Source code for impedance.models.circuits.elements

import numpy as np

initial_state = globals().copy()
non_element_functions = ['element_metadata',
                         'initial_state',
                         'non_element_functions',
                         'typeChecker',
                         'circuit_elements']
# populated at the end of the file -
# this maps ex. 'R' to the function R to always give us a list of
# active elements in any context
circuit_elements = {}


[docs]def element_metadata(num_params, units): """ decorator to store metadata for a circuit element Parameters ---------- num_params : int number of parameters for an element units : list of str list of units for the element parameters """ def decorator(func): def wrapper(p, f): typeChecker(p, f, func.__name__, num_params) return func(p, f) wrapper.num_params = num_params wrapper.units = units wrapper.__name__ = func.__name__ wrapper.__doc__ = func.__doc__ return wrapper return decorator
[docs]def s(series): """ sums elements in series Notes --------- .. math:: Z = Z_1 + Z_2 + ... + Z_n """ z = len(series[0])*[0 + 0*1j] for elem in series: z += elem return z
[docs]def p(parallel): """ adds elements in parallel Notes --------- .. math:: Z = \\frac{1}{\\frac{1}{Z_1} + \\frac{1}{Z_2} + ... + \\frac{1}{Z_n}} """ z = len(parallel[0])*[0 + 0*1j] for elem in parallel: z += 1/elem return 1/z
[docs]@element_metadata(num_params=1, units=['Ohm']) def R(p, f): """ defines a resistor Notes --------- .. math:: Z = R """ R = p[0] Z = np.array(len(f)*[R]) return Z
[docs]@element_metadata(num_params=1, units=['F']) def C(p, f): """ defines a capacitor .. math:: Z = \\frac{1}{C \\times j 2 \\pi f} """ omega = 2*np.pi*np.array(f) C = p[0] Z = 1.0/(C*1j*omega) return Z
[docs]@element_metadata(num_params=1, units=['H']) def L(p, f): """ defines an inductor .. math:: Z = L \\times j 2 \\pi f """ omega = 2*np.pi*np.array(f) L = p[0] Z = L*1j*omega return Z
[docs]@element_metadata(num_params=1, units=['Ohm sec^-1/2']) def W(p, f): """ defines a semi-infinite Warburg element Notes ----- .. math:: Z = \\frac{A_W}{\\sqrt{ 2 \\pi f}} (1-j) """ omega = 2*np.pi*np.array(f) Aw = p[0] Z = Aw*(1-1j)/np.sqrt(omega) return Z
[docs]@element_metadata(num_params=2, units=['Ohm', 'sec']) def Wo(p, f): """ defines an open (finite-space) Warburg element Notes --------- .. math:: Z = \\frac{Z_0}{\\sqrt{ j \\omega \\tau }} \\coth{\\sqrt{j \\omega \\tau }} where :math:`Z_0` = p[0] (Ohms) and :math:`\\tau` = p[1] (sec) = :math:`\\frac{L^2}{D}` """ omega = 2*np.pi*np.array(f) Z0, tau = p[0], p[1] Z = Z0/(np.sqrt(1j*omega*tau)*np.tanh(np.sqrt(1j*omega*tau))) return Z # Zw(omega)
[docs]@element_metadata(num_params=2, units=['Ohm', 'sec']) def Ws(p, f): """ defines a short (finite-length) Warburg element Notes --------- .. math:: Z = \\frac{Z_0}{\\sqrt{ j \\omega \\tau }} \\tanh{\\sqrt{j \\omega \\tau }} where :math:`Z_0` = p[0] (Ohms) and :math:`\\tau` = p[1] (sec) = :math:`\\frac{L^2}{D}` """ omega = 2*np.pi*np.array(f) Z0, tau = p[0], p[1] Z = Z0*np.tanh(np.sqrt(1j*omega*tau))/np.sqrt(1j*omega*tau) return Z
[docs]@element_metadata(num_params=2, units=['Ohm^-1 sec^a', '']) def CPE(p, f): """ defines a constant phase element Notes ----- .. math:: Z = \\frac{1}{Q \\times (j 2 \\pi f)^\\alpha} where :math:`Q` = p[0] and :math:`\\alpha` = p[1]. """ omega = 2*np.pi*np.array(f) Q, alpha = p[0], p[1] Z = 1.0/(Q*(1j*omega)**alpha) return Z
[docs]@element_metadata(num_params=2, units=['H sec', '']) def La(p, f): """ defines a modified inductance element as represented in [1] Notes ----- .. math:: Z = L \\times (j 2 \\pi f)^\\alpha where :math:`L` = p[0] and :math:`\\alpha` = p[1] [1] `EC-Lab Application Note 42, BioLogic Instruments (2019) <https://www.biologic.net/documents/battery-eis-modified-inductance-element-electrochemsitry-application-note-42>`_. """ omega = 2*np.pi*np.array(f) L, alpha = p[0], p[1] Z = (L*1j*omega)**alpha return Z
