Trivelpiece received his Ph.D in Electrical Engineering in 1958 from the California Institute of Technology, with a thesis titled "Space Charge Waves in Cylindrical Plasma Columns," under the direction of Professor Roy Gould. After spending a year in the Netherlands as a Fulbright Scholar, he joined the University of California, Berkeley, as a professor. In 1965 he joined the University of Maryland as a professor of physics. There, he published a textbook, Principles of Plasma Physics, with colleague Nicholas A. Krall, that remains a classic textbook to this day.

In 1972, Robert L. Hirsch became director of the fusion program at the US Atomic Energy Commission (AEC) and launched an effort to enlarge and refocus the program on its energy mission. He reorganized his office to include three assistant directors, one each for the areas of Research, Magnetic Confinement Systems, and Fusion Technology. He recruited Trivelpiece to oversee the Research area, Robert W. Bussard to oversee the Fusion Technology area and Stephen O. Dean to oversee the Magnetic Confinement Systems area. One of Trivelpiece's achievements during his 3 years in that post was to establish a National Controlled Thermonuclear Research Computer Center. Located originally at the Lawrence Livermore National Laboratory (LLNL), it was later expanded to cover all DOE civilian research programs, moved to the Lawrence Berkeley National Laboratory (LBNL), and renamed the National Energy Research Scientific Computing Center. It is currently managed as part of the Department of Energy (DOE) Office of Science.

In 1976, Trivelpiece joined Maxwell Laboratories in San Diego as Vice President for Engineering and Research and, in 1978, he joined Science Applications Inc. (SAI) as Corporate Vice President. He remained there until 1981. During that time he frequently chaired an advisory committee for the Pulsed Power program at Sandia National Laboratories and, was instrumental in providing SAI support for the establishment of Fusion Power Associates. He also served on an informal advisory committee to Congressman Mike McCormack that resulted in Congressional passage of the Magnetic Fusion Energy Engineering Act of 1980, that was signed by President Jimmy Carter in October 1980. That Act called for an accelerated government effort to provide a fusion demonstration power plant by the year 2000.

In 1981, President Ronald Reagan picked Trivelpiece to be Director of the Office of Energy Research (now called the Office of Science) at the US Department of Energy, a position he held until 1987. While at the DOE, Trivelpiece played a key role in several important DOE projects, including building five new major DOE scientific facilities, and preparations for the International Thermonuclear Experimental Reactor (ITER), which was codified at the 1987 Reagan-Gorbachev Summit Meeting.

In April 1987, Trivelpiece left DOE to become Executive Officer of the American Association for the Advancement of Science, publisher of the journal Science. In 1989, he became director of Oak Ridge National Laboratory (ORNL). During eleven years in that post, he implemented many enhancements, including the development of the ORNL Center for Computational Sciences and the construction of the Spallation Neutron Source, while also being given additional responsibilities within the Martin Marietta Corporation (management company of operations of ORNL) as Vice President, Martin Marietta Energy Systems and later as President, Lockheed Martin Energy Research Corporation.

After his retirement in 2000, he provided consulting services, including to the Sandia National Laboratories and to the Lawrence Livermore National Laboratory. In 2012 he provided a lengthy "perspective" contribution to Chapter 12 in the book Search for the Ultimate Energy Source (Springer ISBN 978-1-4614-6036-7). The same book contains a lengthy September 1984 Trivelpiece interview on the aftermath of the passage of Magnetic Fusion Energy Engineering Act of 1980.

Persons who would like to share their memories of Al Trivelpiece should send them to his long-time friend and colleague Jim Decker at: decker@deckergarmansullivan.com