import numpy as np
import matplotlib.pyplot as plt
Here is the circuit that has been analyzed. It is part of a larger circuit designed by Carl Michal for an Earth's-Field NMR apparatus: https://iopscience.iop.org/article/10.1088/0957-0233/21/10/105902
import matplotlib.image as mpimg
img = mpimg.imread('low-pass.png')
plt.figure(figsize = (10, 7))
imgplot = plt.imshow(img)
plt.axis('off');
ff = np.logspace(1, 5, 1000) # Hz
w = 2*np.pi*ff # rad/s
R2 = 47e3 # ohms
R3 = 12e3 # ohms
C2 = 1e-9 # F
Zc2 = 1/(1j*w*C2) # ohms
vRatio = Zc2**2*(1 + R3/R2)/((R2 + Zc2)**2 - Zc2*R3)
vRatio
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abs(vRatio)
array([1.25531343, 1.25531333, 1.25531322, 1.25531311, 1.255313 , 1.25531288, 1.25531277, 1.25531265, 1.25531253, 1.2553124 , 1.25531228, 1.25531215, 1.25531202, 1.25531189, 1.25531175, 1.25531161, 1.25531147, 1.25531133, 1.25531119, 1.25531104, 1.25531089, 1.25531073, 1.25531058, 1.25531042, 1.25531025, 1.25531009, 1.25530992, 1.25530975, 1.25530957, 1.2553094 , 1.25530921, 1.25530903, 1.25530884, 1.25530865, 1.25530845, 1.25530825, 1.25530805, 1.25530784, 1.25530763, 1.25530742, 1.2553072 , 1.25530698, 1.25530675, 1.25530652, 1.25530629, 1.25530605, 1.2553058 , 1.25530556, 1.2553053 , 1.25530505, 1.25530478, 1.25530452, 1.25530424, 1.25530397, 1.25530368, 1.2553034 , 1.2553031 , 1.2553028 , 1.2553025 , 1.25530219, 1.25530187, 1.25530155, 1.25530122, 1.25530089, 1.25530055, 1.25530021, 1.25529985, 1.25529949, 1.25529913, 1.25529876, 1.25529838, 1.25529799, 1.2552976 , 1.25529719, 1.25529679, 1.25529637, 1.25529595, 1.25529551, 1.25529507, 1.25529463, 1.25529417, 1.2552937 , 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1.20024816, 1.19921893, 1.19817057, 1.19710273, 1.19601507, 1.19490721, 1.19377881, 1.1926295 , 1.19145889, 1.19026662, 1.18905229, 1.18781551, 1.18655589, 1.18527303, 1.18396651, 1.18263592, 1.18128085, 1.17990085, 1.17849551, 1.17706438, 1.17560702, 1.17412298, 1.17261179, 1.17107301, 1.16950616, 1.16791077, 1.16628636, 1.16463245, 1.16294853, 1.16123413, 1.15948873, 1.15771183, 1.15590291, 1.15406147, 1.15218697, 1.1502789 , 1.14833671, 1.14635988, 1.14434786, 1.1423001 , 1.14021607, 1.1380952 , 1.13593693, 1.13374073, 1.131506 , 1.12923221, 1.12691877, 1.12456512, 1.12217068, 1.1197349 , 1.11725718, 1.11473696, 1.11217366, 1.10956672, 1.10691556, 1.1042196 , 1.10147828, 1.09869103, 1.09585728, 1.09297648, 1.09004807, 1.08707149, 1.0840462 , 1.08097166, 1.07784734, 1.0746727 , 1.07144722, 1.06817041, 1.06484175, 1.06146076, 1.05802696, 1.05453989, 1.05099908, 1.04740411, 1.04375454, 1.04004997, 1.03629001, 1.03247428, 1.02860243, 1.02467412, 1.02068904, 1.01664688, 1.01254739, 1.00839032, 1.00417544, 0.99990255, 0.99557149, 0.99118211, 0.98673431, 0.98222799, 0.97766311, 0.97303964, 0.9683576 , 0.96361703, 