Printed Circuit Design & Fab - June 2008 - (Page 18)

signal Integrity Applications of Mathcad, Part 2 Mathcad can solve complicated SI expressions including those with complex/ imaginary terms, logarithmic and exponential functions. MATHCAD OFFeRs eFFICIeNCY and flexibility for solving mathematical equations. It can treat analytical expressions having scalar (a single number), vector (column of numbers) and matrix (a rectangular array of number). An array is a general term for vector or matrix. The Mathcad logarithmic features dr. ABE include ln(z) which returns the natural log (ABBAS) riAzi of z (z ≠ 0) and log(z,b) which outputs the base b logarithm of z (z or b ≠ 0). If b is omitted, then log(z,b) equals the base 10 logarithm. Hence, log(z) returns the common logarithm of nonzero z. Let us analyze, applying Mathcad, the concept of coupling even and odd mode impedances. The electric and magnetic fields associated with coupled traces (i.e., traces located in close proximity) will interact in manners that depend on each line’s signal patterns. The effective characteristic impedance and velocity of the transmission will be altered due to such interactions. FiGurE 5 displays the computed (using HyperLynx from Mentor Graphics) odd and even electric (in blue) and magnetic (in red) field distributions for a coupled microstrip. Two coupled traces excited by signals of equal magnitude but opposite polarity (180 degrees phase difference) will produce the odd-mode (also called differential). When these traces are driven by identically phased signals, then the even (or common) mode will result. The relationship between the characteristic impedance of each line Z0, the odd mode impedance Z0o and the even mode impedance Z0e is governed11 by: Z0 = (Z0o * Z0e)^0.5 Equation 1 Equation 1 Equation 1 For any geometry, Z0o and Z0e depend on coupling equivalent magnitude of coupled energy depends on the odd and even mode impedances. FiGurE 6 presents impedance formulae for quarter wavelength matched lines solved with Mathcad. As coupling intensifies, even mode impedance is likely to be high11 since the inductance increases and capacitance diminishes (because field lines are more concentrated between coupled traces and not at the common ground/return). In a Z0 = 60 ohms system for weak coupling (such as k = 0.03) yields Z0e = 61.83 ohms and Z0o = 58.23 ohms. For a strongly coupled case (k = 0.7) with Z0 = 60 ohms outputs Z0e = 142.83 ohms and Z0o = 25.21 ohms. Thus when coupling is strong, Z0e and Z0o can significantly differ from Z0. Frequently, Z0o and Z0e for a coupled pair are plotted against trace separation12. Such graphs reveal that Z0o and Z0e are close to Z0 when trace separation is large (which corresponds to weak coupling). However, when trace separation is small (resulting in strong coupling) there is a significant impedance difference. When ascertaining the correct12 termination for a topology (in order to minimize reflections and maintain signal integrity) it is critical to take into consideration the even and odd mode impedance values. As another example, let us apply Mathcad to a case of coplanar structure, which demonstrates this software’s capability to treat hyperbolic trigonometric such as Tanh( ) and special functions (such as ratios of complete elliptic integrals). Coplanar waveguide (CPW) is a transmission line geometry that includes a central current-carrying trace on top of a dielectric substrate, with side grounds extending beyond a symmetric gap to either side of trace. There are several dif- FiGurE 5. simulated electric field lines and equipotentials for ! a coupled microstrip: (a) odd and (b) even mode. 18 FiGurE 6. Odd and even modes impedance computation. JUNE 2008 printEd circuit dESign & fAB

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Printed Circuit Design & Fab - June 2008

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