Complex Numbers

In this example we manipulate and plot complex numbers. The data correspond to the frequency response of an op-amp based low-pass filter

August 9, 2021

• First, import the usual modules for number crunching (numPy) and plotting (matplotlib)

import numpy as np
import matplotlib.pyplot as plt

• We will attempt to plot both the magnitude and phase of the filter's transfer function.

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');

• The magnitude and phase of the transfer function $v_{out}/v_{in}$ will be plotted as a function of frequency. The frequency range will be 10 Hz to 100 kHz. Set up a list of frequencies evenly spaced on a logarithmic scale.

ff = np.logspace(1, 5, 1000) # Hz
w = 2*np.pi*ff # rad/s

• Enter in component values and calculate the impedance of the capacitor. Notice the use of 1j to represent $\sqrt{-1}$.

R2 = 47e3 # ohms
R3 = 12e3 # ohms
C2 = 1e-9 # F
Zc2 = 1/(1j*w*C2) # ohms

• Here is the transfer function. The voltage ratio $v_{out}/v_{in}$ is complex.

vRatio = Zc2**2*(1 + R3/R2)/((R2 + Zc2)**2 - Zc2*R3)
vRatio

array([ 1.25529677-6.46761144e-03j,  1.25529636-6.52751459e-03j,
        1.25529593-6.58797254e-03j,  1.2552955 -6.64899043e-03j,
        1.25529506-6.71057345e-03j,  1.25529461-6.77272683e-03j,
        1.25529416-6.83545584e-03j,  1.25529369-6.89876584e-03j,
        1.25529322-6.96266218e-03j,  1.25529273-7.02715031e-03j,
        1.25529224-7.09223570e-03j,  1.25529174-7.15792388e-03j,
        1.25529123-7.22422044e-03j,  1.25529071-7.29113101e-03j,
        1.25529018-7.35866127e-03j,  1.25528964-7.42681697e-03j,
        1.2552891 -7.49560389e-03j,  1.25528854-7.56502789e-03j,
        1.25528797-7.63509485e-03j,  1.25528739-7.70581074e-03j,
        1.25528679-7.77718156e-03j,  1.25528619-7.84921337e-03j,
        1.25528558-7.92191231e-03j,  1.25528495-7.99528454e-03j,
        1.25528432-8.06933631e-03j,  1.25528367-8.14407389e-03j,
        1.25528301-8.21950365e-03j,  1.25528234-8.29563199e-03j,
        1.25528165-8.37246539e-03j,  1.25528095-8.45001036e-03j,
        1.25528024-8.52827351e-03j,  1.25527952-8.60726147e-03j,
        1.25527878-8.68698096e-03j,  1.25527803-8.76743876e-03j,
        1.25527727-8.84864170e-03j,  1.25527649-8.93059667e-03j,
        1.25527569-9.01331065e-03j,  1.25527488-9.09679067e-03j,
        1.25527406-9.18104380e-03j,  1.25527322-9.26607722e-03j,
        1.25527237-9.35189814e-03j,  1.2552715 -9.43851387e-03j,
        1.25527061-9.52593175e-03j,  1.25526971-9.61415921e-03j,
        1.25526879-9.70320376e-03j,  1.25526785-9.79307295e-03j,
        1.25526689-9.88377442e-03j,  1.25526592-9.97531588e-03j,
        1.25526493-1.00677051e-02j,  1.25526392-1.01609499e-02j,
        1.25526289-1.02550583e-02j,  1.25526185-1.03500382e-02j,
        1.25526078-1.04458977e-02j,  1.25525969-1.05426449e-02j,
        1.25525859-1.06402882e-02j,  1.25525746-1.07388356e-02j,
        1.25525631-1.08382957e-02j,  1.25525514-1.09386769e-02j,
        1.25525395-1.10399876e-02j,  1.25525274-1.11422366e-02j,
        1.2552515 -1.12454325e-02j,  1.25525024-1.13495840e-02j,
        1.25524896-1.14547000e-02j,  1.25524765-1.15607895e-02j,
        1.25524632-1.16678614e-02j,  1.25524497-1.17759249e-02j,
        1.25524359-1.18849891e-02j,  1.25524218-1.19950633e-02j,
        1.25524075-1.21061568e-02j,  1.25523929-1.22182790e-02j,
        1.25523781-1.23314396e-02j,  1.25523629-1.24456481e-02j,
        1.25523475-1.25609142e-02j,  1.25523318-1.26772477e-02j,
        1.25523158-1.27946584e-02j,  1.25522995-1.29131565e-02j,
        1.25522829-1.30327518e-02j,  1.2552266 -1.31534546e-02j,
        1.25522488-1.32752751e-02j,  1.25522312-1.33982236e-02j,
        1.25522133-1.35223107e-02j,  1.25521951-1.36475468e-02j,
        1.25521766-1.37739426e-02j,  1.25521577-1.39015088e-02j,
        1.25521385-1.40302563e-02j,  1.25521189-1.41601959e-02j,
        1.25520989-1.42913387e-02j,  1.25520786-1.44236959e-02j,
