[SI-LIST] Coupled Lines Modeling

Dear Si-List Members,
I wish all of you a beautiful Holiday season and a happy prosperous New Year!

Procedures for SPICE modeling of coupled transmission lines constitute an 
important
signal integrity topic.
For case of two balanced symmetric lines, the following relations can
prove useful:

Z_even = sqrt[L11 + L12)/(C11- C12)]                 

TD_even = sqrt[(L11 + L12) * (C11 - C12)]         
 
Z_odd = sqrt[(L11- L12)/(C11 + C12)]              

TD_odd = sqrt[(L11- L12) * (C11 + C12)]           

Where,

Z_odd is equivalent impedance for a coupled pair of transmission lines
propagating in  the odd mode pattern;  Z_even is impedance for the
even mode pattern.

similarly, 
TD_odd and TD_even represent equivalent delay for a coupled pair propagating in
odd mode and even mode patterns respectively.

Derivation of above equations are presented in reference 1.

One interesting application of these formulas relates to
obtaining coupled SPICE model from TDR measured data.
Here, the goal being to compute the [L] and [C] matrices
given the following TDR measurements:
  
Differential impedance (i.e.,  Z_diff = 2 * Z_odd for a symmetric pair)

Common mode impedance (i.e., Z_comm = 0.5 * Z_even)

Differential delay (i.e. TD_diff = TD_odd)

Common mode delay  (i.e. TD_comm = TD_even)

The formulas outlined in this post are sufficient for solving L11,
L12, C11 and C12 from the TDR measured  Z_diff, Z_comm,
TD_diff and TD_comm for a coupled pair.

 Furthermore, for case of balanced transmission lines consisting of two 
identical 
conductors symmetrical with respect to planes:

 L11 = L22, L12 = L21, C11 = C22, C12=C21

Subsequently, all elements (including self and mutual inductance and 
capacitance values) of the [L] and [C] matrices can be ascertained to allow
construction of the desired coupled SPICE model.

There exist other techniques for generating coupled lines models based
on TDR measurements, but the method discussed here can be appealing
particularly  to those who enjoy mathematics.

When such a SPICE model is produced, it is in good practice to simulate it 
under 
conditions emulating TDR measurements (i.e. same source and line termination) 
and
 then correlating the simulation results with the TDR measured data.  This can 
provide
useful information for verifying the quality and accuracy of the model.

 Reference
 1. Stephen H. Hall, Garrett W. Hall, and James A. McCall, "High-Speed Digital 
System 
    Design A Handbook of Interconnect Theory and Design Practices", John Wiley 
& Sons,
   Inc.,  2000, PP. 53-57.

Kind Regards,

Abe Riazi
ServerWorks

 

 

 

 

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