From a simple theoretical consideration, we obtained two coupling functions linking upstream solar wind parameters to geomagnetic activity. We took into account (1) a scaling factor related to polar cap expansion while increasing the reconnected magnetic flux in the dayside magnetosphere, and (2) a modified Akasofu function for the reconnected flux for combined IMF B z and B y components. One of these coupling functions may be written as F α = aV sw B yz 1/2 sin α ($theta$/2), where V sw is the solar wind speed, B yz is the magnitude of the IMF vector in the Y-Z plane, $theta$ is the clock angle between the Z axis and IMF vector in the Y-Z plane, a is a coefficient, and the exponent, α, is derived from the experimental data and equals approximately to 2. The F α function is proportional to the square root of B yz , which makes it significantly different from the coupling functions proposed earlier. Nevertheless, the statistical data analysis supports this dependence. For testing the found coupling function, we used the solar wind and IMF data for four years. We computed 2-D diagrams showing the correlation coefficients for the dependence of the polar cap PC geomagnetic activity index on different combinations of solar wind/IMF parameters. The obtained diagrams showed very good agreement with the theoretical coupling function. The correlation coefficient for the dependence of the PC index on the coupling function is about 0.8--0.85, which is significantly higher that that for other commonly used coupling functions for the same time intervals.