1. Time-domain Crosstalk The equations that describe crosstalk in time-domain are derived from those obtained in the frequency-domain under the following assumption: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 1) lossless line 2) three-conductors 3) homogeneous medium around the wires 4) electrically short line (  leads to inductive and capacitive coupling) 5) weak coupling (  equations simplify) 6) low frequency (  equations simplify) The frequency-domain equations are where and are the crosstalk transfer coefficients.  (Analytical solution is possible) (1) (2)

2. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 Transformation into the time-domain gives: which simply states that the crosstalk components and are related to the time-derivative of the input voltage through the coefficients and previously found. Notice that the representation (3) and (4) holds as long as the assumptions of weak coupling and electrically short line are valid. (3) (4)

3. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 Since the expressions for and contain an inductive and a capacitive term, the time-domain crosstalk is the sum of an inductive and a capacitive term. A simple equivalent for the receptor circuit is the following: (5) (6) Figure 1

4. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 The obvious implication of this simple model is that crosstalk exists as long as. Hence, for an input voltage that is a sequence of trapezoidal pulses, the prediction of this simple inductive-capacitive coupling model gives crosstalk contributions during the rise- and fall- times of the input trapezoid. Notice that the near-end crosstalk coefficient is always positive, whereas the far-end crosstalk coefficient is positive if capacitive coupling dominates and is negative otherwise. Figure 2

5. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 Let us analyze the limitations of this model. Only the frequency components below the frequency for which the line is electrically short are correctly processed by this model. Therefore, consider the envelope of the spectrum of a trapezoidal pulse train: Now, let us determine the highest frequency at which this model is valid Figure 3

6. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 The frequency response of the simple capacitive-inductive model of Fig. 1 to the train of trapezoidal pulses of Fig. 2 is given by the graphical superposition of the corresponding frequency responses: Figure 4

7. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 The components of the input spectrum in -40dB/dec region should not contribute significantly to the crosstalk spectrum. Hence only the frequencies such that will be processed correctly. Under which condition constraint (7) is satisfied? At the upper frequency, the line is only a fraction of a wavelength long so that: where is the time-delay of the line. If we choose we obtain: which suggests a simple way to determine which train pulses will be processed correctly by this simple inductive-capacitive model. (7) (8) (9) (10)

8. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 Inclusion of losses Similarly to the frequency domain case, the resistance of the conductors may be included in addition to the inductive and capacitive contributions. This is accomplished through the common-impedance coupling coefficients where (11) (12) (13) (14)

9. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 An exact model for lossless coupled lines It is possible to derive an exact time-domain solution for the multi- conductor transmission line provided that there are no losses. The starting point is the matrix form of the transmission line equations: where (15) (16) (17) (18)

10. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 where (21) (22) To solve these equations we apply the same method used in the frequency-domain case i.e. we decouple them. Hence we introduce two transformations that convert the line voltage and current into mode voltage and current and respectively. (note that (19) and (20) define the transformations as “old matrix”=“ new matrix”) (19) (20)

11. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 Having defined these transformations, we substitute them into the initial equations and obtain: The transformation matrices are chosen so that they diagonalize the matrices and in (23) and (24), respectively. If this is possible we obtain: (23) (24) (25) (26)

12. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 and the equations in the unknown mode voltages and currents decouple according to: Therefore we have obtained two uncoupled two-conductor transmission line equations with characteristic impedances: (27) (28) (29) (30)

13. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 Once these equations have been solved, it is possible to return to the original voltages using (from (19)) and to the original currents (from (20)) (31) (32) (33) (34)

14. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 About the possibility of decoupling the equations The solution of the original equations is based upon the possibility of finding transformations that diagonalize and. It turns out that the matrix is always symmetric (from energy consideration and for any real and symmetric matrix it is always possible to find a matrix such that For a homogeneous medium, (35) (36) Homogeneous Medium

15. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 so and upon substitution into (35) from which we derive Which implies that it is always possible to diagonalize and. Note that (37) (38) (39)

16. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 Inhomogeneous Medium The capacitance matrix is real, symmetric and it can be diagonalized using a matrix that satisfies so that: with a diagonal matrix. Since is positive definite, its eigenvalues are all real and positive. We define a diagonal matrix whose elements are the square root of the elements of. Consider now the matrix The quantity inside parenthesis is diagonalizable by since it is symmetric. Next define: and normalize it according to: where is a diagonal matrix with elements (40) (41) (42) (43)

17. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 The transformation matrices to obtain the mode voltage and current are: Additional properties are Observe that we can diagonalize and since Where and are the same previously introduced in (25) and (26). (44) (45) (46) (47) (48) (49)

18. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 21 Finally, since and are diagonal, the characteristic impedances are: (50) (51)