Example 1.Calculate the mutualinductancebetween an infinitelylong straight line and a rectangular coil.The line and the coil are at thesame plane,andin vacuumSolution: Select cylindrical coordinate systemand let the infinitely long straight line to be at6the z-axis. The magnetic flux density producedIitoby current I, is then12S2aB, =Ho1D2元The magnetic flux linkage Y2i with currentdrI, by current I, is42r = [, B, -dsIf the flowing direction of the current I, is clockwise, ds and Bhavethesamedirection.ThenD+bY, = olja r D+b 1Mola-dr2元D12元U7
Example 1. Calculate the mutual inductance between an infinitely long straight line and a rectangular coil. The line and the coil are at the same plane, and in vacuum. a b dr r D I1 0 I2 z S2 Solution: Select cylindrical coordinate system, and let the infinitely long straight line to be at the z-axis. The magnetic flux density produced by current I 1 is then B e r I 2π 0 1 1 = The magnetic flux linkage 21 with current I 2 by current I 1 is = 2 21 1 d S B S If the flowing direction of the current I 2 is clockwise, dS and B1 have the same direction.Then + + = = D b D D I a D b r r 0 I 1 a 0 1 2 1 ln 2π d 1 2π
Y21_ 40aD+bWehaveM21DbI2元1+If the flowing direction of the current I,12S2ais counter clockwise, then the B, and ds areDopposite, and M2<0drExample 2: Calculate the inductance per unit length of a coaxialline carrying a direct current.Solution: Assume the radius of theinner conductorofthe coaxiallineisa.theinteriorradius of the outerconductoris b.and the exteriorradiusis cuV
We have ln 0 2π 0 1 21 21 + = = D a D b I M If the flowing direction of the current I 2 is counter clockwise, then the B1 and dS are opposite,and M21< 0. Example 2: Calculate the inductance per unit length of a coaxial line carrying a direct current. Solution: Assume the radius of the inner conductor of the coaxial line is a, the interior radius of the outer conductor is b, and the exterior radius is c. b c a O a b dr r D I 0 1 I2 z S2
In the coaxialline,we constructarectangularcircuitfromalongitudinalsection of unitlength,leftside width aand right width (c - b)The current in the inner conductoristhatontheleftside.whilethe currentinthe outer conductor is that on the rightblcLaeddrside.The inductance per unit length of a coaxialline is4where Iis the currentin the coaxialline,and is the magnetic fluxlinkage per unit length with the current I.U
In the coaxial line, we construct a rectangular circuitfrom a longitudinal section of unit length, left side width a and right width (c − b). The current in the inner conductor is that on the left side, while the current in the outer conductor is that on the right side. The inductance per unit length of a coaxial line is I L 1 = where I is the current in the coaxial line, and is the magnetic flux linkage per unit length with the current I. a O I b c r a b c O dr I I e