E and B Decoupling

1. Quasi-TEM Approximation

Assume each field component varies along the line as $e^{i\omega t-\gamma z}$ with $\gamma =\alpha +i\beta$, and split into transverse and longitudinal parts:

The quasi-TEM assumption is that the longitudinal field components are small compared to the transverse parts:

Taking the $\hat{z}$-component of Faraday's law $\mathrm{\nabla }\times \mathbf{E}=-i\omega \mu \mathbf{H}$ gives

Since $H_z\approx 0$, the transverse electric field is approximately curl-free in the cross-section, and can be written in terms of a scalar potential:

To make the correction explicit, compare with the full potential representation in Coulomb gauge ($\mathrm{\nabla }\cdot \mathbf{A}=0$):

With the assumed $z$-dependence, the gauge condition can be written ${\mathrm{\nabla }}_t\cdot {\mathbf{A}}_t=\gamma A_z$. Treating ${\mathrm{\nabla }}_t\sim 1/a$ where $a$ is a characteristic transverse dimension, this gives the scaling

A second scaling relation follows from the quasi-TEM condition on $E_z$:

Combining the two,

So the vector-potential contribution to ${\mathbf{E}}_t$ is suppressed by $(\beta a{)}^2$ relative to the leading scalar-potential piece:

The same argument applied to the magnetic side, using $B_z\approx 0$, gives the transverse magnetic field in terms of a single longitudinal vector potential:

2. Quasi-Static Approximation

Section 1 leaves all higher-order corrections to the transverse fields controlled by a single dimensionless parameter, $\beta a$. The quasi-static limit is the statement that this parameter is small:

or equivalently, that the relevant transverse length scale is much smaller than the guided wavelength,

The wavelength here is the guided wavelength ${\lambda }_g$, not the free-space wavelength ${\lambda }_0=2\pi c/\omega$. For high kinetic inductance materials we might be interested in, this could potentially pose an issue. The phase constant is

where the inductance per unit length could include both geometric and kinetic contributions:

For a clean Nb or Ta CPW with ${\lambda }_L\sim 40$–$100$ nm and $t\sim 100$ nm, $L_k$ is a few percent of $L_g$ and ${\lambda }_g\approx {\lambda }_0/\sqrt{{\epsilon }_{\mathrm{eff}}}$. For a granular-Al film with ${\lambda }_L\sim 600$ nm, $L_k$ can dominate $L_g$ by orders of magnitude, and ${\lambda }_g$ shrinks accordingly. The quasi-static condition is then not necessarily automatic for high-kinetic-inductance devices: it must be checked against the actual $\beta$, with the kinetic contribution included.

3. Decoupling

If we assume the quasi-TEM scalings of Section 1 and the smallness of $\beta a$ from Section 2, the 2D cross-section field equations split into two independent boundary-value problems.

Start from Gauss's law in a charge-free region,

and separate transverse and longitudinal pieces:

The longitudinal term scales as ${\partial }_z(ϵE_z)\sim \gamma ϵE_z$, while the transverse term scales as ${\mathrm{\nabla }}_t\cdot (ϵ{\mathbf{E}}_t)\sim ϵE_t/a$. Their ratio is

where one factor of $\beta a$ comes from $\gamma a$ and another from the quasi-TEM scaling $E_z/E_t=O(\beta a)$. Dropping this correction and substituting ${\mathbf{E}}_t\approx -{\mathrm{\nabla }}_t\varphi$ gives

This is the standard 2D electrostatic equation. Only $\varphi$ appears; $A_z$ has dropped out entirely. For the magnetic side, take the $\hat{z}$-component of Ampere–Maxwell, $\mathrm{\nabla }\times \mathbf{H}=\mathbf{J}+j\omega ϵ\mathbf{E}$:

The displacement contribution $j\omega ϵE_z$ is suppressed relative to the conduction current by the same smallness of $E_z$ and is dropped. Outside the superconductor, $J_z=0$. Inside, the gauge-invariant London equation gives a shift for the supercurrent,

where $C$ is a constant on each isolated conductor that encodes a transport current (set to $C=1$ on the center trace and $C=0$ on each ground plane in the solver (same as the March PHYS7688 midterm paper). Substituting,

This is the magnetostatic equation. Only $A_z$ appears and $\varphi$ has dropped out.

The two boxed equations are decoupled. The electrostatic solve yields $\varphi$, and from it the capacitance per unit length $C$ and the electric participation ratios. The magnetostatic solve yields $A_z$, and from it the total inductance per unit length $L$, the kinetic inductance fraction $\alpha$, and the magnetic participation ratios. The two solves are tied together through the relation $\beta =\omega \sqrt{LC}$, but the profiles are independent and the magnitudes can be rescaled afterwards.

4. Possible Failure of Approximations

Sections 1 and 3 develop their corrections in the same small parameter,

The leading error in ${\mathbf{E}}_t\approx -{\mathrm{\nabla }}_t\varphi$, in ${\mathbf{B}}_t\approx {\mathrm{\nabla }}_t\times (A_z\hat{z})$, and in the decoupling of the two BVPs is $O({\eta }^2)$. $\eta <0.1$ gives sub-percent corrections $\eta \sim 0.3$ gives roughly ten-percent corrections; $\eta \sim 1$ means the cross-section is no longer electrically small.

If it becomes too large, it would push towards revisiting the full-wave solving approach. However, If $\eta$ is large but not too large, a perturbative solution may work to save some computational efficiency. This would involve adding a source to the equations, representing the missing currently omitted $A_t$. I believe this would further suppress to $(\beta {\alpha }_{\text{eff}}{)}^4$, so could be effective depending on how large this is pushed. The size of this perturbation may also be a good way to numerically check how close to satisfying the approximation the setup is. I'm looking into this as an option in more depth.

A full-wave solver may be worth working on on the side regardless to use for validation. However, for doing parameter sweeps/scans, especially with interfaces, it would be best to avoid it.

One last piece to note is this is more likely to be violated by a flip-chip. The gap between the CPW and the lid defines a new characteristic transverse dimension not originally present, so if this is larger than the gap between the center conductor and the ground plane, the condition will fail sooner.

Useful Links (COMSOL Articles)

1. COMSOL Documentation, "Transmission Line Parameters of a Coaxial Cable". This claims that quasi-TEM RLGC extraction assumes longitudinal field components are nonzero but small, and gives the practical condition that signal–return spacing should be much smaller than the wavelength in the medium. This is very similar to the conclusions here.

2. COMSOL Learning Center, "Modeling TEM and Quasi-TEM Transmission Lines". This discusses a voltage definition for TEM and quasi-TEM transmission lines via electric-field line integrals.