0.95881802, 0.95396068, 0.94904516, 0.94407165, 0.93904039, 0.93395163, 0.92880569, 0.92360292, 0.91834369, 0.91302845, 0.90765766, 0.90223184, 0.89675154, 0.89121736, 0.88562996, 0.87999001, 0.87429826, 0.86855546, 0.86276245, 0.85692009, 0.85102928, 0.84509098, 0.83910618, 0.83307591, 0.82700126, 0.82088334, 0.81472331, 0.80852237, 0.80228177, 0.79600278, 0.78968672, 0.78333493, 0.77694881, 0.77052977, 0.76407926, 0.75759877, 0.75108981, 0.74455392, 0.73799267, 0.73140764, 0.72480045, 0.71817274, 0.71152615, 0.70486236, 0.69818305, 0.69148992, 0.68478469, 0.67806908, 0.6713448 , 0.66461361, 0.65787722, 0.65113739, 0.64439585, 0.63765433, 0.63091457, 0.62417829, 0.61744721, 0.61072302, 0.60400743, 0.59730211, 0.59060873, 0.58392893, 0.57726434, 0.57061657, 0.56398719, 0.55737777, 0.55078983, 0.54422489, 0.53768443, 0.53116989, 0.52468269, 0.51822422, 0.51179583, 0.50539885, 0.49903456, 0.49270421, 0.48640902, 0.48015015, 0.47392877, 0.46774596, 0.4616028 , 0.45550031, 0.44943947, 0.44342125, 0.43744654, 0.43151622, 0.42563113, 0.41979204, 0.41399973, 0.40825489, 0.40255822, 0.39691034, 0.39131185, 0.38576333, 0.38026529, 0.37481822, 0.36942257, 0.36407877, 0.35878719, 0.35354817, 0.34836204, 0.34322907, 0.33814951, 0.33312357, 0.32815143, 0.32323325, 0.31836916, 0.31355925, 0.30880358, 0.30410219, 0.2994551 , 0.29486229, 0.29032373, 0.28583935, 0.28140907, 0.27703278, 0.27271035, 0.26844162, 0.26422643, 0.26006458, 0.25595588, 0.25190008, 0.24789694, 0.24394621, 0.24004761, 0.23620084, 0.2324056 , 0.22866158, 0.22496843, 0.22132582, 0.21773339, 0.21419077, 0.21069759, 0.20725345, 0.20385797, 0.20051074, 0.19721135, 0.19395938, 0.1907544 , 0.18759598, 0.18448368, 0.18141706, 0.17839567, 0.17541906, 0.17248677, 0.16959834, 0.1667533 , 0.1639512 , 0.16119156, 0.15847391, 0.15579778, 0.1531627 , 0.15056819, 0.14801377, 0.14549898, 0.14302334, 0.14058637, 0.1381876 , 0.13582654, 0.13350274, 0.13121572, 0.12896501, 0.12675013, 0.12457063, 0.12242604, 0.1203159 , 0.11823973, 0.1161971 , 0.11418754, 0.1122106 , 0.11026582, 0.10835278, 0.10647101, 0.10462009, 0.10279957, 0.10100902, 0.09924802, 0.09751613, 0.09581294, 0.09413802, 0.09249097, 0.09087137, 0.08927882, 0.0877129 , 0.08617324, 0.08465942, 0.08317107, 0.08170779, 0.0802692 , 0.07885493, 0.07746461, 0.07609785, 0.07475431, 0.07343362, 0.07213542, 0.07085936, 0.0696051 , 0.06837228, 0.06716058, 0.06596966, 0.06479918, 0.06364882, 0.06251827, 0.06140719, 0.06031529, 0.05924224, 0.05818775, 0.05715151, 0.05613323, 0.05513261, 0.05414937, 0.05318322, 0.05223388, 0.05130107, 0.05038453, 0.04948397, 0.04859915, 0.04772979, 0.04687564, 0.04603644, 0.04521195, 0.04440192, 0.04360612, 0.04282429, 0.0420562 , 0.04130163, 0.04056034, 0.03983211, 0.03911672, 0.03841396, 0.0377236 , 0.03704544, 0.03637926, 0.03572487, 0.03508206, 0.03445064, 