fix n-port mismatch junction issue in Circuit

[mhuser]

Fix issue #1022
It appears that formula (3) in P. Hallbjörner, Microw. Opt. Technol. Lett. 38, 99 (2003). does not work for mismatched intersections.
This was temporarily fixed for 2-connection intersections and is now generalized to n-connection intersections with mismatched port impedances.

• Fix circuit
• Optimize
• Update test_circuit
• Add mismatched splitter to media and a test case

[mhuser]

I wrote the equations in Media.splitter doc as a reminder.

[jhillairet]

Great work @mhuser !

[jhillairet]

It's no more relevant, but for info here is another implementation of the S-matrix renormalization from a given Z0 to Z0'. The numerical precision is better than the current implementation using s2z(z2s()) but it only works for power waves.

def renormalize_s2(
        s: npy.ndarray, z_old: NumberLike, z_new: NumberLike,
        ) -> npy.ndarray:
    """
    Renormalize a s-parameter matrix given old and new port impedances.
    
    Use equations from [#Reveyrand]_. 
    The numerical precision of this method is better than `renormalize_s`, 
    however, it only supports the power-wave definition.
   
    Parameters
    ----------
    s : complex array of shape `fxnxn`
        s-parameter matrix

    z_old : complex array of shape `fxnxn` or a scalar
        old (original) port impedances

    z_new : complex array of shape `fxnxn` or a scalar
        new port impedances

    Returns
    -------
    :class:`numpy.ndarray`
        renormalized s-parameter matrix (shape `fxnxn`)

    See Also
    --------
    Network.renormalize : method of Network to renormalize s
    renormalize_s
    fix_z0_shape
    s2s

    References
    ----------
    .. [#Reveyrand] T. Reveyrand, "Multiport conversions between S, Z, Y, h, 
        ABCD, and T parameters’, in 2018 International Workshop on Integrated 
        Nonlinear Microwave and Millimetre-wave Circuits (INMMIC)"",
        Brive La Gaillarde: IEEE, Jul. 2018, pp. 1–3.
        doi: 10.1109/INMMIC.2018.8430023.

    """
    nfreqs, nports, nports = s.shape
    z_old = fix_z0_shape(z_old, nfreqs, nports)
    z_new = fix_z0_shape(z_new, nfreqs, nports)
   
    Id = npy.zeros_like(s, dtype='complex')
    R = npy.zeros_like(s, dtype='complex')
    Gold = npy.zeros_like(s, dtype='complex')
    Gnew = npy.zeros_like(s, dtype='complex')
    
    npy.einsum('ijj->ij', Id)[...] = 1
    npy.einsum('ijj->ij', R)[...] = (z_new - z_old)/(z_new + z_old.conj())
    npy.einsum('ijj->ij', Gold)[...] = 1/npy.sqrt(npy.abs(z_old.real))
    npy.einsum('ijj->ij', Gnew)[...] = 1/npy.sqrt(npy.abs(z_new.real))

    A = npy.linalg.inv(Gnew) @ Gold @ (Id - R.conj())

    return npy.linalg.inv(A) @ (s - R.conj()) @ npy.linalg.inv(Id - R@s) @ A.conj()

[mhuser]

It's no more relevant, but for info here is another implementation of the S-matrix renormalization from a given Z0 to Z0'. The numerical precision is better than the current implementation using s2z(z2s()) but it only works for power waves.

Very interesting. Maybe we could use this in renormalize_s and fallback to z2s(s2z() if s_def is not power-wave?
We know that current renormalize_s is weak when transformation between $S$ and $Z$ would yeld the inverse of a singular matrix.

[jhillairet]

It's no more relevant, but for info here is another implementation of the S-matrix renormalization from a given Z0 to Z0'. The numerical precision is better than the current implementation using s2z(z2s()) but it only works for power waves.

Very interesting. Maybe we could use this in renormalize_s and fallback to z2s(s2z() if s_def is not power-wave? We know that current renormalize_s is weak when transformation between S and Z would yeld the inverse of a singular matrix.

Eventually... Let's see that in another PR.