ABCD parameters matrix of unsymmetric coupled lines

[PlanarMetamaterials]

A simplified proof? Not particularly, I think he lays out the derivations of those equations fairly well. Is there something in particular you're trying to accomplish here?

 

[promach]

Why does summation operator exist in equation 3.3b , but not for 3.3a ?

 

[PlanarMetamaterials]

Why does summation operator exist in equation 3.3b , but not for 3.3a ?

I assume you're referring to Clayton Paul's text. In that case, 3.3b contains a sum over all the currents because he is computing voltage drop on the reference conductor, which -- since the total currents of a transmission-line mode must sum to zero -- contains the negative of the sum of the currents on all the other conductors. Equation 3.3a is for computing the voltage drop on a non-reference conductor.

 

[promach]

Why V_i(z + Δz , t) does not have negative sign ?

 

[PlanarMetamaterials]

If you look at Fig. 3.1 and also equation (3.1), you can see that this equation represents the integration of electric field around the loop. Since the distance \( \Delta z\) is assumed to be small, the electric fields at a-a' and b-b' are assumed to be in the same direction. However, since we are integrating around the loop, we are actually integrating in the opposite direction with respect to the electric field for a-a' as we are from b'-b. Therefore, the voltage has opposite signs in the first and third terms.

 

[promach]

but why in equation (3.5) , V_i(z , t) has negative sign while V_i(z + Δz , t) does not have ?

I suppose the opposite situation should be the case instead ?

 

[PlanarMetamaterials]

Fig. 3.1 shows:

And equation 3.1 is:

While equations 3.2 are:

Inserting 3.2 into 3.1 (and others) gives 3.5:

The first term seems to arise from integrating E from a to a', which is in the direction of the loop in Fig. 3.1. According to 3.2a, this term should have a negative sign, as we see in 3.5.

The third term arises from integrating E in the direction of the loop, from b' to b. 3.2b details the integration from b to b', so the third term should be the negative of this (i.e., positive), as we see in 3.5.

 

[promach]

I do not quite understand ψ_i arrow direction (is it right-hand rule notation) inside Figure 3.2 as well as equation (3.7)

 

[PlanarMetamaterials]

I do not quite understand ψ_i arrow direction (is it right-hand rule notation) inside Figure 3.2 as well as equation (3.7)

I'm not sure exactly what you're referring to, but you can see that the surface normal of si is defined as being in the opposite direction of the flux, hence the negative sign.

The direction of the flux of course comes from the direction of the H-vector, which as stated is given by the right-hand rule.

 

[promach]

I mean how to get the final expression of equation (3.7) ?

I suppose I_n is current as in Ampere ?

 

[PlanarMetamaterials]

I suppose I_n is current as in Ampere ?

Yes, those are the currents in each conductor n.

I mean how to get the final expression of equation (3.7) ?

I suppose what isn't directly stated is that the currents in each conductor will contribute to the magnetic flux between the reference and the given conductor, so if you want to know the total flux value, you have to include contributions from all of the currents.

 

[promach]

the surface normal of si is defined as being in the opposite direction of the flux, hence the negative sign.

I do not quite understand the sentence in the book that says :

the direction of the required flux, ψ_i , and the defined unit normal to the surface, a_n , are in opposite directions.

 

[PlanarMetamaterials]

The surface \(s_i\) in this case is in the xz plane, so the normal vector will be in the y-direction, in accordance with the right hand rule. However, we have already established that the flux produced by the current will be passing through this surface in the negative-y direction, as indicated.

 

[promach]

1. Why the inductance matrix L is symmetric ?
2. How to derive the matrix R ?

 

[PlanarMetamaterials]

1. Why the inductance matrix L is symmetric ?

This is an excellent question. Paul touches on this at the end of chapter 3. I'll have to think of a different explanation!

2. How to derive the matrix R ?

[R] generally isn't analytically derivable on its own, since the resistivities will be frequency-dependent in an arbitrary fashion, as well as highly dependent on geometry. Typically, I just solve is as the real part of per-unit-length impedance matrix [Z].

 

[promach]

One question: Would deriving the ABCD matrix of unsymmetric coupled transmission line be able to derive the following balun equations from the paper : New Design Formulas for Impedance-Transforming 3-dB Marchand Baluns ?

 

[PlanarMetamaterials]

Yes, I don't see why not.

It looks like what that derivation does is relate the input/output network impedances to the characteristic impedances of the MTL sections, which you can simply get from a simulation. The relationships between i2, i3, and i4 may change, but as you've indicated you should be able to derive those from the ABCD matrices.

Once thing I should point out is that the derivation seems to assume that the modes have the same propagation constant, such that the electrical length \(\theta\) is the same for both modes. This is a commonly-made approximation, and under some circumstances -- such as a reasonably high dielectric constant (relative permittivity >= 3) -- can be a very poor one. The ABCD formulation I originally suggested takes care of this by modelling the propagation constants of both modes independently.

 

[promach]

The ABCD formulation I originally suggested takes care of this by modelling the propagation constants of both modes independently.

Wait, I am bit confused.
Which post in this thread was about your ABCD formulation ?
Would you be able to give a bit details into this ?

 

[PlanarMetamaterials]

Wait, I am bit confused.
Which post in this thread was about your ABCD formulation ?

Post #4

Would you be able to give a bit details into this ?

Sure, in the modal-terminal domain transformation, the modal properties are specified, along with the mode definitions. In this way, you can directly specify the properties of the modes, such as modal propagation constants. This is done by deriving the ACBD matrix and by invoking Paul's equations 7.8

 

[promach]

Checking through equation (7.109) and earlier part of chapter 7 could not locate the proof to derive the modal terminal transformation (which is equivalent to the expression for ABCD) in post #4

Please correct me if I miss anything.