Inductance of magnetically shielded air-core coils of circular cross section

An expression for the inductance of shielded cylindrical air-core solenoids of arbitrary diameter to length ratio is developed. Three shielding configurations are investigated: 1) the use of a coaxial-concentric magnetic shield, 2) the coaxial magnetic shield with conducting end plates, 3) the coaxial magnetic shield with conducting end rings. Case 2) is solved in closed form by separation of variables, and from this result a method is given for predicting the length of the coaxial magnetic shield required to contain the field for the three cases.