[docs]@element_metadata(num_params=2, units=['Ohm', 'sec']) def G(p, f): """ defines a Gerischer Element as represented in [1] Notes --------- .. math:: Z = \\frac{R_G}{\\sqrt{1 + j \\, 2 \\pi f \\, t_G}} where :math:`R_G` = p[0] and :math:`t_G` = p[1] Gerischer impedance is also commonly represented as [2]: .. math:: Z = \\frac{Z_o}{\\sqrt{K+ j \\, 2 \\pi f}} where :math:`Z_o = \\frac{R_G}{\\sqrt{t_G}}` and :math:`K = \\frac{1}{t_G}` with units :math:`\\Omega sec^{1/2}` and :math:`sec^{-1}` , respectively. [1] Y. Lu, C. Kreller, and S.B. Adler, Journal of The Electrochemical Society, 156, B513-B525 (2009) `doi:10.1149/1.3079337 <https://doi.org/10.1149/1.3079337>`_. [2] M. González-Cuenca, W. Zipprich, B.A. Boukamp, G. Pudmich, and F. Tietz, Fuel Cells, 1, 256-264 (2001) `doi:10.1016/0013-4686(93)85083-B <https://doi.org/10.1016/0013-4686(93)85083-B>`_. """ omega = 2*np.pi*np.array(f) R_G, t_G = p[0], p[1] Z = R_G/np.sqrt(1 + 1j*omega*t_G) return Z
[docs]@element_metadata(num_params=3, units=['Ohm', 'sec', '']) def Gs(p, f): """ defines a finite-length Gerischer Element as represented in [1] Notes --------- .. math:: Z = \\frac{R_G}{\\sqrt{1 + j \\, 2 \\pi f \\, t_G} \\, tanh(\\phi \\sqrt{1 + j \\, 2 \\pi f \\, t_G})} where :math:`R_G` = p[0], :math:`t_G` = p[1] and :math:`\\phi` = p[2] [1] R.D. Green, C.C Liu, and S.B. Adler, Solid State Ionics, 179, 647-660 (2008) `doi:10.1016/j.ssi.2008.04.024 <https://doi.org/10.1016/j.ssi.2008.04.024>`_. """ omega = 2*np.pi*np.array(f) R_G, t_G, phi = p[0], p[1], p[2] Z = R_G/(np.sqrt(1 + 1j*omega*t_G) * np.tanh(phi * np.sqrt(1 + 1j*omega*t_G))) return Z
[docs]@element_metadata(num_params=2, units=['Ohm', 'sec']) def K(p, f): """ An RC element for use in lin-KK model Notes ----- .. math:: Z = \\frac{R}{1 + j \\omega \\tau_k} """ omega = 2*np.pi*np.array(f) R, tau_k = p[0], p[1] Z = R/(1 + 1j*omega*tau_k) return Z
[docs]@element_metadata(num_params=3, units=['Ohm', 'F sec^(gamma - 1)', '']) def TLMQ(p, f): """ Simplified transmission-line model as defined in Eq. 11 of [1] Notes ----- .. math:: Z = \\sqrt{R_{ion}Z_{S}} \\coth \\sqrt{\\frac{R_{ion}}{Z_{S}} [1] J. Landesfeind et al., Journal of The Electrochemical Society, 163 (7) A1373-A1387 (2016) `doi: 10.1016/10.1149/2.1141607jes <http://doi.org/10.1149/2.1141607jes>`_. """ omega = 2*np.pi*np.array(f) Rion, Qs, gamma = p[0], p[1], p[2] Zs = Qs*(1j*omega)**gamma Z = np.sqrt(Rion/Zs)/np.tanh(np.sqrt(Rion*Zs)) return Z
[docs]@element_metadata(num_params=4, units=['Ohm-m^2', 'Ohm-m^2', '', 'sec']) def T(p, f): """ A macrohomogeneous porous electrode model from Paasch et al. [1] Notes ----- .. math:: Z = A\\frac{\\coth{\\beta}}{\\beta} + B\\frac{1}{\\beta\\sinh{\\beta}} where .. math:: A = d\\frac{\\rho_1^2 + \\rho_2^2}{\\rho_1 + \\rho_2} \\quad B = d\\frac{2 \\rho_1 \\rho_2}{\\rho_1 + \\rho_2} and .. math:: \\beta = (a + j \\omega b)^{1/2} \\quad a = \\frac{k d^2}{K} \\quad b = \\frac{d^2}{K} [1] G. Paasch, K. Micka, and P. Gersdorf, Electrochimica Acta, 38, 2653–2662 (1993) `doi: 10.1016/0013-4686(93)85083-B <https://doi.org/10.1016/0013-4686(93)85083-B>`_. """ omega = 2*np.pi*np.array(f) A, B, a, b = p[0], p[1], p[2], p[3] beta = (a + 1j*omega*b)**(1/2) sinh = [] for x in beta: if x < 100: sinh.append(np.sinh(x)) else: sinh.append(1e10) Z = A/(beta*np.tanh(beta)) + B/(beta*np.array(sinh)) return Z
circuit_elements = {key: eval(key) for key in set(globals())-set(initial_state) if key not in non_element_functions} def get_element_from_name(name): excluded_chars = '0123456789_' return ''.join(char for char in name if char not in excluded_chars) def typeChecker(p, f, name, length): assert isinstance(p, list), \ 'in {}, input must be of type list'.format(name) for i in p: assert isinstance(i, (float, int, np.int32, np.float64)), \ 'in {}, value {} in {} is not a number'.format(name, i, p) for i in f: assert isinstance(i, (float, int, np.int32, np.float64)), \ 'in {}, value {} in {} is not a number'.format(name, i, f) assert len(p) == length, \ 'in {}, input list must be length {}'.format(name, length) return