        1.25520579-1.45572786e-02j,  1.25520368-1.46920983e-02j,
        1.25520153-1.48281664e-02j,  1.25519934-1.49654944e-02j,
        1.25519711-1.51040939e-02j,  1.25519484-1.52439768e-02j,
        1.25519253-1.53851550e-02j,  1.25519017-1.55276403e-02j,
        1.25518777-1.56714450e-02j,  1.25518532-1.58165812e-02j,
        1.25518283-1.59630612e-02j,  1.2551803 -1.61108975e-02j,
        1.25517771-1.62601026e-02j,  1.25517508-1.64106892e-02j,
        1.2551724 -1.65626701e-02j,  1.25516967-1.67160581e-02j,
        1.25516689-1.68708663e-02j,  1.25516405-1.70271079e-02j,
        1.25516117-1.71847960e-02j,  1.25515823-1.73439442e-02j,
        1.25515523-1.75045658e-02j,  1.25515218-1.76666745e-02j,
        1.25514908-1.78302840e-02j,  1.25514591-1.79954083e-02j,
        1.25514269-1.81620614e-02j,  1.2551394 -1.83302574e-02j,
        1.25513606-1.85000106e-02j,  1.25513265-1.86713353e-02j,
        1.25512918-1.88442462e-02j,  1.25512564-1.90187578e-02j,
        1.25512204-1.91948851e-02j,  1.25511838-1.93726428e-02j,
        1.25511464-1.95520462e-02j,  1.25511083-1.97331105e-02j,
        1.25510696-1.99158509e-02j,  1.25510301-2.01002830e-02j,
        1.25509899-2.02864225e-02j,  1.25509489-2.04742850e-02j,
        1.25509072-2.06638867e-02j,  1.25508646-2.08552435e-02j,
        1.25508213-2.10483716e-02j,  1.25507772-2.12432875e-02j,
        1.25507323-2.14400077e-02j,  1.25506866-2.16385489e-02j,
        1.25506399-2.18389278e-02j,  1.25505925-2.20411616e-02j,
        1.25505441-2.22452672e-02j,  1.25504948-2.24512621e-02j,
        1.25504446-2.26591637e-02j,  1.25503935-2.28689897e-02j,
        1.25503415-2.30807577e-02j,  1.25502884-2.32944858e-02j,
        1.25502344-2.35101921e-02j,  1.25501794-2.37278949e-02j,
        1.25501233-2.39476125e-02j,  1.25500662-2.41693637e-02j,
        1.25500081-2.43931673e-02j,  1.25499488-2.46190421e-02j,
        1.25498885-2.48470073e-02j,  1.2549827 -2.50770823e-02j,
        1.25497644-2.53092866e-02j,  1.25497006-2.55436397e-02j,
        1.25496357-2.57801616e-02j,  1.25495695-2.60188723e-02j,
        1.25495021-2.62597921e-02j,  1.25494334-2.65029412e-02j,
        1.25493635-2.67483404e-02j,  1.25492923-2.69960104e-02j,
        1.25492197-2.72459721e-02j,  1.25491458-2.74982468e-02j,
        1.25490705-2.77528558e-02j,  1.25489938-2.80098206e-02j,
        1.25489157-2.82691631e-02j,  1.25488362-2.85309051e-02j,
        1.25487551-2.87950688e-02j,  1.25486726-2.90616765e-02j,
        1.25485885-2.93307510e-02j,  1.25485028-2.96023148e-02j,
        1.25484156-2.98763910e-02j,  1.25483267-3.01530027e-02j,
        1.25482362-3.04321735e-02j,  1.2548144 -3.07139268e-02j,
        1.254805  -3.09982866e-02j,  1.25479544-3.12852768e-02j,
        1.25478569-3.15749218e-02j,  1.25477577-3.18672460e-02j,
        1.25476565-3.21622741e-02j,  1.25475536-3.24600311e-02j,
        1.25474486-3.27605423e-02j,  1.25473418-3.30638328e-02j,
        1.25472329-3.33699285e-02j,  1.25471221-3.36788552e-02j,
        1.25470091-3.39906389e-02j,  1.25468941-3.43053060e-02j,
        1.25467769-3.46228832e-02j,  1.25466575-3.49433972e-02j,
        1.2546536 -3.52668750e-02j,  1.25464121-3.55933441e-02j,
        1.2546286 -3.59228320e-02j,  1.25461575-3.62553665e-02j,
        1.25460266-3.65909757e-02j,  1.25458933-3.69296879e-02j,
        1.25457575-3.72715316e-02j,  1.25456192-3.76165358e-02j,
        1.25454783-3.79647296e-02j,  1.25453347-3.83161423e-02j,
        1.25451885-3.86708036e-02j,  1.25450396-3.90287434e-02j,
        1.2544888 -3.93899918e-02j,  1.25447334-3.97545794e-02j,
        1.25445761-4.01225368e-02j,  1.25444158-4.04938951e-02j,
        1.25442525-4.08686856e-02j,  1.25440862-4.12469398e-02j,
        1.25439167-4.16286897e-02j,  1.25437442-4.20139674e-02j,
        1.25435684-4.24028053e-02j,  1.25433893-4.27952362e-02j,
        1.2543207 -4.31912931e-02j,  1.25430212-4.35910094e-02j,
        1.2542832 -4.39944186e-02j,  1.25426392-4.44015549e-02j,
