The plasma potential at a typical substrate position is studied during the positive pulse of a bipolar high-power impulse magnetron sputtering (bipolar HiPIMS) discharge with a Cu target. The goal of the study is to identify suitable conditions for achieving ion acceleration independent on substrate grounding. We find that the time-evolution of the plasma potential during the positive pulse can be separated into several distinct phases, which are highly dependent on the discharge conditions. This includes exploring the influence of the working gas pressure (0.3-2 Pa), HiPIMS peak current (10-70 A corresponding to 0.5-3.5 A cm(-2)), HiPIMS pulse length (5-60 mu s) and the amplitude of the positive voltage U (+) applied during the positive pulse (0-150 V). At low enough pressure, high enough HiPIMS peak current and long enough HiPIMS pulse length, the plasma potential at a typical substrate position is seen to be close to 0 V for a certain time interval (denoted phase B) during the positive pulse. At the same time, spatial mapping of the plasma potential inside the magnetic trap region revealed an elevated value of the plasma potential during phase B. These two plasma potential characteristics are identified as suitable for achieving ion acceleration in the target region. Moreover, by investigating the target current and ion saturation current at the chamber walls, we describe a simple theory linking the value of the plasma potential profile to the ratio of the available target electron current and ion saturation current at the wall.