Your basic problem is that you intend to use an inductive load connected to a switch. The switch can be the H-bridge or a simple MOSFET or transistor. H-bridges can drive the load (coil ) with both current directions, that is all.
Coils do resonate with intrinsic as well as circuit capacitances, but often at quite high frequency. It is better to add a capacitance to make sure you know the resonant frequency (or you can tune the coil with a ferrite or iron core).
Current pattern in such inductive load can be determined if you know all circuit parameters. The equation must include resistance, inductance and capacitance as well as the switching frequency.
Inductive loads to switched current sources generate first an overshoot voltage before the current settles to a current value determined by source voltage and circuit resistance (including source and load).
Any switch including the H-bridge must be designed so that it can supply enough steady current and can withstand the overshoot voltage. With a simple switch the current is limited by an added resistor to coil resistance, and the overvoltage is limited by a parallel diode to the source voltage. In a H-bridge switch the overvoltage will have both positive and negative spike so two diodes would have to be used, with additional switches for each polarity.
This complication is usually too much to add, so I would advise not to use the H-bridge for an inductive load. Instead, use a simple MOSFET switch with a protection diode, and then your current-switched coil will work well.
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Adding a "secondary" coil makes your system a transformer. Coupling between the primary and secondary coils and their turn count will determine the secondary voltage. If loaded, you will adjust the P/S coupling closer to get some power output.