        1.25424429-4.48124523e-02j,  1.25422429-4.52271455e-02j,
        1.25420392-4.56456693e-02j,  1.25418317-4.60680589e-02j,
        1.25416203-4.64943499e-02j,  1.2541405 -4.69245781e-02j,
        1.25411858-4.73587796e-02j,  1.25409624-4.77969908e-02j,
        1.25407349-4.82392487e-02j,  1.25405031-4.86855903e-02j,
        1.2540267 -4.91360532e-02j,  1.25400266-4.95906750e-02j,
        1.25397817-5.00494940e-02j,  1.25395322-5.05125487e-02j,
        1.2539278 -5.09798779e-02j,  1.25390192-5.14515208e-02j,
        1.25387555-5.19275170e-02j,  1.2538487 -5.24079063e-02j,
        1.25382134-5.28927289e-02j,  1.25379347-5.33820255e-02j,
        1.25376509-5.38758371e-02j,  1.25373618-5.43742050e-02j,
        1.25370673-5.48771710e-02j,  1.25367673-5.53847770e-02j,
        1.25364618-5.58970656e-02j,  1.25361506-5.64140796e-02j,
        1.25358335-5.69358623e-02j,  1.25355106-5.74624571e-02j,
        1.25351817-5.79939082e-02j,  1.25348467-5.85302599e-02j,
        1.25345054-5.90715570e-02j,  1.25341578-5.96178446e-02j,
        1.25338037-6.01691684e-02j,  1.25334431-6.07255743e-02j,
        1.25330757-6.12871087e-02j,  1.25327015-6.18538185e-02j,
        1.25323204-6.24257507e-02j,  1.25319321-6.30029531e-02j,
        1.25315367-6.35854737e-02j,  1.25311339-6.41733610e-02j,
        1.25307236-6.47666639e-02j,  1.25303057-6.53654317e-02j,
        1.252988  -6.59697142e-02j,  1.25294464-6.65795616e-02j,
        1.25290047-6.71950244e-02j,  1.25285548-6.78161539e-02j,
        1.25280966-6.84430016e-02j,  1.25276298-6.90756193e-02j,
        1.25271544-6.97140597e-02j,  1.25266701-7.03583755e-02j,
        1.25261768-7.10086201e-02j,  1.25256744-7.16648474e-02j,
        1.25251626-7.23271116e-02j,  1.25246413-7.29954675e-02j,
        1.25241103-7.36699703e-02j,  1.25235694-7.43506758e-02j,
        1.25230185-7.50376401e-02j,  1.25224574-7.57309200e-02j,
        1.25218858-7.64305725e-02j,  1.25213036-7.71366554e-02j,
        1.25207106-7.78492268e-02j,  1.25201065-7.85683454e-02j,
        1.25194913-7.92940703e-02j,  1.25188646-8.00264612e-02j,
        1.25182262-8.07655783e-02j,  1.2517576 -8.15114823e-02j,
        1.25169137-8.22642343e-02j,  1.25162391-8.30238961e-02j,
        1.2515552 -8.37905300e-02j,  1.25148521-8.45641987e-02j,
        1.25141392-8.53449656e-02j,  1.25134131-8.61328945e-02j,
        1.25126734-8.69280497e-02j,  1.25119201-8.77304962e-02j,
        1.25111527-8.85402996e-02j,  1.25103711-8.93575257e-02j,
        1.25095749-9.01822411e-02j,  1.2508764 -9.10145131e-02j,
        1.2507938 -9.18544092e-02j,  1.25070967-9.27019978e-02j,
        1.25062397-9.35573476e-02j,  1.25053669-9.44205280e-02j,
        1.25044778-9.52916090e-02j,  1.25035722-9.61706611e-02j,
        1.25026498-9.70577554e-02j,  1.25017102-9.79529637e-02j,
        1.25007532-9.88563581e-02j,  1.24997785-9.97680116e-02j,
        1.24987856-1.00687998e-01j,  1.24977744-1.01616390e-01j,
        1.24967443-1.02553264e-01j,  1.24956951-1.03498694e-01j,
        1.24946265-1.04452757e-01j,  1.2493538 -1.05415528e-01j,
        1.24924293-1.06387086e-01j,  1.24913   -1.07367506e-01j,
        1.24901498-1.08356869e-01j,  1.24889782-1.09355253e-01j,
        1.24877849-1.10362737e-01j,  1.24865694-1.11379401e-01j,
        1.24853314-1.12405327e-01j,  1.24840704-1.13440595e-01j,
        1.2482786 -1.14485289e-01j,  1.24814778-1.15539489e-01j,
        1.24801454-1.16603281e-01j,  1.24787882-1.17676748e-01j,
        1.24774058-1.18759974e-01j,  1.24759978-1.19853044e-01j,
        1.24745636-1.20956046e-01j,  1.24731029-1.22069064e-01j,
        1.24716151-1.23192187e-01j,  1.24700997-1.24325502e-01j,
        1.24685562-1.25469097e-01j,  1.24669841-1.26623063e-01j,
        1.24653828-1.27787488e-01j,  1.24637519-1.28962464e-01j,
        1.24620907-1.30148080e-01j,  1.24603987-1.31344430e-01j,
        1.24586754-1.32551606e-01j,  1.24569201-1.33769700e-01j,
        1.24551324-1.34998806e-01j,  1.24533114-1.36239019e-01j,
