1. Crosstalk Crosstalk is the electromagnetic coupling among conductors that are close to each other. Crosstalk is an EMC concern because it deals with the design of a system that does not interfere with itself. Crosstalk may affect the radiated/conducted emission of a product if, for example, an internal cable passes close enough to another cable that exits the product. Crosstalk occurs if there are three or more conductors; many of the notions learnt for two-conductor transmission lines are easily transferred to the study of multi-conductor lines. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

2. Crosstalk in three-conductor lines Consider the following schematic: reference conductor generator conductor receptor conductor near end terminal far end terminal The goal of crosstalk analysis is the prediction of the near and far end terminal voltages from the knowledge of the line characteristics. There are two main kinds of analysis Figure 1 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

3. This analysis applies to many kinds of three-conductor transmission lines. Some examples are: receptor wire reference wire generator wire receptor wire generator wire reference conductor (ground plane) receptor wire reference conductor shield generator wire generator conductor receptor conductor reference conductor Figure 2 (a)(b) (c) (d) University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

4. Similarly to the case of two-conductor transmission lines, the knowledge of the per-unit length parameters is required. The per-unit length parameters may be obtained for some of the configurations shown as long as: 1) the surrounding medium is homogeneous; 2) the assumption of widely spaced conductors is made. Assuming that the per-unit-length parameters are available, we can consider a section of length of a three-conductor transmission line and write the corresponding transmission line equations. It turns out that, by using a matrix notation, the transmission line equations for a multi-conductor line resemble those for an ordinary two-conductor transmission line. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

5. Let us consider the equivalent circuit of a length of a three-conductor transmission line. The transmission line equations are: (1) (2) Figure 3 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

6. The meaning of the symbols used in (1) and (2) is: and (7) (8) University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

7. Per-unit-length parameters We will consider only structures containing wires; PCB-like structures can only be investigated using numerical methods. The internal parameters such as r G, r R, r 0 do not depend from the configuration, if the wires are widely separated. Therefore we only need to compute the external parameters L and C. It is important to keep in mind that for a homogeneous medium surrounding the wires, two important relationships hold: and (9) (10) University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

8. The elements of the L matrix are found under the assumption of wide separation of the wires. In this condition the current distribution around the wire is essentially uniform. We recall a previous result for the magnetic flux that penetrates a surface of unit length limited by the edges at radial distance and as in the following. (11) + surface 1m Figure 4 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

9. Then we consider a three-wire configuration: For this configuration we can write: (12) or (13) Figure 5 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

10. Using the result of (11), we obtain (14) (15) (16) And from these elements, we obtain the capacitance using the relationship: (17) University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

11. Per-unit-length capacitance parameters may also be derived directly as in the following. We recall that under the TEM approximation, the voltage between two points at radial distances R l and R 2 such as: is Note that for a positive charge distribution on the wire, the voltage is positive at the point closer to the wire. (18) Figure 6 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

12. For the equivalent circuit of Fig. 3, we found the following relationship between charges and voltages with respect to the reference conductor. (19) It is easy to invert the previous relation and write: (20) The elements of the P matrix are given by the following ratios (21) University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

13. The self terms are computed from the following from which it is obtained and, similarly (22) (23) Figure 7 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

14. The mutual terms are obtained referring to the following geometry for the three-wire line: (24) It is easy to verify that the entries in the capacitance matrix C computed from and are equivalent. Figure 8 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

15. Some numerical examples Consider a ribbon cable made up of three-wire such as The inductances are: (25) and (26) (27) Figure 9 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

16. Consider the case of two wires above an infinite ground plane. The ground plane may be removed by introducing the images of the wires In order to compute the magnetic flux, we must include only the field linked above the ground plane. Hence, we obtain: (28) (29) Figure 10 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

17. For the mutual inductance we consider: (30) (31) Figure 11 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18

18. The characteristic impedance of each isolated circuit is The per-unit length capacitances are: Which lead to a characteristic impedance This shows that the presence of the ground plane affects the circuit. University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 18