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March 2017 • The PCB Design Magazine 39 where they are easier to solve, and find the so- lution. For instance, a circle can be transformed into a square (Figure 2). Or an open geometry like that of planar traces (traces referenced to planes), on a PCB, can be transformed into a closed geometry. The CMT equivalent is a coor- dinate transformation and is applicable to both planar and non-planar transmission lines. CMT are ideal for analyzing coplanar waveguides (CPW) particularly those that lack a ground ref- erence plane. All electrical systems function based on the action of electric fields produced by charges, and magnetic fields produced by currents. To understand the working principle of these sys- tems, the field lines, that envelop them, must be evaluated, allowing a spatial visualization of the phenomena. These maps typically rep- resent flux, equipotential surfaces and densities distributions, having information about field intensity, potential difference, energy storage, charges and current densities. The conformal mapping or transformation of two intersecting curves from the z-plane to the w-plane (fifth dimension), preserves the angles between every pair of curves. That is, if two curves in the z-plane intersect at a particu- lar angle, the corresponding transformed curves will also intersect at the same angle, although the curves in the w-plane may not have any re- semblance to the original curves. The z-plane (x,y) coordinate system is an or- thogonal one. And for an analytical function, the w-plane's (u,v) coordinate system, is also or- thogonal. So, an elliptical electromagnetic field, can be transformed into a more useable geom- etry and still maintain consistency as in Figure 3. This is an example of a set of curves mapped into a set of straight lines which greatly simpli- fies the analysis. The electrostatic energy in both the (x,y) and (u,v) coordinate systems remains space-time in- variant. Consequentially, the capacitance, of a system of conductors, remains unchanged on the transformation of the arrangement of con- ductors. Under the conformal mapping trans- formation, there is a change in the geometrical shape of the conductor's arrangement without any change in the capacitance. This is a very important property for the analysis of the trans- mission line parameters. CPW expressions are derived using these conformal mapping techniques and elliptic in- tegrals to calculate the impedance of strip con- figurations. A conventional CPW on a dielectric substrate consists of a center strip conductor with semi-infinite ground planes on either side. This structure supports a quasi-TEM (resembling the transverse electromagnetic wave) mode of propagation. A quasi-TEM wave only exists in a microstrip line–on the outer surface of a PCB. In this mode, electric fields and magnetic fields are perpendicular to each other and perpendicular to the direction of propagation. The CPW offers several advantages over a conventional microstrip transmission line: • Simplifies fabrication • Facilitates easy shunt as well as series surface mounting of active and passive devices • Eliminates the need for via holes and wraparound (ground plating on the edge of a substrate to provide a low inductance path) • Reduces radiation loss at very high microwave frequencies Furthermore, the impedance is determined by the ratio of trace width to clearance, so size reduction is possible without limit, the only penalty being higher losses. In addition, a vir- tual ground plane exists between any two ad- jacent lines, as there is no field at that point. Hence crosstalk effects, between adjacent lines, are very weak. MICROSTRIP COPLANAR WAVEGUIDES Figure 3: Electromagnetic fields in the z-plane, left, mapped to the w-plane, right (source: Gibbs).