        1.24514568-1.37490435e-01j,  1.24495678-1.38753148e-01j,
        1.24476438-1.40027255e-01j,  1.24456841-1.41312854e-01j,
        1.24436882-1.42610041e-01j,  1.24416553-1.43918916e-01j,
        1.24395847-1.45239578e-01j,  1.24374758-1.46572126e-01j,
        1.24353279-1.47916660e-01j,  1.24331402-1.49273283e-01j,
        1.2430912 -1.50642094e-01j,  1.24286426-1.52023198e-01j,
        1.24263311-1.53416696e-01j,  1.24239769-1.54822693e-01j,
        1.24215791-1.56241293e-01j,  1.2419137 -1.57672600e-01j,
        1.24166497-1.59116721e-01j,  1.24141164-1.60573762e-01j,
        1.24115362-1.62043829e-01j,  1.24089083-1.63527030e-01j,
        1.24062318-1.65023474e-01j,  1.24035059-1.66533269e-01j,
        1.24007295-1.68056524e-01j,  1.23979018-1.69593349e-01j,
        1.23950219-1.71143856e-01j,  1.23920888-1.72708155e-01j,
        1.23891014-1.74286358e-01j,  1.23860589-1.75878577e-01j,
        1.23829602-1.77484926e-01j,  1.23798042-1.79105518e-01j,
        1.237659  -1.80740468e-01j,  1.23733164-1.82389889e-01j,
        1.23699824-1.84053897e-01j,  1.23665869-1.85732608e-01j,
        1.23631287-1.87426139e-01j,  1.23596067-1.89134606e-01j,
        1.23560197-1.90858126e-01j,  1.23523666-1.92596817e-01j,
        1.23486461-1.94350799e-01j,  1.23448569-1.96120188e-01j,
        1.2340998 -1.97905106e-01j,  1.23370679-1.99705671e-01j,
        1.23330653-2.01522004e-01j,  1.2328989 -2.03354225e-01j,
        1.23248377-2.05202455e-01j,  1.23206098-2.07066816e-01j,
        1.23163042-2.08947429e-01j,  1.23119192-2.10844417e-01j,
        1.23074535-2.12757902e-01j,  1.23029057-2.14688007e-01j,
        1.22982741-2.16634855e-01j,  1.22935574-2.18598569e-01j,
        1.22887539-2.20579273e-01j,  1.22838621-2.22577091e-01j,
        1.22788804-2.24592146e-01j,  1.22738072-2.26624564e-01j,
        1.22686407-2.28674468e-01j,  1.22633793-2.30741983e-01j,
        1.22580213-2.32827233e-01j,  1.22525649-2.34930344e-01j,
        1.22470084-2.37051440e-01j,  1.22413499-2.39190647e-01j,
        1.22355876-2.41348088e-01j,  1.22297196-2.43523888e-01j,
        1.2223744 -2.45718174e-01j,  1.22176589-2.47931068e-01j,
        1.22114623-2.50162695e-01j,  1.22051522-2.52413180e-01j,
        1.21987266-2.54682647e-01j,  1.21921833-2.56971219e-01j,
        1.21855202-2.59279021e-01j,  1.21787353-2.61606174e-01j,
        1.21718263-2.63952803e-01j,  1.21647909-2.66319028e-01j,
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       -0.02972678-8.39886465e-03j, -0.02921148-8.17371154e-03j,
       -0.02870459-7.95452242e-03j, -0.02820601-7.74114256e-03j,
       -0.02771561-7.53342113e-03j, -0.02723329-7.33121106e-03j,
       -0.02675891-7.13436897e-03j, -0.02629238-6.94275506e-03j,
       -0.02583357-6.75623308e-03j, -0.02538237-6.57467018e-03j,
       -0.02493867-6.39793689e-03j, -0.02450236-6.22590698e-03j,
       -0.02407334-6.05845745e-03j, -0.02365148-5.89546839e-03j,
       -0.02323669-5.73682296e-03j, -0.02282885-5.58240726e-03j,
       -0.02242787-5.43211032e-03j, -0.02203364-5.28582399e-03j,
       -0.02164605-5.14344287e-03j, -0.02126501-5.00486427e-03j,
       -0.02089041-4.86998813e-03j, -0.02052216-4.73871693e-03j,
       -0.02016016-4.61095569e-03j, -0.0198043 -4.48661185e-03j,
       -0.0194545 -4.36559523e-03j, -0.01911066-4.24781799e-03j,
       -0.01877268-4.13319453e-03j, -0.01844048-4.02164147e-03j,
       -0.01811396-3.91307761e-03j, -0.01779304-3.80742381e-03j,
       -0.01747761-3.70460302e-03j, -0.01716761-3.60454015e-03j,
       -0.01686293-3.50716208e-03j, -0.01656349-3.41239760e-03j,
       -0.01626922-3.32017733e-03j, -0.01598002-3.23043370e-03j,
       -0.01569582-3.14310092e-03j, -0.01541653-3.05811489e-03j,
       -0.01514208-2.97541320e-03j, -0.01487238-2.89493506e-03j,
       -0.01460735-2.81662129e-03j, -0.01434693-2.74041422e-03j,
       -0.01409104-2.66625773e-03j, -0.01383959-2.59409716e-03j,
       -0.01359253-2.52387926e-03j, -0.01334977-2.45555221e-03j,
