We use 65,133 hourly averages of transpolar voltage (Φ_PC) from observations made over 25 years by the SuperDARN radars, along with simultaneous interpolated am geomagnetic index values, and study their optimum interplanetary coupling functions. We find lags of 18min. and 31min. for Φ_PC and am, respectively, and fit using a general coupling function with four free fit exponents. To converge to a fit, we need to average interplanetary parameters and then apply the exponent which is a widely-used approximation: we show how and why this is valid for all interplanetary parameters, except the factor quantifying the effect of the clock angle of the interplanetary magnetic field, sin^d(θ/2), which should be computed at high time resolution and then averaged. We demonstrate the effect of the exponent d on the distribution, and hence weighting, of samples and show it causes d to be best determined from the requirement that the coupling function is a linear predictor, which yields d=2.50±0.07 for Φ_PC and d=3.00±0.22 for am. To check for overfitting, fits are made to half the available data and tested against the other half. Ensembles of 1000 fits are used to study the effect of the number of samples on the distribution of errors in individual fits and on systematic biases in the ensemble means. We find only a weak dependence of solar wind density for Φ_PC but a significant one for am. The optimum coupling functions are shown to be significantly different for Φ_PC and am.