Closed-Form Expressions for the Electromagnetic Radiation of Microstrip Signal Traces

An analytical study of the unintentional electromagnetic radiation of a microstrip signal trace is presented. Based on transmission-line theory and appropriate far-field Green's functions, a closed-form solution for the electric field is developed, including the contribution from the vertical terminal connections. It allows calculating the frequency and directional response of the radiated emission, depending on the material and geometrical parameters and terminal load impedances. For the special case of matched termination, the general solution assumes a compact form, allowing to set up a formula for the peak-amplitude frequencies. It is shown that in the higher frequency region, where the wavelength is comparable or shorter than the line length, the frequency-response envelope continuously rises with 20 dB/dec. Based on appropriate simplifications, the asymptotic solution for the electrically short line is deduced, including a practical formula for the maximum electric field. It is found that the usual small-current-loop (magnetic-dipole) estimation is incomplete due to the omission of the electric-dipole contribution.