       -0.01311124-2.38906554e-03j, -0.01287688-2.32437010e-03j,
       -0.01264661-2.26141804e-03j, -0.01242037-2.20016276e-03j,
       -0.0121981 -2.14055891e-03j, -0.01197971-2.08256231e-03j,
       -0.01176515-2.02612996e-03j, -0.01155436-1.97122000e-03j,
       -0.01134727-1.91779165e-03j, -0.01114382-1.86580525e-03j,
       -0.01094395-1.81522214e-03j, -0.01074759-1.76600473e-03j,
       -0.01055469-1.71811638e-03j, -0.0103652 -1.67152147e-03j,
       -0.01017904-1.62618528e-03j, -0.00999617-1.58207403e-03j,
       -0.00981653-1.53915485e-03j, -0.00964006-1.49739571e-03j,
       -0.00946671-1.45676546e-03j, -0.00929643-1.41723377e-03j,
       -0.00912916-1.37877110e-03j, -0.00896486-1.34134872e-03j,
       -0.00880347-1.30493866e-03j, -0.00864494-1.26951368e-03j,
       -0.00848923-1.23504728e-03j, -0.00833628-1.20151365e-03j,
       -0.00818604-1.16888770e-03j, -0.00803848-1.13714497e-03j,
       -0.00789354-1.10626169e-03j, -0.00775117-1.07621469e-03j,
       -0.00761135-1.04698145e-03j, -0.00747401-1.01854002e-03j,
       -0.00733912-9.90869070e-04j, -0.00720663-9.63947818e-04j,
       -0.00707651-9.37756045e-04j, -0.00694871-9.12274074e-04j,
       -0.00682319-8.87482752e-04j, -0.00669991-8.63363442e-04j,
       -0.00657883-8.39898004e-04j, -0.00645992-8.17068784e-04j,
       -0.00634314-7.94858604e-04j, -0.00622844-7.73250745e-04j,
       -0.00611579-7.52228934e-04j, -0.00600517-7.31777340e-04j,
       -0.00589652-7.11880552e-04j, -0.00578982-6.92523577e-04j,
       -0.00568503-6.73691820e-04j, -0.00558212-6.55371083e-04j,
       -0.00548106-6.37547547e-04j, -0.00538181-6.20207763e-04j,
       -0.00528434-6.03338648e-04j, -0.00518862-5.86927467e-04j,
       -0.00509462-5.70961830e-04j, -0.0050023 -5.55429678e-04j,
       -0.00491165-5.40319279e-04j, -0.00482263-5.25619215e-04j,
       -0.0047352 -5.11318378e-04j, -0.00464935-4.97405957e-04j,
       -0.00456505-4.83871432e-04j, -0.00448226-4.70704569e-04j,
       -0.00440096-4.57895408e-04j, -0.00432113-4.45434257e-04j,
       -0.00424273-4.33311688e-04j, -0.00416574-4.21518524e-04j,
       -0.00409015-4.10045837e-04j, -0.00401591-3.98884942e-04j,
       -0.00394302-3.88027385e-04j, -0.00387144-3.77464943e-04j,
       -0.00380115-3.67189613e-04j, -0.00373212-3.57193609e-04j,
       -0.00366435-3.47469355e-04j, -0.0035978 -3.38009481e-04j,
       -0.00353244-3.28806814e-04j, -0.00346827-3.19854377e-04j,
       -0.00340526-3.11145380e-04j, -0.00334339-3.02673217e-04j,
       -0.00328263-2.94431460e-04j, -0.00322298-2.86413858e-04j,
       -0.0031644 -2.78614325e-04j, -0.00310688-2.71026941e-04j,
       -0.0030504 -2.63645948e-04j, -0.00299494-2.56465741e-04j,
       -0.00294049-2.49480869e-04j, -0.00288702-2.42686026e-04j,
       -0.00283452-2.36076054e-04j, -0.00278297-2.29645929e-04j,
       -0.00273235-2.23390768e-04j, -0.00268265-2.17305818e-04j,
       -0.00263385-2.11386454e-04j, -0.00258593-2.05628178e-04j,
       -0.00253888-2.00026613e-04j, -0.00249269-1.94577501e-04j,
       -0.00244733-1.89276698e-04j, -0.00240279-1.84120175e-04j,
       -0.00235906-1.79104011e-04j, -0.00231612-1.74224389e-04j,
       -0.00227396-1.69477599e-04j, -0.00223257-1.64860030e-04j,
       -0.00219193-1.60368169e-04j, -0.00215202-1.55998599e-04j,
       -0.00211284-1.51747993e-04j, -0.00207437-1.47613118e-04j,
       -0.0020366 -1.43590826e-04j, -0.00199951-1.39678057e-04j,
       -0.00196309-1.35871831e-04j, -0.00192734-1.32169251e-04j,
       -0.00189224-1.28567497e-04j, -0.00185777-1.25063828e-04j,
       -0.00182393-1.21655576e-04j, -0.0017907 -1.18340144e-04j,
       -0.00175808-1.15115007e-04j, -0.00172605-1.11977710e-04j,
       -0.0016946 -1.08925862e-04j, -0.00166373-1.05957138e-04j,
       -0.00163341-1.03069276e-04j, -0.00160365-1.00260077e-04j,
       -0.00157442-9.75273989e-05j, -0.00154573-9.48691603e-05j,
       -0.00151756-9.22833349e-05j, -0.0014899 -8.97679520e-05j,
       -0.00146275-8.73210944e-05j, -0.00143609-8.49408969e-05j])

