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Patent
A patent is a set of exclusive rights granted by a state to a patentee (the inventor or assignee) for a fixed period of time in exchange for the regulated, public disclosure of certain details of a device, method, process or composition of matter (substance) (known as an invention) which is new, inventive, and useful or industrially applicable.
Royalties
Sometimes simply referred to as 'royalty' is typically the sum of money paid to the proprietor or Licensor of Intellectual Property (IP) Rights for the benefits derived, or sought to be derived, by the user (the Licensee) through the exercise of such rights. Royalties may be paid for the use of copyright, patent, registered design, knowhow or trademark or a combination of them. However, the term has also much wider application and can cover mining royalties, performance of art royalties, etc. The discussion here is limited to IP.
Patent royalties
In the United States, a patent owner, or patentee, is legally entitled to no less than a "reasonable" royalty from an entity that infringes said owner's patent. If the patent owner has lost profits due to infringement, and if said lost profits are more than a reasonable royalty, then the patent owner is entitled to a royalty up to the amount of lost profits.
Normally, a patent owner and a patent infringer will negotiate a mutually agreeable license so that the patent owner will give the patent infringer the legal right to make, user or sell the patented invention, and the patent infringer will compensate the patent owner in some manner.
If they cannot come to agreement, the patent owner may sue the patent infringer in a U.S. federal court. 99% of the cases that are filed are settled before the court reaches a decision. If the case does go to trial, however, then evidence will be presented related to validity of the patent, whether or not it was infringed, what a "reasonable" royalty is in the field of the patented invention and what the patent owner's lost profits are. Evidence will also be presented as to whether or not the alleged infringer "willfully" infringed the patent. If the court finds that the infringement was willful, then punitive damages may be assessed. Punitive damages may be up to three times actual damages.