0.03383041, 0.03322118, 0.03262275, 0.03203495, 0.03145759, 0.03089049, 0.03033348, 0.02978637, 0.029249 , 0.02872121, 0.02820281, 0.02769366, 0.02719358, 0.02670243, 0.02622005, 0.02574628, 0.02528098, 0.02482399, 0.02437517, 0.02393438, 0.02350148, 0.02307633, 0.0226588 , 0.02224874, 0.02184604, 0.02145055, 0.02106216, 0.02068074, 0.02030616, 0.01993831, 0.01957706, 0.0192223 , 0.01887392, 0.01853181, 0.01819584, 0.01786592, 0.01754193, 0.01722378, 0.01691135, 0.01660455, 0.01630328, 0.01600743, 0.01571692, 0.01543164, 0.01515151, 0.01487643, 0.01460631, 0.01434107, 0.01408061, 0.01382486, 0.01357373, 0.01332713, 0.01308498, 0.01284721, 0.01261374, 0.01238449, 0.01215938, 0.01193834, 0.01172131, 0.01150819, 0.01129894, 0.01109347, 0.01089172, 0.01069362, 0.01049911, 0.01030812, 0.01012059, 0.00993646, 0.00975566, 0.00957814, 0.00940384, 0.0092327 , 0.00906465, 0.00889966, 0.00873766, 0.0085786 , 0.00842242, 0.00826907, 0.00811851, 0.00797068, 0.00782553, 0.00768302, 0.00754309, 0.00740571, 0.00727082, 0.00713837, 0.00700834, 0.00688066, 0.00675531, 0.00663223, 0.00651139, 0.00639274, 0.00627625, 0.00616188, 0.00604959, 0.00593934, 0.00583109, 0.00572481, 0.00562046, 0.00551801, 0.00541743, 0.00531867, 0.00522171, 0.00512651, 0.00503305, 0.00494128, 0.00485119, 0.00476273, 0.00467589, 0.00459062, 0.00450691, 0.00442472, 0.00434402, 0.0042648 , 0.00418702, 0.00411065, 0.00403567, 0.00396206, 0.00388979, 0.00381884, 0.00374918, 0.00368079, 0.00361364, 0.00354771, 0.00348299, 0.00341945, 0.00335706, 0.00329581, 0.00323568, 0.00317664, 0.00311868, 0.00306177, 0.0030059 , 0.00295105, 0.0028972 , 0.00284433, 0.00279243, 0.00274147, 0.00269144, 0.00264232, 0.00259409, 0.00254675, 0.00250027, 0.00245463, 0.00240983, 0.00236585, 0.00232267, 0.00228027, 0.00223865, 0.00219779, 0.00215767, 0.00211828, 0.00207961, 0.00204165, 0.00200438, 0.00196779, 0.00193187, 0.0018966 , 0.00186198, 0.00182798, 0.00179461, 0.00176185, 0.00172968, 0.0016981 , 0.0016671 , 0.00163666, 0.00160678, 0.00157744, 0.00154864, 0.00152036, 0.0014926 , 0.00146535, 0.0014386 ])
np.angle(vRatio)
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-0.72903526, -0.73561798, -0.74225495, -0.74894642, -0.75569261, -0.76249377, -0.7693501 , -0.77626183, -0.78322915, -0.79025226, -0.79733135, -0.80446659, -0.81165814, -0.81890617, -0.82621081, -0.8335722 , -0.84099046, -0.84846569, -0.85599799, -0.86358744, -0.87123412, -0.87893807, -0.88669933, -0.89451793, -0.90239387, -0.91032715, -0.91831775, -0.92636561, -0.93447068, -0.94263289, -0.95085213, -0.95912829, -0.96746123, -0.97585081, -0.98429683, -0.99279911, -1.00135742, -1.00997153, -1.01864117, -1.02736605, -1.03614586, -1.04498027, -1.05386892, -1.06281142, -1.07180737, -1.08085633, -1.08995785, -1.09911143, -1.10831657, -1.11757272, -1.12687932, -1.13623578, -1.14564147, -1.15509575, -1.16459793, -1.17414733, -1.18374319, -1.19338477, -1.20307128, -1.21280189, -1.22257578, -1.23239205, -1.24224982, -1.25214816, -1.26208611, -1.27206269, -1.2820769 , -1.29212769, -1.30221402, -1.3123348 , -1.3224889 , -1.33267521, -1.34289256, -1.35313976, -1.36341563, -1.37371891, -1.38404838, -1.39440276, -1.40478076, -1.41518108, -1.42560239, -1.43604335, -1.44650261, -1.45697879, -1.46747051, -1.47797637, -1.48849496, -1.49902485, -1.50956463, -1.52011284, -1.53066805, -1.54122881, -1.55179365, -1.56236112, -1.57292976, -1.5834981 , -1.59406469, -1.60462805, -1.61518674, -1.6257393 , -1.63628428, -1.64682024, -1.65734575, -1.66785937, -1.67835968, -1.6888453 , -1.69931481, -1.70976685, -1.72020004, -1.73061304, -1.74100451, -1.75137313, -1.7617176 , -1.77203665, -1.78232901, -1.79259345, -1.80282875, -1.8130337 , -1.82320715, -1.83334793, -1.84345493, -1.85352705, -1.8635632 , -1.87356234, -1.88352344, -1.89344552, -1.90332759, -1.91316872, -1.92296798, -1.9327245 , -1.94243741, -1.95210587, -1.96172909, -1.97130628, -1.98083671, -1.99031963, -1.99975438, -2.00914027, -2.01847667, -2.02776297, -2.03699859, -2.04618298, -2.0553156 , -2.06439596, -2.07342357, -2.082398 , -2.0913188 , -2.1001856 , -2.10899801, -2.11775568, -2.12645829, -2.13510555, -2.14369716, -2.15223288, -2.16071247, -2.16913573, -2.17750246, -2.18581249, -2.19406569, -2.20226192, -2.21040107, -2.21848305, -2.2265078 , -2.23447526, -2.24238539, -2.25023819, -2.25803363, -2.26577175, -2.27345257, -2.28107614, -2.28864251, -2.29615176, -2.30360398, -2.31099926, -2.31833773, -2.32561949, -2.3328447 , -2.3400135 , -2.34712605, -2.35418252, -2.36118308, -2.36812794, -2.37501727, -2.38185131, -2.38863025, -2.39535432, -2.40202375, -2.40863879, -2.41519968, -2.42170666, -2.42816001, -2.43455998, -2.44090684, -2.44720087, -2.45344236, -2.45963157, -2.46576882, -2.47185438, -2.47788856, -2.48387166, -2.48980398, -2.49568583, -2.50151754, -2.50729939, -2.51303173, -2.51871485, -2.5243491 , -2.52993478, -2.53547222, -2.54096175, -2.5464037 , -2.5517984 , -2.55714617, -2.56244734, -2.56770226, -2.57291124, -2.57807463, -2.58319276, -2.58826595, -2.59329455, -2.59827888, -2.60321929, -2.60811609, -2.61296963, -2.61778024, -2.62254824, -2.62727398, -2.63195777, -2.63659996, -2.64120086, -2.64576081, -2.65028013, -2.65475915, -2.65919818, -2.66359757, -2.66795762, -2.67227866, -2.67656101, -2.68080498, -2.68501089, -2.68917906, -2.6933098 , -2.69740342, -2.70146024, -2.70548055, -2.70946467, -2.7134129 , -2.71732554, -2.72120291, -2.72504529, -2.72885298, -2.73262628, -2.73636549, -2.7400709 , -2.7437428 , -2.74738147, -2.75098721, -2.75456029, -2.75810101, -2.76160964, -2.76508646, -2.76853175, -2.77194578, -2.77532883, -2.77868116, -2.78200306, -2.78529477, -2.78855658, -2.79178874, -2.79499151, -2.79816516, -2.80130994, -2.8044261 , -2.80751391, -2.8105736 , -2.81360544, -2.81660966, -2.81958651, -2.82253625, -2.8254591 , -2.82835531, -2.83122511, -2.83406875, -2.83688645, -2.83967845, -2.84244498, -2.84518627, -2.84790254, -2.85059401, -2.85326092, -2.85590347, -2.8585219 , -2.86111641, -2.86368722, -2.86623455, -2.8687586 , -2.8712596 , -2.87373773, -2.87619322, -2.87862626, -2.88103706, -2.88342582, -2.88579274, -2.88813802, -2.89046184, -2.89276442, -2.89504593, -2.89730657, -2.89954653, -2.901766 , -2.90396515, -2.90614419, -2.90830328, -2.91044262, -2.91256237, -2.91466272, -2.91674384, -2.91880592, -2.92084911, -2.9228736 , -2.92487955, -2.92686713, -2.92883652, -2.93078786, -2.93272134, -2.93463711, -2.93653533, -2.93841617, -2.94027977, -2.94212631, -2.94395593, -2.94576878, -2.94756503, -2.94934483, -2.95110831, -2.95285564, -2.95458695, -2.9563024 , -2.95800213, -2.95968629, -2.96135501, -2.96300843, -2.9646467 , -2.96626996, -2.96787834, -2.96947198, -2.97105101, -2.97261556, -2.97416578, -2.97570178, -2.9772237 , -2.97873167, -2.98022582, -2.98170627, -2.98317314, -2.98462657, -2.98606667, -2.98749357, -2.98890738, -2.99030823, -2.99169624, -2.99307152, -2.99443419, -2.99578436, -2.99712215, -2.99844768, -2.99976105, -3.00106237, -3.00235176, -3.00362933, -3.00489518, -3.00614942, -3.00739216, -3.0086235 , -3.00984355, -3.01105241, -3.01225018, -3.01343696, -3.01461286, -3.01577798, -3.01693241, -3.01807625, -3.0192096 , -3.02033255, -3.0214452 , -3.02254765, -3.02363998, -3.0247223 , -3.02579468, -3.02685723, -3.02791003, -3.02895318, -3.02998676, -3.03101085, -3.03202555, -3.03303094, -3.03402711, -3.03501414, -3.03599212, -3.03696112, -3.03792124, -3.03887254, -3.03981512, -3.04074905, -3.04167442, -3.04259129, -3.04349976, -3.04439988, -3.04529175, -3.04617544, -3.04705102, -3.04791857, -3.04877816, -3.04962986, -3.05047375, -3.05130989, -3.05213837, -3.05295924, -3.05377258, -3.05457846, -3.05537694, -3.0561681 , -3.056952 , -3.05772871, -3.05849829, -3.05926081, -3.06001633, -3.06076492, -3.06150665, -3.06224156, -3.06296974, -3.06369123, -3.0644061 , -3.06511442, -3.06581623, -3.06651161, -3.0672006 , -3.06788327, -3.06855968, -3.06922988, -3.06989394, -3.0705519 , -3.07120382, -3.07184976, -3.07248977, -3.07312391, -3.07375223, -3.07437479, -3.07499163, -3.07560282, -3.07620839, -3.07680841, -3.07740292, -3.07799198, -3.07857563, -3.07915393, -3.07972692, -3.08029465, -3.08085718, -3.08141454, -3.08196678, -3.08251396])
plt.loglog(ff, abs(vRatio), 'k-', linewidth = 2.5)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Voltage Ratio')
plt.axis((10, 1e5, 1e-3, 2));
plt.semilogx(ff, np.angle(vRatio)/np.pi, 'r-', linewidth = 1.5)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Phase divided by $\pi$')
plt.axis((10, 1e5, -1, 0.05));