_comp2.s11.s) gamma_s_all = (B1 - np.sqrt(np.square(B1) - 4*sqabs(C1) + 0j))/(2*C1) gamma_l_all = (B2 - np.sqrt(np.square(B2) - 4*sqabs(C2) + 0j))/(2*C2) gamma_s = gamma_s_all[idx_900mhz, 0, 0] gamma_l = gamma_l_all[idx_900mhz, 0, 0] z_s = net.s2z(np.array([[[gamma_s]]]))[0,0,0] z_l = net.s2z(np.array([[[gamma_l]]]))[0,0,0] bjt_comp2.plot_s_smith() z_s, z_l
Out[77]:
((16.221499138301905+24.689163340981278j), (31.8441752257032+52.299573751504738j))
In [78]:
n = net.Network(name="g_ms", s=gamma_s_all) n.plot_s_smith() n = net.Network(name="g_ml", s=gamma_l_all) n.plot_s_smith()
In [79]:
K3 = ((1 - sqabs(bjt_comp2.s11.s) - sqabs(bjt_comp2.s22.s) + sqabs(delta3))/(2*np.absolute(bjt_comp2.s12.s*bjt_comp2.s21.s)))[:,0,0] plt.plot(bjt_comp2.f, K3)
Out[79]:
[<matplotlib.lines.Line2D at 0x7f307360f160>]
In [80]:
x_s_1, x_s_2 = calc_matching_network_vals(np.conj(z_s), 50) x_l_1, x_l_2 = calc_matching_network_vals(np.conj(z_l), 50) x_s_1, x_s_2, x_l_1, x_l_2
Out[80]:
(48.097237217769305j, (1.2005786761170096e-07-34.64936761496535j), 76.3444747802025j, (4.3848428288954676e-07-66.21814576757241j))
In [81]:
c_s_shunt = np.real(1/(2j*np.pi*915e+6*x_s_2)) l_s_ser = np.real(x_s_1/(2j*np.pi*915e+6)) l_l_ser = np.real(x_l_1/(2j*np.pi*915e+6)) c_l_shunt = np.real(1/(2j*np.pi*915e+6*x_l_2)) #c_s_shunt, c_s_ser, c_l_shunt, l_l_ser c_s_shunt, l_s_ser, c_l_shunt, l_l_ser
Out[81]:
(5.020000089138901e-12, 8.366025193739305e-09, 2.626769845327086e-12, 1.3279344851392096e-08)
In [82]:
input_network3 = tem.shunt_capacitor(c_s_shunt) ** tem.inductor(l_s_ser) output_network3 = tem.inductor(l_l_ser) ** tem.shunt_capacitor(c_l_shunt) amplifier3 = input_network3 ** tem.resistor(10.) ** bjt_comp ** output_network3 amplifier3.plot_s_smith()
In [83]:
amplifier3.s21.plot_s_db()
In [84]:
10*np.log10(amplifier3.nf(50.)[idx_900mhz])
Out[84]:
1.4853529861561126
In [85]:
amplifier3.s11.plot_s_db() amplifier3.s22.plot_s_db()
In [86]:
# let's see how it works cascaded cascaded = amplifier3 ** amplifier3 cascaded.plot_s_smith()
In [87]:
cascaded.s21.plot_s_db()
In [88]:
cascaded.s11.plot_s_db() cascaded.s22.plot_s_db() 10*np.log10(cascaded.nf(50.)[idx_900mhz])
Out[88]:
1.5114710750219467
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