This notebook contains an interactive visualization of the frequency dependent impedance of a simple electrical circuit (usually referred to as an equivalent circuit, or an equivalent circuit model in electrochemistry), -R-(RC)-. The data is displayed in a complex plane plot (a Nyquist plot) and a Bode plot. The latter is usually a plot of the modulus of the impedance, and the phase angle, vs the logarithm of the frequency, but in this case I have chosen to plot the (negative of the) imaginary part of the complex impedance vs the logarithm of the frequency.
The notebook is written for use with Python 3 and uses ipywidgets for interactivity.
In [1]:
#python dependencies
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from ipywidgets import interact, FloatSlider, RadioButtons
#change the default font set (matplotlib.rc)
mpl.rc('mathtext', fontset='stixsans', default='regular')
#increase text size somewhat
mpl.rcParams.update({'axes.labelsize':12, 'font.size': 12})
In [2]:
#set up notebook for inline plots
%matplotlib inline
In [3]:
#generate an angular frequency range
def ang_freq_range(fstart = 9685000, fstop = 0.0968, pts_per_decade = 12):
"""returns an angular frequency range as a numpy array,
between fstart [Hz] and fstop [Hz], with a set number
of points per decade (defaults to 12)"""
decades = np.log10(fstart)-np.log10(fstop)
pts_total = np.around(decades*pts_per_decade)
frange = np.logspace(np.log10(fstop),np.log10(fstart), pts_total, endpoint=True)
return 2*np.pi*frange
w = ang_freq_range()
In [4]:
#define function that returns impedance of -R-(RC)- circuit
def Z_R_RC(R0, R1, C1, w):
"""Returns the impedance of a -R-(RC)- circuit.
Input
=====
R0 = series resistance (Ohmic resistance) of circuit
R1 = resistance of parallel connected circuit element
C1 = capacitance of parallel connected circuit element
w = angular frequency, accepts an array as well as a single number
Output
======
The frequency dependent impedance as a complex number."""
Z_R0 = R0
Z_R1 = R1
Z_C1 = -1j/(w*C1) #capacitive reactance
Z_RC = 1/(1/Z_R1 + 1/Z_C1) #parallel connection
return Z_R0 + Z_RC #Z_R0 and Z_RC connected in series
In [5]:
#define 2D plot function(Nyquist / Complex Plane)
def plot_nyquist(R0,R1,C1):
Z = Z_R_RC(R0,R1,C1,np.logspace(7,0,7*12))
#set up a figure canvas with two plot areas (sets of axes)
fig,ax = plt.subplots(nrows=2, ncols=1)
#add a Nyquist plot (first plot)
ax[0].plot(Z.real, -1*Z.imag, marker='o',ms=5, mec='b', mew=0.7, mfc='none')
ax[0].set_xlim(0,60)
ax[0].set_ylim(0,25)
ax[0].set_aspect('equal')
ax[0].set_xlabel('Z$_{real}$ [$\Omega$]')
ax[0].set_ylabel('-Z$_{imag}$ [$\Omega$]')
#add a Bode plot with
ax[1].plot(np.logspace(7,0,7*12)/2*np.pi,-1*Z.imag, marker='o',ms=5, mec='b', mew=0.7, mfc='none')
ax[1].set_xscale("log")
ax[1].set_xlim(min(w),max(w))
ax[1].set_ylim()
ax[1].set_ylabel('-Z$_{imag}$ [$\Omega$]')
ax[1].set_xlabel('frequency [Hz]')
plt.tight_layout()
In [6]:
interact(plot_nyquist, R0=(1,20), R1=(1,40), C1=(1e-6, 1e-4, 1e-6))
Out[6]:
JR Johansson's excellent version information extension (see here for install and usage instructions):
In [7]:
%load_ext version_information
In [8]:
%version_information numpy, matplotlib, ipywidgets
Out[8]: