Differential and single-ended impedances
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 ξ is a parameter that indicates the strength of coupling:
We will derive the relation between differential and single-ended impedances.
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 ξ:
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 Cs≦C and 0<Cm,C because of physical implications, then
Solving eq. (2) for the positive ξ:
This parameter ξ 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,ξ:
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 ξ:
Then we can simply express the even and odd mode impedances in terms of Z0 and ξ :
Noting 0≦ξ<1 because of physical implications, then
Conversely, Z0 is the arithmetic average of Z+ and Z−, and ξ is the relative difference between Z+ and Z−：