• We can use abs(...) to find the magnitude of the complex numbers. If $z = a+j\,b$, abs(z) is equivalent to $\sqrt{a^2 + b^2}$.

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 , 1.25529323, 1.25529275, 1.25529226,
       1.25529176, 1.25529125, 1.25529073, 1.2552902 , 1.25528966,
       1.25528911, 1.25528855, 1.25528798, 1.2552874 , 1.25528681,
       1.25528621, 1.2552856 , 1.25528497, 1.25528434, 1.25528369,
       1.25528303, 1.25528236, 1.25528167, 1.25528097, 1.25528026,
       1.25527954, 1.2552788 , 1.25527805, 1.25527729, 1.25527651,
       1.25527572, 1.25527491, 1.25527408, 1.25527324, 1.25527239,
       1.25527152, 1.25527063, 1.25526973, 1.25526881, 1.25526787,
       1.25526692, 1.25526595, 1.25526496, 1.25526395, 1.25526292,
       1.25526188, 1.25526081, 1.25525972, 1.25525862, 1.25525749,
       1.25525634, 1.25525518, 1.25525398, 1.25525277, 1.25525154,
       1.25525028, 1.255249  , 1.25524769, 1.25524636, 1.25524501,
       1.25524363, 1.25524222, 1.25524079, 1.25523933, 1.25523785,
       1.25523633, 1.25523479, 1.25523322, 1.25523162, 1.25522999,
       1.25522833, 1.25522664, 1.25522492, 1.25522317, 1.25522138,
       1.25521956, 1.25521771, 1.25521582, 1.2552139 , 1.25521194,
       1.25520995, 1.25520791, 1.25520584, 1.25520373, 1.25520159,
       1.2551994 , 1.25519717, 1.2551949 , 1.25519259, 1.25519023,
       1.25518783, 1.25518539, 1.2551829 , 1.25518036, 1.25517778,
       1.25517515, 1.25517247, 1.25516974, 1.25516696, 1.25516413,
       1.25516124, 1.2551583 , 1.25515531, 1.25515226, 1.25514915,
       1.25514599, 1.25514277, 1.25513948, 1.25513614, 1.25513274,
       1.25512927, 1.25512573, 1.25512213, 1.25511847, 1.25511473,
       1.25511093, 1.25510705, 1.2551031 , 1.25509908, 1.25509499,
       1.25509081, 1.25508656, 1.25508224, 1.25507783, 1.25507334,
       1.25506876, 1.2550641 , 1.25505935, 1.25505452, 1.25504959,
       1.25504458, 1.25503947, 1.25503426, 1.25502896, 1.25502356,
       1.25501806, 1.25501245, 1.25500674, 1.25500093, 1.25499501,
       1.25498897, 1.25498283, 1.25497657, 1.25497019, 1.2549637 ,
       1.25495708, 1.25495034, 1.25494348, 1.25493649, 1.25492937,
       1.25492211, 1.25491472, 1.25490719, 1.25489953, 1.25489172,
       1.25488376, 1.25487566, 1.2548674 , 1.254859  , 1.25485043,
       1.25484171, 1.25483282, 1.25482377, 1.25481455, 1.25480516,
       1.25479559, 1.25478585, 1.25477592, 1.25476581, 1.25475551,
       1.25474502, 1.25473433, 1.25472345, 1.25471236, 1.25470106,
       1.25468956, 1.25467784, 1.2546659 , 1.25465374, 1.25464136,
       1.25462874, 1.25461589, 1.2546028 , 1.25458947, 1.25457589,
       1.25456205, 1.25454796, 1.2545336 , 1.25451898, 1.25450409,
       1.25448891, 1.25447346, 1.25445772, 1.25444168, 1.25442535,
       1.25440871, 1.25439176, 1.2543745 , 1.25435692, 1.254339  ,
       1.25432076, 1.25430217, 1.25428324, 1.25426396, 1.25424432,
       1.25422431, 1.25420393, 1.25418317, 1.25416202, 1.25414048,
       1.25411853, 1.25409618, 1.25407342, 1.25405023, 1.2540266 ,
       1.25400254, 1.25397803, 1.25395306, 1.25392763, 1.25390172,
       1.25387533, 1.25384845, 1.25382107, 1.25379318, 1.25376477,
       1.25373583, 1.25370636, 1.25367633, 1.25364574, 1.25361459,
       1.25358285, 1.25355052, 1.25351759, 1.25348405, 1.25344988,
       1.25341508, 1.25337962, 1.25334351, 1.25330672, 1.25326925,
       1.25323108, 1.2531922 , 1.2531526 , 1.25311225, 1.25307116,
       1.2530293 , 1.25298666, 1.25294323, 1.25289898, 1.25285391,
       1.252808  , 1.25276124, 1.25271361, 1.25266508, 1.25261566,
       1.25256531, 1.25251402, 1.25246178, 1.25240856, 1.25235436,
       1.25229914, 1.25224289, 1.2521856 , 1.25212723, 1.25206778,
       1.25200722, 1.25194553, 1.2518827 , 1.25181869, 1.25175348,
       1.25168707, 1.25161941, 1.25155049, 1.25148029, 1.25140877,
       1.25133593, 1.25126172, 1.25118614, 1.25110914, 1.25103071,
       1.25095081, 1.25086942, 1.25078652, 1.25070207, 1.25061605,
       1.25052842, 1.25043915, 1.25034822, 1.2502556 , 1.25016124,
       1.25006513, 1.24996722, 1.24986749, 1.24976589, 1.2496624 ,
       1.24955698, 1.24944959, 1.2493402 , 1.24922877, 1.24911525,
       1.24899962, 1.24888183, 1.24876184, 1.24863961, 1.2485151 ,
       1.24838826, 1.24825906, 1.24812744, 1.24799337, 1.24785679,
       1.24771767, 1.24757594, 1.24743157, 1.2472845 , 1.24713469,
       1.24698207, 1.24682661, 1.24666824, 1.24650692, 1.24634258,
       1.24617517, 1.24600463, 1.24583091, 1.24565394, 1.24547366,
       1.24529001, 1.24510293, 1.24491236, 1.24471822, 1.24452045,
       1.24431899, 1.24411376, 1.24390469, 1.24369172, 1.24347476,
       1.24325375, 1.2430286 , 1.24279924, 1.24256559, 1.24232757,
       1.24208511, 1.2418381 , 1.24158647, 1.24133014, 1.24106901,
       1.240803  , 1.240532  , 1.24025594, 1.23997471, 1.23968822,
       1.23939637, 1.23909905, 1.23879617, 1.23848762, 1.23817329,
       1.23785308, 1.23752687, 1.23719456, 1.23685602, 1.23651115,
       1.23615982, 1.23580191, 1.23543729, 1.23506585, 1.23468745,
       1.23430197, 1.23390926, 1.2335092 , 1.23310165, 1.23268646,
       1.2322635 , 1.23183261, 1.23139365, 1.23094647, 1.23049091,
       1.23002682, 1.22955403, 1.22907238, 1.22858172, 1.22808186,
       1.22757264, 1.22705387, 1.2265254 , 1.22598702, 1.22543856,
       1.22487982, 1.22431062, 1.22373076, 1.22314004, 1.22253826,
       1.22192521, 1.22130068, 1.22066446, 1.22001633, 1.21935607,
       1.21868345, 1.21799825, 1.21730022, 1.21658913, 1.21586475,
       1.21512681, 1.21437508, 1.2136093 , 1.2128292 , 1.21203452,
       1.211225  , 1.21040035, 1.20956031, 1.20870458, 1.20783288,
       1.20694493, 1.2060404 , 1.20511902, 1.20418046, 1.20322442,
       1.20225057, 1.20125859, 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 ])

• We can use np.angle(...) to find the phase of the complex numbers. If $z = a+j\,b$, np.angle(z) is equivalent to $\arctan(b/a)$.

np.angle(vRatio)

array([-0.00515221, -0.00519993, -0.00524809, -0.0052967 , -0.00534576,
       -0.00539528, -0.00544525, -0.00549568, -0.00554659, -0.00559796,
       -0.00564981, -0.00570214, -0.00575495, -0.00580826, -0.00586205,
       -0.00591635, -0.00597115, -0.00602645, -0.00608227, -0.00613861,
       -0.00619546, -0.00625285, -0.00631076, -0.00636921, -0.00642821,
       -0.00648774, -0.00654784, -0.00660848, -0.00666969, -0.00673147,
       -0.00679382, -0.00685674, -0.00692025, -0.00698435, -0.00704904,
       -0.00711433, -0.00718022, -0.00724672, -0.00731385, -0.00738159,
       -0.00744996, -0.00751896, -0.0075886 , -0.00765889, -0.00772983,
       -0.00780142, -0.00787368, -0.00794661, -0.00802021, -0.0080945 ,
       -0.00816947, -0.00824514, -0.0083215 , -0.00839858, -0.00847637,
       -0.00855488, -0.00863411, -0.00871408, -0.0087948 , -0.00887626,
       -0.00895847, -0.00904144, -0.00912519, -0.00920971, -0.00929501,
       -0.0093811 , -0.00946799, -0.00955568, -0.00964419, -0.00973352,
       -0.00982367, -0.00991466, -0.01000649, -0.01009917, -0.01019271,
       -0.01028712, -0.0103824 , -0.01047856, -0.01057562, -0.01067357,
       -0.01077243, -0.01087221, -0.01097291, -0.01107454, -0.01117712,
       -0.01128064, -0.01138512, -0.01149058, -0.011597  , -0.01170442,
       -0.01181283, -0.01192224, -0.01203266, -0.01214411, -0.01225659,
       -0.01237012, -0.01248469, -0.01260033, -0.01271703, -0.01283482,
       -0.0129537 , -0.01307368, -0.01319477, -0.01331698, -0.01344032,
       -0.01356481, -0.01369045, -0.01381725, -0.01394523, -0.0140744 ,
       -0.01420475, -0.01433632, -0.01446911, -0.01460312, -0.01473838,
       -0.01487489, -0.01501266, -0.01515171, -0.01529205, -0.01543369,
       -0.01557664, -0.01572091, -0.01586652, -0.01601348, -0.0161618 ,
       -0.01631149, -0.01646257, -0.01661505, -0.01676894, -0.01692426,
       -0.01708101, -0.01723922, -0.01739889, -0.01756004, -0.01772269,
       -0.01788684, -0.01805251, -0.01821971, -0.01838847, -0.01855879,
       -0.01873068, -0.01890417, -0.01907926, -0.01925598, -0.01943433,
       -0.01961433, -0.019796  , -0.01997936, -0.02016441, -0.02035117,
       -0.02053967, -0.02072991, -0.02092191, -0.0211157 , -0.02131127,
       -0.02150866, -0.02170788, -0.02190894, -0.02211186, -0.02231667,
       -0.02252337, -0.02273198, -0.02294253, -0.02315502, -0.02336949,
       -0.02358594, -0.0238044 , -0.02402488, -0.0242474 , -0.02447198,
       -0.02469864, -0.02492741, -0.02515829, -0.02539131, -0.02562649,
       -0.02586384, -0.0261034 , -0.02634517, -0.02658918, -0.02683545,
       -0.02708401, -0.02733486, -0.02758804, -0.02784356, -0.02810145,
       -0.02836173, -0.02862442, -0.02888955, -0.02915712, -0.02942718,
       -0.02969974, -0.02997482, -0.03025245, -0.03053265, -0.03081545,
       -0.03110087, -0.03138892, -0.03167965, -0.03197307, -0.03226921,
       -0.03256809, -0.03286974, -0.03317418, -0.03348144, -0.03379155,
       -0.03410453, -0.03442041, -0.03473922, -0.03506097, -0.03538571,
       -0.03571346, -0.03604424, -0.03637808, -0.03671502, -0.03705507,
       -0.03739828, -0.03774467, -0.03809426, -0.03844709, -0.03880319,
       -0.03916259, -0.03952531, -0.0398914 , -0.04026087, -0.04063377,
       -0.04101012, -0.04138996, -0.04177332, -0.04216022, -0.04255071,
       -0.04294482, -0.04334257, -0.04374401, -0.04414917, -0.04455808,
       -0.04497078, -0.0453873 , -0.04580767, -0.04623195, -0.04666015,
       -0.04709231, -0.04752848, -0.04796869, -0.04841297, -0.04886137,
       -0.04931392, -0.04977067, -0.05023164, -0.05069689, -0.05116644,
       -0.05164034, -0.05211863, -0.05260135, -0.05308854, -0.05358024,
       -0.0540765 , -0.05457735, -0.05508284, -0.05559301, -0.05610791,
       -0.05662758, -0.05715205, -0.05768139, -0.05821563, -0.05875481,
       -0.05929899, -0.05984821, -0.06040252, -0.06096196, -0.06152658,
       -0.06209643, -0.06267155, -0.06325201, -0.06383784, -0.06442909,
