Extra Resources¶
Distributed Elements¶
Semi-infinite (planar infinite length) Warburg¶
Describes linear diffusion from a medium with length which can be approximated as infinite.
(1)¶\[Z_{W} = \frac{A_W}{\sqrt{w}}(1-j)\]
Or
\[Z_{W} = \sqrt{\frac{R_d}{s C_d}} (\sqrt{R_{d}~sC_{d}})\]
Where \(s = j \omega\) with \(j\) being the imaginary unit and \(\omega\) the angular frequency. \(A_{W}\) has units of \(\Omega s^{-0.5}\), \(R\) has units of Ohms (\(\Omega\)) and \(C\) has units of Farads (\(F\)).
And
\[A_W = \frac{RT}{F^{2}C_{o}\sqrt{D_o}}\]
w = 2 * jnp.pi * freq
s = 1j * w
Zw = Aw/jnp.sqrt(w) * (1-1j)
# Or
Zw = jnp.sqrt(Rd/s*Cd) * (jnp.sqrt(Rd * s*Cd))
Finite length diffusion with reflective boundary¶
Describes the reaction of mobile active species distributed in a layer with finite length, terminated by an impermeable boundary.
(2)¶\[Z_{Wo} = \sqrt{\frac{R_d}{s C_d}} \coth(\sqrt{R_{d}~sC_{d}})\]
Or
\[Z_{Wo} = R \frac{coth(j \omega \tau)^{\phi}}{(j \omega \tau)_{\phi}}\]
Where \(\phi\) = 0.5
w = 2 * jnp.pi * freq
s = 1j * w
ZWs = jnp.sqrt(Rd/s*Cd) * 1/jnp.tanh(jnp.sqrt(Rd * s*Cd))
Finite length diffusion with transmissive boundary¶
Describes the reaction of mobile active species distributed in a layer with finite length, terminated by an impermeable boundary.
(3)¶\[Z_{Ws} = \sqrt{\frac{R_d}{s C_d}} \tanh(\sqrt{R_{d}~sC_{d}})\]
Or
\[Z_{Ws} = R \frac{tanh(j \omega \tau)^{\phi}}{(j \omega \tau)_{\phi}}\]
Where \(\phi\) = 0.5
w = 2 * jnp.pi * freq
s = 1j * w
ZWs = jnp.sqrt(Rd/s*Cd) * jnp.tanh(jnp.sqrt(Rd * s*Cd))