Differential and single-ended impedances

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Assuming that the propagation speed of TEM wave is constant:

the even and odd mode impedances are:

where Z0 is the characteristic impedance of each line:

and is a parameter that indicates the strength of coupling:

Conversely,

We will derive the relation between differential and single-ended impedances.

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First, we will derive the relation among capacitances Cs, Cm and C.

Assuming the line-1 exists alone with line-charge q1, its voltage is:

Superposing the electric field of the line-2 charged with q2, the line-1 voltage can be expressed with a parameter :

Because the line-1 and line-2 are identical, the line-2 voltage is similarly expressed:

If the line-1 and line-2 voltages are held at v1 and v2, the line-charges are:

Comparing the coefficients of two equations, we obtain the relation among Cs, Cm and C:

Noting that CsC and 0<Cm,C because of physical implications, then

Solving eq. (2) for the positive :

This parameter increases according to Cm/C, so it indicates the strength of coupling.

***

In the previous section, we have expressed Z} and u} in terms of L,Lm,Cs,Cm:

Inserting (1) and (2), we obtain the expression of Z} and u} in terms of L,Lm,:

where Z0 and u is the characteristic impedance and propagation speed of each line:

Although u+ and u| look different from each other, the propagation speed of TEM wave must be a constant, that is the light speed. So assuming

, the inductances are also expressed by the parameter :

Then we can simply express the even and odd mode impedances in terms of Z0 and :

Noting 0<1 because of physical implications, then

Conversely, Z0 is the arithmetic average of Z+ and Z|, and is the relative difference between Z+ and Z|F


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