       -0.06502582, -0.06562807, -0.06623591, -0.06684937, -0.06746851,
       -0.06809339, -0.06872406, -0.06936056, -0.07000296, -0.07065131,
       -0.07130566, -0.07196607, -0.0726326 , -0.0733053 , -0.07398423,
       -0.07466945, -0.07536101, -0.07605898, -0.07676341, -0.07747437,
       -0.07819191, -0.07891609, -0.07964698, -0.08038464, -0.08112912,
       -0.08188051, -0.08263885, -0.08340421, -0.08417666, -0.08495626,
       -0.08574308, -0.08653719, -0.08733865, -0.08814754, -0.08896391,
       -0.08978784, -0.09061941, -0.09145867, -0.0923057 , -0.09316058,
       -0.09402337, -0.09489415, -0.095773  , -0.09665998, -0.09755517,
       -0.09845866, -0.09937051, -0.1002908 , -0.10121961, -0.10215703,
       -0.10310312, -0.10405797, -0.10502167, -0.10599428, -0.1069759 ,
       -0.10796661, -0.1089665 , -0.10997564, -0.11099412, -0.11202203,
       -0.11305946, -0.11410649, -0.11516322, -0.11622973, -0.11730611,
       -0.11839246, -0.11948887, -0.12059542, -0.12171222, -0.12283936,
       -0.12397693, -0.12512504, -0.12628377, -0.12745323, -0.12863351,
       -0.12982472, -0.13102695, -0.13224032, -0.13346491, -0.13470084,
       -0.13594821, -0.13720713, -0.13847769, -0.13976002, -0.14105422,
       -0.14236039, -0.14367865, -0.14500912, -0.14635189, -0.1477071 ,
       -0.14907484, -0.15045525, -0.15184843, -0.1532545 , -0.15467358,
       -0.15610579, -0.15755126, -0.1590101 , -0.16048244, -0.1619684 ,
       -0.16346811, -0.1649817 , -0.16650929, -0.16805102, -0.16960701,
       -0.17117739, -0.1727623 , -0.17436187, -0.17597624, -0.17760554,
       -0.17924992, -0.1809095 , -0.18258444, -0.18427486, -0.18598092,
       -0.18770276, -0.18944052, -0.19119435, -0.1929644 , -0.19475082,
       -0.19655376, -0.19837336, -0.20020979, -0.2020632 , -0.20393374,
       -0.20582158, -0.20772687, -0.20964976, -0.21159044, -0.21354904,
       -0.21552576, -0.21752073, -0.21953415, -0.22156617, -0.22361696,
       -0.22568671, -0.22777557, -0.22988373, -0.23201137, -0.23415866,
       -0.23632578, -0.23851292, -0.24072026, -0.24294797, -0.24519626,
       -0.2474653 , -0.2497553 , -0.25206643, -0.25439889, -0.25675288,
       -0.25912859, -0.26152623, -0.26394598, -0.26638806, -0.26885267,
       -0.27134001, -0.27385028, -0.27638371, -0.27894049, -0.28152085,
       -0.28412499, -0.28675313, -0.28940549, -0.29208229, -0.29478374,
       -0.29751008, -0.30026153, -0.30303832, -0.30584067, -0.30866881,
       -0.31152299, -0.31440343, -0.31731036, -0.32024404, -0.32320469,
       -0.32619257, -0.32920791, -0.33225096, -0.33532197, -0.33842119,
       -0.34154888, -0.34470528, -0.34789066, -0.35110526, -0.35434936,
       -0.35762321, -0.36092708, -0.36426123, -0.36762593, -0.37102146,
       -0.37444808, -0.37790606, -0.38139569, -0.38491724, -0.38847099,
       -0.39205722, -0.39567622, -0.39932827, -0.40301366, -0.40673268,
       -0.41048562, -0.41427278, -0.41809444, -0.42195091, -0.42584248,
       -0.42976945, -0.43373214, -0.43773083, -0.44176584, -0.44583748,
       -0.44994605, -0.45409186, -0.45827523, -0.46249647, -0.4667559 ,
       -0.47105383, -0.47539058, -0.47976647, -0.48418183, -0.48863697,
       -0.49313222, -0.4976679 , -0.50224434, -0.50686187, -0.51152081,
       -0.5162215 , -0.52096426, -0.52574943, -0.53057733, -0.53544829,
       -0.54036266, -0.54532076, -0.55032292, -0.55536949, -0.56046078,
       -0.56559714, -0.5707789 , -0.57600639, -0.58127994, -0.58659989,
       -0.59196656, -0.59738029, -0.60284141, -0.60835025, -0.61390713,
       -0.61951237, -0.62516631, -0.63086927, -0.63662156, -0.6424235 ,
       -0.64827541, -0.65417761, -0.66013039, -0.66613408, -0.67218896,
       -0.67829536, -0.68445355, -0.69066383, -0.69692649, -0.70324182,
       -0.70961009, -0.71603158, -0.72250655, -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])

• Here's a plot of the magnitude of the voltge ratio.

plt.loglog(ff, abs(vRatio), 'k-', linewidth = 2.5)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Voltage Ratio')
plt.axis((10, 1e5, 1e-3, 2));

• Here's a plot of the phase in radians scaled by $\pi$.

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));