University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation of the exact results of cross-talk analysis We are going to consider some particular cases that help understanding the phenomenon of cross-talk Assumption 1) electrically short line Assumption 2) weakly coupling between generator and receptor circuits When, When, we obtain (1) (2) and

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 As a consequence of approximations 1) and 2) the exact results for the near-end and far-end voltages become: (3) (4) This results suggests that (principle of superposition of inductive and capacitive coupling) the cross-talk is a linear combination of the contributions due to the mutual inductance (inductive coupling) and the mutual capacitance (capacitive coupling).

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Assumption 3) sufficiently small frequency, which yields (3) (4)

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 This result can be obtained from the following equivalent circuit. Observe that: 1) The voltage source represents the emf induced in the receptor circuit, according to Faraday’s law, where is the total mutual inductance between the generator and receptor circuits. 2) The current source represent the charge induced in the receptor circuit, where is the total mutual capacitance between the generator and receptor circuits. Figure 1

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Therefore, the key to understand cross-talk is the notion of superposition between inductive coupling and capacitive coupling. Intuitively, one would expect that inductive coupling dominates for low-impedance loads (high currents, low voltages) and that capacitive coupling dominates for high-impedance loads (low currents, high voltages). Inductive coupling dominates in if and in if (notice that the inductive coupling components are opposite in sign and different in magnitude) (7) (8)

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Capacitive coupling dominates in if and in if (notice that the inductive coupling components have the same sign and are equal in magnitude) (9) (10) The total coupling is the sum of the individual components, hence (11) (12) Notice that the termination impedances can be chosen so that the capacitive and inductive coupling cancel each other. This is at the basis of the directional coupler in microwave circuits.

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Cross-talk Transfer Functions Cross-talk can be viewed as transfer function between the input voltage and the output voltages and. The corresponding transfer functions are: where (13) (14) (15) (16) (17) (18)

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Effect of losses - Common - impedance coupling The assumption of lossless line previously considered is valid for electrically short lines when the frequency is below 1GHz. When a conductor is not lossless there may be cross-talk at lower frequencies. Let us refer to the following figure: Usually the resistance of the reference conductor is such that so most of the current comes back through the reference conductor. (19) Figure 2

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Now, the current through the reference conductor produces a voltage drop given by: where lumps the total resistance of the reference conductor. The voltage appears in the receptor circuit producing contributions to the transfer functions given by: (20) (21) (22) (23)

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 One should observe that the common-impedance coupling provides a frequency independent floor as shown in the following diagram. Finally, the total coupling is approximately the sum the three contributions considered so far: (24) (25) Figure 3